The History of Discrete Math and Linear Algebra
Almost nobody in this story was building the mathematics
your course teaches. They were counting poems, arguing about dice,
sorting bridges, and trying to win a fight about free will.
Start hereWelcome, and how to read this book
This is the story of where the mathematics in your course came from, written to be read alongside it.
Your course has two names because it is two subjects. Discrete math is Units 1 and 2: sets, counting, chance, and the language for talking about things you can list. Linear algebra is Unit 3: vectors, matrices, eigenvalues, and the machinery for handling many numbers at once. Unit 4, Markov chains, is where they meet, and that is not a curriculum convenience. It is what happened.
Here is the version this book argues for, in one sentence. Almost nobody in this story was building the mathematics that machine learning now runs on. They were doing something else, and this fell out. Pingala was counting the rhythms of Sanskrit poetry. Cardano was trying to win at dice. A clerk in Han China was dividing up bundles of rice on a ruled board and doing Gaussian elimination two thousand years before it had that name. Euler was answering a puzzle about a walk through a town. Markov was trying to win an argument about free will.
The second thing this book argues is that the words are much younger than the mathematics. The word matrix is younger than the railway. Almost all of Unit 1's vocabulary entered English between 1909 and 1926. If the vocabulary feels ancient and forbidding, that is a fact about typography, not about the subject.
This book is a companion, not a replacement. Read a chapter before the unit it belongs to, or arrive from a link in your course book and read one section. Both ways work, and nothing here assumes you have already read the rest.
Three parts, and they are used differently:
Part I, the story. Nine chapters, in rough chronological order, roughly {hours:.0f} hours of reading end to end and about {mins_avg} minutes a chapter. Each one opens with a question worth guessing at and closes with one worth answering.
Part II, the master timeline. {events} dated events across ten eras. Nobody reads this front to back. You arrive at it to place somebody.
Part III, reference. Eleven appendices and a glossary: {people} people, {terms} words and glyphs with their origins, {conflicts} logged disagreements between sources, {sources} sources, and every calculation in the book re-run in code so you can check it yourself.
Every factual sentence in this book carries a pointer to the source it came from, like this: (S001). Click one and it takes you to the full reference in Appendix J. Where two sources disagree, the book says so and shows every position rather than picking a favorite and sounding confident. {open_conflicts} of the {conflicts} logged disagreements are still open. That is not a defect. An open disagreement recorded is scholarship; an open disagreement forgotten becomes a fact.
Where something could not be checked, the book says that too, in plain words, rather than quietly leaving it out or quietly asserting it.
A date is a claim somebody made in a document, not a fact from nowhere. Two careful historians reading the same manuscript can date it differently and both be reasonable. When you see the Disputed label in this book, that is what it means: not "nobody knows anything", but "the evidence genuinely underdetermines the answer, and here is the shape of the argument."
Chapter 4 makes this concrete straight away. Ask when the Nine Chapters on the Mathematical Art was written and four respectable sources give four different answers, spanning three centuries, and none of them is being careless.
Every figure in this book was drawn for this book, by a script, in this book's own colors. Nothing was downloaded, traced or copied, so you may reuse any of them.
That also means no figure here is a photograph of a historical document, and the book never captions one as though it were. The map of Konigsberg in Chapter 7 is our map, not Euler's plate. Where a real manuscript page exists and its rights are clear, Appendix H describes it and links to the archive that holds it, rather than reproducing a file nobody here has looked at.
There are no portraits in this book, and that is deliberate. Not one portrait of anybody in this history is cleared for reuse, so you get a name, a place and a date instead of a face.
A few reading tools, available everywhere: every section has a Listen button to hear it read aloud, and every underlined term shows its meaning on hover.
It uses your device's own voice, not a recording. That has three consequences worth knowing.
It remembers where you stopped. Start a section, stop part way, come back later, and a Resume listening button appears on that section and picks up from the paragraph you were on.
It keeps going with the screen off, and your lock screen shows which section is playing, with pause and play controls.
There is no audio file to download, and this book is not going to pretend otherwise. Because the voice is generated on your device, the Listen button works with no signal at all, but only while the page is open. If you need audio somewhere with no reception, open the book once while you still have signal and leave the tab open.
The top bar holds the rest: Aa sets text size, line spacing, and the reading voice, Night flips to a dark theme, and ↓ Print / PDF prints the whole book or one section. To search, use the Search box in the sidebar, or your browser's own find-on-page (Ctrl+F, or Cmd+F on a Mac).
Everything in this book works offline: open it once with internet and it keeps working without one. The links out to Discrete Math and Linear Algebra are the one exception, since they go to another book.
Quick questions, quick answers
Do I have to read it in order?
No. Part I rewards reading in order, because the two halves of the subject keep colliding and the collisions build up. The other two parts are built to be arrived at from a link. Every chapter stands on its own.
Where does this fit with my course?
Appendix I maps all {placements} sections of your Discrete Math and Linear Algebra course to the history that belongs beside each one, and every section number there is a link you can click.
Is any of this on the test?
Ask your teacher. This book exists to make the mathematics make sense, not to add memorization. Knowing that matrix means womb will not be graded, but it may make the subject feel less like it arrived from space.
Why do some claims have a Disputed label?
Because the sources disagree and the disagreement is not settled. Appendix F lists all {conflicts} of them with every position side by side, and says what document would settle each one.
Can I check the math for myself?
Yes, and you should. Appendix E works every calculation in this book and prints the output of the scripts that checked it, unedited.
My textbook says something different.
It might be right, and it might be one of the {s['myths'] if 'myths' in s else 122} claims in Appendix G that turn out not to survive contact with the documents. Check there first, then tell your teacher either way.
I think I found a mistake.
Tell your teacher. A sharp catch helps every reader after you, and this book is more checkable than most, on purpose.
Jump straight to Chapter 1, which starts with a book finished in 1848 by a man who did not live to see it printed, and which contains the definition of a set your course opens with. The style key just below is here for whenever you want to know what a mark means.
Front matterThe style key: what every mark means
This book uses color and shape on purpose, not for decoration. Here is the full key. Every mark below is one this book actually uses, and a check in the build fails if one of them stops appearing.
The goals box. One opens each of the three parts with a short list of what you are about to be able to do. Navy bar.
A pointer to the place to begin, or a scene set before any names and dates arrive. Filled soft sky-blue panel.
A plain statement about the book itself: how it is put together, what the pictures are, how the Listen button works. Tangerine bar.
A claim about the evidence, with the numbers behind it. 21 of this book's 48 logged disagreements between sources are still open, and Appendix F lists every one with what would settle it. Steel-blue bar on a pale panel.
A question placed before the passage that answers it, using a wrong answer somebody competent once published as the distractor. Guess first, then open the reveal. 9 of these, one per chapter. Guessing wrong and finding out is how the answer sticks.
The chapter's last word: one open recall question, with the answer and a link into the part of your course it serves. 9 of these too.
A caution: a story you have probably heard that is wrong, or a limit on what the evidence can carry. Amber bar, the book's one caution color. Appendix G is 122 of these, checked against the documents.
The marks inside a sentence
| Mark | What it means |
|---|---|
| S001 | A source pointer. Every factual sentence carries one. Click it and you land on the full reference in Appendix J, where 266 sources are listed. It is a box, not a superscript, so it is big enough to tap on a phone. |
| 1740 to 1755 Disputed | The sources disagree and it is not settled. The word is written out rather than left to italics, because a screen reader does not announce italics and the large-print setting removes them. 27 rows in this book carry it. |
| dates unknown Not known | Nobody is arguing; the record is simply silent. That is a different thing from a dispute, and it is much the commoner of the two: 199 rows carry this one. An empty cell in this book means nobody here knows, never that nobody looked. |
| Bold navy | The point of the sentence, or a new technical term at the moment it is defined. |
| Italic | A title, a word in another language, or ordinary emphasis. Italic does not mean disputed. It used to in an earlier draft; the visible label carries that job now. |
| Math set in a serif | Every symbol and equation is typeset, never written as source code, and every fraction stacks. This book carries 385 of them, for example , the line Hamilton cut into Broom Bridge. |
Dates, and why some of them look strange
England kept the Julian calendar until 1752, while most of Europe had moved to the Gregorian one in 1582. For most of the seventeenth century that puts English dates ten days behind continental ones, and the English year began on 25 March rather than 1 January. This matters for Newton. A letter dated 1 February 1665 in London and one dated 11 February 1665 in Paris can be the same day, and a London document dated February 1665 may belong to what we would call 1666.
Where a date is given in this book in the form 1665/6, that is the convention for a date between 1 January and 24 March, when the English and modern year numbers disagree. Where it matters to an argument, the chapter says which calendar it is using.
Abbreviations, spelled out
| Short | Full form |
|---|---|
| BCE / CE | Before the Common Era / Common Era. The same years as BC and AD, named without the theology. |
| c. | From circa, "around". An approximate date. |
| fl. | From floruit, "flourished". Used when birth and death years are unknown but the person is documented working in a period. |
| O.S. / N.S. | Old Style / New Style: the Julian and Gregorian calendars, as above. |
| IPA | International Phonetic Alphabet, the pronunciation notation in Appendix C. |
| MS / MSS | Manuscript / manuscripts. A handwritten document, as opposed to a printed one. |
A note on dashes: this book uses commas, colons, and parentheses instead of long dashes on purpose, so a dash is never confused with a minus sign.
A note on spelling: US spelling throughout, except inside a direct quotation, inside an archive's own wording of its rights terms, and in a proper name. Those three keep whatever the original used.
Front matter
Table of Contents
One place to jump anywhere. Open a part to see what is in it; every line is a link. Part I is the story and rewards reading in order. Parts II and III are reference, built to be arrived at from a link rather than read front to back.
Front Matterhow to read this book▸
Part I · The Story9 chapters, about 3 hours end to end▸
- Part I Opener & Objectives
- Chapter 1. The words are younger than you are
- Chapter 2. The triangle with five names, and the one thing Pascal added
- Chapter 3. Chance: from a gambler's notebook to five axioms
- Chapter 4. The algorithm came first, the word came last
- Chapter 5. The planets, a half-translated word, and the machine age
- Chapter 6. Built to win an argument
- Chapter 7. The picture, and the table underneath it
- Chapter 8. The people, and the twelve doors out of the math room
- Chapter 9. A womb, an archer, and one man's exclamation point
Part II · The Master Timeline583 dated events, ten eras▸
- Part II Opener & Objectives
- How to read the timeline
- Before 500: counting, chance, and the first systems
- 500 to 1400: India, the Islamic world, and China
- 1400 to 1650: print, gamblers, and the algebraic turn
- 1650 to 1800: the problems get their shape
- 1800 to 1850: elimination, determinants, and a new word
- 1850 to 1880: logic becomes algebra, and algebra gets a matrix
- 1880 to 1910: foundations, cardinals, and the first chains
- 1910 to 1935: axioms, spaces, and a settled probability
- 1935 to 1960: war, computers, and the machine age
- 1960 to now: networks, search, and machine learning
Part III · Reference306 people, 142 terms, 266 sources▸
- Part III Opener & Objectives
- Appendix A. People register
- Appendix B. Symbols, words, and where they came from
- Appendix C. Pronunciation quick reference
- Appendix D. The doors out of the math room
- Appendix E. Every number in this book, re-derived and checked in code
- Appendix F. Disagreements between sources
- Appendix G. Stories that check out, and stories that do not
- Appendix H. Pictures, and who owns them
- Appendix I. Where each story goes in your course
- Appendix J. Bibliography
- Appendix K. What is still open, and what to request
- Glossary
Back Matterhow this book was built▸
The story: two subjects, nine chapters, and the people who were not trying to build them
Almost nobody in this story was building the mathematics your course teaches. Pingala was counting the rhythms of Sanskrit poetry. Cardano was trying to win at dice. A clerk in Han China was dividing up bundles of rice. Euler was answering a puzzle about a walk through a town. Markov was trying to win an argument about free will. The discrete half and the linear algebra half of your course grew up apart, kept colliding, and finally merged in Unit 4.
- Say where the words in Unit 1 came from, and roughly how old they are.
- Name at least four places outside Europe where this mathematics was worked out first.
- Explain what Pascal added to the triangle that carries his name, and what he did not.
- Say why the algorithm in Section 3.6 is two thousand years older than its name.
- Explain what argument Markov built his chains to win, and how they won it.
- Say what is in Euler's founding paper on graphs, and what is not.
- Tell a story about this mathematics arriving in another subject, with its source.
- Connect a historical episode to the section of your own course where it belongs.
Chapter 1
The words are younger than you are
You are about to learn a vocabulary that feels ancient and is not. The word "union," in English, in the sense your book uses it, is from 1912. "Complement" is from 1914. "Empty set" turns up in 1919 in a paper about quadratic forms, with nobody bothering to explain it. The phrase "set theory" reaches English in 1926, which means your great-grandparents were alive before the subject had its English name. Meanwhile the one part of Unit 1 that students find hardest, counting infinite collections, is the part that broke the people who invented it. This chapter tells you where each word and each symbol came, who chose it, and what it cost them.
Sets got a modern definition in a manuscript Bolzano finished in the summer of 1848 and did not live to see printed. Boole and De Morgan, working in England in the same year, 1847, turned logic into algebra and gave us the complement and the universe it lives inside. The overlapping circles were already 200 years old when Venn improved them in 1880, and he said so. Cantor and Dedekind, writing to each other from 1872, discovered that infinity comes in sizes, and Dedekind made the paradox into the definition. Then Russell posted one paragraph to Frege in 1902 and the foundations gave way. Zermelo's repair in 1908, seven rules, one of which says you may only cut a new set out of an old one, is why your Unit 1 is safe. The words and symbols were all fixed afterwards, by a handful of named people, mostly between 1889 and 1939.
Unit 1's vocabulary looks ancient. Union, intersection, complement, the empty set. Guess the year the phrase "set theory" first appears in English.
I have a guess▸
1926, in the Annals of Mathematics, written by Orrin Frink (S055).
The rest of the vocabulary is no older. "Union" in the sense your book uses it is 1912 (S055). "Complement" is 1914 (S055). "Empty set" turns up in 1919 in a paper about quadratic forms, with nobody bothering to explain it (S055).
If you guessed the Greeks, or the 1600s, you guessed what the typography suggests. Your great-grandparents were alive before this subject had its English name.
A country house in Bohemia, and a book nobody read
In 1847 Bernard Bolzano, whom the Austrian authorities had already removed from his university post, went to stay at Villa Liboch near Melnik and started writing about infinity. He finished in the summer of 1848, at 67, and it was the last summer of his life. Three years later a friend, Frantisek Prihonsky, edited the manuscript into a book and got it printed in Leipzig by C. H. Reclam sen. The title page says it plainly: Herausgegeben aus dem schriftlichen Nachlasse des Verfassers, edited from the author's written remains. On page 3, in section 4, Bolzano defines a Menge as "ein Inbegriff gewisser Dinge" in which "die Art der Verbindung oder Anordnung ihrer Teile gleichgueltig ist," a totality of certain things in which the manner of connection or arrangement of the parts is a matter of indifference. That is your Definition 1.1.1, seventy-odd years early. Sections 20 and 21 look straight at infinite collections that can be matched, one to one, with proper parts of themselves.
Why this matters in your course. Course section 1.1, "Sets, the Empty Set, and Subsets," opens with a definition that says a set is a collection where order does not matter. Bolzano wrote that sentence in 1848. And Course section 1.4, "Cardinality," is built on comparing sizes by matching, which is the exact move Bolzano was staring at when he called it a paradox.
The receipt. Paradoxien des Unendlichen (Leipzig: C. H. Reclam sen., 1851), title page and section 4, p. 3, read from the Internet Archive scan (S044). The 1847 start at Villa Liboch and the summer 1848 completion at the age of 67 come from Prihonsky's own foreword (S044).
So what? Being first is not the same as being heard. Bolzano wrote the modern definition of a set in German, posthumously, from a small Leipzig house, and the subject went on without him for forty years. The one person who read him carefully and said so in print was Dedekind, in 1888, who cited "Bolzano (Paradoxien des Unendlichen, section 20, 1851)" by section number (S043, pp. 19-20).
Two books, one year, and a law that may not be his
1847 produced two English books on the algebra of logic within months of each other. George Boole, who had no university degree and was running a boarding school in Lincoln because his father's shoe shop had failed when Boole was 16, published an 82 page pamphlet called The Mathematical Analysis of Logic. In it he uses 1 for the Universe and 1 - x for everything that is not x. From that he derives the law the whole system rests on: "The result of a given act of election performed twice, or any number of times in succession, is the result of the same act performed once... Thus we have xx = x, or x squared equals x."
The same year produced the second book. Augustus De Morgan was born in Madurai, blind in one eye since infancy, and made a professor at 21 of a university that had only just been invented. He published Formal Logic, or, The Calculus of Inference, Necessary and Probable with Taylor and Walton in London. On page 38 he writes: "Every name is treated in connection with its contrary or contradictory name; the distinction between these words not being made."
Now the awkward part. The two laws that carry De Morgan's name are not in that book. Jeff Miller's first-use compilation puts the explicit statement in a paper of 1850, in the Transactions of the Cambridge Philosophical Society, volume 9, pages 79-127, in the form "The contrary of an aggregate is the compound of the contraries of the aggregants." Miller, citing William and Martha Kneale's The Development of Logic (1962), also reports that the same laws "occur explicitly" in William of Ockham's Summa Totius Logicae, five centuries earlier. And the phrase "De Morgan's Laws" itself does not appear in print until a 1945 index to the Journal of Symbolic Logic, 74 years after De Morgan died.
Why this matters in your course. Course section 1.2, "Unions, Intersections, and Complements," carries two named boxes: "The algebra of sets" and "De Morgan's laws." Boole is the ancestor of the first, and the U in your notation table, the universal set, is De Morgan's idea that you cannot say "not A" until you have said what universe you are. "Not a prime number" means something different if the universe is the whole numbers under 20 or all the real numbers.
The receipt. Boole's 1 for the Universe and 1 - x for the complement, and the x squared equals x derivation, are at pp. 15 and 16 of the Project Gutenberg PDF of the 1847 pamphlet (S045). De Morgan's title page and his pairing of every name with its contrary are at the title page and p. 38 of the 1847 Internet Archive scan (S047). The 1850 location, the Ockham attribution through Kneale and Kneale, and the 1945 first printing of the phrase are all from Miller's DE MORGAN'S LAWS entry (S055, d.html). The paper's exact position in volume 9 was confirmed from the volume's own contents page, though the paper itself was not read in this book, so this rests on the volume's catalog record and not on De Morgan (S069, volume contents page).
So what? A named law is a bookmark, not a birth certificate. De Morgan did state the laws, in a paper whose title is 27 words long and ends with probability. Whether he was first is unsettled, and the name is a twentieth century convenience.
Everybody's circles
John Venn was a lecturer in the Moral Sciences at Gonville and Caius College, Cambridge, which is to say a philosopher, and in July 1880 he published "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine. The book version, Symbolic Logic, followed from Macmillan in 1881. Its contents page describes chapter V as "Diagrammatic Representation. Defects of the familiar Eulerian scheme; and proposal of a new scheme."
Read that again: the familiar Eulerian scheme. Venn is not claiming to have invented anything. He is reforming a tool everybody already used. And the tool was old. Circles for testing the validity of a syllogism enter print in Johann Christoph Sturm's Universalia Euclidea in 1661. Leibniz drew circle and ellipse diagrams around 1686 in an 18 page manuscript, De Formae Logicae Comprobatione per Linearum Ductus, which sat unprinted until 1903, over 200 years later. Euler put essentially the same circles into a bestseller: 234 letters written from 1760 to 1762 to Princess Charlotte Ludovica Luisa of Anhalt-Dessau, published as a three volume book, translated into most European languages, with 12 printings of the French edition alone. Deborah Bennett, whose 2015 chapter is the best account of the priority chain, is blunt about Euler: "he did not claim originality; in fact, the diagrams were contained in study materials intended to represent the state of current knowledge." Bennett also checked a popular claim about Giulio Pace and killed it: "a thorough examination of a 1619 edition of Pace's translation and commentary revealed no Venn-like diagrams."
Venn even did the survey himself. Of 60 logic treatises from the previous century that he consulted, 34 already used diagrams, nearly all of them Eulerian.
So what did Venn do? One thing, and it is the thing your book teaches. Bennett puts it in six words: "every one of Venn's diagrams began with the same drawing." You draw the full overlapping frame first, before you know anything, and then you shade in what you learn. Venn's own objections to the old circles were that a single proposition can be drawn more than one way, that you can only draw the Euler picture after you have already solved the problem, and that the scheme cannot handle disjunctive statements.
Why this matters in your course. Your book's method box "Reading a Venn diagram word problem" in Course section 1.2 works exactly the way Venn insisted it should. You draw all the overlapping regions first, then fill in the numbers you are given, then deduce the rest. That procedure is 1880, not 1768, and it is the whole of Venn's contribution.
The receipt. Sturm 1661, Leibniz c. 1686 and its 1903 printing, Euler's letters and their reception, the Pace debunking, the 34 of 60 count, Venn's three objections, and the same-drawing insight are all Bennett, pp. 106-113 (S051). "The familiar Eulerian scheme" and "a new scheme of diagrammatic notation" are Venn's own words, from the contents page and p. xxix of the 1881 first edition (S048). Euler's Letter CIII, "Of Syllogisms, and their different Forms," was confirmed in Hunter's second English edition, London 1802 (S049).
So what? The diagram is not Venn's. The method is. And the name was somebody else's decision entirely: the Oxford English Dictionary's earliest citations, through Miller, are J. N. Keynes writing "Dr Venn's diagrams" in 1884 and C. I. Lewis writing "the Venn diagrams" in 1918 (S055, v.html).
Five days in December 1873
Georg Cantor and Richard Dedekind met in Switzerland in 1872 and started writing to each other. The Stanford Encyclopedia of Philosophy entry on Dedekind says the letters "amount to a joint exploration of the notions of set and infinity." On 2 December 1873 Cantor wrote to Dedekind that he did not know whether the real numbers could be matched one to one with the counting numbers. On 7 December, five days later, he wrote again with the proof that they cannot.
The paper came out in 1874 in Crelle's Journal fur die Reine und Angewandte Mathematik, volume 77, pages 258-262, under the title "Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen," which translates roughly as "On a property of the collection of all real algebraic numbers." It sounds like filing. Inside are two results: first, that all the real algebraic numbers can be arranged in a single sequence, even though they are everywhere dense; second, that given any sequence of distinct reals and any interval, you can find a real in that interval that is not in the sequence. Put those together and you have both the existence of transcendental numbers and the fact that the reals cannot be listed.
Dedekind's part gets forgotten. He supplied Cantor with the proof that the algebraic numbers are countable, which is half the 1874 paper. And in 1888, in Was sind und was sollen die Zahlen?, Dedekind wrote down the definition that a smart 16 year old can use tomorrow. Definition 64: "A system S is said to be infinite when it is similar to a proper part of itself; in the contrary case S is said to be a finite system." Then he did the thing almost nobody does. He named who got there first: "The property which I have employed as the definition of the infinite system had been pointed out before the appearance of my paper by G. Cantor (Ein Beitrag zur Mannigfaltigkeitslehre, Crelle's Journal, Vol. 84, 1878)," and "as also by Bolzano (Paradoxien des Unendlichen, section 20, 1851)."
Why this matters in your course. Course section 1.4 is "Cardinality." For finite sets, cardinality is counting. Cantor's 1874 paper is the moment somebody proved that counting keeps working for infinite sets and gives different answers. Dedekind's Definition 64 is the cleanest statement of what "infinite" even means, and it is stated entirely in terms of subsets and one to one matching, which are Unit 1 tools.
The receipt. The two December 1873 letters are from Dauben (S053). The 1874 paper's title, journal, volume, pages, and both proofs were read from a facsimile of the Crelle printing (S040). Definition 64 and the credits to Cantor and Bolzano are at pp. 31 and 19-20 of Beman's English translation (S043). The description of the correspondence and Dedekind's supply of the countability proof are from the Stanford Encyclopedia (S059).
So what? Zermelo, who was in the room for the next act, said modern set theory was "created by Cantor and Dedekind." Two names (S059). Cantor gets the documentary; Dedekind gets a footnote and wrote the better definition.
The proof you have seen is not the proof that happened
Search the 1874 paper for the word "diagonal" and it is not there. The 1874 argument is a nested-intervals argument: you take a supposed list of all the reals, and you trap a missing one inside a shrinking sequence of intervals. The diagonal argument, the grid of decimals where you walk down the main diagonal and change every digit, came seventeen years later, in 1891.
Dauben adds a reading of the flat title that is worth telling and worth labeling as a reading. Kronecker, who had been one of Cantor's teachers in Berlin, sat on the editorial board of Crelle's Journal. Dauben writes: "Had Cantor been more direct with a title like 'The set of real numbers is non-denumerably infinite,' he could have counted on a strongly negative reaction." That is Dauben's interpretation of a tactical decision. The paper itself says nothing about it.
Why this matters in your course. Course section 1.4 asks you to compare sizes of sets. The order in which the ideas arrived tells you something useful about how mathematics is done: the flashy argument is usually a later, cleaner rewrite of a clumsier first proof that already worked. Nested intervals in 1874 did the job. The diagonal in 1891 did the job beautifully.
The receipt. The absence of "diagonal" from the 1874 paper, and the nested-intervals structure of the argument, were read from the facsimile at pp. 260-262 (S040). Kronecker's position on the Crelle board and the quoted sentence about the title are Dauben's (S053).
So what? This book could not open Cantor's 1891 diagonal paper, Uber eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der DMV 1, pp. 75-78. No open scan was reachable. So this book will tell you the 1891 paper exists and what date it carries, and will not quote a line of it as unverified.
The letter that broke Frege, and the seven rules that fixed it
Cantor's definition of a set, in the 1895 Beitrage, is generous. In James Meyer's 2024 English rendering it reads: "By a 'set' we understand any collection into a whole M of definite and separate objects m of our intuition or our thought." Any collection. You get to decide what goes in the bag.
Bertrand Russell noticed what that permits. Sometime in the spring of 1901 (he said June in 1944, spring in 1959, and May in 1969, so he could not keep his own story straight) he found the contradiction. On 16 June 1902 he wrote to Gottlob Frege, who had a two volume work at the printer whose whole foundation was exactly this kind of unrestricted set formation. Russell's letter: "Let w be the predicate of being a predicate that cannot be predicated of itself. Can w be predicated of itself? From either answer follows its contradictory." And then, in the language of classes: "there is no class (as a whole) of those classes which, as wholes, are not members of themselves."
Frege replied with both dismay and admiration, and conceded in writing that "the transformation of the generality of an identity into an identity of ranges of values... is not always permissible, that my law V... is false."
The repair came from Ernst Zermelo in 1908, in Untersuchungen uber die Grundlagen der Mengenlehre I. Seven axioms: Extensionality, Elementary Sets, Separation, Power Set, Union, Choice, Infinity. He wrote them in words rather than symbols, because the quantifier notation then in fashion annoyed him. Two of the seven are load-bearing for your Unit 1. Elementary Sets is the axiom that asserts the empty set exists, along with singletons and unordered pairs: the empty set is in your course because somebody put it there on purpose. Separation says that "for any given set and any given 'definite' property of elements, one can 'separate' out from" the set those that satisfy it. That is the whole fix. You may not conjure a set out of a property alone. You may only cut a piece out of a set you already have. Zermelo then proved, inside his own system, that "Every set M possesses at least one subset M_0 that is not an element of M," which blocks the paradox.
Why this matters in your course. Every time your book writes "the set of all outcomes where...," it is carving a subset out of the sample space S, which already exists. That is Separation, and it is why Unit 1 never falls over. Definition 1.1.2, the empty set, is Zermelo's Elementary Sets axiom in student clothing.
The receipt. The dating confusion, the letter of 16 June 1902, both quoted passages from Russell, and Frege's reply are from the Stanford Encyclopedia entry on Russell's paradox, read in full (S056). The seven axioms, the wording of Separation, and the blocking theorem are from the Stanford Encyclopedia entry on Zermelo's axiomatization, read in full (S057). "In words rather than symbols" is MacTutor's phrase (S065). Cantor's definition is quoted from Meyer's 2024 translation of section 1 of the Beitrage (S042).
So what? Frege's book was at the printer. He published anyway, with the contradiction acknowledged. That is what intellectual honesty costs and looks like.
A Latin booklet in Turin, and a Norwegian letter in Paris
In 1889 Giuseppe Peano, then a young assistant at the University of Turin, published a booklet called Arithmetices principia, nova methodo exposita. The whole thing is in Latin. MacTutor calls that choice "an act of sheer romanticism." On page vi he states the plan: "Ideas omnes quae in arithmeticae principiis occurrunt, signis indicavi, ita ut quaelibet propositio his tantum signis enuncietur." I have indicated by signs all the ideas that occur in the principles of arithmetic, so that any proposition is stated with these signs alone. He lists ten logical signs. On page x he explains one of them in six words: "Signum epsilon significat est." The sign epsilon means "is."
That is the symbol on row 1 of your Appendix B.1. It is a Greek letter chosen to be the first letter of a Latin verb, by an Italian who later invented an artificial language called Latino sine flexione, Latin without the grammar, and published his life's work in it, which is part of why almost nobody read it. Russell, meeting Peano's notation at the Paris Congress in 1900, said "his notation afforded an instrument of logical analysis such as I had been seeking for years." Peano's own students said his work was "above our heads."
The empty set symbol has a different and better-documented story. It first appears in print in Bourbaki's Elements de mathematique, Paris, 1939, on page 4, in the phrase "la partie vide de E," the empty part of E. Bourbaki was not a person: it was the collective pseudonym of a group of French mathematicians. Fifty-three years later one of them, Andre Weil, claimed the glyph personally in his autobiography, at page 114: "The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar." Weil also recorded that the group felt "it was high time to fix these notations once and for all."
So the symbol is not a zero and it is not a Greek phi. It is a letter of the Norwegian and Danish alphabet, and it is in your textbook because one Frenchman in a semi-secret writing collective could read Norwegian.
Cantor made a comparable choice with a comparable reason. Aleph-null first appears in Mathematische Annalen XLVI (1895), p. 492, printed "Alef-null." In a letter of 30 April 1895 Cantor explained the choice of a Hebrew letter: "it seemed to me that for this purpose, other alphabets were [already] over-used."
Why this matters in your course. Four rows of your notation table land in this scene: the membership epsilon (1889), the empty set glyph (1939), the braces for sets that Zermelo printed in 1908 at p. 263, and the cardinality idea behind aleph-null. These are not eternal marks. Each was picked by a named person for a stated reason inside a 50 year window.
The receipt. The Latin quotations and the ten signs were read directly from the 1889 booklet, pp. vi to xi (S050). Bourbaki 1939 p. 4, Weil's two quoted sentences, the aleph-null page reference, Cantor's letter of 30 April 1895 in Martin Davis's translation, and Zermelo's braces are all from Miller's symbols compilation (S054). "An act of sheer romanticism," Russell's praise, and the students' verdict are MacTutor's (S064).
So what? Notation is a set of decisions, and decisions have authors. Also, one caution this book will not gloss over: the standard one-liner that Peano invented the union and intersection symbols in the 1888 Calcolo geometrico rests on Cajori, reported by Miller. In the 1889 booklet, which was read directly here, the same two glyphs are glossed in Latin as et and vel, that is, logical "and" and "or," not union and intersection of sets. No open copy of the 1888 book was reachable, so this book says the question is open (S054 against S050; see 1.10, item 2).
Halle, and what the record says
Here is the version that circulates online: Kronecker's persecution drove Cantor mad. Joseph Dauben, who wrote the standard scholarly biography, says that is wrong, and says so in one sentence: "Although frustration over his lack of progress on the Continuum Hypothesis or stress from Kronecker's attack may have helped to trigger the breakdown, it now seems clear that such events had little to do with its underlying cause." A psychiatrist who treated Cantor diagnosed cyclic manic depression, an illness with its own course. Dauben's wider objection is worth quoting too: "The more sensational accounts of Cantor's life distort the truth by trivializing the genuine intellectual concerns that motivated some of the most thoughtful contemporary opposition to Cantor's theory."
This book states what the scholarship states and stops there. It does not diagnose, it does not dramatize, and it does not treat an illness as a plot device.
The facts around it are these. Cantor's first serious breakdown was in May 1884 and lasted somewhat more than a month. There were further episodes, and MacTutor records leave in the winters of 1902-03, 1904-05, and 1907-08. His youngest son died on 16 December 1899. He was hospitalized in 1908, and at the Halle Nervenklinik from 1913 to 1918. He entered the sanatorium for the last time in June 1917 and died on 6 January 1918.
Kronecker's attacks were real and ugly. He called Cantor a "scientific charlatan," a "renegade," and a "corrupter of youth," and he delayed publication of Cantor's 1878 paper on dimensional invariance. They were also, Dauben insists, serious mathematics rather than spite.
And the man himself is stranger and more interesting than the myth. Cantor believed the transfinite numbers had come to him as a message from God and wrote to Pope Leo XIII about the infinite. He wrote: "My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer." Dauben reports that Cantor viewed his episodes as divinely inspired, and that the enforced isolation of hospital gave him uninterrupted time to think. In between, he founded the Deutsche Mathematiker-Vereinigung in 1890 and chaired its first meeting at Halle in September 1891.
Why this matters in your course. Unit 1.4 is where students meet the idea that some infinities are bigger than others, and it is the point in the course where the mathematics feels genuinely strange. It is worth knowing that it felt strange to the people who found it too, and that the resistance was mathematical, not merely personal.
The receipt. Both Dauben quotations, the diagnosis, the May 1884 breakdown, the 1908 and 1913-1918 hospitalizations, Kronecker's three insults, the delay of the 1878 paper, the letter to Pope Leo XIII, and "my theory stands as firm as a rock" are all from Dauben's paper, read in full (S053). The 1899 death of his son, the winters of leave, the June 1917 admission, the 6 January 1918 death, and the DMV dates are from MacTutor (S062).
So what? The Kronecker story is popular because it is tidy: villain, victim, tragedy. The record is less tidy and more human. Say what the sources say and no more.
The symbol for the empty set is not a zero and not a Greek phi. What is it, and how did it get into your textbook?
Show the answer▸
It is a letter of the Norwegian and Danish alphabet. It first appears in print in Bourbaki's Elements de mathematique, Paris, 1939, on page 4, in the phrase "la partie vide de E" (S054).
Bourbaki was not a person. It was the collective pseudonym of a group of French mathematicians. Fifty-three years later one of them, Andre Weil, claimed the glyph personally, at page 114 of his autobiography: "The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar" (S054).
So the mark sitting in Course section 1.1 is there because one Frenchman in a semi-secret writing collective could read Norwegian. Notation is a set of decisions, and decisions have authors.
Chapter 2
The triangle with five names, and the one thing Pascal added
You are about to spend a unit computing , , and , and somewhere in it a triangle of numbers will appear with Pascal's name on it. That name is a mistake, or at least an accident. The table was written down in India, in China, in Baghdad, in Marrakesh, and in Germany and Italy before Blaise Pascal was born, and Pascal never called it his. What he did add is worth more than a name: he proved things about it with an argument that starts at one row and steps forward forever. This chapter gives you the receipts for all of that, including a sixth century arithmetic error you can catch yourself in under a minute.
Counting problems come from odd places: Sanskrit poetry, perfume recipes, silk threads, a stopped game. The array we call Pascal's triangle was written down in at least five civilizations first. Halayudha describes it in tenth century India as meru-prastara, the Mount Meru spread. Al-Karaji had it in Baghdad before 1029 and al-Samaw'al copied twelve rows around 1150. Ibn Mun'im used it in Marrakesh before 1228 to count Arabic words. Jia Xian had it in China around 1050, and by 1303 it was printed there as "the Old Method." Stifel printed it at Nuremberg in 1544. Pascal wrote his treatise in 1654, printed 1665; his contribution was a proof by two lemmas running "to infinity." The name arrived in 1708.
The array of numbers your course calls Pascal's triangle was written down in at least five civilizations before Pascal was born. He knew that, and said so. Name the one thing he added that none of them had.
I have a guess▸
A proof. Two lemmas, stated to run "to infinity," which is one of the earliest recognizable arguments by induction (S016).
Not the array: Halayudha describes it in tenth century India as meru-prastara, the Mount Meru spread (S011, S013). Not the rule for building it, not the connection to counting, not the uses in algebra. Al-Karaji had it in Baghdad before 1029, Ibn Mun'im used it in Marrakesh before 1228 to count Arabic words, Jia Xian had it in China around 1050, and Stifel printed it at Nuremberg in 1544 (S011, S013).
If you guessed "the triangle" you gave the answer most textbooks give, and it is the one thing Pascal did not do. The name arrived in 1708, fifty-six years after he wrote.
A poet counts syllables, and a fact is born by repetition
Sometime in the second century BCE, in India, a writer called Pingala set out the rules of Sanskrit meter in the Chandahsastra. A Sanskrit meter is a string of syllables, each of them either light or heavy. Ask how many meters of syllables exist, and you are asking how many binary strings of length exist. Ask how many have exactly light syllables, and you are asking for a binomial coefficient. Pingala's tradition made six standard problems out of this and gave them names: prastara (list every form), nashta (recover a form from its serial number), uddishta (find the serial number of a form), lagakriya (count the forms with a given number of light syllables), sankhya (the total count), and adhvayoga (how much space the whole list takes to write out). Jayant Shah calls it "mathematics for its own sake." A thousand years later, in the tenth century, a commentator named Halayudha explains how to build the triangular array: one square cell at the top, then two cells below it extending half way on both sides, and so on, each cell the sum of the two above it. He says this construction comes from Pingala's final sutras, 8.34 and 8.35. (S002)
Why this matters in your course. Lagakriya is exactly , which is Definition 2.1.3 in your book, arriving from poetry rather than from gambling. And prastara, listing every arrangement, is the multiplication principle of Course section 2.1 done by hand.
The receipt. Shah goes back to the sutras themselves and finds that they will not carry the weight put on them. Sutras 8.34 and 8.35 ("pare purnam", "pare purnam iti") express the doubling recursion for the total number of meters, not a recipe for a triangle. His verdict: "the sūtra by itself has no information for carrying out the construction described by Halāyudha." He then identifies a different Pingala sutra, "ekottarakramasah | purvaprkta lasamkhya", as the one that does describe the binomial coefficients by repeated partial sums, and points out that Halayudha does not quote it. (S002)
So what? "Pingala invented Pascal's triangle around 200 BCE" is the single most repeated sentence in this whole subject, and it rests on one commentator's attribution written a thousand years after the fact, which makes it a claim rather than a fact. Jeff Miller's reference page states the traditional version and credits Roger Cooke and A. W. F. Edwards for it, so you now have two respectable sources pointing opposite ways. (S011) (S002)
Sixteen perfumes, four at a time, and a mistake that survived 1,300 years
In the sixth century CE, the astronomer Varahamihira wrote the Brhat Samhita, and chapter 77 of it is titled gandhayukti, the preparation of perfumes. Verses 13 and 14 name sixteen aromatic substances: Ghana, Valaka, Saileyaka, Karpura, Usira, Nagapuspa, Vyaghranakha, Sprkka, Agaru, Madanaka, Nakha, Tagara, Dhanya, Karcura, Cola, and sandal. Choose any four of them. Give each of the four a different strength, one part, two parts, three parts, or four parts. How many perfumes is that? Verse 17 answers: "The process gives us 174720 (4000+70000+100000+720) different varieties of perfumes." (S010)
Why this matters in your course. This is a textbook 2.1 problem with a real product at the end of it: choose from , then arrange them. It uses the combination rule and the permutation rule in the same breath, which is the exact skill the section is testing.
The receipt. In 1884, N. Chidambaram Iyer translated the chapter into English, and in a note after verse 21 he says the arithmetic is wrong: Varahamihira counted 96 arrangements for each set of four substances where the correct count is 24. Iyer gives the corrected total as 43680, or 28392 once you also rule out combinations of substances that must not be mixed. (S010, translator's note) The full working is in section 2.7 below.
So what? A famous astronomer got a counting problem wrong, nobody noticed for over a thousand years, and then a translator with a pencil caught it; you have the same pencil. It is also still open rather than settled, because "error" is Iyer's judgment and not Varahamihira's confession.
Mahavira writes down , then tests it on flavors
In the ninth century, in southern India, the Jain mathematician Mahavira wrote the Ganita-sara-sangraha. In chapter VI, "Mixed Problems", rule 218 is headed "The rule regarding the (possible) varieties of combinations (among given things)". It begins: "Beginning with one and increasing by one, let the numbers going up to the given number of things be written down". You write the numbers in order in an upper row, in reverse order in a lower row, then divide the product of the rightmost numbers on top by the product of the matching numbers below. That is , laid out as a physical procedure with two rows of digits. Rangacharya's 1912 footnote spells it out as over (S009)
Why this matters in your course. Rule 218 is your combination formula, written six hundred years before Europe had it, and written as an algorithm rather than as a formula. Students who find abstract often find Mahavira's two-row layout obvious, and it is the same thing.
The receipt. Example 219 asks, in Rangacharya's English, "Tell (me) now, O mathematician, the combination varieties as also the combination quantities of the tastes," and lists six of them: astringent, bitter, sour, pungent, saline, and sweet. Examples 220 and 221 apply the same rule to varieties of diamond necklaces and to flower garlands made from ketaki, asoka, campaka, and nilotpala blossoms. The verse immediately before the rule, verse 217, is a logic puzzle about five men each told by the same woman that she likes only him, and asks how many of her statements are true. (S009) The Indian line does not stop there: in 1356 Narayana Pandita finished the Ganitakaumudi, close to 930 verses, and gave chapter 13, the Ankapasa or "net of numbers", entirely to combinatorics, dating the book himself as Saka 1278. (S020)
So what? The combinatorics in your syllabus is not a European invention with an exotic prehistory. It is a worldwide subject, and one of its clearest statements sits inside a chapter of commercial arithmetic, between a puzzle about five liars and a section on splitting profits among merchants.
Baghdad: "you place on a board one and one below it"
Around 1150, in Baghdad, a man of about nineteen named al-Samaw'al ibn Yahya al-Maghribi wrote al-Bahir fi al-Jabr, The Splendid Book of Algebra. In it he prints the table of binomial coefficients down to its twelfth row. He does not claim it. He says where he got it, quoting his predecessor al-Karaji, who died around 1029 and whose own book is lost: "Al-Karajī says: in order to achieve that, you place on a board one and one below it". (S003, PDF p. 17)
Why this matters in your course. Twelve rows is not a doodle. It is a table deep enough to expand , and it is five hundred years before the Traite. Al-Samaw'al also proves the identity we write as for and , which is the kind of index law students use without noticing all through Unit 2.
The receipt. Bajri, Hannah, and Montelle published the first complete English translation of this section in 2015, working from the Ahmad and Rashed edition made from two Istanbul manuscripts, Aya Sofia 2718 (copied 1324) and Esat Efendi 3155. They say it is also the first publication and study of the diagrams in Aya Sofia 2718. They argue that his propositions follow the classical Euclidean five-part structure, that he demonstrates the passage from case to case , and that the diagrams themselves carry the inductive force, conveying "the similarity of all the proofs n = k to n = k + 1". (S003)
So what? Al-Samaw'al wrote on medicine, mathematics, astronomy, religion, theories of love, and erotica, so medieval mathematicians were not narrow people. And notice what an honest attribution looks like: he names his source, in the text, in a sentence you can still read.
Marrakesh: counting words with colored silk
In Marrakesh, in the twelfth and thirteenth centuries, Ahmad Ibn Mun'im al-Abdari, born in Denia in Andalusia and dead in 1228, asked how many words you can build from the twenty-eight letters of the Arabic alphabet. Only one of his mathematical books survives, Fiqh al-hisab, The Science of Calculation, and it contains "a chapter of 19 pages that contains the important combinatorial propositions and trends which will be rediscovered, in Europe, only in the 16th and 17th century." His model for the problem was physical: "the concrete model of threads of silk of different colours." He then used "the method of the arithmetic triangle, to enumerate all words which are possible to pronounce when one utilises the 28 letters of the Arab alphabet." (S015)
Why this matters in your course. Picking threads of distinct colors out of available colors is a combination, full stop. If your class ever needs a physical manipulative for Course section 2.1, it was invented in a textile city eight hundred years ago.
The receipt. The AMUCHMA newsletter carries the three quoted phrases above and names the scholarly literature behind them, all of it by Ahmed Djebbar. (S015) Djebbar's own survey adds that Fiqh al-hisab "is the first book in the history of mathematics to devote a whole chapter to combinatorial problems," and that a generation later, in the same city, Ibn al-Banna al-Marrakushi made "the announcement and the demonstration, for the first time to our knowledge, of the formula of factorials," and also demonstrated the relationship between figurate numbers and combinations. (S016)
So what? Ibn al-Banna is credited with the first statement and proof of the thing you write as . There is also a trap sitting in the middle of this story, because Ibn Mun'im of Marrakesh is routinely confused with a different mathematician, Muhammad ibn 'Abd al Mun'im, who worked at the court of Roger II in Sicily. (S016)
China: the table that was already called "the Old Method"
Around 1050, under the Song dynasty, Jia Xian devised the triangular array of binomial coefficients now called the Jia Xian triangle, carrying coefficients for equations up to the sixth degree, and used it for extracting roots of polynomials of degree higher than three. In 1261 Yang Hui, working at Qiantang in Zhejiang, wrote the Xiangjie jiuzhang suanfa, which absorbed about two thirds of Jia Xian's problems. In 1303 Zhu Shijie printed the array in the Siyuan yujian, the Precious Mirror of the Four Elements. And by then it had a name already: Britannica records that in Zhu's book "it was already called the 'Old Method.'" (S013, S014, S017)
Why this matters in your course. The 1303 table tabulates the binomial coefficients up to the eighth power, which is the row that you will generate in class. (S011) It is the same object as your , in a different script, three and a half centuries early.
The receipt. Jia Xian's own book survives only partially. The transmission chain is three near misses in a row: Yang Hui quoted him in 1261, the imperial Yongle dadian encyclopaedia copied Yang Hui in 1408, and a printed edition appeared in 1842. (S014) Break any one of those links and "Jia Xian's triangle" is a phrase nobody has ever heard.
So what? "The Old Method" is the best line in the whole naming story: in 1303, three hundred and twenty years before Pascal was born, this table was already old news in China. Be careful with one thing, though, because MAA Convergence states flatly that the knowledge "passed from China to India, then via Arab sources to Europe by the 16th century," and that sentence does not survive contact with the other sources: Indian prosodic combinatorics predates Jia Xian by centuries, and al-Karaji died around 1029, before Jia Xian flourished. (S017 vs S002, S003)
Provence, 1321, and the two Europeans before Pascal
In the spring of 1321, at the age of thirty three, Levi ben Gershon finished Maaseh Hoshev, "The Art of Calculation." Part 1 is "sixty-eight theorems and proofs in Euclidean style about arithmetic, algebra, and combinatorics," several of them permutation and combination results, and Shai Simonson credits him with "the earliest rigorous use of mathematical induction." That is three hundred years before Pascal. Levi wrote only in Hebrew and read neither Greek, Latin, nor Arabic; his Euclid came through Hebrew translations. He spent his whole life in Provence, wrote roughly a book a year from 1321 until his death in 1344, and his philosophy book Milhamot Adonai, "Wars of the Lord," was mockingly renamed by critics Milhamot im Adonai, "Wars with God." (S004)
Two centuries later the array surfaces in print in Europe. Michael Stifel, 1487 to 1567, "one of the best-known German cossists of the sixteenth century," put it in the Arithmetica Integra at Nuremberg in 1544, and he was not counting anything with it: he used it for root-finding. (S018) And in Italy the array carries the name of Niccolo Fontana, born in Brescia in 1499, who in the French sack of the city in 1512 had "his jaws and palate cleft by a sabre." He stammered for the rest of his life, people called him Tartaglia, the Stammerer, and he adopted the name and signed his books with it. He died in Venice on 13 December 1557. (S019)
Why this matters in your course. Every proof you will meet of Pascal's rule is an induction, and induction was being done on combinatorics problems in 1321 by a man writing in Hebrew in the south of France. Stifel matters for a different reason: the same array, used for a completely different job. A table is not a theorem, and knowing what a table is for is part of the work.
The receipt. Simonson's quoted phrases come from his 2000 article in Mathematics Teacher, read in full alongside his preprint. He notes that the early use of induction in Levi's work was first pointed out by Rabinovitch in 1970, and that Lange's 1909 critical edition with a German translation omits the section of problems entirely. (S004) Britannica supplies both Tartaglia quotations. (S019)
So what? Italy names this triangle after a man whose own name is an injury. And Levi is the answer to "why have I never heard of him": Simonson gives three reasons, that the work was ahead of its time, that the readership was limited to readers of Hebrew, and that his rationalist philosophy made him controversial, so language walls mathematics off as effectively as secrecy does.
Paris: two lemmas, and a name that arrives too late for its owner
Pascal's book is called the Traite du triangle arithmetique avec quelques autres petits traitez sur la mesme matiere, printed by G. Desprez in Paris in 1665, 177 pages. Pascal's own name for the figure, used throughout, is the triangle arithmetique, and his word for a row is a base. Its sections are headed "for numeric orders," "for combinations," and, third, for determining "the partis qu'on doit faire entre deux ioueurs qui ioient en plusieurs parties," the division of stakes between two players in an interrupted game. On page 7 he states the Douziesme Consequence, a ratio between two neighboring cells in the same base, and then does something new. (S001)
Why this matters in your course. This is the hinge between Unit 2's counting and Unit 2's probability. Pascal is using combinations to answer a gambling question, which is the same move your course makes when it goes from Course section 2.1 to Course section 2.2.
The receipt. Here is the argument, in Pascal's own French as the 1665 scan carries it, with its OCR errors preserved and a plain gloss under each line. (S001, OCR p. 7)
"Quoy que cette propofitiôn ait vne infinité de cas, i'eii donncray ▼nedemonftration bien courte, enfuppofant z lemme." "Although this proposition has an infinity of cases, I will give a very short demonstration of it, by assuming two lemmas."
"Le I. qui efteuident de foy-mefme, que cette proportion fè rencontre dans laicconde bafe" "The first, which is evident of itself, that this proportion is found in the second base."
"Le 2. que fi cette proportion fc trouue dans vne bafe quelconque, cUefe trouueranecenairement dans la bafe fuiuante." "The second, that if this proportion is found in any base whatever, it will necessarily be found in the following base."
"elle eft dans la féconde bafe, par le premier lemme, donc par le fécond elle eft dans la troifiefme bafe, donc dans la quatriefme, èc à l'infiny." "it is in the second base, by the first lemma; therefore by the second it is in the third base, therefore in the fourth, and so on to infinity."
Base case, inductive step, conclusion. In words. In ordinary French. With no symbols at all.
Then the naming, which Pascal had no part. In 1708 Pierre Remond de Montmort, in the Essay d'analyse sur les jeux de hazard, wrote "Table de M. Pascal pour les combinaisons", the first time the name is attached to the table. In 1730 Abraham De Moivre Latinized it in the Miscellanea analytica as "Triangulum Arithmeticum PASCALIANUM". In 1876 Edouard Lucas published a "Note sur le triangle arithmetique de Pascal et sur la serie de Lame". The plain English phrase "Pascal's triangle" only turns up in 1886, in Algebra by George Chrystal. (S011, following Edwards) Elsewhere it kept other names entirely: Italy says Tartaglia's triangle, China says Yang Hui's triangle, and India said meru-prastara, the Staircase of Mount Meru. (S011, S013)
So what? Miller quotes Edwards's verdict that "that the Arithmetical Triangle should bear Pascal's name cannot be disputed," because the Traite "brought together all the different aspects of the numbers," and that is a defensible position worth arguing about in class. What is not defensible is thinking Pascal drew it first. (S011)
Before it was anybody's triangle, what was the array called in India, and what does that name mean?
Show the answer▸
Meru-prastara, the spread of Mount Meru: the sacred mountain of Hindu cosmology, whose stepped shape the array copies (S011, S013). Halayudha uses it in his tenth century commentary on Pingala.
China calls it Yang Hui's triangle. Iran calls it Khayyam's triangle. Italy calls it Tartaglia's triangle. Every culture that met this array named it after the person they happened to be reading, which is the pattern this whole book keeps running into, and none of those people invented it either.
The counting you do in Course section 2.1 is older than every one of the names attached to it.
Chapter 3
Chance: from a gambler's notebook to five axioms
You are about to learn five axioms, three rules, and one famous formula, and you will learn them in about a week. It took the human race roughly three hundred years, from a doctor scribbling about dice in Milan to a 62 page book in German in 1933. This chapter shows you the wrong turns, because the wrong turns are the same ones you will make: confusing an average with a probability, splitting a pot by the score instead of by what is left to play, and believing that a name on a rule tells you who found it. By the end you will be able to say who wrote what, in what year, and which of the names on your formula sheet are attached to the wrong person.
Girolamo Cardano wrote the first mathematics of dice and left it in a drawer; it was printed in 1663, a century too late to matter. Galileo settled a gamblers' argument by counting orderings instead of combinations. In 1654 Pascal and Fermat, in four months of letters, worked out how to divide an interrupted stake. Huygens priced a chance in 1657, Graunt counted London's plague dead in 1662, Jacob Bernoulli proved the law of large numbers around 1689 and published nothing, and de Moivre worked out of a London coffee house. Bayes never published his essay; Richard Price read it to the Royal Society in 1763. The formula you will learn is Laplace's, from 1814. Cournot named it after Bayes in 1843. Kolmogorov axiomatized the lot in 1933.
Your course teaches a formula called Bayes' Rule. Guess who wrote the version your course teaches, and roughly when.
I have a guess▸
Pierre-Simon Laplace, in 1814, in the Essai philosophique sur les probabilites (S086). The name was attached to it in 1843, by Antoine Augustin Cournot (S088).
Thomas Bayes wrote an essay that contains the idea and never published it. Richard Price read it to the Royal Society in 1763, two years after Bayes died.
This happens constantly in this chapter. The person who first has an idea, the person who first prints it, and the person whose name ends up on it are three different people more often than they are one.
The book in the drawer
Girolamo Cardano was born in Pavia on 24 September 1501, the illegitimate son of a lawyer whom Leonardo da Vinci consulted on geometry. Illegitimacy barred him from Milan's College of Physicians, so he supported himself by gambling and later took over his father's mathematics post at the Piatti Foundation. Somewhere in there he wrote the Liber de ludo aleae, about fifteen pages long, in thirty-two chapters. In chapter 14 he wrote down what Gorroochurn calls "the first definition of classical (or mathematical) probability": you "consider the whole circuit, and the number of those casts which represents in how many ways the favorable result can occur." That is exactly the formula in your Course section 1.5. In chapter 13 he had already solved the three-dice problem that was later put to Galileo, and he used the arithmetic triangle in his Opus novum de proportionibus of 1570, before Pascal. Then nothing happened. The book was printed in 1663, in the first volume of his collected works, "long after Cardan's death, which took place in 1576." (S094, S072, S070)
Why this matters in your course. Course section 2.2 gives you the properties of probability as a tidy list. Cardano had most of the list and could not make it stick, which tells you that a correct rule is only half the job. He also made the single most common error in a first probability course: he computed an expected value of one half and read it as a probability of one half, which Ore named "reasoning on the mean." (S093, S073)
The receipt. Todhunter treats the book as "a gambler's manual" and refuses to date the manuscript, saying only that it was "published in 1663, in the first volume of the edition of Cardan's collected works" (S070, arts. 3 and 4). Bowman reports Ore's three reasons for its obscurity: no modern symbols, no organization, and two incompatible methods used side by side; and quotes the verdict that it "tended to collect dust on the back of library shelves" (S073). Gorroochurn supplies chapters 13, 14 and 15 and the 1570 triangle (S072).
So what? Being right is not enough. Cardano wrote a self-help book for gamblers, never labeled which of his statements were mistakes he had already corrected, and had no algebra to compress the argument. Readability is part of mathematics.
Galileo counts to 216
Dice players insisted that with three dice a total of 10 comes up more often than 9, and 11 more often than 12. On the face of it they were wrong: 9, 10, 11 and 12 each have exactly six combinations. Galileo's piece, which the York transcription titles "Concerning an Investigation on Dice" in E. H. Thorne's translation, starts from the observation that 3 and an 18 "can only be made in one way" and builds three rules: three equal numbers give one way, two equal and one different give three ways, three different numbers give six ways. Count all the orderings and there are 216. Ten comes up 27 ways, nine comes up 25. The players were right. (S071)
Why this matters in your course. This is the difference between a combination and a permutation (2.1), the naive probability formula (1.5), and the multiplication principle, all in two pages and all before anyone had the vocabulary. It is also the cleanest demonstration in the chapter that the sample space has to be built out of equally likely things, which is what Course section 2.2 is really insisting on.
The receipt. Todhunter: "out of 216 possible cases 27 are favourable to the appearance of the number 10, and 25 are favourable to the appearance of the number 9," and he calls the treatment "a careful and accurate analysis" (S070, art. 8). The counts are reproduced from scratch in verify/domainC.py Section 1 and again in verify/ch03.py Section 1, all checks passing.
So what? The gap is 27 against 25 out of 216, which is one throw in 108. The gamblers felt a difference of nine tenths of one percent by playing. Hold on to that number: it comes back in Four months of letters, and one very famous story that will not survive, where a much smaller gap is claimed for a much more famous gambler.
Four months of letters, and one very famous story that will not survive
In the summer and autumn of 1654 Blaise Pascal in Paris and Pierre de Fermat in Toulouse, who never met, exchanged letters about how to divide a stake when a game stops early. Pascal's letter of 29 July sets out the 32-pistole argument: "I am sure of 32 pistoles, for even a loss gives them to me," and divides the remaining 32 equally, so the leader takes 48 and the other 16. On 24 August Pascal enumerates the 27 futures of a three-player game and gets 17, 5 and 5; Fermat confirms it on 29 August and, in the same letter, states a conjecture about primes that is false. On 27 October Pascal writes "I admire your method for the problem of the points" and tells Fermat to "find someone elsewhere to follow you in your discoveries concerning numbers." Four weeks later, on 23 November 1654, Pascal has the religious experience that ends his mathematical life. (S074, S075)
Why this matters in your course. Fermat's method is a sample space argument and Pascal's is a recursion, which is to say a conditional probability argument, so Course section 2.3 has both of its parents in one exchange of letters. The three-player division is the multiplication principle used in anger.
The receipt. The York transcription of Vera Sanford's translation carries the letters in order with their dates, the 32-pistole passage, and the 17, 5 and 5 division (S074). Both divisions are recomputed in verify/domainC.py Sections 4 and 5: 48 and 16 out of 64, and 17, 5 and 5 out of 27.
Now the story you have heard. Every textbook says the whole thing started when a gambler, the Chevalier de Mere, noticed that betting on at least one double six in 24 throws of two dice was losing him money. Oystein Ore did the arithmetic. The probability at 24 throws is 0.4914 and at 25 throws it is 0.5055, and to detect a gap that size from play you would need "at least 100 sequences of trials, which in turn would involve several thousand individual throws." Ore's verdict on the discovery-by-play story: "This, as we shall, seems very unlikely" (S075, pp. 411 to 412). Ore also objects to the caricature itself: Antoine Gombaud, chevalier de Mere, lived 1607 to 1684, was a courtier and moral philosopher at the court of Louis XIV who wrote on ethics, and would, in Ore's phrase, "turn in his grave" (S075, p. 409). The letters themselves say only that de Mere "has never been able to find the just value of the problem of the points nor has he been able to find a method" (S074). The phrase that fixed the caricature in every later book is Todhunter's, from 1865: "the Chevalier de Mere (a reputed gamester)" (S070, art. 11).
So what? Compare the two gaps. The three-dice gamblers were detecting a difference of about 0.0093 in a game people played constantly for generations. De Mere is supposed to have detected 0.0141 on his own. It is not impossible. It is just that nobody has a document saying he did, and the man who checked, checked properly.
Two books, five years apart: a price on a chance, and a pile of the dead
Christiaan Huygens drafted a short treatise in Dutch in 1656 to 1657 called Van Rekeningh in Spelen van Geluck. His old teacher Frans van Schooten had been pressing him since April 1656 to let it be turned into Latin, and it went out that way, as De ratiociniis in ludo aleae, inside van Schooten's Exercitationum mathematicarum, Leiden, 1657. It has fourteen propositions and five problems. It never uses the word probability. It asks what a chance is worth, in money. The postulate, in W. Browne's 1714 English, reads: "my Chance or Expectation to win any thing, is worth just such a Sum, as wou'd again procure me the same Chance." Proposition III is the weighted mean, which is the modern definition of expected value with the notation taken out. (S076)
Five years later, in London, a haberdasher named John Graunt published Natural and Political Observations... upon the Bills of Mortality. The weekly lists of London's dead had been running continuously since, in Graunt's own words, "the Twenty ninth of December 1603, being the first year of King James his Reign,"
His Observation 2 says why: "the rise of keeping these Accompts was taken from the Plague." The causes of death were determined by Searchers, sworn women who went and looked at the bodies and reported to the Parish Clerk; the Clerk sent the records on, they were compiled on Wednesday and published on Thursday, and a subscription cost four shillings a year. Graunt read about seventy years of them. (S077, S078)
Why this matters in your course. Huygens is where the habit of attaching a number to an uncertain situation starts, which is the whole of Course section 2.2. Graunt is the other tradition, the one your course's cover promises: not equally likely cases, but counted data with a pattern in it.
The receipt. Browne's 1714 translation prints the postulate and Propositions I to III verbatim, and the Leiden reference page dates the Dutch composition, van Schooten's Latin, and the fourteen propositions (S076) and. The three propositions are checked against the modern expectation formula in verify/domainC.py Section 6. Graunt's dates and the Searchers come from Chapter 1 of the 1676 fifth edition (S077); the ASA classroom activity gives his life dates, 1620 to 1674, and calls him "a tradesman, a haberdasher" (S078).
So what? Statistics starts with plague, and the data collection was done by working women whose names nobody wrote down. Neither half of the subject came out of a university.
Twenty-four years of not publishing, and an inequality named after the wrong man
Jacob Bernoulli developed his probability ideas between 1684 and 1690 and finished the proof of what we call the law of large numbers around 1689. He wrote: "I prize this discovery more highly than if I had given the very quadrature of the circle: for even if this were discovered, it would be of little use." Then he did not publish it. He blamed his "innate lethargy, formidably exacerbated by the unstableness of my health." He could not get the data he wanted, hunting for mortality tables and the works of John Graunt and Johan de Witt without success. He was in a long public feud with his brother Johann over priority in the calculus. He died in 1705. His widow, Judith Stupanus, "seems to have been a shrewd business woman, but she neither understood nor cared about intellectual property," and kept the papers from both the brother and the nephew. Their son was in Italy studying art and only authorized publication after two people talked him round. The family hired as editors "a recently promoted doctor of law, then an unemployed minister," who are "said not to have understood a lot of what they were preparing for publication." Ars Conjectandi appeared in August 1713. The nephew Nicolaus, called its editor by many later writers, contributed three pages: an errata list and a preface. (S079, S080)
The subject also got its name here. Bernoulli defined it in Part IV, chapter 2: "To conjecture some thing means to measure its probability: so we define the Art of Conjecturing or Stochastics to be the art of measuring as exactly as possible the probabilities of things." The root is the Greek stochos, a target, and Mattmuller identifies Plato's Philebus 55e as the specific source, arguing Bernoulli chose it to "give his brainchild a classical ancestry." The title Ars Conjectandi copies Ars Cogitandi, the Latin Port Royal Logic of 1662. Both choices are branding. (S079, S080)
The tail of this story is the tool that makes the law of large numbers provable in one line. Bienayme, an inspector general in the French Ministry of Finances, derived the inequality in 1853. Chebyshev derived it independently and published in 1867, in French and Russian simultaneously; his paper does not mention Bienayme anywhere, and closes by noting that "in the particular case where the probability of the event remains the same during all the trials, we have the theorem of Bernoulli." The editor of Liouville's journal knew the history: he reprinted Bienayme's 1853 paper immediately before Chebyshev's. (S092, S091, S094, S080)
Why this matters in your course. The law of large numbers is the bridge between the axioms of 2.2 and the intuition that probability means long-run frequency. It is also the reason your 2.4 posterior settles down as evidence piles up.
The receipt. Mattmuller, read in full, supplies the 1689 proof, the quotations from Bernoulli about lethargy and the quadrature of the circle, the Stupanus episode, the editors, and the Plato derivation of "stochastic" (S079, pp. 278 to 288). Seneta dates the appearance to August 1713 and states that Bienayme "first proved" the inequality in 1853 and that Chebyshev obtained it independently in 1867 (S080). Chebyshev's silence about Bienayme is a fact about the text of the 1867 paper, read in full in Pulskamp's translation (S091). The bound is computed at several values in verify/domainC.py Section 11.
So what? Fourteen years is not a coincidence and it is not theft. Two people found the same thing, and the world remembered the one whose name was easier to attach. You will see the same pattern twice more before this chapter ends.
The refugee at the coffee house table
"Abraham De Moivre was born at Vitry in Champagne on May 26, 1667." In 1685 Louis XIV revoked the Edict of Nantes and banned Protestant worship, and de Moivre, a Huguenot, left France; on 28 August 1687 he and his brother Daniel presented themselves at the Savoy Church in London to be admitted as Huguenots. As a foreigner he never got a university post. He tutored rich men's sons and did consulting mathematics at a table in Slaughter's Coffee House, "a favorite meeting place of French emigres," which he used as his professional address. He became friends with Halley in 1692 and then with Newton, who "took delight in his company," took him home, and "had absolute confidence in Mr. De Moivre for thirty years." On 30 November 1697 the Royal Society elected him a Fellow. He dedicated the 1718 Doctrine of Chances to Newton and meant it. In the 1756 preface he turned the whole subject into natural theology: the doctrine that finds chance where chance really is can also prove "that where Uniformity, Order and Constancy reside, there also reside Choice and Design." He died on 27 November 1754. (S082, S081)
Why this matters in your course. Course section 2.5 defines independence, and the first English definition is de Moivre's, in the 1738 edition: "Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other." The product rule for independent events is in the 1718 Introduction: "if a Fraction expresses the Probability of an Event, and another Fraction the probability of another Event, and those two Events are independent; the Probability that both those Events will Happen, will be the Product of those two Fractions." So does the definition of expectation you met in Two books, five years apart: a price on a chance, and a pile of the dead: "The Expectation of obtaining any Thing, is estimated by the Value of that Thing multiplied by the Probability of obtaining it." (S081, S088)
The receipt. Bellhouse and Genest's annotated translation of Maty's near-contemporary biography carries the birth line, the Savoy Church record, Slaughter's, the Newton friendship, and the election date (S082, pp. 2, 5, 6, 8, 18, 19). The 1718 quotations were read from the Internet Archive scan (S081, Introduction pp. 2 to 4). The 1738 definition of independence is quoted by Miller from the Doctrine of Chances (S088, i.html).
Then there is the famous story. MacTutor prints it like this: "He found that he was sleeping 15 minutes longer each night and summing the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. He was right!" MacTutor cites nothing for it, and it appears nowhere in the annotated translation of Maty's biography, which is the standard scholarly life. Tag it. What was tried, and failed, was a search for an earlier printed source; only modern retellings came back. The earliest attestation is unknown. (S094, S082)
So what? A good anecdote outlives its evidence. If you cannot find who first told a story, you have found out something about the story.
The paper Bayes never published, and what is not in it
Thomas Bayes was a Nonconformist minister, barred from Oxford and Cambridge, who studied at Edinburgh instead. Nobody knows when he was born: the best a modern archival historian can do is "probably between July of 1701 and April of 1702." He was elected a Fellow of the Royal Society on 4 November 1742 without having published a mathematical paper under his own name; the nomination certificate, signed 8 April 1742, called him "well skilled in Geometry and all parts of Mathematical and Philosophical Learning." He signed his will on 12 December 1760 and died suddenly on 7 April 1761 at Tunbridge Wells. Two London newspapers of that week, The Public Advertiser and the Whitehall Evening Post, report the death and say only that it was sudden. He was buried in the family vault at Bunhill Fields, and the only recorded funeral expense was fourteen shillings to open the vault. Three letters and one notebook survive in his hand. (S084)
Two years later his friend Richard Price sent the Essay to the Royal Society with a covering letter headed "Newington-Green, Nov. 10, 1763." The article carries the note "Read Dec. 23, 1763." The heading names the author as "the Late Rev. Mr. Bayes," so the paper announces on its own face that its author is dead. (S083)
Why this matters in your course. Open the paper and the formula from Course section 2.4 is not there. There is no vertical bar, because the vertical bar had not been invented. What is there is a problem in one sentence: "Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named." There is a definition of probability by way of expectation, and there is Proposition 3, which is your multiplication rule from 2.3 written without symbols: "The probability that two subsequent events will both happen is a ratio compounded of the probability of the 1st, and the probability of the 2d on supposition the 1st happens." (S083)
The receipt. All four quotations were read from the Internet Archive scan of Philosophical Transactions 53, pages 370 to 418 (S083). The biography is Bellhouse's, read in full, including the death-date evidence (S084, pp. 4, 17, 27). Print 7 April 1761. MacTutor's 17 April carries no source at all, and a ten-day gap is exactly what a transcription slip looks like (S094, and the settlement recorded in), resolved.
There is one more twist, and Stephen Stigler found it by going to three libraries and looking at physical covers. About fifty offprints were produced in June 1764, and they carry a different title from the journal's: "A Method of Calculating the Exact Probability of All Conclusions founded on Induction." Stigler argues the choice was Price's, "for he would have been paying the bill." A paper about gambling in the journal; a paper about knowledge on the cover. (S085)
So what? The person who edits your work decides how the world reads it. And a rule can be named after someone who never wrote it, which is the subject of the next scene.
Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom
In 1814 Pierre-Simon Laplace published the Essai philosophique sur les probabilites, and numbered his rules First Principle through Tenth. The First Principle is your Course section 1.5: probability is "the ratio of the number of favorable cases to that of all the cases possible." The Sixth Principle is your Course section 2.4, in a sentence: "The probability of the existence of any one of these causes is then a fraction whose numerator is the probability of the event resulting from this cause and whose denominator is the sum of the similar probabilities relative to all the causes." Read that denominator clause again. "The sum of the similar probabilities relative to all the causes" is the law of total probability, printed in 1814, without a single symbol. (S086)
Twenty-nine years later Antoine Augustin Cournot, in the Exposition de la theorie des chances et des probabilites of 1843, wrote "la regle de Bayes" with its modern meaning. Jeff Miller, who tracks these things, notes flatly that the rule "is not in the Essay but comes from Laplace." The older phrase "Bayes' theorem," meaning Bayes's own inverse problem, is in Lubbock and Drinkwater-Bethune in 1830 and in Todhunter in 1865. The word "Bayesian" only enters circulation around 1950. (S088)
The notation is younger still. Harold Jeffreys wrote P(p given q) with a vertical stroke in Scientific Inference in 1931, and Feller made Pr{A given B} standard in 1950. The letter P for probability, as a function you can write as P(A), is Kolmogorov's, from Axiom III of the Grundbegriffe in 1933. (S088, S090)
And the axioms. Kolmogorov's book is 62 pages long. Chapter I, sections 1 to 4, contains the whole of your Unit 2 spine in order: five axioms on pages 2 and 3, conditional probability defined as P(AB)/P(A) on page 6, the theorem on total probability as formula (13) on page 7, and the theorem of Bayes as formula (14) on the same page. In his own preface Kolmogorov wrote: "In the pertinent mathematical circles it has been common for some time to construct probability theory in accordance with this general point of view." Shafer and Vovk, who traced every source the book drew on, put it this way: "Like any textbook, its mathematics was novel for most of its readers, but its real originality was rhetorical and philosophical." Frechet, whose work Kolmogorov credits by name in that same preface, said in 1937 that everything needed had come together by 1909, and added: "This is what Mr. Kolmogorov did. This is his achievement. (And we do not believe he wanted to claim any others, so far as the axiomatic theory is concerned)." (S089, S090)
Why this matters in your course. Sections 2.2, 2.3 and 2.4 of your book are Chapter I of a 1933 monograph, in the same order, with better typesetting. Knowing that is worth something when the axioms look like they fell from the sky.
The receipt. The Laplace quotations were read from the Truscott and Emory translation (S086). Cournot 1843, Jeffreys 1931, Feller 1950, and P(A) in 1933 are all from Miller's dated entries (S088). The Kolmogorov page and formula numbers were read from Morrison's 1950 English translation (S090, pp. 2, 3, 6, 7). The Shafer and Vovk judgments and the Frechet quotation were read in full text (S089, pp. 1, 2, 13, 24 to 28).
So what? Three of the names on your formula sheet point at the wrong person or at nobody in particular: Bayes' rule is Laplace's, the Chebyshev inequality is Bienayme's, and the law of total probability has no traceable namer at all. That is not a scandal. It is how naming works, and noticing it is a skill.
Girolamo Cardano wrote the first real mathematics of dice. Why did it change nothing?
Show the answer▸
He left it in a drawer. The Liber de ludo aleae was printed in 1663, in the first volume of his collected works, more than a century after he wrote it (S070).
By then Pascal and Fermat had already had the 1654 correspondence, Huygens had already published in 1657, and the field had moved on without him. Todhunter, reading it two centuries later, called it "a gambler's manual" (S070).
Being right is not enough, and neither is being first. Cardano had no algebra to compress his argument, never marked which of his own statements were mistakes he had corrected, and wrote for gamblers rather than for mathematicians. Readability is part of mathematics, which is a thing worth knowing when you write up your own work.
Chapter 4
The algorithm came first, the word came last
You are about to learn a procedure for solving systems of linear equations that your textbook calls Gaussian elimination, and a word, matrix, that sounds like it has been around forever. Neither is what it looks like. The procedure was being run on a board of bamboo sticks in Han China roughly two thousand years before Gauss was born, and Gauss himself called it "common". The word matrix is younger than the railway: James Joseph Sylvester coined it in 1850, and he meant it to mean womb. By the end of this chapter you will be able to do the elimination three ways, you will know which names on your worksheet are honest and which are accidents, and you will have a defensible answer when somebody asks who invented any of it.
Chapter 8 of the Nine Chapters on the Mathematical Art poses eighteen problems that are linear systems, lays them out as columns of counting rods on a ruled board, and solves them by the elimination taught in Course section 3.6. To keep that elimination legal, the Chinese also wrote the first surviving rules for negative numbers. Europe takes another millennium and a half. Seki reaches the determinant in Edo in 1683, Leibniz reaches it in a 1693 letter nobody reads for 157 years, Cramer prints the rule in an appendix in 1750, Cauchy fixes the word determinant in 1815, Jacobi makes the subject common property in 1841, Sylvester names the matrix in 1850, and Cayley writes its algebra in 1858. The phrase "Gaussian elimination" arrives last, in 1953.
Course section 3.6 teaches you to solve a system by elimination, and somebody probably told you it is called Gaussian elimination. Guess how long the procedure existed before it got that name.
I have a guess▸
About two thousand years. The oldest surviving text that performs it is Chapter 8 of the Nine Chapters on the Mathematical Art, laid out as columns of counting rods on a ruled board (S120, S123). The phrase "Gaussian elimination" appears to be George Forsythe's, in 1953 (S120, p. 788).
Gauss did use the method, in 1809 and 1810, on asteroid orbits, and he made a real contribution to it. He did not invent it, and 144 years passed between his Latin and the English name (S120, p. 785).
The algorithm came first. The word came last. That sentence is the whole chapter.
Rice, rods, and a book with no date
Start with the thing most books get wrong. Ask when the Nine Chapters on the Mathematical Art was written and you will get four answers from four respectable places. MacTutor's matrices topic says 200 BCE to 100 BCE (S122). MacTutor's own Nine Chapters topic says 200 BCE to 50 CE, and argues for shortly after 200 BCE from the units the text uses (S123). Lam Lay Yong, one of the leading historians of Chinese mathematics, writes only that "it is generally accepted that it was written between 100 B.C. and 100 A.D." and adds that some parts are probably older (S126). Grcar says "over 2000 years ago" (S120). They are not so much contradicting each other as answering a badly posed question, because the Nine Chapters is a compilation assembled over generations. Liu Hui's own preface says that after the Qin book burning of 213 BCE, Zhang Cang and his team "produced a new version removing the poor parts and filling in the missing parts" (S123). The one firm date in the whole story is 263 CE, when Liu Hui wrote his commentary (S123, S124, S125).
Now the object. A clerk has a board ruled into squares and a bag of small sticks, usually bamboo, sometimes bone, wood, iron, ivory, or jade (S126). The problem is about paddy. Three bundles of top grade, two of medium, and one of low yield 39 dou of grain; two top, three medium, and one low yield 34; one top, two medium, and three low yield 26 (S124, quoting the Shen, Crossley, and Lun translation). He lays each equation out as a column, and he fills the columns from right to left (S124). Then he multiplies a whole column through by a number and subtracts the rightmost column from it as many times as possible, until the top entry goes to zero (S122). He repeats. When one number is left in the last column, he reads the answer off the board and works backwards. That is Gaussian elimination. Grcar, writing in the Notices of the AMS, does not hedge: "The solution was by Gaussian elimination. The right column was paired with each of the other columns to remove their top numbers" (S120, p. 783). Chapter 8 holds eighteen problems of this kind (S120, S123), inside a book of 246 problems about engineering, surveying, trade, and taxation (S123).
Run the procedure for two minutes and you hit the obstacle: you have to take a bigger number away from a smaller one. On the board, 3 minus 5. The Nine Chapters does not dodge it. Chapter 8 states the zhengfu rule, 正負, positive and negative, which Guo Shuchun calls the "sign procedure" (S125, p. 67), and the board handled it physically: "Rods with a red dot were used for positive numbers, and those with a black dot for negative numbers" (S124). That is the oldest surviving set of rules for signed arithmetic anywhere, and Europe would not treat negatives as ordinary quantities for another fifteen hundred years.
Why this matters in your course. Appendix A.4 calls the permitted steps "the legal moves". Course section 3.6 calls the same steps row operations and performs them on an augmented matrix. The board is the augmented matrix, in wood. MacTutor puts it in one line: chapter 8 uses "elementary column operations as is done today in the method of Gaussian elimination", with coefficients in columns rather than rows (S123). Rows or columns is a bookkeeping choice and nothing more. Liu Hui even saw the condition that makes a system solvable, defining the fangcheng situation as one in which "rows do not depend on those besides them" (S125, p. 69), which is linear independence, stated in Chinese, seventeen centuries before it got a European name.
The receipt. Grcar, S120, p. 783: "composed in China over 2000 years ago", and "The solution was by Gaussian elimination." MacTutor, S123, on chapter 8: "elementary column operations as is done today in the method of Gaussian elimination", and chapter 8 "includes rules to compute with" negative numbers. Schwartz, S124, quoting Shen, Crossley, and Lun: "Now given 3 bundles of top grade paddy, 2 bundles of medium grade paddy, [and] 1 bundle of low grade paddy. Yield: 39 dou of grain..." Guo Shuchun, S125, p. 67 and p. 69. Lam Lay Yong, S126: "written between 100 B.C. and 100 A.D." (S120, S125, S126; REFERENCE | S122, S123, S124)
So what? Two answers. Negative numbers were not invented by philosophers arguing about whether nothing can be less than nothing; they were invented by people who needed an algorithm to keep running. And the dating matters as a habit of mind: the strongest argument on the table is about weights and measures, since the text uses the basic decimal units that became standard around 200 BCE and not the finer subdivisions introduced after about 250 CE (S123). That is how a date gets defended, with evidence from inside the object. A book that tells you the Nine Chapters was written in a particular year is guessing, and now you can catch it.
Seki in a closed country, Leibniz in a letter
Seki Takakazu was born in March 1642 in Fujioka, in Kozuke, into a samurai family, and was adopted by the Seki Gorozaemon household, whose name he took (S128). A servant introduced him to mathematics when he was nine, and he taught himself the rest (S128). Japan under the Tokugawa had cut its contact with Europe almost entirely, and Japanese mathematical schools kept their methods secret, which is why attribution inside wasan is so hard (S128). In 1683 Seki wrote up his method for fukudai, "concealed problems". Smith and Mikami, in the first Western monograph on Japanese mathematics, describe what he did: he expanded the array of coefficients, "practically the determinant that is the eliminant of the equations", he removed "a constant literal factor in any row or column, exactly as we remove a factor from a determinant today", and he "knew that the number of terms in the expansion of a determinant of the nth order was n!" (S127, p. 138). Their verdict, in 1914: "it is rather remarkable that no predecessor of Seki's discovered the idea of the determinant" (S127, p. 139).
Ten years later, on 28 April 1693, Leibniz wrote to the Marquis de l'Hopital and explained that he sometimes writes numbers instead of letters for coefficients: 10 for a, 11 for b, and so on, so that the first digit names the equation and the second names the unknown (S129, scan pp. 24 to 26). Then he gave the condition under which three equations in two unknowns are consistent, which is the vanishing of a 3 by 3 determinant. He told nobody else. Gerhardt printed the correspondence in 1850, 157 years later, and even then, Muir records, the passage "escaped observation" until Dirichlet noticed what it meant (S129, introduction).
Why this matters in your course. Course section 3.5 hands you a determinant and a test: if it is zero, something has gone wrong with the system. That test is exactly what Leibniz wrote to l'Hopital, and Seki's array is the same object approached from the other side of the world.
The receipt. Smith and Mikami, S127, p. 138 and p. 139, read in the 1914 Internet Archive scan. Muir, S129, scan p. 24, for the date 28 April 1693, and the introduction for Gerhardt 1850 and Dirichlet. Muir's first period runs "Leibnitz, Fontaine, Cramer, Bezout, Vandermonde, Laplace, Lagrange" (S129, scan p. 24). (S127, S129)
So what? The honest priority answer is: it depends what you count. MacTutor states flatly that Seki "was the first person to study determinants in 1683" and that "Seki's version was the more general" than Leibniz's, and gives no source for that judgment (S128). Smith and Mikami never mention Leibniz at all, so they are not making a priority claim (S127). Muir starts the European story with Leibniz and never mentions Japan (S129). Three sources, three silences, and the paper that would settle it, Hayashi's "The 'Fukudai' and Determinants in Japanese Mathematics", could not be opened in this book. Say the dates, say that Seki's is earlier, and stop there.
The most famous rule in the course, printed in an appendix
Gabriel Cramer's Introduction a l'analyse des lignes courbes algebriques came out in Geneva in 1750. It has thirteen chapters, thirty-three plates of figures, and three appendices (S131, p. 239). The rule everybody now calls Cramer's rule is in one of the appendices, at printed pages 656 to 659, with no proof (S129, S130, S131). Cramer needed it because he was fitting an algebraic curve through a set of given points, which is a different problem entirely (S122, S131). His section is headed "De l'evanouissement des inconnues", the vanishing of the unknowns, and he writes that algebra supplies rules for this "dont le succes est infaillible", whose success is infallible (S130, scan leaf 749). The general rule is announced as "L'examen de ces Formules fournit cette Regle generale. Le nombre des equations & des inconnues etant n...", and the common denominator "a autant de termes qu'il y a de divers arrangements de n choses differentes", as many terms as there are arrangements of n different things (S130, scan leaf 751). The word "determinant" never appears. Cramer has the object and not the name.
He does name one thing, and it is the piece students find hardest. To decide whether a term gets a plus or a minus, count the pairs that are out of order: "Quand un exposant est suivi dans le meme terme, mediatement ou immediatement, d'un exposant plus petit que lui, j'appellerai cela un derangement" (S130, scan leaf 751).
Two more Paris names belong in this scene, because they finish the job. On 12 January 1771 Alexandre-Theophile Vandermonde read his Memoire sur l'elimination to the Academy (S129, scan p. 18). He had come to mathematics late, at 35, having been pushed toward music by his father, and the violin was his instrument (S430); he wrote four mathematical papers in his life, all between 1771 and 1772 (S430). His notation for coefficients is "essentially the same as that of Leibnitz, writing where Leibnitz wrote 12 or [1,2]" (S129, scan pp. 18 and 40). In the same 1772 Academy volume, at pages 267 to 376, Laplace published the expansion by minors that still carries his name (S129, scan p. 17). Laplace called the methods of Cramer and Bezout impractical and used the word "resultant", apparently unaware that Leibniz had used it first (S122).
Why this matters in your course. Course section 3.6 offers "solving with the inverse" as an alternative to elimination. Cramer's rule is the third method, and it is the one that made determinants famous. It is also the one that shows students why nobody solves large systems this way: the denominator has n! terms, which is 6 for three unknowns, 24 for four, and 120 for five (verified in verify/ch04.py section 4). Laplace's expansion, meanwhile, is how Course section 3.5 asks you to compute a 3 by 3 determinant: pick a row, multiply each entry by the determinant of what is left when you delete its row and column, and alternate the signs. The point of it is that the choice does not matter. Every one of the six rows and columns of a 3 by 3 gives the same number, checked in verify/domainD.py section 7.
The receipt. Cramer, S130, scan leaves 749 to 752, read in the 1750 Geneva edition on the Internet Archive. Joffredo, S131, p. 237, on where the rule sits in the book. Muir, S129, for the pages 656 to 659, for the 12 January 1771 reading at scan p. 18, and for Laplace's pages at scan p. 17. MacTutor, S430, for Vandermonde's life. (S129, S130; SCHOLARLY | S131; REFERENCE | S122, S430)
So what? The rival claim is worth airing. MacTutor says Maclaurin's Treatise of Algebra, written in the 1730s and published in 1748, two years after his death, "contained the first published results on determinants", proving the rule for 2 by 2 and 3 by 3 and indicating 4 by 4 (S122). The copy of Maclaurin that this book could open is a later edition, not the 1748 first edition, and it carries a running head reading "CONTAINING some GENERAL THEOREMS FOR THE EXTERMINATING UNKNOWN QUANTITIES IN GIVEN EQUATIONS" (S132, scan p. 98). Muir's volume 1 does not mention Maclaurin at all (S129). Maclaurin proved the small cases and Cramer stated the general one. Which of those counts as first is a question about you, not about the mathematics. And keep an eye on Vandermonde, who is the cleanest case in this book of a name that means nothing: the determinant every course calls the Vandermonde determinant appears nowhere in his four papers, and MacTutor puts the naming down to a misreading of his notation (S430). The project's own script prints the warning above the calculation: "Vandermonde's 1771 memoir does not contain this determinant" (verify/domainD_output.txt, section 9).
Gauss calls it common
In 1801 Gauss printed the word determinans in the Disquisitiones Arithmeticae, and it meant something else. One warning before the sentence itself: the 1801 Latin was read here through the optical character recognition of a scan, and that OCR mangles long s, ligatures, and u for v, so every Latin word in this scene carries a flag and none of it should be typeset from this reading without somebody looking at the page image, which is cleared as IMG-023 (S133). He was talking about the binary quadratic form he wrote as (a, b, c), and the number he named was bb minus ac (S133, S134). That is not the determinant of a matrix. It is the negative of the determinant of the array [[a, b], [b, c]], which the project checked (verify/domainD.py, section 10).
Then, in 1809, in Theoria motus corporum coelestium, at page 214, Gauss described the elimination and called it eliminatio vulgaris, common elimination (S134). Grcar, in the peer reviewed version of the argument, quotes his Latin as per eliminationem vulgarem, "by common elimination" (S121). Grcar's abstract puts the whole story in one sentence: this is the method "that Newton did not want to publish, that Euler did not recommend, that Legendre called 'ordinary,' and that Gauss called 'common'" (S121). What Gauss did contribute is real and much narrower: a bracket notation, [xy], for the coefficients of the normal equations of least squares, with recursive formulas such as [xy, 1] = [xy] minus [ax][ay] divided by [aa]. That scheme "halved the work of schoolbook elimination" and let human computers throw away the algebra and work with numbers (S120, p. 785).
The name arrives 144 years after the Latin. George Forsythe, in 1953, "appears to have been the first to call it 'Gaussian elimination'", and in doing so, Grcar says, Forsythe "misattributed 'high school' elimination to Gauss" (S120, p. 788).
Why this matters in your course. Course section 3.6 names the procedure after a man who called it ordinary and who never claimed it. The glossary headword "Gaussian elimination" is a naming accident with a date on it.
The receipt. Grcar, S120, p. 788: Forsythe "appears to have been the first to call it 'Gaussian elimination'" and "misattributed 'high school' elimination to Gauss". Grcar, S121, abstract, for "that Gauss called 'common'", and for per eliminationem vulgarem. Miller, S134, for eliminatio vulgaris at Theoria motus p. 214. (S120, S121; REFERENCE | S134)
So what? The other Jordan is waiting in the same sentence. "Gauss-Jordan" honours Wilhelm Jordan, 1842 to 1899, a geodesist who put the method into the third edition of his Handbuch der Vermessungskunde in 1888 so that survey crews could squeeze the error out of their angle measurements (S134, S142, S144). The foreword to that edition is dated May 1888 and signed by Jordan, which is how Althoen and McLaughlin corrected Householder's claim that the book was posthumous (S144). It has nothing to do with Camille Jordan, 1838 to 1922, the Paris mathematician, who gets the Jordan normal form and who Althoen and McLaughlin say attracts "a natural tendency" to be given the credit (S144). A Belgian named Clasen published a comparable method in Brussels in the same year and got nothing at all (S134, S142, S144). Even the initials are contested: Miller says "J. B.", Knill says "B.-I." (S134, S142).
Two men, one afternoon in 1812
On 30 November 1812 Cauchy addressed a memoir to the Institut in Paris with a title long enough to be a paragraph: "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs egales et de signes contraires par suite des transpositions operees entre les variables qu'elles renferment" (S134, S129). Binet presented a proof of the multiplication theorem on the same occasion (S122, S134). Muir, who spent a career on this, writes: "Here we have the record of only one year and of only two authors to deal with; but the authors, Binet and Cauchy, are of supreme importance" (S129). He says the multiplication theorem "was discovered independently... by Binet and Cauchy... about the same time" (S129, scan p. 127). MacTutor calls Binet's proof the less satisfactory of the two (S122).
In that memoir Cauchy took Gauss's word and pointed it at a different object: "The writings of Laplace, Vandermonde, Bezout, and Gauss are referred to, and from the latter the name 'determinant' is adopted" (S129). Muir's verdict is a sentence worth reading twice: "Cauchy relaid the foundation, rebuilt the whole, and initiated new enlargements; the result being an edifice which the architects of to-day may still admire" (S129).
Then nothing much happens for thirty years, until 1841. In that year Jacobi published three memoirs in volume 22 of Crelle's Journal. Stackel, editing them in 1896, says what changed: determinants, "this important instrument of research, which, as Baltzer puts it, had until then remained in the possession of a chosen few, became the common property of mathematicians" (S138, scan p. 70).
Why this matters in your course. Course section 3.3 teaches matrix multiplication and Course section 3.5 teaches the determinant. The fact that ties them together, det(PQ) equals det(P) times det(Q), is Binet's and Cauchy's, from 1812. It is checked on a 3 by 3 pair in verify/domainD.py section 8, and in the rectangular Cauchy-Binet form on the same page.
The receipt. Muir, S129, on 1812 and on Cauchy's borrowing of the word. Miller, S134: the memoir was "addressed on 30 November 1812 and first published in Journal de l'Ecole Polytechnique, XVIIe Cahier, Tome X, Paris, 1815". Stackel, S138, scan p. 70. (S129, S138; REFERENCE | S134)
So what? Watch the phrasing whenever you see a date for "determinant". MacTutor says Cauchy first used it in the modern sense in 1812 (S122); Miller says the memoir was addressed in 1812 and printed in 1815 (S134). Both are true, of different events. The correct sentence is "read in 1812, printed in 1815", and any source that picks one and hides the other is cutting a corner.
The man they would not let graduate
James Joseph Sylvester was born in London on 3 September 1814. In 1837 he came second in the Cambridge mathematical tripos, which was the hardest examination in Britain, and the university refused him his degree: "it was necessary for a student to sign up to the Thirty-Nine Articles of the Church of England before graduating and Sylvester, being Jewish, naturally refused" (S128). He got a BA and an MA from Trinity College Dublin in 1841, once the law there changed, and in the same year, aged 27, he took the mathematics chair at the University of Virginia (S128). It lasted months. On 1 February 1842 he complained about the behavior of a first-year student. The student was reprimanded and no more. Sylvester objected on 19 March and resigned three days later (S128). He sailed for England on 20 November 1843, and spent years as an actuary and a tutor; one of his pupils was Florence Nightingale (S128). He was 62 when Johns Hopkins gave him a chair in 1877, and in 1878 he founded the American Journal of Mathematics, "the first mathematical journal in the United States" (S128). He supervised nine doctoral students in seven years there, became Savilian professor of Geometry at Oxford in 1883 at 68, and died in London on 15 March 1897 (S128).
Somewhere in the middle of that, in 1850, he wrote a paper about something else entirely and dropped a word into it. The sentence, at printed page 150 of his collected papers, is about an oblong arrangement of terms: "This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants" (S135, p. 150). A year later, at page 247, he said plainly what he had meant: "I have in previous papers defined a 'Matrix' as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent" (S135, p. 247).
Why this matters in your course. Course section 3.2 opens with the definition of a matrix. It is Sylvester's, almost word for word: in his own glossary he writes "Matrix. - A square or rectangular arrangement of terms in lines and columns" (S135, scan leaf 601). He is already comfortable with a matrix that is not square, telling a reader to "form the rectangular matrix consisting of n rows and (n + 1) columns" (S135, scan leaf 227), and he counts rows before columns, exactly as your shape notation does.
The receipt. Sylvester, S135, printed pages 150 and 247, read in the 1904 collected papers on the Internet Archive. Miller, S134, dates the coinage to the Philosophical Magazine, 1850, pp. 363 to 370, and notes that the word had long meant "the place from which something else originates". Latin matrix is "womb, breeding female", from mater, "mother" (S306, via the project etymology table). Sylvester's ages are recomputed in verify/ch04.py section 5. (S135; REFERENCE | S128, S134, S306)
So what? Not everybody liked it. Charles Dodgson, who wrote Alice's Adventures in Wonderland under the name Lewis Carroll, objected to the word in his Elementary Treatise on Determinants of 1867 and wanted us to say "block" instead (S134). He lost. And notice what the word tells you about 1850: the rectangle was named for what you could pull out of it, because determinants were the interesting objects and the array was only their parent. The array became the object later, and the word outlived the reason for the word.
"I have not thought it necessary"
Cayley's A Memoir on the Theory of Matrices was received by the Royal Society on 10 December 1857 and read on 14 January 1858 (S136, p. 17). It is short, twenty-one pages, and it does the things Course section 3.2 to Course section 3.5 do. It defines a matrix as "a set of quantities arranged in the form of a square" (S136, art. 1). It says the point of the whole exercise: "matrices (attending only to those of the same order) comport themselves as single quantities" (S136, p. 17). It gives the multiplication rule as composition, line against column (S136, art. 11). It gives the inverse through the partial derivatives of the determinant (S136, art. 17). It defines the transpose without using that word: "Two matrices such as (a, b / c, d) and (a, c / b, d) are said to be formed one from the other by transposition, and this may be denoted by the symbol tr." (S136, art. 36). And it warns you about the thing Course section 3.3 warns you about: "matrices are not in general convertible" (S136, p. 17), convertible being Cayley's word for what you call commuting.
Then it states the theorem that every matrix satisfies "an algebraical equation of its own order, the coefficient of the highest power being unity" (S136, pp. 17 to 18), verifies it for a 2 by 2, says "I have verified the theorem, in the next simplest case, of a matrix of the order 3" (S136, p. 23), and stops: "but I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree" (S136, art. 23, p. 24).
Why this matters in your course. This memoir is where the arithmetic of Course section 3.3 is written down as an arithmetic rather than as a shorthand. Cayley's own checks are recomputed in verify/domainD.py sections 4 and 5, and both come out as the zero matrix. He was right. Frobenius supplied the general proof about twenty years later, and then, in 1896, on discovering the memoir at last, adopted the word "matrix" and handed the credit back to Cayley (S122).
The receipt. Cayley, S136, read in a scan of the Philosophical Transactions printing, article by article. MacTutor quotes the same refusal (S122). (S136)
So what? Here the textbook story and the specialist disagree, and the specialist is worth hearing. Thomas Hawkins, who gave the invited address on this subject at the 1974 International Congress in Vancouver, writes that "the significance of Cayley's memoir on matrices of 1858 has been grossly exaggerated" (S137, p. 566), and that "Cayley's celebrated memoir went generally unnoticed, especially outside of England, until the 1880's" (S137, p. 561). His reasons: Gauss's Disquisitiones of 1801 already contains "the idea of composing two linear substitutions to form a third", which is matrix multiplication, and Eisenstein in 1844 had already seen that linear substitutions "can be added and multiplied much as ordinary numbers" (S137, p. 562). Present both versions and let the class argue.
One more thing about that memoir, and it is the sort of detail a search can settle in a second and a legend cannot survive. A full-text search of the scan finds no occurrence of "Hamilton" and no occurrence of "quaternion" anywhere in it (S136). Hamilton had proved a case of the theorem for quaternions in his Lectures on Quaternions of 1853, at page 566 (S134), and MacTutor describes it as a 4 by 4 special case (S122), but the two men did not collaborate on this and Cayley does not cite him. The joint name is not theirs. Bocher printed "Hamilton-Cayley equation" in 1907, and Turnbull printed "the Cayley-Hamilton theorem" in 1929, seventy-one years after the memoir (S134).
A cipher, and a machine to run it
In the June-July 1929 issue of the American Mathematical Monthly, a teacher at Hunter College in New York published the first cipher built on matrix algebra (S140). Lester Hill sets up arithmetic on the alphabet modulo 26, transforms blocks of n letters at a time, and states the condition for a key to be usable: "We say that T is a 'normal' transformation if its determinant is primary" (S140, p. 309), where primary means the determinant is coprime to 26. He never uses the word matrix once in the paper (S140). He is pleased with himself, and says so: "We have all the apparatus of an extraordinarily effective polygraphic (n-graphic) cipher system" (S140, p. 310). He also teases the reader: "If polygraphic ciphers... prove to be of real interest, we shall indicate a surprising way in which these ciphers may be manipulated easily and quickly, even for fairly large values of n (say n = 8, 9, or 10)" (S140, p. 310).
The surprising way already existed on paper. On 14 February 1929, months before that issue, Hill and his colleague Louis Weisner had filed a patent for a hand-cranked machine of gears, shafts, and printing wheels that computes the product and prints the answer. Weisner is the first-named inventor. The specification opens: "This invention relates to what may be termed a message protector, and has for an object to provide a mechanism for grouping symbols to be used on bank checks, code messages or other messages or on any communication to indicate its authenticity and accuracy" (S141). US Patent 1,845,947 was granted on 16 February 1932 (S141).
Why this matters in your course. Hill's test for a workable key is the test Course section 3.5 teaches, with one twist. Course section 3.5 asks whether the determinant is nonzero. Hill asks whether the determinant is invertible modulo 26, which is the same question asked inside a finite alphabet. The project's script works a full 3 by 3 example: the key has determinant 441, which is 25 modulo 26, and 25 is its own inverse modulo 26, so the key is usable; ACT encrypts to POH and decrypts back to ACT (verify/domainD.py, section 12).
The receipt. Hill, S140, pp. 306, 309, and 310, read in a scanned PDF. Weisner and Hill, S141, US Patent 1,845,947, read on Google Patents. The interval from the filing to the earliest date the June-July issue could carry, 1 June 1929, is 107 days, computed in verify/ch04.py section 5. (S140, S141)
So what? Two cautions. First, do not print Hill's own letter-to-number table from the available scan: its OCR is unreliable and internally inconsistent, since it reads n = 13 and 13 is not coprime to 26, so it could not be one of Hill's own primary letters (S140). The worked example in 4.7 uses a standard modern key for that reason, and says so. Second, the machine is usually called Hill's. The patent names Weisner first.
James Joseph Sylvester coined the word "matrix" in 1850. A year later he explained what he meant by it. What did he mean?
Show the answer▸
A womb. His own words, in 1851: "I have in previous papers defined a 'Matrix' as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent" (S135, p. 247).
That is not a metaphor he reached for casually. In 1850 determinants were the important object and the array was just the thing you took determinants out of (S135, p. 150). The rectangle was the container; the determinant was the point. Your course has it the other way round, and it took Cayley's 1858 memoir to turn the container into the subject.
Every time you write a capital in Course section 3.2, you are using a word that means the thing other things are born from.
Chapter 5
The planets, a half-translated word, and the machine age
Almost every word in Course section 3.1 and Course section 3.7 was invented by somebody you can name, on a date you can check, and most of them are younger than the railway. "Vector" and "scalar" were coined in a magazine article in July 1846. "Eigenvalue" did not exist in English until a letter to Nature in 1927, and it is half a German word that nobody finished translating. The mathematics is older than the words, and the mathematics you can run on a computer is younger than both: the algorithms behind the eigenvalues your calculator prints were written between 1947 and 1965, by people whose names are not in your textbook. By the end of this chapter you will know where each word came, why eigenvalues were invented to answer a question about the solar system, and who to thank the next time a machine hands you an answer in under a second.
A schoolteacher in Stettin writes the founding book of linear algebra in 1844; about 600 copies are pulped as waste paper in 1864. Hamilton names the scalar and the vector in July 1846, the tensor and the versor in October 1846. Peano writes the vector space axioms in 1888, in a book explaining Grassmann, unnoticed. Astronomers asking whether the planets will stay put produce the secular equation: Cauchy calls the polynomial characteristic, Sylvester calls the roots latent in 1883, Hilbert calls them Eigenwerte in 1904. Perron proves the theorem behind Google in 1907 and thinks it a technicality. Then the machines arrive: von Neumann and Goldstine in 1947, Turing in 1948, Taussky on aircraft flutter, QR in 1961, invented twice.
In 1844 a schoolteacher in Stettin published the book that founded linear algebra. Guess what happened to the copies.
I have a guess▸
About 600 of them were pulped as waste paper in 1864, twenty years after publication, because the publisher could not sell them (S181).
Hermann Gunther Grassmann was a Gymnasium teacher who never held a university post in mathematics. His Ausdehnungslehre was almost unreadable, almost unread, and about forty years ahead of everybody. He eventually gave up and became a distinguished Sanskrit scholar instead.
Vectors, linear combinations, span, linear independence, the dimension of a subspace: all of it is in that book, and Course section 3.1 teaches it because Peano read a copy that had not been pulped.
The schoolteacher, and 600 copies of waste paper
Hermann Gunther Grassmann was born on 15 April 1809 in Stettin, Prussia, now Szczecin in Poland, the third of twelve children (S181). He went to the University of Berlin from 1827 to 1830 and studied theology, classical languages, philosophy, and literature. He took no mathematics course there at all (S181). He taught himself, passed his teaching examinations in December 1831, and qualified only to teach the lower forms of a gymnasium (S181). He was an assistant teacher in Stettin from spring 1832, spent a year in Berlin from autumn 1834 replacing Jacob Steiner at the Gewerbeschule, and then went home to teach mathematics, physics, German, Latin, and religious studies at Stettin's new Otto Schule. He married Therese Knappe on 12 April 1849 and they had eleven children, seven of whom reached adulthood (S181).
He wrote Die lineale Ausdehnungslehre between spring 1842 and autumn 1843, and Otto Wigand of Leipzig published it in 1844, the year Grassmann turned 35 (S181, S171). It is the founding book of linear algebra. Dorier locates the modern concept of a vector in its section 14, linear dependence in section 18, and dimension in sections 16 and 20, and points out that Grassmann caught both faces of dimension at once, the smallest number of generators and the largest number of independent elements, a distinction later writers blurred (S172). He had an exchange lemma, an Austauschsatz, which gives you the uniqueness of the size of a basis (S172).
Nobody read it. You do not have to take a historian's word for that, because Grassmann says it himself in the preface to the second edition: "Das Werk, dessen zweite Auflage ich hiermit der Oeffentlichkeit uebergebe, hat in den ersten 23 Jahren nach seinem ersten Erscheinen nur eine geringe und meist nur gelegentliche Beachtung gefunden," which is "The work whose second edition I hereby hand to the public found, in the first 23 years after its first appearance, only slight and mostly only occasional attention" (S171). Twenty three years from 1844 lands on 1867. No journal proposed a review, so Grassmann had to write a summary of his own book in Grunert's Archiv in 1845 (S172). Mobius saw value in it but admitted he could not follow the philosophical part and refused to write a critique (S172). Richard Baltzer said the book made him dizzy and that "tout devient bleu-ciel devant mes yeux," everything goes sky blue before my eyes (S172). Hamilton, in Dublin, wrote that to read the thing he would have to "learn to smoke" (S177). Crowe counts only three published comments on Grassmann's work before the 1860s (S177).
Then the ending. Grassmann rewrote the book in conventional mathematical form in 1862, stripping out the philosophy; Crowe says that version "met with even less attention than the first" and that its 300 copies were privately printed at Grassmann's own expense (S177). And in 1864, twenty years after publication, about 600 copies of the 1844 book "were in 1864 used for waste paper" (S177).
Why this matters in your course. Everything Course section 3.1 asks you to do is in this book: add vectors, scale them, take a linear combination, count dimensions, and ask whether a set of vectors is independent. Dorier's summary is exact: Grassmann's theory "contained the bases for a unified theory of linearity, as it introduced... linear dependence, basis, and dimension" (S173, p. 243). Grassmann also tells you where the whole thing came, and it is something you already know: "Den ersten Anstoss gab mir die Betrachtung des Negativen in der Geometrie; ich gewoehnte mich, die Strecken AB und BA als entgegengesetzte Grossen aufzufassen," the first impulse came from considering the negative in geometry, and getting used to treating the segments AB and BA as opposite quantities (S171). A vector is a directed segment. He took that seriously and would not stop.
The receipt. Grassmann, S171, title page and both prefaces, read through the OCR of the 1878 Internet Archive scan. Crowe, S177, for the 600 copies, the 300 copies, the three published comments, and "learn to smoke". Dorier, S172, for the silence, the Grunert summary, Mobius, and Baltzer. Dorier, S173, p. 243, for the content summary and for Kummer. MacTutor, S181, for every date in the first paragraph. (S171; SCHOLARLY | S172, S173, S177; REFERENCE | S181)
So what? Two answers, and the second is the strange one. First, the popular version of this story is that jealous professors could not see what he had, and the record does not support it. Mobius said plainly that he could not follow the philosophy. Dorier's explanation is the useful one for a student: readers were "discouraged by the strict Euclidean organization, which did not permit a partial reading" (S173, p. 243). You could not dip into chapter 6. Every idea depended on the one before it. Modern textbooks are cut into pieces you can enter at any point for exactly this reason.
Second, Grassmann gave up. He learned Sanskrit and Gothic, wrote the Worterbuch des Rig-Veda, was elected to the American Oriental Society, and took an honorary doctorate from the University of Tubingen in 1876 for that work, not for the mathematics (S181). What is now called Grassmann's Law in historical linguistics comes out of that second career (S181), and it was still a live research object a century later: in 1966 D. Terence Langendoen published a three page paper in Language, the journal of the Linguistic Society of America, whose entire purpose was to state a restriction on it (S203). The man who invented linear algebra is famous in a different building on the same campus, for something else. MacTutor quotes A. C. Lewis: "It seems to be Grassmann's fate to be rediscovered from time to time, each time as if he had been virtually forgotten" (S181).
Four words, two magazine articles, three months apart
Hamilton's "On Quaternions; or on a new System of Imaginaries in Algebra" ran in eighteen installments in The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, third series, volumes xxv to xxxvi, from 1844 to 1850 (S174). Two of those installments are where your vocabulary comes.
In article 18, in the installment of July 1846, Philosophical Magazine volume xxix, pages 26 to 31, Hamilton splits a quaternion into two pieces and names them both. Of the algebraically real part, which "may receive... all values contained on the one scale of progression of number from negative to positive infinity," he writes: "we shall call it therefore the scalar part, or simply the scalar of the quaternion" (S174, art. 18). Of the algebraically imaginary part: it "may be called the vector part, or simply the vector of the quaternion" (S174, art. 18). He gives you the etymology of "scalar" inside the coining sentence: it is the part that lives on a scale.
Three months later, in article 19, in the installment of October 1846, pages 326 to 328, he names two more. "If then we make , and if we suppose TQ to be always a real and positive or absolute number, which we may call the tensor of the quaternion Q" (S174, art. 19). And when such an imaginary is divided by its own tensor, "we propose to call this quotient the versor" (S174, art. 19).
Behind all four words is the walk. Hamilton's own account, in a letter to his son the Reverend Archibald H. Hamilton dated 5 August 1865, gives the date as 16 October 1843: "An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come." He had no paper. "I cut with a knife on a stone of Brougham Bridge... the fundamental formula with the symbols, i, j, k; namely, " (S175). On the same day he asked the Council of the Royal Irish Academy for leave to read a paper on quaternions, and he read it 28 days later, on 13 November 1843 (S175).
Why this matters in your course. Course section 3.1 defines a vector and a scalar in its first two definition boxes. Both words entered mathematics in one paragraph of one magazine article in July 1846, and two independent sources agree on the year without meaning to: Merriam-Webster gives 1846 as the first known use of both nouns in English (S202). Merriam-Webster also gives the etymologies you can hand a class: "vector" is borrowed from Latin vector, "carrier, conveyer," from vehere, "to convey, carry"; "scalar" is from Latin scalaris, from scalae, "stairs, ladder" (S202). A vector carries you somewhere. A scalar is a rung.
The receipt. Hamilton, S174, articles 18 and 19, read in David Wilkins's transcription for Trinity College Dublin, with the Philosophical Magazine volume and page numbers from Wilkins's editorial note. Hamilton, S175, the letter of 5 August 1865, transcribed from Graves's Life, volume II, chapter XXVIII. Merriam-Webster, S202, for the first-use dates and the Latin. The quaternion relations themselves are checked in a 2 by 2 complex matrix representation in verify/domainE.py, section 7, which also confirms . (S174, S175; REFERENCE | S202)
So what? Handle the bridge honestly, because it is the best-loved story in the subject and it teaches a habit. The evidence is one document, and it is Hamilton's own letter, written 7,964 days after the walk, which is twenty one years and about ten months (verify/ch05.py, section 4). That is not nothing, and it is not an independent witness either. He also spells the place "Brougham Bridge"; modern Dublin calls it Broom Bridge (S175). No source opened for this book confirms that any cut in the stone survives (S175). So: documented, retrospective, self-reported, and worth telling with all three of those words attached.
And notice the price he paid. To get the formula he had to give up . That is exactly the warning Course section 3.3 gives you about matrix multiplication, and it was first paid for on a canal towpath in 1843. One more twist for a class: Hamilton's "tensor" meant the length of a quaternion, the stretching factor, and nothing else (S174, art. 19). The modern object that carries that name is a later borrowing of the word. Hamilton would not recognize what a machine learning library does with it.
The axioms, in Italian, in a book about somebody else's book
Giuseppe Peano was born on 27 August 1858 in Cuneo and took his doctorate at the University of Turin on 29 September 1880 (S183). In 1888, the year he turned 30, he published Calcolo geometrico, secondo l'Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva, which translates as Geometrical calculus, according to the Ausdehnungslehre of H. Grassmann, preceded by the operations of deductive logic (S182). The preface is dated February 1888 (S182). The book's main job was to make Grassmann readable, which tells you how bad the situation was forty four years after 1844.
Chapter 9 contains four numbered definitions. Kennedy's translation gives them as: an equality relation on the entities, satisfying the usual logical equations; a sum that is closed, commutative, and associative, ", "; a product defined first for a positive whole number and then extended, since "We shall assume that a meaning is given to the expression , whatever the real number "; and a zero, "there exists an entity of the system, that we shall designate by 0", with (S182). Then Peano writes the sentence: "Systems of entities for which the definitions (1)-(4) are given, so as to satisfy the conditions imposed, are said to be linear systems" (S182).
That is your Course section 3.1 board work, in Italian, in 1888. Kennedy's verdict: "one of the most remarkable features of this book historically is that here, for the first time, axioms are presented for a vector space" (S182). MacTutor calls it "the first definition of a vector space given with a remarkably modern notation and style" and adds that Peano defined dimension as "the maximal number of linearly independent objects in the system" and proved that finite dimensional spaces have bases (S183, S179).
Why this matters in your course. Course section 3.1 hands you vectors, addition, scalar multiplication, the zero vector, linear combination, and dimension. Every one of those is on Peano's list or follows from it within a page. Weyl added the dimensional axiom in 1918: "There are n linearly independent vectors, but every n + 1 are dependent" (S173, p. 247). Banach set out the fully axiomatic approach in his 1920 doctoral dissertation (S179), and in 1922 published the version that is usually quoted, in which he postulates "sets of elements of which I will postulate certain properties" (S173, p. 254). Those are two different documents, thirty two and thirty four years after Peano, and a textbook must not merge them.
The receipt. Kennedy, S182, for the four definitions and the "linear systems" sentence, read in the free 2002 Definitive Edition ebook. MacTutor, S179, for the four axiom groups and the definition of dimension. MacTutor, S183, for Peano's dates and the judgment on the 1888 book. Dorier, S173, pp. 232, 246, 247, and 254, for the independence definition, the reception, Weyl, and Banach. (S173, S182; REFERENCE | S179, S183)
So what? First, the honest note, because it changes what you are allowed to say. Nobody working on this book has opened the 1888 Italian original. Google Books lists a full-view scan, digitized from Sapienza University of Rome, and it returned metadata only; the Edizione Nazionale server at mathematica.sns.it presents a self-signed certificate; Kennedy's English translation is in copyright and available only through restricted lending (S182). Two fetches of Kennedy's own PDF returned two different page numbers for the same passage, "141-142" and "38-39" (S182). So this chapter states that the axioms are in chapter 9 of the 1888 book, and it does not give you a section number inside that chapter, because nobody here has seen one. That gap is item 8 in 5.10. When you read a textbook that gives you a precise section number for this, ask where the writer got it.
Second, the milestone was invisible. Kennedy notes the achievement "aroused no attention at the time" (S182). Dorier is blunter: "this definition was not quickly followed by many new developments, and his approach was not taken up immediately" (S173, p. 246). It took roughly thirty years and Weyl and Banach for the axioms to become the way people think. Being first and being read are different achievements, and this book keeps finding cases where one person got both and another got neither.
"Behold how these vectorists love one another"
From 1890 to 1894 professional physicists and mathematicians fought in public, across eight journals, twelve scientists, and thirty eight publications, about how vectors should be written (S177). On one side were the quaternionists, led by Peter Guthrie Tait, 1831 to 1901, of Edinburgh, who had corresponded with the ageing Hamilton from 1858 (one letter runs to 96 pages), published about seventy quaternion papers, and whose Elementary Treatise of Quaternions of 1867 gave heavy treatment to the operator , which is why Maxwell called Tait the "Chief Musician upon Nabla" (S177). On the other were Josiah Willard Gibbs at Yale, who privately printed the first half of his Elements of Vector Analysis in 1881 and the second in 1884, and Oliver Heaviside in England, who began introducing vectorial methods in 1883, gave his first unified presentation in 1885, received Gibbs's pamphlet around 1888, and published a 173 page chapter called "The Elements of Vectorial Algebra and Analysis" in his Electromagnetic Theory of 1893 (S177).
The tone was not academic. Tait called Gibbs's pamphlet "a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann," and called Gibbs "one of the retarders of Quaternion progress" (S177). Gibbs replied that if his offense had been solely in notations, "it would have been less accurate to describe my productions as a monstrosity, than to characterize its dress as uncouth" (S177). Late in the fight Tait compared coordinate geometry to a steam hammer and quaternions to "the elephant's trunk, ready at any moment for anything, be it to pick up a crumb or a field gun, to strangle a tiger, or to uproot a tree" (S177). Lord Rayleigh watched the whole thing and wrote, after Tertullian, "Behold how these vectorists love one another" (S177). Lord Kelvin, who had resisted quaternions since the 1867 Treatise on Natural Philosophy he wrote with Tait, said in 1901 that the two of them "had a thirty-eight years' war over quaternions" (S177).
Why this matters in your course. The dot product in Course section 3.1 is a war reparation. The quaternion product is a single operation that mixes a scalar part and a vector part; Gibbs and Heaviside made the practical decision to split it into two separate products, and that split is exactly what your textbook teaches as the dot product and the cross product (S177). By 1910 the Gibbs and Heaviside system had won and the other two contenders were dead (S177). One small warning for anyone reading a nineteenth century text: Tait's scalar product carried the opposite sign from the modern convention (S177).
The receipt. Crowe, S177, read in full: the debate's numbers, every quotation above, the publication counts, and Crowe's own explanation of the outcome. (S177)
So what? The losing side was not the weaker mathematics, and it was not the smaller side either. Crowe counts 594 publications in the quaternion tradition from the 1840s to 1900 against 217 in the Grassmannian tradition, a lead of 377, and Hamilton alone wrote 109 of the roughly 150 quaternion papers that appeared before his death in 1865 (S177). What beat all of that, on Crowe's account, was a bundle of unromantic advantages: the link to the fast-growing field of electrical science through Heaviside, textbooks that went into multiple editions, the rising authority of Gibbs and Heaviside as scientists, Maxwell's ambivalence about quaternionic methods, and the practical decision to split the product in two (S177). Notation wins on usability, not on beauty. That is worth arguing about in class, because it is still true of programming languages.
Will the solar system fall apart?
The word "secular" in "secular equation" is Latin for a long age, and it means the very slow drift of the planets' orbits over centuries. Miller records that the term recalls the long-run behavior of the solar system investigated by Laplace and Lagrange, and that it turns up in Jacobi's eigenvalue algorithm paper and in Cauchy's 1829 work on symmetric determinants (S176). This is where eigenvalues come. Not from data, not from vibration in the abstract, but from astronomers asking whether the arrangement of the planets is stable or whether it is quietly drifting apart.
Cauchy is the hinge. MacTutor says that in 1826, working with quadratic forms, he used the word "tableau" for the array of coefficients, "found the eigenvalues and gave results on diagonalisation of a matrix," introduced similar matrices, proved that similar matrices have the same characteristic equation, and "proved that every real symmetric matrix is diagonalisable" (S180). In 1840, in the "Memoire sur l'integration des equations lineaires," he wrote l'equation caracteristique (S176). MacTutor also notes that d'Alembert had met eigenvalue ideas eighty years earlier while studying a vibrating string with masses attached to it, and that Sturm generalized the eigenvalue problem to systems of differential equations (S180).
Then Sylvester names them, and the title of the paper gives the game away. In 1883, in "On the equation to the secular inequalities in the planetary theory," he writes: "It will be convenient to introduce here a notion... namely that of the latent roots of a matrix, latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf" (S178, quoting Sylvester 1883; the date is confirmed independently at S176).
Twenty one years later, in Gottingen, Hilbert used a different word. The first of his six Mitteilungen, "Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen," appeared in the Nachrichten of the Gottingen society in 1904 at pages 49 to 91, and that is where Eigenwert and Eigenfunktion enter mathematics (S176). The pagination is confirmed from the front matter of the 1912 Teubner collected volume, whose Vorwort opens "Im vorliegenden Buche bringe ich meine sechs Mitteilungen...", "In the present book I present my six communications..." (S184). The six run to 298 printed pages in total (verify/ch05.py, section 4). Hilbert's Spektrum arrives in the fourth Mitteilung of 1906, possibly borrowed, Dieudonne suggested, from Wirtinger's "Bandenspectrum" and the optical spectra of molecules (S176).
Why this matters in your course. Course section 3.7 gives you two words in its two definition boxes, "eigenvalue" and "characteristic polynomial," and they come from two different centuries and two different languages. "Characteristic" is Cauchy's, French, 1840. "Eigen" is Hilbert's, German, 1904. Merriam-Webster spells out what happened next: "eigenvalue" is a "partial translation of German Eigenwert, from eigen own, peculiar + Wert value," and its first known use in English is 1927 (S202). Somebody translated half a word and stopped. That is the vocabulary you are being examined on.
The receipt. Miller, S176, e.html and s.html, for Cauchy 1840, Sylvester 1883, Hilbert 1904, Courant and Hilbert 1924, Dirac 1926, Eddington 1927, Halmos, and Spektrum. Higham, S178, for the full Sylvester quotation and the title of the 1883 paper. MacTutor, S180, for Cauchy's 1826 results, d'Alembert, and Sturm. Hilbert, S184, for the pagination of all six Mitteilungen and the Vorwort. Merriam-Webster, S202, for the etymology and the 1927 date. (S184; SCHOLARLY | S178; REFERENCE | S176, S180, S202)
So what? Because the argument about the word lasted sixty years and somebody kept the receipts. Miller records the earliest English "eigenfunctions" in Dirac's 1926 paper "On the Theory of Quantum Mechanics," and the earliest English "eigenvalue" in a letter Arthur Stanley Eddington sent to Nature on 23 July 1927, which is charming because Eddington is explaining Schrodinger to people who cannot follow him: readers "trying to acquire a general acquaintance with Schrodinger's wave mechanics" who "find their mathematical equipment insufficient to follow his first great problem, to determine the eigenvalues and eigenfunctions for the hydrogen atom" (S176). Von Neumann wrote "proper values" in English. Halmos, in Finite Dimensional Vector Spaces in 1958, page 102, complained that "Almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature," and campaigned for years: "For many years I have battled for proper values, and against the one and a half times translated German-English hybrid that is often used to refer to them" (S176). In A Hilbert Space Problem Book in 1967 he gave up in four words: "eigenvalues have won it" (S176). From Eddington's letter in 1927 to the compilation of this book in 2026 is 99 years. The standard vocabulary of Course section 3.7 is not yet a century old in English, and it won an argument rather than an election.
One thing this book could not settle, and you should know about it. Cauchy's eigenvalue work is dated 1826 by MacTutor and 1829 by Miller, and the standard citation in numerical linear algebra is his 1829 memoir on the secular inequalities of planetary motion. Those are probably two papers in one body of work, but nobody here could open either: Gallica is unreachable from this environment, and the definitive scholarly treatment, Thomas Hawkins's "Cauchy and the spectral theory of matrices" of 1975, is only on ScienceDirect, which is blocked (S180, S176). See 5.10, item 1.
A footnote about continued fractions, and the twenty five billion dollar eigenvector
Oskar Perron's Habilitationsschrift was published in 1907 as "Grundlagen fur eine Theorie des Jacobischen Kettenbruchalgorithmus," Mathematische Annalen 64, pages 11 to 76. It is about the convergence of Jacobi's multidimensional continued fractions. In section 14 he considers real nonnegative partial quotients and needs a fact about matrices whose entries are all positive, so he states it and proves it. MacCluer's description of what Perron thought he was doing is the whole story in six words: he proved "what he apparently thought to be a mere technical lemma" (S185).
Here is the lemma, in MacCluer's statement: "The eigenvalue of largest absolute value of a positive (square) matrix A is both simple and positive and belongs to a positive eigenvector. All other eigenvalues are smaller in absolute value" (S185). Five years later, Georg Frobenius extended it. His papers "Uber Matrizen aus positiven Elementen" appeared in the Berlin Sitzungsberichte in 1908 and 1909, and "Uber Matrizen aus nicht negativen Elementen" in 1912, pages 456 to 477 (S185). Frobenius's extension covers matrices that are merely nonnegative, provided they are what he called unzerlegbar, indecomposable, which we now translate as irreducible: a matrix that cannot be permuted into block upper triangular form (S185, S186). MacTutor puts his contribution plainly: "He introduced the concept of irreducibility for matrices and the papers which he wrote containing this theory around 1910 remain today the fundamental results in the discipline" (S187).
Why this matters in your course. This is the theorem that makes Unit 4 a well-posed question. Higham's modern statement is short enough to check on a 2 by 2: if A is nonnegative and irreducible then its spectral radius is an eigenvalue, it is strictly positive, it has a strictly positive eigenvector, and it is a simple eigenvalue; and if A is positive, no other eigenvalue can match it in size (S186, theorems 2 and 3). Read that as a sentence about a Markov chain and it says: there is exactly one long-run distribution, every state gets a positive share of it, and repeated multiplication converges to it. The normalized positive eigenvector even has a name now, the Perron vector, and the eigenvalue is the Perron root (S186). Course section 3.7 tells you an eigenvector is a direction a matrix does not turn. Perron-Frobenius tells you that for a whole important class of matrices, exactly one such direction points into the positive corner.
The receipt. MacCluer, S185, for the 1907 and 1912 citations, for the statement of Perron's theorem, for unzerlegbar, for the "mere technical lemma" judgment, and for the applications list. Higham, S186, for the three modern statements and for "Perron root" and "Perron vector". MacTutor, S187, for Frobenius's dates and for the irreducibility sentence. (S185; REFERENCE | S186, S187)
So what? Two payoffs, both worth real money, and one of them is checked line by line in 5.7.
The first is search. Sergey Brin and Lawrence Page's 1998 paper, written at Stanford about a 1997 crawl of 24 million pages carrying 259 million anchors and 322 million links, states in its own words that PageRank "can be calculated using a simple iterative algorithm, and corresponds to the principal eigenvector of the normalized link matrix of the web" (S198). Their model is a person clicking at random: "The probability that the random surfer visits a page is its PageRank" (S198). Bryan and Leise, writing for exactly your reading level in SIAM Review in 2006, prove that the damped matrix always has a one dimensional eigenspace for eigenvalue 1, so the ranking is unique and positive, and they say where the proof comes from: it "is basically a special case of the Perron-Frobenius theorem" (S197). One caution for anyone reading around: Brin and Page use a damping factor "usually set to 0.85" and Bryan and Leise use . Those are the same choice in complementary conventions, , and they are not a disagreement (S197, S198). Perron's technical lemma of 1907 is 99 years older than Bryan and Leise's paper, and MacCluer's 1998-era survey of the theorem's applications does not mention Google at all (S185).
The second is economics. Wassily Leontief, born in St Petersburg on 5 August 1906 and working at Harvard, received the 1973 Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel "for the development of the input-output method and for its application to important economic problems" (S199). The method is a matrix A of how much of everything you need to make one unit of everything else, and then the equation : solve it and you know how much the whole economy must produce so that consumers get the final demand . It is one of MacCluer's listed applications, and it only gives sensible answers when the spectral radius of A is below 1, which is a Perron-Frobenius condition (S185, S199). Somebody won a Nobel Prize for inverting a matrix. That is a slight simplification, but only slight.
Aircraft that shook themselves apart, and the year the machines arrived
Olga Taussky was born on 30 August 1906 in Olmutz, then in the Austro-Hungarian Empire and now Olomouc in the Czech Republic. She took her doctorate at the University of Vienna in 1930, aged 24, and spent 1931 to 1932 at Gottingen editing the number theory volume of Hilbert's collected works alongside Wilhelm Magnus and Helmut Ulm (S192). In the Second World War she was put on a problem with a body count. She describes it herself: "I was assigned to the study of flutter in supersonic aircraft, which leads to boundary value problems in hyperbolic partial differential equations," and she defines the phenomenon in the same article: "In flight the interaction between the elastic forces in the airframe and the aerodynamic forces induces self-excited vibration which, above a certain speed, is unstable. This phenomenon is called flutter" (S191). She worked on it at the National Physical Laboratory at Teddington from 1943 to 1946 (S192).
The problem reduced to "a certain 6 x 6 matrix of the form , where , the flutter parameter, is taken as 1," and to the question whether it had any real eigenvalue to the left of certain small circles (S191). And then the detail that every student should hear: "By a mere accident I had heard about the Gersgorin theorem, whose statement is given in a Zentralblatt review" (S191). She had read a review, in a review journal, of a 1931 paper by S. Gersgorin in the Izvestiya of the Soviet Academy of Sciences, pages 749 to 754, and it turned out to be the tool that bounded where the eigenvalues could sit (S191).
Meanwhile, at Princeton, the machines were being invented and nobody agreed how much precision they needed. John von Neumann and Herman Goldstine presented "Numerical inverting of matrices of high order" on 5 September 1947; the editors received it 26 days later on 1 October, and it filled 79 pages of the Bulletin of the American Mathematical Society for November 1947, pages 1021 to 1099 (S188). Its purpose is one sentence long: "The purpose of this paper is to derive rigorous error estimates in connection with the inverting of matrices of high order" (S188). The authors say why it was needed: the new computing devices had revived interest in numerical methods, and there was wide disagreement about how much precision it takes to invert a matrix of order 10 or more (S188). Their headline result, for elimination with proper scaling: matrices of orders 15, 50, and 150 can usually be inverted with a relative precision 8, 10, and 12 decimal digits worse, respectively, than the number of digits you carry (S188).
A year later Alan Turing, at the National Physical Laboratory in Teddington, published "Rounding-off errors in matrix processes" in the first volume of the Quarterly Journal of Mechanics and Applied Mathematics, pages 287 to 308 (S189).
Why this matters in your course. This is where linear algebra stops being algebra and becomes engineering, and it is where two words your software prints at you were born. Dopico's correction is the important one, because the folklore has it backwards: von Neumann and Goldstine introduced the LU factorization in 1947, and Turing stated the condition for its existence and uniqueness and introduced the name "LU" for the general decomposition (S189). On the other term, Dopico is precise about what is and is not known: "It is not easy to determine who discovered the 'condition number'. No question that the name was introduced by Turing (1948)" (S189). Course section 3.5's question, is this matrix invertible, becomes on a machine a different question, how close to non-invertible is it, and that second question is what a condition number answers.
The receipt. Taussky, S191, her own 1988 memoir in the American Mathematical Monthly: the flutter assignment, the definition, the 6 by 6 matrix, and the Gersgorin sentence with its bibliography entry. Luchins and McLoughlin, S192, for every biographical date. Von Neumann and Goldstine, S188, read at the AMS open archive for the front matter, the stated purpose, and the digit-loss figures; the 79 pages of analysis were not read line by line. Dopico, S189, read in full, for LU, for the condition number, and for the Wilkinson quotations. Turing's 1948 paper itself was NOT read: the Oxford Academic copy serves a landing page and an abstract only. Two bibliographic facts about it are safe and come from an open-access record: the journal received it on 4 November 1947 and Turing's affiliation on it is the National Physical Laboratory, Teddington, Middlesex (S390, via). (S188, S191; SCHOLARLY | S189, S192)
So what? Because the person who made all of it usable is the one nobody has heard of. James Hardy Wilkinson, born 27 September 1919 in Strood, Kent, took his Cambridge degree in 1939 and spent 1940 to 1946 on the thermodynamics of explosions, ballistics, supersonic flow, and shell fragmentation, moving from analytic methods to numerical ones on mechanical calculating machines (S195, S196). In May 1946 he became Turing's assistant at the NPL on the Automatic Computing Engine, and wrote floating-point arithmetic subroutines for a machine that did not exist yet. When Turing left for Manchester in 1948 Wilkinson took over, and the Pilot ACE ran in May 1950; it was famous enough in Britain that "A cartoon about Pilot ACE even appeared in the Daily Mirror!" (S196). His idea was backward error analysis, and it is a flip a 16 year old can appreciate. Instead of asking how wrong your answer is, ask what problem your answer is the exact solution to. If it is the exact answer to a question very close to the one you asked, your algorithm is fine and your data was the problem. He wrote Rounding Errors in Algebraic Processes in 1963 and The Algebraic Eigenvalue Problem in 1965, was elected FRS in 1969, and won the Turing Award in 1970. George Forsythe's verdict in 1960: "Wilkinson is single-handedly responsible for the creation of almost all of the current body of scientific knowledge about the computer solution of the problems of linear algebra" (S195).
And Taussky did something else that is easy to miss. She and John Todd moved to the USA in 1947, and in her own words: "At that time I picked up the torch. Matrix theory had become a subject for me. Matrices were not any longer just used, they were algebraic structures like rings, groups, lattices...." (S191). She organized the first matrix theory conference at the National Bureau of Standards in 1951, and went to Caltech in 1957 as the first woman on its mathematics faculty. She arrived as a Research Associate, with permission but no obligation to teach. Tenure came in 1963, the year she turned 57, and a full professorship in 1971, the year she turned 65, making her the first woman at Caltech at that rank. She published about 300 papers (S192). Her line about credit belongs on a classroom wall: "We mathematicians are too quick to credit the developer and forget the explorer" (S192).
One more from the same decade. In December 1952 Magnus Hestenes of the National Bureau of Standards and UCLA and Eduard Stiefel of UCLA and the ETH in Zurich published "Methods of Conjugate Gradients for Solving Linear Systems" in the Journal of Research of the National Bureau of Standards 49(6), pages 409 to 436. Their opening line is the whole reason this chapter exists: "One of the major problems in machine computations is to find an effective method of solving a system of n simultaneous equations in n unknowns, particularly if n is large." Read their own caveat with a class, because it is the honest sentence: "The cg-method is an iterative method which terminates in at most n steps if no rounding-off errors are encountered" (S190). Real machines have rounding errors.
Two strangers, one algorithm, and a decomposition named after the wrong people
In 1959 a young man at a British government body called the National Research and Development Corporation, employed by Christopher Strachey, wrote a new way to compute the eigenvalues of a matrix. He wrote it in assembly language, for a computer called Pegasus. He submitted part 1 to The Computer Journal on 29 October 1959, resubmitted it with part 2 on 6 June 1961, and the two papers appeared in October 1961 at pages 265 to 271 and 332 to 345 (S193). His name was John G. F. Francis, and in 1961 he moved to Ferranti Ltd to write a commercial compiler for the Orion computer, and left the subject (S193).
Two hundred and fifty days after Francis's first submission, on 5 July 1960, Vera Nikolaevna Kublanovskaya submitted "Certain algorithms for the solution of the complete eigenvalue problem" to the Doklady of the Soviet Academy of Sciences, where it appeared in volume 136 at pages 26 to 28 (S193, S194). Both had started from the same source, Heinz Rutishauser's 1958 paper on the LR algorithm (S193). Neither knew the other existed.
Kublanovskaya's route there was not a conventional one. Born on 21 November 1920 in a village in Vologda Oblast, she went to secondary school in Belozersk, trained as a primary teacher, and entered the Gertzen Pedagogical Institute in Leningrad in 1939. When her mother fell seriously ill during the war she went home, which is how she missed the Siege of Leningrad; she taught mathematics locally and helped build the village defenses. She resumed her studies in 1945, took her degree at Leningrad State University in 1948, joined the Steklov Institute's Leningrad branch, worked with Leonid Kantorovich on secret computational projects connected with the atomic bomb until 1955, and took her candidate's degree that year (S194). She was 39 years old on the day she submitted the QR paper.
Then the strange part. Dongarra and Sullivan placed the QR algorithm sixth on their list of the top ten algorithms of the twentieth century in 2000; Beresford Parlett called it "a genuinely new contribution to the field of numerical analysis" and Nick Higham "one of the jewels in the crown of matrix computations" (S193). And around the year 2000, "nobody in the numerical analysis community of mathematicians had any recollection of ever having seen John Francis himself" (S193). Frank Uhlig and Gene Golub went looking for him separately, met by chance to compare notes "in the Steklov Institute coffee room during the Second International Conference on Matrix Methods and Operator Equations in Moscow in July 2007," and found him. Golub visited Francis at his home in England on 7 August 2007, forty five years and ten months after the papers were published. Uhlig went in late July 2008 (S193).
Why this matters in your course. When Course section 3.7 asks you to find the eigenvalues of a 2 by 2 matrix by hand, you factor a quadratic. That does not scale: there is no formula for the roots of a general polynomial of degree five or more, so for a 5 by 5 matrix the characteristic polynomial is a dead end as a computational method. QR is what replaced it, and it is what runs inside every eigenvalue command you will ever type. Golub and Uhlig are unusually clear about the credit: "John Francis developed the whole complex of the QR algorithm on his own, apart from the obvious influence of Rutishauser's paper (Rutishauser, 1958). John did not collaborate with anybody. The ideas, theorems and implementation for and of QR were all his" (S193). And, in the same paper, that Kublanovskaya began developing her version in 1958 after reading the same Rutishauser paper (S193).
The receipt. Golub and Uhlig, S193, read in full: all the submission dates, the Pegasus and the assembly language, the Moscow coffee room, the visits, and both quotations. MacTutor, S194, for Kublanovskaya's life. The 250 days between the two submissions and the interval to Golub's visit are computed in verify/ch05.py, section 4. (S193; REFERENCE | S194)
So what? Then finish with the decomposition that carries the wrong name, because it is the one a machine learning course cares about most.
The singular value decomposition was first published in 1873 by Eugenio Beltrami, in "Sulle funzioni bilineari" in the Giornale di Matematiche ad Uso degli Studenti Delle Universita, volume 11, pages 98 to 106 (S170). Read that journal title: a journal for the use of university students. Its purpose was to get undergraduates comfortable with bilinear forms (S170). Beltrami was 38, and he was not a comfortable professor: he had run out of money in 1856 and left the University of Pavia, worked as secretary to a railway engineer between Verona and Milan, and only got an academic post six years later, at Bologna in 1862 (S201). Camille Jordan derived the same decomposition independently in 1874, by a variational argument with deflation that avoids the degeneracies complicating Beltrami's approach. Stewart's verdict is the fair one: "Together Beltrami and Jordan are the progenitors of the singular value decomposition, Beltrami by virtue of first publication and Jordan by the completeness and elegance of his treatment." Sylvester published three items on it in 1889, and Autonne extended it to complex matrices in 1913 (S170).
Now the naming. In 1907, in Mathematische Annalen 63, pages 433 to 476, Erhard Schmidt proved that the best rank-k approximation to a matrix, measured in the Frobenius norm, is its truncated singular value decomposition. Stewart calls this "the crowning glory of Schmidt's work" (S170). Hermann Weyl gave another proof in 1912 and built the perturbation theory around it, and Stewart's closing judgment is that "With Weyl's contribution, the theory of the singular value decomposition can be said to have matured" (S170). Then, in 1936, Carl Eckart and Gale Young published in the psychology journal Psychometrika, volume 1, pages 211 to 218. They extended the decomposition to rectangular matrices, and they "rediscovered Schmidt's approximation theorem, which is often (and incorrectly) called the Eckart-Young theorem" (S170). Miller records the standard attribution to Eckart and Young and adds, in five words, "The result, however, is much older" (S176). Note that this book could not open the 1936 paper itself; Stewart is the authority for what is in it.
And the gap between having the theorem and being able to use it was 92 years. In 1965 Gene Golub and William Kahan published the method that computes the SVD without first forming , using an augmented matrix and a reduction to a tridiagonal problem with Sturm sequences (S200). That detail is the whole point: squaring the matrix squares its condition number, so the obvious method throws away half your digits. Golub and Businger applied QR with implicit shifts in a 1967 Stanford technical report, and Golub and Reinsch published the version that went into libraries in 1970 (S200). Even the name "singular value" is borrowed: it came out of the literature on integral equations, where Bateman used it in 1908 for the reciprocals of a kernel's eigenvalues, and the modern sense settled with Smithies in 1938 (S170, S176).
Compression, principal component analysis, and recommendation systems all run on that 1907 theorem, computed by that 1965 method, under a name given to it by two people in 1936 who were rediscovering it. Course section 3.7 lists Principal Component Analysis as a glossary headword. This is its family tree.
The word "eigenvalue" is half German and half English. Who wrote the German half, when, and what did English speakers call these numbers before that?
Show the answer▸
David Hilbert, in 1904, in the first of his six papers on linear integral equations, where Eigenwert and Eigenfunktion enter mathematics (S176).
Before that, English had a perfectly good word of its own. In 1883 Sylvester called them latent roots: "latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf" (S178). Cauchy had already fixed characteristic for the polynomial (S180).
So the hybrid you write in Course section 3.7 is what happens when a language borrows half a compound word and translates the other half. Nobody decided this. It settled.
Chapter 6
Built to win an argument
Unit 4 asks you to build a transition matrix, classify its states, solve for a stationary distribution, and test a chain for reversibility. Every one of those four skills exists because a fifty year old number theorist in St Petersburg wanted to destroy another man's argument about free will, and needed a counterexample to do it. By the end of this chapter you will know what the argument was, what Markov built to break it, how he tested it on twenty thousand letters of a poem counted by hand, and how the same object ended up ranking web pages, transcribing speech, and running most of modern statistics. You will also be able to catch four or five things your textbook probably got wrong, including the one about the church.
Pavel Nekrasov claimed that the law of large numbers requires independent events, that social statistics obey the law, and therefore that human choices are free. Markov, who called the work an abuse of mathematics, built sequences that are strongly dependent and obey the law anyway, in a paper of 1906, and in 1913 he tested the idea on the first 20,000 letters of Eugene Onegin, sorted into 8,638 vowels and 11,362 consonants by hand. The Ehrenfests had already used the same kind of chain in 1907 to settle an argument in physics. Kolmogorov made it analytic in 1931, Doeblin pushed it further in a notebook nobody read until 2000, Shannon used it to model English in 1948 without ever reading Markov, and Metropolis and the Rosenbluths turned it inside out in 1953 so you could design a chain to land wherever you wanted.
Andrei Markov built the chains in Unit 4 to win an argument. Guess what the argument was about.
I have a guess▸
Free will. Pavel Nekrasov had argued that the law of large numbers requires independent events, that social statistics obey the law of large numbers, and therefore that human choices must be independent, which is to say free (S218).
Markov thought this was an abuse of mathematics. His answer was not a philosophical rebuttal. He built a sequence of events that are strongly dependent and obey the law anyway, which destroys the middle step and takes the conclusion with it.
If you guessed weather, or language, or Google, those all came later and none of them was the point. The transition matrix in Course section 4.1 exists because two Russian mathematicians disagreed about whether people are free.
A proof of free will, and the man who broke it
Pavel Alekseevich Nekrasov, 1853 to 1924, graduated from a Russian Orthodox seminary before he entered Moscow University, and in politics, Oscar Sheynin writes, "he associated himself with reactionary elements" (S218, foreword p. 2). Around 1900 his writing became, in Sheynin's word, "unimaginably verbose", and he began tying mathematics to religion and politics (S218). In 1902 he published a paper of 138 pages in Matematicheskii Sbornik, volume 23, pages 463 to 600, with the title "Philosophy and logic of the science of mass expressions of human activities" (S218).
Here is the chain of reasoning he needed, reconstructed from a peer reviewed specialist article rather than from a popular summary. Social statistics, things like crime rates and marriage rates, are strikingly regular from year to year. The law of large numbers explains that kind of regularity. Nekrasov claimed that the law of large numbers requires independent events. Human choices, being independent of one another, are therefore free. Basharin, Langville, and Naumov put the middle step in his own reported words: Nekrasov "erroneously claimed that 'independence is a necessary condition for the law of large numbers'" (S212, preprint p. 11), and they set out the motive plainly: "Moscow members like Nekrasov tried to enlist statistics and probability to provide a foundation for their doctrine of free will" (S212, preprint p. 6).
Markov attacked the weakest link. He did not argue about theology. He built a counterexample: sequences of events that are emphatically not independent, each one depending on the one before, which obey the law of large numbers anyway. That is the 1906 paper, "Extension of the law of large numbers to dependent quantities", Izvestiia of the Physico-Mathematical Society of Kazan University, second series, volume 15, number 4, pages 135 to 156 (S211, ref. 27; S212, ref. 15). He was fifty years old, and he had spent his career on number theory and continued fractions (S211, p. 1; checked in verify/ch06.py section 7). His conclusion is one sentence: "independence of quantities does not constitute a necessary condition for existence of the law of large numbers" (S220).
Why this matters in your course. Definition 4.1.1 gives you the Markov property: what happens next depends on where you are now, not on how you got here. That definition is the exact midpoint between independence and full memory, and it was invented to occupy that midpoint. Merriam-Webster's dictionary definition is the same idea in one sentence for a general reader: "the probabilities of occurrence of various future states depend only on the present state of the system or on the immediately preceding state and not on the path by which the present state was achieved" (S244).
The receipt. Basharin, Langville, and Naumov, S212, preprint pp. 6 and 11, read in full in the author-side preprint. Sheynin, S218, foreword pp. 2 and 3, on Nekrasov's seminary training and politics, and for the 1902 citation. Encyclopedia of Mathematics, S220, for Markov's conclusion. Seneta, S211, p. 1, for the 1906 paper and Markov's age. (S211, S212, S218; REFERENCE | S220, S244)
So what? Two honest cautions. First, only Nekrasov's side of the correspondence survives, so we have his insults and not Markov's replies (S218, foreword p. 2). Second, Seneta is careful to write that Markov "interpreted" Nekrasov as making the necessity claim (S211), which leaves open whether Markov read him fairly. Nobody in this book has read Nekrasov's own words on free will. The paper that carries them, Seneta's 2003 article on Quetelet and Nekrasov, is behind a paywall that this book could not open (S218, refs). What is documented is the logical structure, and it is enough: if dependent events obey the law too, then observing the law tells you nothing about freedom.
Twenty thousand letters, counted by hand
On 23 January 1913 O.S. (5 February N.S.) Markov read a paper to the physical-mathematical faculty of the Academy of Sciences in St Petersburg. Its title, in the Custance and Link translation, is "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains" (S214, title and header). He had taken the first 20,000 letters of Pushkin's novel in verse, thrown out the punctuation, the spaces, and the Russian hard and soft signs, and sorted every remaining letter into vowel or consonant (S213, S214).
The physical labor is the part worth telling. He divided the 20,000 letters into 200 groups of one hundred, arranged each group in a square table of ten rows and ten columns, combined the columns in pairs, wrote the five sums underneath each other in a vertical column, and produced 40 such tables, "which fill an entire page" (S213). Out came four numbers: 8,638 vowels and 11,362 consonants, giving ; 1,104 vowel-vowel pairs, giving ; and , the chance that a vowel follows a consonant, which reconstructs from his counts as 7,534 over 11,362. Markov's own line for the gap between the two conditional rates is "delta is 0.128 - 0.663 = -0.535" (S214). Then he did the whole thing again on 100,000 letters of Aksakov (S213).
Why this matters in your course. One of Unit 4's named methods is "building a transition matrix from a word problem". This is where that method comes, and it is not a word problem: it is data, counted by a 56 year old academician with a pen. The four numbers give you the matrix directly, and is the second eigenvalue of it, which is what controls how fast the chain forgets. All of that is worked in 6.7.2.
The receipt. Markov in translation, S214: the counts, , , , and the delta line. Link, S213, for the 200 groups, the ten by ten tables, and the Aksakov study. Von Hilgers and Langville rank this application first of five and quote "the stationary vowel probability is p = 0.432" (S243). The internal consistency of Markov's four numbers is exact on the integer counts, at F-1.1 to F-1.9 of verify/domainF_output.txt. (S214; SCHOLARLY | S213, S243)
So what? Be careful with the retellings. Brian Hayes, in American Scientist, gives derived pair counts of 1,104 vowel-vowel, 3,827 consonant-consonant, and "the remaining 15,069 pairs" as mixed (S210, p. 95). A run of 20,000 letters contains 19,999 overlapping pairs, not 20,000, and 19,999 minus 1,104 minus 3,827 is 15,068. Hayes's printed figure needs the text treated as a closed loop. Both arithmetics are in verify/ch06_output.txt section 2, and every count they rest on is printed in verify/book_numbers.py section 1:
`` [PASS] mixed pairs on the 19,999 convention computed = 15068 claimed = 15068 [PASS] Hayes's printed 15,069 needs a 20,000-pair loop computed = 15069 claimed = 15069 [PASS] the two conventions differ by exactly one pair computed = 1 claimed = 1 ``
Use Markov's own four numbers, which are exactly consistent, and leave the derived pair counts alone.
The church that would not throw him out
In February 1912 O.S., Andrei Andreevich Markov wrote to the Most Holy Governing Synod of the Russian Orthodox Church and asked to be excommunicated. He gave three reasons: he did not believe the Biblical stories, he saw no essential difference between icons and idols, and he did not sympathize "with any religion which, like the Orthodoxy, is supported, and in turn lends its support to fire and sword" (S217).
The Synod refused. It resolved instead that Markov had seceded from God's Church, and a comment on one of its internal letters records why the refusal: excommunication "would be too honourable for Markov" (S217). He was refused even the dignity of being expelled.
That was not an isolated gesture. In 1902 Maxim Gorky's election to the Academy was annulled on the tsar's orders, and Markov protested and refused the honors awarded him the following year (S219, S210). In June 1907 O.S., when Nicholas II dissolved the Second Duma, Markov publicly repudiated his membership of the electorate; MacTutor notes he "might have expected to suffer severe consequences but the authorities chose not to make an example of an elderly and distinguished academician" (S219). In 1913, when the Romanovs celebrated three hundred years in power, Markov "showed his disapproval of the celebration but holding celebrations of his own", as MacTutor puts it: "he celebrated 200 years of the Law of Large Numbers" (S219). Bernoulli's Ars Conjectandi was printed in 1713, so the bicentenary was real, and it landed on the tercentenary (both intervals checked in verify/ch06.py section 7). He timed the third edition of his textbook to the same year (S211). In September 1917 O.S. he asked the Academy to send him to a disadvantaged town in the Russian interior and was posted to Zaraisk, where he taught mathematics in the secondary school "without receiving any remuneration" (S219). On 5 March 1921 he wrote to the Academy to say that, for lack of footwear, he could not attend its meetings (S212).
Why this matters in your course. Not at all, mathematically, and that is the point of putting it. The chain in Course section 4.1 was built by a person, in a place, at a cost, and the same combativeness that produced "an abuse of mathematics" produced the Synod letter. Basharin, Langville, and Naumov record the nicknames: "Andrew the Furious", and "the militant academician" (S212).
The receipt. Sheynin, S217, read in full: "In February 1912, the eminent Russian mathematician Andrei Andreevich Markov (1856 - 1922) sent his request to the Most Holy Governing Synod of that Church"; "His request was not granted; the Synod resolved that Markov had seceded from God's Church"; "excommunication would be too honourable for Markov". Sheynin is working from L. I. Emeliakh's archival study of 1954, which quotes the documents (S217, refs). MacTutor, S219, for Gorky, the Duma, the counter-celebration, and Zaraisk. (S212, S217; REFERENCE | S219)
So what? This is the single best example in the book of a story everybody tells wrong. Hayes prints it as "When the Russian church excommunicated Leo Tolstoy, Markov asked that he be expelled also. (The request was granted.)" (S210, p. 93). Two things in that sentence are shaky. The request was refused, on documented archival evidence. And the Tolstoy framing is a compression: Tolstoy was excommunicated in 1901 and died in 1910, so the 1912 timing needs another explanation, and Sheynin proposes the Beilis blood-libel prosecution in Kiev, noting that Markov actively protested the antisemitic campaign (S217). That proposal is Sheynin's inference and not a documented statement of Markov's motive, so treat the neat Tolstoy version as a legend and the refusal as fact.
Two authors, one urn, and why heat only flows one way
Volume 8, issue 9 of Physikalische Zeitschrift, dated 6 May 1907, lists in its own contents: "P. u. T. Ehrenfest, Uber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem. S. 311" (S224). P. und T. is Paul and Tatiana. The paper answers the two famous objections to Boltzmann's H-theorem, Loschmidt's reversibility objection and Zermelo's recurrence objection (S221, S224), with a model you can run with counters on a table: put 2R numbered balls in two boxes, pick a number at random, and move that ball to the other box.
Mark Kac, writing in 1947, credits "P. and T. Ehrenfest" in his text (S221). Felix Klein, commissioning the follow-up encyclopedia article in 1906, invited both of them, "whom he explicitly included in his invitation" (S223). Margriet van der Heijden's judgment on which half is whose is worth quoting in full: "The urn model aligns with Afanassjewa's inclination for mathematics, her fondness for logical reasoning, and her lifelong interest in probability. The flea model is clearly a product of Ehrenfest" (S223). Tatiana Alexeyevna Afanassjewa, born in Kiev on 19 November 1876, had not been allowed to join the Gottingen mathematics club because it excluded women, until Paul got the rule changed (S222). In 1926 Paul wrote that "It would be entertaining if all publications by my wife and myself could at one point all be [...] printed together in chronological order" (S223).
Kac's answer to the physics argument is a number. With R = 10000 and one move per second, the expected time to return to the all-in-one-box state is seconds, "of the order of years" (S221). Meanwhile the expected return time to the balanced state is 177.25 seconds, which Kac rounds to about 175 (S221; the exact value is recomputed in verify/ch06_output.txt). Both objections are correct, and both are irrelevant, and the reason is the size of a stationary probability.
Why this matters in your course. The Ehrenfest urn is Course section 4.4 in physical form. Its stationary distribution is binomial, it satisfies detailed balance on every pair, and the expected return time to a state is one divided by that state's stationary probability. Section 6.7.3 runs the whole thing with four balls and exact fractions.
The receipt. The contents page of the 1907 issue, S224, read as Internet Archive OCR, with the page number confirmed against Kac's independent citation (S221, ref. 7). Kac, S221, for the model, for Zermelo's objection ("Thus, argued Zermelo, the irreversibility postulated in thermodynamics and the 'recurrence' properties of dynamical systems are irreconcilable"), and for the recurrence times. Van der Heijden, S223. MacTutor, S222, for Afanassjewa's life. (S224; SCHOLARLY | S221, S223; REFERENCE | S222)
So what? Do two things with this. Call it the Ehrenfest urn and not "Ehrenfest's model", because the journal's own contents page names both authors and has done since 1907. And check Kac's numbers rather than repeating them. His seconds is seconds, which is about years, so the printed is an order of magnitude and not a computation. His 175 seconds for the balanced state, on the other hand, is nearly exact: the stationary probability of the balanced state with 20,000 balls is , and one divided by that is 177.25 seconds (verify/ch06_output.txt, section 7). Either of the big figures dwarfs the age of the universe, which is around years.
The envelope
Wolfgang Doeblin was born in Berlin on 17 March 1915, the son of Alfred Doblin, the neurologist and novelist who wrote Berlin Alexanderplatz (S233, S234). He was expelled from the School of Political Science in Berlin in 1932 for socialist propaganda and anti-Nazi activity; the family fled when the Reichstag burned in 1933 and took French citizenship (S232, S234). He studied mathematics in Paris from 1934 to 1939, worked at the Institut Henri Poincare from 1935 to 1938, took his doctorate under Maurice Frechet in 1938, and in a career of about five years published 13 papers and 13 notes (S232, S233).
Then he was a soldier, a duty telephonist in a French infantry regiment on the Lorraine front (S232). On 26 February 1940 he posted a sealed envelope to the Academie des Sciences in Paris. Under the Academy's pli cachete rules a sealed deposit stays shut until the author, or his heirs, ask for it. It was filed as number 11668 (S232, S234). On 21 June 1940, 116 days later, with German troops at the village of Housseras in the Vosges and no way out, "W. Doeblin burnt his remaining papers and committed suicide in a barn" (S234). He was twenty-five (S232).
The envelope was opened in May 2000, at the request of his family. Inside was an exercise book containing a handwritten memoir called "Sur l'equation de Kolmogoroff" (S233, S234). Bernard Bru, who ran the operation, says it "gives for the first time a general theory of the case of Markov motions (whose time laws satisfy the Kolmogorov equation)", and that "It anticipated stochastic calculus as it is practiced not only in the universities of the world, but also in major banks to calculate costs and profits of financial products subject to market uncertainties" (S232). MacTutor puts the same point concretely: the notebook contained early stochastic calculus, including a version of Ito's formula (S233). It was printed complete in the Comptes rendus in December 2000 (S232).
Why this matters in your course. Doeblin pioneered the coupling method and worked on Markov chains with general state spaces (S233). Coupling is the modern machinery behind the sentence your textbook writes as "the chain settles down", and the name a student meets as "the Doeblin condition" belongs to a boy who died at twenty-five with his best work in a filing cabinet in Paris.
The receipt. Bru, S232, p. 61, read in full: "In the spring of 2000, at the request of the legatees, the Commission of 'sealed documents' (plis cachetes) of the Academie des Sciences of Paris opened file 11668, deposited 26 February 1940." MacTutor, S233, for May 2000, the doctorate, and Ito's formula. The Zentralblatt feature, S234, for the burning of the papers and the barn. The 116 day interval is computed in verify/ch06.py section 7. (S232; REFERENCE | S233, S234)
So what? Watch two numbers. The Zentralblatt feature says the envelope was opened in 1991 (S234); Bru, who was in the room, says the spring of 2000, and MacTutor says May 2000. May is in spring, so Bru and MacTutor agree, and this book prints May 2000 and never 1991. Second, Bru's own sentence reads "Two months later, 21 June 1940, the soldier Doblin killed himself" (S232), but the two dates he gives are 116 days apart, which is nearly four months. Print the dates, not the interval word.
An equation with two names, one of them borrowed
Kolmogorov's "Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" is dated 26 July 1930 and appeared in Mathematische Annalen 104 in 1931, pages 415 to 458 (S236, S246). Its focus, Shafer and Vovk write, is "the equation that ties together the transition probabilities... now called the Chapman-Kolmogorov equation", and "The article launched the study of this equation by purely analytical methods, a study that kept probabilists occupied for fifty years" (S236). MacTutor adds a separate item, and the order of the dates is worth watching: in 1931, on a trip to Lake Sevan in Armenia with Pavel Aleksandrov, Kolmogorov worked on Markov processes with continuous states and continuous time, work that MacTutor says "mark the beginning of diffusion theory" (S237). That trip is a year later than the July 1930 date on the Mathematische Annalen paper, so the two are not the same piece of work, whatever the shared year of publication suggests.
Now the other half of the name. Sydney Chapman lived from 1888 to 1970, was born in Eccles near Manchester, and died in Boulder, Colorado. His fields, in MacTutor's summary, were "gas dynamics, geomagnetism and the ionosphere", and he won the Adams Prize in 1928 for an essay on geomagnetism (S238). MacTutor's biography of him does not mention the Chapman-Kolmogorov equation at all. Neither do Shafer and Vovk cite any Chapman paper while using the name (S236).
Why this matters in your course. Course section 4.3 gives you , the n-step forecasting formula. The Chapman-Kolmogorov equation is the general statement behind it: to get from now to later, add up all the ways of going through a middle time. Multiplying transition matrices is that sum, written as matrix multiplication.
The other engine under Course section 4.3 is Perron-Frobenius, and it belongs to Chapter 5, which is where the eigenvalue machinery lives. One line is enough here. For a nonnegative irreducible matrix, the spectral radius is a simple positive eigenvalue with a positive eigenvector attached (S186, Theorem 2). Apply that to a transition matrix, whose spectral radius is 1, and the positive eigenvector, normalized to sum to 1, is the stationary distribution. That is why has exactly one probability solution when the chain is irreducible. Chapter 5 does the proof and the history; this chapter uses the result.
The receipt. Shafer and Vovk, S236, read for the 1931 sections. EuDML, S246, for the volume and page range, independently matching Miller (S216). MacTutor, S237 and S238. Higham, S186, Theorem 2, for the statement of Perron-Frobenius. (S236; REFERENCE | S237, S238, S246, S186)
So what? The Chapman half is unverified at the primary level in this book. Chapman's 1928 paper, "On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid", Proceedings of the Royal Society A 119(781), 34 to 54, could not be opened: the publisher's site returns 403 to this book's fetcher (S238). So the position is this: half of one of the central equations of probability is named after a geophysicist whose own mathematical biography does not connect him to it, and whose paper we have not read. Say that, rather than repeating the name as though it were an argument.
Detailed balance, and a rule one line long
Detailed balance is not a probability idea that wandered into physics. It is a physics idea that wandered into probability. Boltzmann proved the H-theorem in 1872 on the principle that at equilibrium the rate of forward collisions equals the rate of the reverse collisions (S230). Lorentz objected that inverse collisions are impossible for polyatomic molecules, and in 1887 Boltzmann came back, in the same journal, with a weaker "cyclic balance" condition and proved the H-theorem still holds under it (S230). Wegscheider formulated detailed balance for chemical kinetics in 1901; Einstein used it for the quantum theory of radiation in 1916; Onsager generalized the nineteenth century reciprocal relations of Kelvin and Helmholtz in 1931 and tied them, in his own papers, to "detailed balancing of elementary processes: at equilibrium, each elementary transaction should be equilibrated by its inverse transaction" (S231).
Eighty-one years after Boltzmann, at Los Alamos, five authors turned the condition around. "Equation of State Calculations by Fast Computing Machines" was received on 6 March 1953 and printed in the Journal of Chemical Physics in June. The author line is Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller (S226). The rule is one line: accept a move that lowers the energy; accept a move that raises it with probability (S226, p. 1088). "The time per cycle on the Los Alamos MANIAC is approximately three minutes, and a given point on the pressure curve was obtained in four to five hours of running" (S226).
Why this matters in your course. Course section 4.4 teaches you to check detailed balance and to conclude that a distribution is stationary. Metropolis and the Rosenbluths ran that implication backwards: choose the distribution you want first, then build a chain that satisfies detailed balance with respect to it, and the chain is guaranteed to settle there. That is the Course section 4.4 glossary headword "Markov chain Monte Carlo (MCMC)", and 6.7.5 works the smallest honest case.
The receipt. Metropolis et al., S226, read in full in a university-hosted scan, for the author line, the acceptance rule, the ergodicity paragraph, and the MANIAC timing. Gorban, S230, and Yablonsky et al., S231, for the physics chronology. Hitchcock, S227, and Robert and Casella, S228, for the aftermath. (S226; SCHOLARLY | S227, S228, S230, S231)
So what? Three corrections, in order of how often they are needed.
First, the paper does not contain the phrase "detailed balance" or the word "reversible". Its own justification is ergodicity: "Since a particle is allowed to move to any point within a square of side 2a with a finite probability, it is clear that a large enough number of moves will enable it to reach any point in the complete square... Hence, the method is ergodic" (S226, p. 1088). The detailed-balance reading came later, and reversibility is made explicit in Hastings (1970), according to Robert and Casella (S228).
Second, the name is wrong. Marshall Rosenbluth, one of the five authors, stated that Metropolis "played no role in its development other than providing computer time", and Arianna Rosenbluth told James Gubernatis that Mici Teller "started the computer code for the equation of state work, but she took it over and wrote from scratch the one used" (S256, p. 057303-2). Marshall said he and Arianna did all the work (S256). Arianna Rosenbluth, 15 September 1927 to 28 December 2020, had a Harvard PhD in physics at twenty-two, and her obituary states that she "developed the implementation of the algorithm for the MANIAC I hardware, making her the first person to ever implement the Markov chain Monte Carlo method" (S257). Two cautions on the correction itself: Marshall Rosenbluth gave those recollections at a fiftieth anniversary conference in 2003, which is memory at fifty years' distance, and the printed page renders Arianna's name as "Adriana" in the key sentence, apparently in error (S256).
Third, W. Keith Hastings published the better, more general version in 1970, in Biometrika, a top statistics journal, in the language statisticians speak, and statisticians ignored it for two decades (S227). It took Gelfand and Smith in 1990, who "solidified the theory behind Gibbs sampling and some related methods, and, crucially for statistical practitioners, gave examples of common Bayesian analyses which could be greatly enhanced by these methods" (S227), to make it ordinary. Being right, being early, and being published in the right place are three different things.
Shannon opens a book at random
Section 4 of Part I of A Mathematical Theory of Communication is headed "GRAPHICAL REPRESENTATION OF A MARKOFF PROCESS", with the old ff spelling. Shannon writes: "Stochastic processes of the type described above are known mathematically as discrete Markoff processes and have been extensively studied in the literature" (S225). To show what such a process produces, he built one by hand, with a book: "one opens a book at random and selects a letter at random on the page. This letter is recorded. The book is then opened to another page and one reads until this letter is encountered. The succeeding letter is then recorded, etc." (S225).
His outputs are the ancestors of every autocomplete a student has used. Zero order, symbols independent and equiprobable: "XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCQSGHYD QPAAMKBZAACIBZLHJQD." Second order at the word level: "THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHARACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED." (S225). His own comment: "The resemblance to ordinary English text increases quite noticeably at each of the above steps" (S225).
Why this matters in your course. Markov analyzed text with a chain; Shannon generated text with one (S215). That is the same matrix used in the two directions, and it is the cleanest way to show a class that a transition matrix is a model, not a summary.
The receipt. Shannon, S225, read in full for Part I, in the corrected Bell Labs reprint: the section heading, the Markoff sentence, the construction method, and the sample texts, all quoted verbatim. Link, S215: "Shannon did not reference one of Markov's articles, but Frechet's book Methode des fonctions arbitraires from 1938." (S225; SCHOLARLY | S215)
So what? Here is the tie to modern language models, stated exactly, because the loose version is everywhere and it is wrong.
What is true: an n-gram model of order k is a Markov chain on the space of k-grams. Not "like" one, not "a kind of" one. It is one, exactly, and Shannon's 1948 samples are draws from it (S225). His second-order word approximation is a bigram model, which is a Markov chain whose states are words.
What is not true: that a modern large language model is a Markov chain in any useful sense. A transformer conditions on its whole context window, weights positions differently, and shares parameters across contexts, so it is not a table of transition probabilities between states. You can force a formal statement out of it, namely that a fixed context window plus a sampling step makes the sequence of windows a Markov chain on an astronomically large state space, but that statement is technically true and nearly empty, and no source read for this book asserts it, so it carries as unverified and does not belong in a chapter.
What is true and more interesting than either: researchers deliberately train transformers on data generated by known Markov chains, so they can check whether the network recovers the chain. Makkuva and colleagues report that "when trained on first-order Markov chains, transformers with two or more layers consistently develop an induction head mechanism to estimate the in-context bigram conditional distribution", while "single-layer transformers, unable to form an induction head, directly learn the Markov kernel but often face a surprising challenge: they become trapped in local minima representing the unigram distribution" (S240, abstract). Only the abstract of that paper was read for this book, so nothing from its body is cited here. The honest sentence for a classroom is: the thing Shannon built by hand is a Markov chain; the thing on your phone is not, and Markov chains are now used as the test bench for studying it.
One more caution, and it is the one that makes the whole chapter honest: Shannon never read Markov. He cited Frechet's 1938 book (S215). Ideas travel through translations, conference corridors, and one person telling another. Link's account puts the main transfer at the International Congress of Mathematicians in Bologna in 1928 (S215).
In 1913 Markov tested his idea on a real text. Which text, how much of it, and how did he count?
Show the answer▸
The first 20,000 letters of Pushkin's Eugene Onegin, counted by hand. He struck out the punctuation and the spaces, split the letters into 200 groups of a hundred, arranged each group in a ten by ten table, and combined the columns in pairs, producing 40 tables "which fill an entire page" (S213).
Out came four numbers: 8,638 vowels and 11,362 consonants, and 1,104 vowel-vowel pairs (S213). Those four numbers give you the transition matrix directly. Add them: 8,638 plus 11,362 is exactly 20,000, which is the first check anybody should run on a historical count and one this book ran in code.
He was fifty-six years old and he did it with a pen. Any class can repeat this experiment in its own language in an afternoon, and get different numbers.
Chapter 7
The picture, and the table underneath it
In Unit 4 you will draw a state diagram, which is dots joined by arrows, and then write it down as a transition matrix, which is a table of numbers. That move looks like a bookkeeping trick and it is not. It is the oldest idea in this chapter, it has a date and a person attached, and once you see it you can never unsee it: squaring the table counts two-step routes through the picture, whether the entries are bridges or probabilities. This chapter gives you the history of that move, from a man in St Petersburg who thought the problem was beneath him to a table of weather probabilities that behaves exactly the same way. It also gives you two of the best arguments in mathematics to have in a classroom: whether a proof a human cannot read is a proof, and what happens to a famous statistic when you only count the experiments that finished.
Euler read a paper on seven bridges to the St Petersburg Academy on 26 August 1735, named the subject "the geometry of position" after Leibniz, and drew no graph. The word "graph" arrives in 1878, from chemistry, via Sylvester. Between and after, six things happen. Kirchhoff turns a circuit into a matrix and takes its determinant. A student's question to De Morgan on 23 October 1852 starts a 124-year argument that a computer ends in 1976, and Kempe's proof of 1879 lasts eleven years of it. Ramsey dies at 26 leaving a lemma that becomes a field. Esther Klein asks about five dots. Erdos proves things exist without building them. And three statisticians demolish a guess of Euler's on the front page of the New York Times.
Euler's paper on the seven bridges of Konigsberg is the founding document of graph theory. Guess how many graphs are in it.
I have a guess▸
None. Not one. There is a map of the town and there are strings of letters, and there is no picture of dots joined by lines anywhere in the paper (S420).
The word "graph" for this kind of object does not exist until 1878, when Sylvester used it in a note called "Chemistry and Algebra" in Nature, borrowing it from the way chemists drew molecules (S302, letter g). That is 142 years after Euler read his paper aloud.
The picture came after the theorem, not before it. Euler solved the problem by counting how many bridges touch each piece of land, which is a bookkeeping argument, and he called the subject "the geometry of position" after a phrase of Leibniz's.
The founding paper of graph theory has no graph in it
Konigsberg in Prussia sits on a river that splits into two branches around an island. Euler names the island in his Latin by its German name, "der Kneiphof", says there are seven bridges, and refers to them by the lowercase letters a to g (S420). The question is whether you can walk across every bridge exactly once. What almost nobody tells you is how the paper opens. Before he mentions Konigsberg at all, Euler says that besides the geometry of magnitudes there is another branch, still almost unknown, which Leibniz was the first to mention and which he called the geometry of position: "alterius partis etiamnum admodum ignotae primus mentionem fecit Leibnitzius, quam Geometriam situs vocauit" (S420). Konigsberg is offered as a specimen of that branch, a demonstration piece for a subject that did not exist (S420).
Then he refuses to brute force it. He says that listing every possible route would be unbearable and that he suspected a much simpler method would exist: "talem enim methodum multo simpliciorem fore sum suspicatus" (S420). What he produced instead is bookkeeping. He labels the four land masses A, B, C, and D and the seven bridges a to g, and he counts how many times each capital letter has to appear in a written route (S422). A land mass reached by an odd number k of bridges must appear (k+1)/2 times (S422). Add those up and you need nine letters. A route across seven bridges is written with eight. Nine will not fit into eight, so there is no route. That argument is worked out in full in 7.7.
He also thought the whole thing was beneath him, and said so twice in one year, to two different people. To Carl Ehler in Danzig: "Thus you, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else..." (S422). And then, on 13 March 1736, in letter OO1468 to Giovanni Marinoni in Vienna (S421), the admission that gives the game away: "This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it" (S422; the square brackets are the source's own).
Two codas. In 1875 the townspeople built an eighth bridge joining B to C (S422), which is exactly the bridge that makes the walk possible. The machine confirms it: with eight bridges only A and D have an odd count, and one route that crosses every bridge exactly once runs A to B to A to C to A to D to B to C to D (verify/domainJ.py J-1.15, J-1.16). Then the town was flattened. The Convergence article gives 1944 to 1945 for the wartime destruction and says many of the bridges went with it (S422). Konigsberg became Kaliningrad in 1946 (S445). How many of the seven still stand is a number this book could not source, so this book prints no number for it (S422).
Why this matters in your course. Everything in Unit 4 begins the way E053 begins: a picture of places with letters on it. Course section 4.2 asks whether one state is accessible from another, which is a question about routes through that picture, and Euler is the first person to answer a routing question by counting rather than by trying. Course section 4.1 then asks you to write the picture down as a table, which is 7.7.
The receipt. Euler, S420, opening paragraph, for the Leibniz sentence and the word specimen; second paragraph for Kneiphof, the seven bridges, the lettering a to g, and the refusal to enumerate. S422 for the Ehler and Marinoni quotations, the A to D lettering, the (k+1)/2 rule, the eighth bridge of 1875, and the 1944 to 1945 destruction. S421 for the date 13 March 1736 and the catalog number OO1468. (S420; REFERENCE | S421, S422)
So what? Two things. First, the picture in your textbook, four dots and seven curved lines, is not Euler's. He drew a map with letters on it, and every source in this book that describes his figures agrees that dots and lines are a later invention (S420, S422). Second, do not say "Euler's 1736 paper" without qualification. It was read on 26 August 1735 and printed in 1741, in the volume dated 1736 (S421, S422). Read in 1735, printed in 1741, in the volume for 1736: that is the honest sentence, and it takes no longer to say.
The word "graph" comes out of a chemistry lab
For 142 years the subject had no name for its own object. Then, on 7 February 1878, Sylvester published a note called "Chemistry and Algebra" in Nature 17, at page 284, and used the word "graph" in what Jeff Miller's Earliest Uses records as its first modern sense (S302, letter g). Sylvester's own sentence, as Miller quotes it with a locator in the Collected Mathematical Papers III, pp. 103 to 104, is this: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph" (S302). Read it twice. The parent of the word is "chemicograph", Sylvester's own coinage in the same sentence, and the model is a Kekule structural diagram, the thing on the wall of a chemistry classroom with atoms as letters and bonds as lines (S302). Mathematics did not invent that picture. It borrowed it from chemists and kept the short form of the word.
The other vocabulary arrives in the same neighborhood. "Tree" is attested by the OED, through Miller, to Cayley's 1857 paper "On the Theory of the analytical Forms called Trees" (S302, letter t). Miller also records a rival claim that Sylvester coined "tree", sourced only to an unnamed internet site, so this book prints Cayley 1857 and flags the rival at 7.10 (S302). "Topology" is older than either and comes from a different direction: Listing introduced Topologie in German in 1847 in Vorstudien zur Topologie, and the English word turns up in Nature in February 1883, described there as a term Listing introduced "to distinguish what may be called qualitative geometry" (S302, letter t). And the first systematic book on the subject is Denes Konig's Theorie der endlichen und unendlichen Graphen, Leipzig, 1936, two hundred years after Konigsberg (S302, letter g).
Why this matters in your course. Appendix B.2 is a glossary, and a glossary hides the fact that words have parents. "Graph" in Course section 4.1, meaning a state diagram, and "graph" in Appendix A, meaning the curve of a function, are two different words that happen to be spelled the same. Miller separates them cleanly: the function sense is Chrystal's Algebra I. 307, 1886, "This curve we may call the graph of the function", and the verb "to graph" is Perry's Applied Mechanics, 1898, p. 21 (S302). The network sense came first, in 1878, and came from chemistry.
The receipt. All of the above is from Jeff Miller's Earliest Uses, letters g and t, read on 20 August 2026 (S302). Miller cites OED2 for the topology, tree, and function-graph entries, and cites an internet site for the Sylvester "graph" attribution while quoting Sylvester's own sentence with a page locator (S302). (S302)
So what? Ask where the chemistry went. The research behind this book records a Cayley paper of 1875 on counting isomers, which is the same idea running in the other direction: count the tree-shaped molecules with a given formula by counting the trees. Nobody in this book opened that paper, and nobody opened the Nature note of 1878 either, because the Nature page served only a publisher's summary, which is an abstract and not the note. So the honest position is: the word came from chemistry, Miller has the sentence and the locator, and the chemistry papers themselves are on the list of things this book has not read.
Eleven years of a wrong proof, and then a proof nobody could read
On 23 October 1852 a question reached Augustus De Morgan in London. Francis Guthrie, a student of his, had noticed that four colors seemed to be enough to color any map so that no two neighboring regions share a color, and had passed the question through his brother Frederick (S423). De Morgan wrote to William Rowan Hamilton in Dublin the same day, and his opening line is the best sentence in this chapter: "A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet" (S423). Hamilton replied three days later and would not touch it: "I am not likely to attempt your quaternion of colour very soon" (S423).
Twenty-seven years later Alfred Bray Kempe, a London barrister who had studied under Cayley and who would spend his professional life on ecclesiastical law, announced a proof in Nature on 17 July 1879 (S423, S424). It was believed. Kempe was elected a Fellow of the Royal Society and was knighted in 1912 (S423, S424). Then in 1890 Percy John Heawood found the hole, in the reducibility argument for regions with five or fewer neighbors, and in the same paper proved constructively that five colors always suffice (S423, S424, S425). Kempe stood up at the London Mathematical Society and acknowledged the error (S423). De la Vallee Poussin found the same mistake independently in 1896 (S423). Two of Kempe's ideas survived and are still in the modern proof: unavoidability and reducibility, and the device now called a Kempe chain (S423, S424).
Heawood is worth a paragraph of his own. G. A. Dirac described him: "In his appearance, manners and habits of thought, Heawood was an extravagantly unusual man. He had an immense moustache and a meagre, slightly stooping figure" (S425). Heawood was born in 1861 and died in 1955, and he worked on map coloring for nearly sixty years without finishing it, inventing along the way the idea of an "empire", a country made of several disjoint pieces that all have to share a color (S425).
The answer came in 1976. Appel and Haken's announcement, received by the Bulletin of the American Mathematical Society on 26 July 1976, is two pages long. Its theorem is one line: "Every planar map can be colored with at most four colors" (S426, p. 711). Its method is an unavoidable set of "fewer than 2000 configurations, each of ring size fourteen or smaller", checked with several computer programs written by John Koch and by the authors (S426, p. 711). It does not say how many hours the computer ran and it does not name the machine (S426). The argument started at once. Georges Gonthier, who later rebuilt the proof inside a machine checker, describes the mood without flinching: "Appel and Haken's 1976 triumph had a hint of defeat: they'd had a computer do the proof for them!" (S427, p. 1382). In 1979 Thomas Tymoczko wrote down the four objections a sharp sixteen year old raises anyway. A computer proof, he argued, is not a priori, not certain, not surveyable, and not checkable by human mathematicians, where a surveyable proof is one that "can be looked over, reviewed, verified by a rational agent" (S428, quoting Tymoczko 1979, p. 59). Gonthier's answer was not to shorten the proof but to make the checker a machine that has to be convinced of every step, English sentences included: "The only solution to that conundrum was to formalize the entire proof...This we finally achieved in 2005" (S427, p. 1382).
Why this matters in your course. A map coloring is a partition of the regions into four classes, which is Course section 1.3. More importantly, this is the course's clearest case study in checking. This book has a whole verification apparatus behind it, and Heawood is what that apparatus is for: someone who reads the argument line by line and finds where it fails.
The receipt. De Morgan and Hamilton quotations, dates 23 and 26 October 1852, Kempe's Nature announcement of 17 July 1879, Heawood 1890, and Heesch's discharging of 1969, all from MacTutor (S423). Kempe's life dates, 6 July 1849 to 21 April 1922, and the eleven-year gap, from S424. The Dirac description from S425. The theorem statement, the "fewer than 2000 configurations" phrase, and the 26 July 1976 receipt date from the announcement itself (S426, p. 711). Gonthier's two sentences from S427, p. 1382. Tymoczko's four objections and the definition of surveyability from S428, Course section 3.3. The problem stood from 1852 to 1976, which is 124 years (verify/domainJ.py J-13.5). (S426; SCHOLARLY | S427; REFERENCE | S423, S424, S425, S428)
So what? Watch what happens to the numbers. The authors say "fewer than 2000 configurations". MacTutor's topic page says "approximately 1500 configurations" and adds 1200 hours of computer time, a figure that is nowhere in the announcement. MacTutor's biography of Kempe says the proof "analyzed 1,936 cases" (S423, S424, S426). Three sources, three numbers, and they are not counting the same thing. The 1977 papers that would settle it are behind a challenge page this book could not get past. So quote the authors' own phrase, say who says what else, and do not print a single tidy figure. That habit is worth more than the answer.
The second bridge from a picture to a matrix, from the same small town
Gustav Robert Kirchhoff was born in Konigsberg on 12 March 1824 (S429, S445). The seven bridges were still standing and Euler's paper about them was already 88 years old. In 1845 Kirchhoff announced the two circuit laws every physics student learns: the current into a node equals the current out of it, and around any loop the electromotive forces balance the potential drops (S429). Those laws turned a tangle of wires into a system of linear equations, which is Course section 3.6 doing physics. He graduated from Konigsberg in 1847 and moved to Berlin, and the paper in which spanning trees and the determinant formula now called the matrix tree theorem appear is dated to that same year (S429).
Here is the honest part, and it is not a small one. Nobody in this book opened Kirchhoff's 1847 paper. It was chased through an OpenAlex lookup that returned HTTP 429 twice, Wiley was outside the reachable set, and the Internet Archive timed out on every attempt (S429). Worse, MacTutor's biography of Kirchhoff does not mention trees, spanning trees, or graph theory anywhere at all, so it cannot be used to source the attribution either (S429). What this book has is the mathematics, verified, and a date it cannot vouch for. So: the theorem is real and is checked below; the story that it is in a paper of 1847 in the Annalen der Physik und Chemie 72, pp. 497 to 508, is tagged as unverified here and should be checked against the journal before anyone prints it as fact.
Why this matters in your course. Course section 3.5 teaches you to compute a determinant and gives you almost nothing to do with one except test whether a system has a unique solution. Kirchhoff's theorem is the best answer to "what is a determinant for" that this book can offer at this level. Take the picture, build the same table as before, subtract it from a diagonal of degrees, cross out any one row and its matching column, take the determinant of what is left, and the number you get is the number of spanning trees. It is verified against brute force on six graphs, including the simple version of Konigsberg, which has 8 (verify/domainJ.py J-4.1 to J-4.6), and it reproduces Cayley's formula for the complete graphs through (J-4.C2 to J-4.C6). It is worked longhand in 7.7.4.
The receipt. Birth in Konigsberg on 12 March 1824, death in Berlin on 17 October 1887, the circuit laws of 1845, the 1847 graduation and move to Berlin, and the absence of any graph theory in the biography: all S429. Pronunciation and the 1946 renaming of Konigsberg to Kaliningrad: S445. The spanning tree counts: verify/domainJ_output.txt, section J-4. (S429, S445; the 1847 attribution is UNVERIFIED)
So what? Two of the founding ideas of graph theory come out of the same small Prussian town, a century apart, and neither man thought he was doing graph theory. Euler thought he was doing the geometry of position. Kirchhoff thought he was doing electricity (S420, S429). Nobody sets out to found a field. They set out to solve the thing in front of them, and the field is what gets left behind.
Paths older than the picture: knights, an Icosian game, and a ring of noughts and ones
A knight's tour is a path that visits every square of a chessboard exactly once by legal knight moves, which is a Hamiltonian path on a graph with 64 vertices. The earliest ones in this book's evidence are not European and not modern. An Arabic manuscript of about 840, Nuzhat al-arbab al-'aqul fi'sh-shatranj al-manqul, which Jelliss renders "The delight of the intelligent", carries two full 8 by 8 tours, one by Ali C. Mani and one by al-Adli ar-Rumi, and al-Adli's is reentrant, meaning it comes back to where it started (S431). Around 900 the Kashmiri poet Rudrata wrote a half-board tour into the Kavyalankara as what Jelliss calls a cryptotour: a verse whose syllables can be read straight down the page, or in the order a knight visits them (S431). We know about it through Nami of Guzerat's commentary of 1069 and Hermann Jacobi's analysis of 1896 (S431). Euler wrote on knight's tours in the 1750s and Vandermonde in 1771, in a paper he called "Remarques sur des problemes de situation", borrowing Euler's own phrase for the subject (S430, S431). Both were about nine centuries late.
The other path problem in this scene is a ring. A de Bruijn sequence is a cyclic string in which every word of a given length appears exactly once. For binary words of length three the answer is 00010111: read the eight overlapping three-character windows around the ring and you get all eight of 000 through 111, each once (verify/domainJ.py J-12.1). The standard way to construct one is to build a small directed graph and take an Euler circuit through it, which is Euler's 1736 argument doing paid work two hundred and ten years later. MacTutor dates de Bruijn's sequences to 1946 and gives his life as 9 July 1918 to 17 February 2012 (S444). De Bruijn also spent his later career building AUTOMATH, a language "so designed that it is incorrect to state a theorem without first 'constructing' a proof of the theorem" (S444), which is the same instinct that produced Gonthier's Coq proof in scene 7.3.
Now the caveat this scene exists to carry. You will read that these sequences appear in Sanskrit prosody as the mnemonic yamatarajabhanasalagam, ten syllables that encode all eight light-heavy triples. The arithmetic is verified here and it works. Writing light as 0 and heavy as 1, the usual modern transcription is 0111010001, and its eight overlapping three-syllable windows are 011, 111, 110, 101, 010, 100, 000, and 001, which is all eight binary triples exactly once (verify/domainJ.py J-12.4, J-12.5). The history is not verified. No manuscript was opened, no dated attestation was found, and no scholarly edition was read. Subhash Kak's note in the Indian Journal of History of Science 35(2) (2000), 123 to 127, is the usual citation and it 404s at the path tried; Rachel Hall's "Math for Poets and Drummers" was opened and covers Pingala and Hemacandra but does not mention the mnemonic at all (S444). So the rule for this book is: print the arithmetic, print that the story is often repeated, and print that it is not verified here.
Why this matters in your course. Course section 4.2 asks whether you can get from one state to another and whether you can get back. A knight's tour is that question with a chessboard for a state diagram, and a de Bruijn ring is that question with the answer used as an engineering component.
The receipt. The c. 840 manuscript, the two tours, al-Adli's reentrancy, Rudrata's cryptotour of c. 900, the reign of Sankaravarman 884 to 903, Nami 1069, and Jacobi 1896: all Jelliss, S431, whose own authority is Murray's A History of Chess (1913) and Murray's unpublished manuscript in the Bodleian. Vandermonde's 1771 paper and his phrase: MacTutor, S430. De Bruijn's dates, the 1946 sequences, and the AUTOMATH quotation: MacTutor, S444. De Bruijn arithmetic: verify/domainJ_output.txt, J-12. (S430, S431, S444)
So what? Ask about Hamilton, and then be disappointed on purpose. A Hamiltonian cycle is named after William Rowan Hamilton and the usual story attaches it to his Icosian game of 1857, a puzzle he is said to have sold to a games dealer. This book could reach none of it: MacTutor's Hamilton biography says nothing beyond an image link, and the Trinity College Dublin pages could not be read for this book. So everything about the Icosian game in this book is unverified, and the name on the cycle sits on a story we cannot check. That is a useful thing for a student to see: the gap between what is famous and what is documented is not always small, and the honest move is to say which side of it you are standing on.
Esther Klein asks about five dots, and Erdos proves things exist without building them
Frank Plumpton Ramsey read a paper called "On a problem of formal logic" to the London Mathematical Society on 13 December 1928 (S432). It was about determining the consistency of logical formulas, and the combinatorial theorem inside it was a lemma he needed on the way (S432). He died on 19 January 1930, aged 26, after surgery at Guy's Hospital for jaundice, and the paper was published that year (S432; the age is checked at verify/domainJ.py J-13.9). By then he had also written two of the founding papers of modern economics and several of modern philosophy of language (S432). The lemma became a field.
Three years after his death, in a Budapest park, a group of Jewish students met to do mathematics for fun. Esther Klein noticed that five points in the plane, no three on a line, always contain four that form a convex quadrilateral, and asked whether the pattern continues: is there, for every n, a number N(n) such that any N points in general position contain n forming a convex polygon (S433, p. 463; S434)? Erdos and Szekeres answered her in Compositio Mathematica in 1935, and they name her in the text, in print, as "Miss Esther Klein" (S433, p. 463). They record , , and , conjecture , and give two proofs, the first through Ramsey's theorem and the second through monotone sequences (S433, pp. 464 to 470). The conjecture's arithmetic is checked at J-7.3 to J-7.6, and Klein's own observation is checked by brute force over 1668 general-position five-point subsets of a grid (J-7.K).
George Szekeres married Esther Klein on 13 June 1937, and Erdos named the problem the Happy Ending problem for exactly that reason (S434). In 1939, facing Nazi persecution, Szekeres took a job in Shanghai as a leather chemist and stayed until 1948 (S434). He had trained as a chemical engineer because his family expected it, graduating in 1933, and spent six years as an analytical chemist in Budapest while doing the mathematics that made him famous (S434). George and Esther died on the same day, 28 August 2005, within an hour of each other, in a nursing home in Adelaide, after 68 years of marriage (S434).
The third act belongs to Erdos alone. In 1947 he published three pages in the Bulletin of the AMS and changed how existence gets proved. Theorem I gives for at least 3, where is what we now call the Ramsey number , and the proof is a count (S435, p. 292). Count all the two-colorings of the complete graph on N points. Count the ones that contain a monochromatic clique of size k. Show the second count is smaller than the first. Therefore a coloring with no monochromatic clique exists. Erdos never produces one. Nobody has produced one since (S435). The inequality behind the count, , is checked for k = 3 to 10 at verify/domainJ.py J-6.1 to J-6.8, and it holds every time.
Why this matters in your course. This is the reason probability is worth learning, stated in three pages. Course section 2.2 gives you the properties of probability and Course section 2.5 gives you independence; Erdos uses them to prove something about a finite object that contains no randomness at all. In the probability version of his argument, you color each edge red or blue independently with probability one half, compute the expected number of monochromatic cliques as , and note that if the expectation is below 1 then some outcome must have zero (J-6.exp). The theorem the whole thing rests on is checkable by hand: R(3,3) = 6, meaning that among any six people three are mutual acquaintances or three are mutual strangers, and five is not enough. Both halves are settled by exhaustive search, 1024 colorings of K5 and 32,768 of K6 (J-5.1 to J-5.5).
The receipt. Ramsey's dates, the 13 December 1928 reading, and the jaundice: S432. "Miss Esther Klein", the N0 values, and the conjecture: S433, pp. 463 to 464, read in the Numdam scan of the original printing. The marriage on 13 June 1937, the Happy Ending name, Shanghai, and the deaths on 28 August 2005: S434. Theorem I and the absence of any exhibited coloring: S435, p. 292 and p. 293. (S433, S435; REFERENCE | S432, S434)
So what? Print her name. The problem is Esther Klein's, she is named in the 1935 paper, and she is routinely dropped from retellings that keep both men. And then sit with the uncomfortable half of Erdos's argument. He proved that a needle is in the haystack by proving that the haystack is mostly needles, and to this day nobody can point at one. If that feels like cheating, you are having the same argument the four color theorem started, from the other end: there, the object was produced and the proof could not be read; here, the proof can be read in an afternoon and the object has never been produced.
Networks snap, and the most famous experiment in social science mostly failed
Erdos and Renyi asked what a graph chosen at random looks like. Their first paper, received on 19 November 1958, fixes n labeled points and N edges drawn uniformly at random, and proves a threshold: with N = floor(() n log n + cn) edges, the probability that the graph is connected tends to exp(-exp(-2c)) (S436, Theorem 1). Read that as a switch. At c = 0 the limit is exp(-1), about 0.367879; at c = 2 it is 0.981851 (verify/domainJ.py J-8.1, J-8.2). Two years later, in a paper received on 28 December 1959 and titled with the authors' own framing word, "On the evolution of random graphs", they showed what happens earlier and it is stranger. With N = cn edges and c below one half, almost every point sits in a small isolated tree. Above one half, a single giant component appears, of size about G(c) n where G(c) = 1 - x(c)/(2c) and x solves x exp(-x) = 2c exp(-2c) (S437). At exactly one edge per vertex, 79.7 percent of the whole graph is in one lump (J-8.6). There is no gentle middle. The network is dust, and then it is one thing.
Now the other half of the story, which is about people rather than points. Frigyes Karinthy got there first, in fiction. In 1929, in a short story called "Chain-Links" in the collection Everything is Different, his narrator proposes a bet: "We should select any person from the 1.5 billion inhabitants of the Earth - anyone, anywhere at all", and claims that "using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances" (S439, in Adam Makkai's translation). Note the arithmetic, because it is where the famous phrase comes from and it is off by one from what people think: five people in between is six links (J-10.1). The phrase "six degrees of separation" is not in the story (S439).
Thirty-eight years later Stanley Milgram tried it with letters, and the result is the most quoted number in popular network science. In March 2002 Judith Kleinfeld went to Milgram's papers in the archives at Yale and reported what was in the boxes. In an unpublished pilot study, "only three of 60 letters - 5 percent - made it" (S440). In the published studies, "less than 30 percent of the folders got through" (S440). Three of sixty is exactly one twentieth (J-10.2, J-10.3). The famous "about six" is a median computed over the minority of chains that finished, and it says nothing whatever about the majority that died.
The mathematics of it did arrive, twenty-nine years after Milgram. Watts and Strogatz, in Nature on 4 June 1998, take a ring lattice of n vertices each joined to its k nearest neighbors and rewire each edge at random with probability p (S441, p. 440). At p = 0 you have a lattice; at p = 1 a random graph; in between, and over a wide range of p, you get the small-world signature: path length about what a random graph would give, clustering far above it (S441, p. 441). Their Table 1 puts three real networks side by side. Film actors: characteristic path length 3.65 against 2.99 for a random graph, clustering 0.79 against 0.00027. The western United States power grid: 18.7 against 12.4, and 0.080 against 0.005. The neural network of the nematode C. elegans: 2.65 against 2.25, and 0.28 against 0.05 (S441, Table 1). In ratios: the actor network is 2,925.9 times more clustered than chance while being only 22 percent further across (J-9.1, J-9.2).
Why this matters in your course. Sharp thresholds are the shape of surprise this course keeps meeting. The birthday problem is the same shape with a different parameter: at 22 people a shared birthday is below even odds, at 23 it is above, and the digest's script computes both side by side deliberately (J-8.7, J-8.8). Course section 2.2 will hand you the birthday problem; Erdos and Renyi are what it grows into.
The receipt. Theorem 1 and the 19 November 1958 receipt date: S436, from the offprint's footer. The giant component, G(c), and the 28 December 1959 receipt date: S437. Karinthy's two sentences: S439, in the Makkai translation. Kleinfeld's two figures: S440, quoted verbatim. Watts and Strogatz's model sentence, the small-world signature, and Table 1: S441, pp. 440 to 441. (S436, S437, S439, S441; SCHOLARLY | S440)
So what? This is survivorship bias with a household name attached, and it is the best example a probability course will ever get for free. If you compute an average only over the trials that finished, you have measured the trials that finished. Ask the class what number Milgram would have reported if the 57 lost letters in the pilot had been counted as chains of infinite length, and watch the room work out that the question has no good answer, which is the point. Note also what this book could not do: Travers and Milgram 1969 and Milgram 1967 are image scans with no text layer, so this book prints Kleinfeld's archival figures and not Milgram's own medians (S440).
Euler is wrong for the first time in this chapter, and it makes the front page
Euler presented E530 to the St Petersburg Academy on 8 March 1779 (S421). The question in it, as it is usually told, is about 36 officers: six ranks drawn from six regiments, one officer of each combination, to be paraded in a 6 by 6 square with no rank and no regiment repeated in any row or column. He could not do it. He conjectured, without proof, that the same failure happens at every order of the form 4k+2: 2, 6, 10, 14, and onwards (S421, S442). Bose and Shrikhande date the conjecture to 1782 (S442).
For a century and a half people tried to prove him right. Peterson, Wernicke, and MacNeish each published a proof between 1901 and 1922, and every one of them was wrong (S442). Then Bose and Shrikhande built a pair of orthogonal Latin squares of order 22, and infinitely many more of the form 36w + 22, in a paper received by the Transactions of the American Mathematical Society on 10 April 1959 (S442). Ernest Tilden Parker joined them. On Sunday 26 April 1959 the New York Times put it on the front page, and the three picked up a nickname that stuck: Euler's spoilers (S443).
The satisfying part is that Euler was right about the case he looked at himself. The 36 officers problem really is impossible, and this book settled it by machine rather than by citation. Existence of an orthogonal partner survives the operations that turn one Latin square into another, so it is enough to test the reduced squares, of which there are 9408 at order 6. All 9408 were generated, every transversal of each was computed, the counts came out as 0, 8, 24, or 32 and never anything else, 2100 of the squares have no transversal at all, and not one of the 9408 has the six pairwise disjoint transversals an orthogonal partner would need (verify/domainJ.py J-11.R6, J-11.6, J-11.6T, J-11.6D). Euler's conjecture holds at 2 and at 6, and fails at every larger 4k+2, which is exactly what Bose, Shrikhande, and Parker proved (S442).
Why this matters in your course. A pair of orthogonal Latin squares is a combinatorial design, and designs are the mathematics underneath experimental statistics: they are how you arrange treatments so that the effects you want to measure do not get tangled together. Course section 2.1 counts arrangements; this is what counting arrangements is for. Bose went on to co-discover the BCH error-correcting codes in 1960 with Ray-Chaudhuri and Hocquenghem, and the Bose-Mesner algebra carries his name (S443).
The receipt. E530's presentation date of 8 March 1779 and the archive's summary of the conjecture: S421. The 1782 dating, the failed proofs of 1901 to 1922, the order 22 construction, the 36w + 22 family, and the 10 April 1959 receipt date: S442, read in the AMS PDF. The front page of 26 April 1959, the nickname, Bose's dates of 19 June 1901 to 31 October 1987, and BCH codes: S443. (S442; REFERENCE | S421, S443)
So what? Two corrections and one gap. First, MacTutor calls the conjecture "175-year-old"; 1782 to 1959 is 177 years (J-13.8). Prefer the arithmetic, or write "the best part of two centuries". Second, the Euler Archive's own catalog page says the conjecture "was later disproven in 1970", which is simply wrong, and this book does not cite S421 for that date (S421, S442, S443). Third, the gap: nobody in this book has seen the New York Times of 26 April 1959. The claim rests on MacTutor alone, and the headline and the reporter's name are unverified here (S443).
A student's question in 1852 started an argument that ran for 124 years. What was the question, and how did it end?
Show the answer▸
Francis Guthrie noticed that four colors seemed to be enough to color any map so that no two neighboring regions share a color, and asked his teacher Augustus De Morgan whether it was true. The question reached De Morgan on 23 October 1852 (S423).
Kempe published a proof in 1879. It stood for eleven years before Percy Heawood found the hole in it, and Heawood then spent nearly sixty years on the problem without closing it (S425).
It ended in 1976, when Appel and Haken finished a proof that required a computer to check about 1,900 configurations, one at a time (S426, p. 711). No human being has ever read the whole thing.
That is worth sitting with: a question a student can ask in one sentence, an answer no person can hold in their head. Coloring a map is the same kind of object as the adjacency matrix in Course section 3.4.
Chapter 8
The people, and the twelve doors out of the math room
Two jobs. The first is to introduce you to twelve people, and one person who never existed, whose work is inside this course and whose names mostly are not. You will meet a man who wrote 105 job applications because 105 was the number of colleges that would consider him, a nun who invented an algorithm and then went back to teaching undergraduates, and a physicist who wrote the code for the most used algorithm in computational science and got none of the name. The second job is to hand you and your teacher twelve connections between this course and another subject. Each one is specific, sourced and ready to use. They reach into history, biology, music, economics, literature, design and philosophy, and one of them is a hard conversation about gambling. If you have ever asked what this is for, this chapter is the answer, and every claim in it has a source you can check.
Between 1829 and 2025 the mathematics in this course kept leaving the mathematics room. Guerry drew a polar area chart of the wind in 1829 and Nightingale aimed one at the British Army in 1858. Snow argued cholera out of a water pump in September 1854 and only drew the famous map in December. Hardy wrote a page of algebra to biologists in April 1908 and apologized for it. Du Bois and five Atlanta University students hand-drew 72 items, on 56 boards, for Paris in 1900. Markov counted Pushkin's letters to win an argument about free will, Turing invented a unit for evidence and named it after a town in Oxfordshire, Hiller made a computer write a string quartet in 1956, Leontief turned the American economy into a matrix and the Air Force paid for it, and Lester Hill turned the alphabet into one. The people who did this work were, very often, the people the institutions of their day would not employ.
In September 1854 John Snow argued that a cholera outbreak was coming from one water pump on Broad Street, and won the argument. The map of that outbreak is one of the most reproduced graphics in the history of statistics. Guess when he drew it.
I have a guess▸
December 1854, after the argument was already over. The pump handle came off in September. The map came months later, for the report (S277 and the sources in Appendix D under Health).
The map did not persuade anybody of anything at the time. It is a record of an argument, drawn afterwards, which is the opposite of the story usually told about it.
Keep that in mind whenever somebody shows you a famous visualization. The picture that made history and the picture that won the fight are frequently not the same picture, and sometimes not even the same year.
One hundred and five letters
David Harold Blackwell was born in Centralia, Illinois on 24 April 1919, to Grover Blackwell, who worked for the Illinois Central Railroad, and Mabel Johnson (S250). He took a BA in 1938, an MA in 1939, and a PhD in 1941 at 22, supervised by Joseph Doob, on Markov chains (S250, S252). He won a Rosenwald Fellowship to the Institute for Advanced Study at Princeton for 1941 to 1942, and something went wrong there that this book cannot pin down, because the three sources tell it three ways and each is worse than the last. Peter Bickel, in a peer reviewed PNAS retrospective, writes only that Doob "had to intervene to ensure him privileges at Princeton University" (S251, p. 2). MacTutor says Princeton's president objected to Blackwell being made an honorary faculty member and wrote to the Institute's Director that the Institute was "abusing the hospitality of the University" (S250). Scott Williams's Mathematicians of the African Diaspora page says the president wrote that the Institute was "abusing the University's hospitality by admitting a black" and then "organized a great protestation" against extending the appointment (S252). Take the mildest sourced version as the fact and the other two as reported, because the last of the three is not peer reviewed and cites no archive box for the letter. Blackwell himself, MacTutor records, was largely unaware of the dispute at the time (S250).
Then, in 1942, he wrote to all 105 Black colleges in the United States, because those were the institutions that would consider hiring him (S252). He taught at Southern University in Baton Rouge, then Clark College in Atlanta, then Howard University from 1944 to 1954, where he went from Instructor to full Professor and Department Head inside three years (S250). Bickel's sentence about those ten years is the one to sit with: "During the 10 years he spent at Howard, while carrying a heavy teaching load and serving as Department Chair for seven years, Blackwell published 20 papers and a monograph" (S251, p. 1). That is two papers a year, while chairing a department for seven of the ten (verify/ch08.py, section 6). In 1965, twenty-four years after the doctorate, he became the first Black member of the National Academy of Sciences (S250, S251, S252).
Why this matters in your course. The Rao-Blackwell theorem, which Bickel calls "a fundamental improvement scheme in estimation" (S251, p. 1), is Course section 2.3 stated as a promise: conditioning on the right information never makes your estimate worse. His 1941 thesis is Course section 4.1, Markov Chain Properties and Transition Matrices, before that section had a textbook. He also gave dynamic programming a rigorous foundation and introduced what are now called Blackwell optimal policies (S251, p. 2), and he spent three summers at RAND from 1948 to 1950 on duels, in which two players walk toward each other with loaded pistols and each must choose when to fire (S250).
The receipt. MacTutor, S250, for the birth, the degrees, the Rosenwald Fellowship, the Howard and Berkeley appointments, and the "abusing the hospitality of the University" wording. Bickel, S251, p. 1 and p. 2, for the twenty papers, the seven years as Chair, the Rao-Blackwell description, and the milder Princeton account. Williams, S252, for the 105 colleges and the harshest wording. (S251; REFERENCE | S250, S252)
So what? Print his own sentence and let it sit there: "Basically, I'm not interested in doing research and I never have been. I'm interested in understanding and that's quite a different thing" (S251, p. 2). That is a whole philosophy of studying mathematics, from a man who had to write 105 letters to get a job doing it.
A doctorate with two dates
Mary Celine Fasenmyer was born on 4 October 1906 in Crown, Pennsylvania, and graduated from St Joseph's Academy in Titusville in 1923 (S253, S254). She then taught school for ten years before starting a degree (S253). She entered the Sisters of Mercy in 1933, taking the name Sister Celine, and took an AB from Mercyhurst College in Erie in the same year (S253, S254). An MA from the University of Pittsburgh followed in 1937, mathematics major with a physics minor (S254). Her doctoral study ran from the autumn of 1942 to June 1946 at the University of Michigan, under Earl Rainville, and the thesis was called "Some Generalized Hypergeometric Polynomials" (S254). She was 39 (verify/ch08.py, section 6). She published two papers, one in the Bulletin of the AMS in 1947 and one in the American Mathematical Monthly in 1949, went back to teaching undergraduates at Mercyhurst, and never published again (S253, S254).
The date is not clean, and this book will not pretend otherwise. Petkovsek, Wilf, and Zeilberger, who are the reason anyone outside Erie knows her name, write in A=B that she "developed, in her doctoral dissertation of 1945, the first computerizable method for finding recurrence relations that are satisfied by sums" (S255, p. 55), and their bibliography key is [Fase45] (S255, p. 18). The Agnes Scott biography, which cites its own sources, gives June 1946 at Michigan (S254). MacTutor gives 1946 and reads as though the degree came from Pittsburgh (S253). The likely reconciliation is a dissertation dated 1945 and a degree conferred in June 1946, one year apart, but that is an inference and not a document (8.10, item 1). Thirty-two years after the degree, in 1978, Doron Zeilberger read one of her two papers and saw what was in it (S253).
Why this matters in your course. Course section 2.1 Counting is full of identities that look like magic tricks: a sum of binomial coefficients that collapses to something short. Fasenmyer wrote down the procedure a machine can follow to find the recurrence that such a sum satisfies, which turns each of those tricks from a puzzle into a computation. The field that grew out of it is called WZ theory, for Wilf and Zeilberger (S253). The method is called Sister Celine's method, and chapter 4 of A=B is named after her (S255, p. 55).
The receipt. MacTutor, S253, for the life dates, the ten years of school teaching, the two papers, and Zeilberger in 1978. Agnes Scott, S254, for the University of Michigan, June 1946, the autumn 1942 start, and the thesis title. A=B, S255, pp. 18 and 55, read in the authors' own PDF, for the 1945 dating and the "computerizable method" quotation. (S255; REFERENCE | S253, S254)
So what? Two papers. That is the whole publication record, and one of them started a field. The other lesson is about how citations decay: the men who developed her idea got the field named after them within a decade, and the woman who had the idea got a chapter heading fifty years later. Both namings are honest. Neither is symmetric.
Aircraft flutter, a review journal, and an algorithm invented twice
Olga Taussky was born on 30 August 1906 in Olmutz, now Olomouc in the Czech Republic, took her Vienna doctorate in 1930 at 24, and spent 1931 to 1932 at Gottingen editing the number theory volume of Hilbert's collected works (S192). In the Second World War she was put on a problem with a body count, at the National Physical Laboratory at Teddington from 1943 to 1946, and she describes it herself: "I was assigned to the study of flutter in supersonic aircraft, which leads to boundary value problems in hyperbolic partial differential equations" (S191). Her own definition of the phenomenon is the clearest sentence anyone has written about it: "In flight the interaction between the elastic forces in the airframe and the aerodynamic forces induces self-excited vibration which, above a certain speed, is unstable. This phenomenon is called flutter" (S191). The problem came down to a 6 by 6 matrix and to whether it had a real eigenvalue in a particular place, and she solved it because of a piece of luck she recorded honestly: "By a mere accident I had heard about the Gersgorin theorem, whose statement is given in a Zentralblatt review" (S191). She read a review, in a review journal, of a 1931 Russian paper, and it turned out to be the tool that bounded where the eigenvalues could sit. Chapter 5, Aircraft that shook themselves apart, and the year the machines arrived, works the mathematics; this chapter is here for the person.
Two more names, and one date that should be on a poster. On 29 October 1959 John G. F. Francis, working for the National Research and Development Corporation, submitted a new way to compute the eigenvalues of a matrix, written in assembly language for a computer called Pegasus (S193). Two hundred and fifty days later, on 5 July 1960, Vera Nikolaevna Kublanovskaya submitted the same algorithm to the Doklady of the Soviet Academy of Sciences, having started from the same 1958 paper by Rutishauser and knowing nothing of Francis (S193, S194; the interval is recomputed at verify/ch08.py, section 6). Kublanovskaya was born on 21 November 1920 in a village in Vologda Oblast, trained as a primary teacher, went home to nurse her mother during the war and so missed the Siege of Leningrad, took her degree in 1948, and worked with Kantorovich on secret computational projects connected with the atomic bomb until 1955 (S194). Francis moved to Ferranti in 1961 to write a commercial compiler and left the subject; around the year 2000, "nobody in the numerical analysis community of mathematicians had any recollection of ever having seen John Francis himself" (S193). Chapter 5, Two strangers, one algorithm, and a decomposition named after the wrong people, tells the rest, including the coffee room in Moscow where two people looking for him compared notes.
Why this matters in your course. Course section 3.7 asks you to find eigenvalues by factoring a quadratic. That works for a 2 by 2 and stops working immediately afterwards, because there is no formula for the roots of a general polynomial of degree five or more. QR is what replaced it, and it is what runs inside every eigenvalue command you will ever type. Taussky's Gersgorin disks are the other half of the same section: a cheap way to say where the eigenvalues must be without computing them at all.
The receipt. Taussky, S191, her own 1988 memoir in the American Mathematical Monthly, for the flutter assignment, the definition, the 6 by 6 matrix, and the Zentralblatt sentence. Luchins and McLoughlin, S192, for the biographical dates. Golub and Uhlig, S193, for both submission dates and the sentence about nobody having met Francis. MacTutor, S194, for Kublanovskaya's life. All four were read by Domain E and are carried into this chapter unchanged. (S191; SCHOLARLY | S192, S193; REFERENCE | S194)
So what? Taussky's own line about credit belongs on a classroom wall: "We mathematicians are too quick to credit the developer and forget the explorer" (S192). She was the first woman on the Caltech mathematics faculty, arriving in 1957 as a Research Associate with permission but no obligation to teach, granted tenure in 1963, and made a full professor in 1971, the first woman at Caltech at that rank (S192). She published about 300 papers.
The woman who wrote the code
Arianna Wright was born in Houston, Texas on 15 September 1927 (S257). She took a BS at Rice in 1946, an MA at Radcliffe in 1947, and a PhD in physics at Harvard in 1949 at 22, under John Hasbrouck Van Vleck, on "Some Aspects of Paramagnetic Relaxation" (S257). She won the Texas women's foil fencing championship and the Houston men's championship; the Olympics fell through on cancellations and money (S257). At Los Alamos she wrote, from scratch, the code that ran the 1953 equation-of-state calculation on the MANIAC, which her obituary describes as "making her the first person to ever implement the Markov chain Monte Carlo method" (S257). After the birth of her first child she left professional research, kept doing mathematics at home on knot theory, and died on 28 December 2020 (S257).
The credit question is settled on the record by one of her co-authors. At the algorithm's fiftieth anniversary conference in 2003, Marshall Rosenbluth stated that Nicholas Metropolis "played no role in its development other than providing computer time", and that he and Arianna did all the work (S256, p. 057303-2). Arianna told James Gubernatis that Augusta "Mici" Teller "started the computer code for the equation of state work, but she took it over and wrote from scratch the one used" (S256, p. 057303-2). Two cautions, and the book prints both: Marshall was recalling events at fifty years' distance, and the printed page renders Arianna's name as "Adriana" in that very sentence, apparently in error (S256). Chapter 6, Detailed balance, and a rule one line long, works the algorithm itself and does not need repeating here.
Why this matters in your course. Course section 4.3 asks you to find the stationary distribution of a chain somebody hands you. The 1953 paper runs that implication backwards: name the distribution you want first, then build a chain that settles there. Arianna Rosenbluth is the person who first made a computer do it.
The receipt. Betancourt's IMS obituary, S257, for every date, degree, and the fencing. Gubernatis, S256, p. 057303-2, peer reviewed in Physics of Plasmas, for both quotations and for the "Adriana" misprint. (S256; REFERENCE | S257)
So what? Ask the class what the algorithm should be called, and then make them defend a rule rather than a name. Whatever rule they choose, apply it to the Ehrenfest urn and to Gaussian elimination, both of which Chapters 6 and 4 show are named for the wrong people or for only half of them.
Note G, and the argument that will not end
In October 1842 Luigi Federico Menabrea, "of Turin, Officer of the Military Engineers", published a French account of Charles Babbage's Analytical Engine in the Bibliotheque Universelle de Geneve, number 82 (S260). In 1843 an English translation appeared in Scientific Memoirs, volume 3, with translator's Notes running roughly twice the length of the article, signed only with the initials "A. A. L." (S260). Those are the initials of Ada Augusta, Countess of Lovelace, whose name is not printed. One sentence in Note A is the best description of computing anyone wrote for the next hundred years: "The Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves" (S260, Note A). Note G carries the table that is, in the careful phrasing of three historians, "a table often described as 'the first computer program'" (S259).
The authorship argument is real and this book cannot end it. Dorothy Stein's 1985 biography argued that Lovelace was mathematically weak and Babbage wrote the Notes; Thomas Misa in 2016 and Hollings, Martin, and Rice in 2017 argue that the 1843 paper was the product of an intense intellectual collaboration between Lovelace and Babbage (S258, p. 204). Those three historians read every surviving letter between Lovelace and her tutor Augustus De Morgan and concluded something less dramatic than either camp: by early 1842 she had worked through coordinate geometry, functions, inequalities, logarithmic and exponential functions, infinite series, the concepts of the calculus, and the algebra of complex numbers, with a weakness in algebraic manipulation that they attribute to her patchy earlier education (S258, p. 227). They judge she was probably not yet ready for research in 1842, but within a year or two of being so (S258, p. 227). De Morgan wrote to her mother on 21 January 1844 that had she been a man at Cambridge she would have become "an original mathematical investigator, perhaps of first rate eminence" (S258, p. 203, archive reference LB 339). One honest gap: nobody in this book has read Note G. The transcription fetch stopped at Note C, twice (S260; 8.10, item 6).
Why this matters in your course. Course section 3.2 defines a matrix as an arrangement of terms in rows and columns. Note G's table is that object doing a job: rows of operations, columns of working storage, read in order. Unit 4's whole idea, a process that repeats a rule step by step, is the same idea approached from the other end.
The receipt. The 1843 text, S260, read in transcription for the title page, the "A. A. L." signature, and the Jacquard sentence in Note A. Hollings, Martin, and Rice, S258, pp. 203, 204, and 227, read in full in the Edinburgh repository copy; and S259, for the hedged description of the Note G table. (S260; SCHOLARLY | S258, S259)
So what? Notice where the famous sentence lives. The loom line is in Note A, not Note G (S260), and half the retellings put it in the wrong place. If a story cannot survive somebody checking which note it is, it was never a story about the evidence.
No school, no post, no toilet
Alicia Boole Stott, born in Cork on 8 June 1860, received no formal schooling at all (S266). Her mother, Mary Everest Boole, taught her on the principle that "children should be led up to find it out for themselves by successive questions" (S266). At eighteen her brother-in-law Charles Howard Hinton showed her a set of wooden cubes he was using to think about the fourth dimension, and she turned out to be able to see it (S266). With no training and no university she identified all six regular four-dimensional polytopes, cut three-dimensional cross-sections of each by purely Euclidean methods, built the sections out of cardboard, and coined the word "polytope" (S266). Groningen gave her an honorary doctorate in 1914 and she did not go to collect it (S266). In 1936, when Coxeter left for Toronto, she gave him an antique stained-glass Archimedean solid lampshade and wrote: "I wonder where you will get to in it! How I wish I could follow" (S266).
Emmy Noether was born in Erlangen on 23 March 1882, into a country where, from 1900 to 1902, women could study at a university only unofficially and only with each professor's individual permission; she was one of two women at Erlangen and could only sit in on courses (S278). The rules changed on 24 October 1904 (S278). Hilbert and Klein invited her to Gottingen in 1915, and the faculty refused her a lecturing post for four years (verify/ch08.py, section 6), so Hilbert put her courses in the timetable under his own name. The printed listing survives: "Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4-6, no tuition" (S278). She got the habilitation in 1919 and the unsalaried title of Privatdozent. In April 1933 the Nazis had her dismissed, with no pension and no compensation; she sailed for Bryn Mawr in October and died there on 14 April 1935, aged 53 (S278).
Florence Nightingale David was born in Ivington on 23 August 1909 and became Karl Pearson's research student around 1931 (S265). She produced his tables of the correlation coefficient by hand on a Brunsviga calculating machine and estimated that she "turned that hand Brunsviga roughly 2 million times" (S265). During the Blitz she analyzed bomb damage for the Ministry of Home Security to answer the question "where's the safest place to be when the bomb falls", and when the V-bombs came she fitted a bivariate normal surface to the strikes and found the direction of its major axes in order to point back at the firing positions (S265). She said R. A. Fisher would not answer her questions because she was a woman, and that she was turned down for jobs on the grounds that the institution had no toilet facilities for women (S265). Neyman "always called me the Duchess" (S265). In 1962 she wrote Games, Gods and Gambling, and her reason is quotable: "I had lessons in Greek when I was young and I got rather bored with people talking about dice" (S265).
Why this matters in your course. Boole Stott is the answer to the question every student asks in Course section 3.1 when a vector acquires n components: what does four dimensions look like? It looks like a stack of cardboard slices, and somebody built them. Noether is the reason Course section 3.2's matrix is an element of an algebraic structure and not just a table. F. N. David is the person who wrote the history that Chapter 3 draws on, and her wartime work is the least abstract answer available to "when would I use this".
The receipt. MacTutor, S266, for Boole Stott's birth, the cubes at eighteen, the six polytopes, the coinage, the 1914 doctorate in absentia, and both quotations. MacTutor, S278, for Noether's dates, the 1900 to 1902 rule, 24 October 1904, the course listing, the April 1933 dismissal, and the October sailing. Laird's Statistical Science interview, S265, for every David quotation and for the V-bomb work. (S265; REFERENCE | S266, S278)
So what? One more name belongs here and this chapter will not retell her story, because Chapter 7 already does it properly. Esther Klein noticed that five points in the plane, no three on a line, always contain four forming a convex quadrilateral, and asked whether the pattern continues. Erdos and Szekeres answered her in Compositio Mathematica in 1935 and named her in the printed text as "Miss Esther Klein" (Chapter 7, Esther Klein asks about five dots, and Erdos proves things exist without building them, quoting the 1935 paper at S433, which is listed in Chapter 7's sources). She is routinely dropped from retellings that keep both men. Go and read Chapter 7, Esther Klein asks about five dots, and Erdos proves things exist without building them rather than a summary of it.
A mathematician who never existed, and a man with no address
Around 1923, at the Ecole Normale Superieure in Paris, a senior student named Raoul Husson put on a false beard and an obscure accent and delivered a fake lecture full of invented theorems named after French generals. The last and most absurd was "Bourbaki's theorem", after Charles Bourbaki, a general of the Franco-Prussian War of 1870 to 1871 (S275, S276). Eleven years later, on Monday 10 December 1934, six mathematicians met in the Cafe Capoulade on the Boulevard Saint-Michel, annoyed by the analysis textbook they had to teach from: Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonne, Rene de Possel, and Andre Weil (S276). They remembered the joke and adopted the name. Their author acquired a first initial, a Russian nationality, manuscripts allegedly held in Leningrad, a death by "acute lead poisoning [i.e., gunshot] during the revolution", and a homeland, Poldavia, "recently wiped off the map of Europe" (S275). In 1949 Nicolas Bourbaki applied for individual membership of the American Mathematical Society. The Secretary, John Kline, objected to the prank and drew up a two-column table setting the applicant's biographical details against each other to show they were inconsistent (S275). The group got an institutional membership instead, which is the funniest available outcome.
The joke nearly killed a man. In November 1939 Andre Weil was arrested in Finland while visiting Rolf Nevanlinna and Lars Ahlfors. What aroused suspicion was a set of letters in Russian from Pontryagin about research, and a pocketful of calling cards reading "Nicolas Bourbaki, member of the Royal Academy of Poldavia" (S277). Nevanlinna secured deportation instead of execution, by his own later account (S277; the "nearly shot" element rests on that memoir alone, 8.10, item 8). Weil was released on 12 December 1939, ended up in prison in Rouen, and there proved the Riemann hypothesis for curves over finite fields (S277). His sister was Simone Weil, the philosopher and Resistance figure, 1909 to 1943 (S277).
Paul Erdos, 1913 to 20 September 1996, owned almost nothing: "He had no home, no family, no job, and no permanent address" (S263, p. 31). He turned up at colleagues' houses, worked for a few days, and moved on. In his private vocabulary children were "epsilons", husbands were "slaves", wives were "bosses", and people with families were "captured" (S263, p. 31). He wrote about 1500 papers with, by the same article's two different counts, "almost 500" co-authors on page 1 and 485 on page 32 (S263). The Erdos Number Project, running since 25 May 1995, gives 514 direct co-authors and 13,782 people at distance 2 as of August 2025 (S264). Those numbers are not in conflict, they are dated: joint papers kept appearing after his death.
Why this matters in your course. Bourbaki is the reason the empty set has the symbol it has, printed in Elements de mathematique in 1939, which Chapter 1 tells in full. The Erdos numbers are a real graph with real vertices and real edges, and computing your distance from a nominated person is breadth-first search done by hand. And 514 plus 13,782 is 14,296, while the project's own page says 14,297: the extra person is Erdos, who has Erdos number 0 (verify/domainG_output.txt, section 1).
The receipt. Barany, S275, peer reviewed in History of Science and published CC BY 4.0, for the prank lecture, the fabricated biography, Poldavia, and Kline's table. MacTutor, S276, for the date, the cafe, and the six names; S277 for the Finnish arrest, the calling card text, and Simone Weil. Erdelyi and Vertesi, S263, pp. 1 and 31 to 32. Grossman, S264, for the August 2025 counts. (S263, S275; REFERENCE | S264, S276, S277)
So what? Three of the best Bourbaki and Erdos stories are not in any source this book opened: the "Betti Bourbaki" wedding announcement, Hilbert's line about the university senate not being a bathing establishment, and Erdos's greeting "my brain is open". All three were searched for and none was found (S263, S275, S276, S278). They are in 8.4 as legends, and the documented substitutes are better: a real two-column table, a real course listing, and a real calling card that nearly got a man shot.
Seventy-two drawings on fifty-six boards
In 1900 W. E. B. Du Bois, one recent Atlanta University graduate named William Andrew Rogers, and four students still enrolled in his year-long sociology course, Henry Napoleon Lee, Lula Iola Mack, Edward Lee Simon, and William George Westmoreland, hand-drew a set of charts about Black American life and shipped them to the Paris Exposition Universelle (S281). The Library of Congress holds them as LOT 11931 and describes them in its own catalog as "72 items: pen, ink, wash and photographs; on 56 poster boards 71 x 56 cm" (S282). Seventy-two items on fifty-six boards means sixteen boards carry more than one item, and the catalog says why: two records read "Chart drawn on opposite side of LOT 11931, no. 29" and "Drawn on opposite side of LOT 11931, no. 38" (S380, S381). They had a deadline, a boat, and not enough board. The count of 63 that you will see in a great deal of good writing, including at S281, is counting the charts a different way; it is not a factual dispute. The title of the second series, which the cataloguer took from the board itself, is "A series of statistical charts illustrating the condition of the descendants of former African slaves now in residence in the United States of America" (S381).
Now the design. One chart curls a bar into a spiral because the numbers grew too fast for a straight line to fit the board (S281; the board is 3976 square centimeters, about 0.4 of a square meter, checked at verify/ch08.py, section 6). The colors look like 1970s graphic design. And then look at every axis in every chart: "not a single bar or area chart does any measure related to his subject trend down" (S281). That is a rhetorical choice, made deliberately, aimed at European visitors who had been told a different story about Black American life since emancipation.
Why this matters in your course. Every single idea in Unit 1 is on those boards. "Slaves and free Negroes" is a complement, Course section 1.2. "Age distribution of Georgia Negroes compared with France" is a partition, Course section 1.3. "The states of the United States according to their Negro population" is cardinality drawn as width, Course section 1.4. "Darien, McIntosh Co., Ga. Distribution of 1000 Negro inhabitants" fixes the total at exactly 1000 so that every count reads directly as a probability, sections 1.3 and 1.5. "Conjugal condition of American Negroes according to age periods" is a conditional distribution: read across one age band and you are reading Course section 2.3. Chapter 1 already prints three of these boards; this chapter prints the rest.
The receipt. Library of Congress item records, S380 and S381, read as JSON, for the medium, the two "opposite side" notes, and the second series title. The collection record, S382, for "72 drawings charting the condition of African Americans at the turn of the century" and for the exhibit including 500 photographs. S383 for the rights field repeated verbatim across five separate LOT 11931 items. Data by Design, S281, for the five named students and for the axis-direction observation. (S282, S380, S381, S383; SCHOLARLY | S281; REFERENCE | S382)
So what? Every one of these boards is catalogued by the Library of Congress with the Rights Advisory line "No known restrictions on publication", and the record's access_restricted field is false (S380, S381, S383). That is why they are printed in this book and why you can print them in a classroom handout. It is also worth knowing that you will never see the originals: the catalog note reads "Use digital images. Originals are too fragile to be served" (S282).
W. E. B. Du Bois sent a set of hand-drawn data charts to the Paris Exposition of 1900. How many, and who drew them?
Show the answer▸
The Library of Congress holds them as LOT 11931 and describes the set in its own catalog as "72 items: pen, ink, wash and photographs; on 56 poster boards 71 x 56 cm" (S282).
Seventy-two items on fifty-six boards is not the same as seventy-two boards, and a great many retellings say seventy-two boards. This book quotes the catalog rather than the retelling.
They were drawn by Du Bois and a team of Atlanta University students, by hand, in ink, with no computer and no printing press (S281). The rights line on the Library of Congress record reads "No known restrictions on publication," so a class can print them tomorrow.
Every technique in Course section 2.2 about presenting proportions honestly is in those boards, fifty years before anybody wrote the rules down.
Chapter 9
A womb, an archer, and one man's exclamation point
Every word on your syllabus was chosen by somebody, on a particular day, for a reason, and most of those people are more recent than you think. A matrix is a womb. A stochastic process is named after the pillar an archer shoots. The exclamation point in n! was one man's suggestion in 1808, and it beat a rival sign invented by a Cambridge graduate in 1827. By the end of this chapter you will know where your own vocabulary came, you will be able to tell a documented origin from a good story, and you will be able to trace a word yourself with the same three websites this book used.
Mathematical notation is not ancient and it was not handed down. It is a set of decisions, most of them taken in the last two hundred years, many of them close-run. Kramp introduced n! in 1808 and eight years later a colleague was still complaining that nobody had adopted it. Ettingshausen printed the binomial bracket in Vienna in 1827. Sylvester named the matrix in 1850 and explained a year later that he meant a womb. Hilbert wrote Eigenwert in 1904 and English kept half of it. Almost all of Unit 1's vocabulary dates from 1909 to 1926. The bar in P(A|B) arrives in 1931. The empty set symbol is a Norwegian letter, claimed by one man in a memoir written fifty-three years later.
Christian Kramp introduced the exclamation point for factorial in 1808, with a one-sentence justification: "I use the very simple notation n! to denote the product" (S301). Guess how long it took to catch on.
I have a guess▸
Longer than you would think. Eight years later a colleague was still complaining in print that nobody had adopted it (S301).
The reason Kramp gave was not elegance or meaning. It was that the exclamation point was a piece of type a printer already had in the case and nobody else was using (S301, vol. 2, sect. 448, p. 73).
That is most of this chapter in one story. Notation wins because it is available, compact and easy to set in type, not because it is right. There is no committee. There is a person, a printer, and a few decades of other people slowly agreeing.
The word that means womb
In 1850 James Joseph Sylvester was writing an addition to two earlier papers, one on a new class of theorems and one on Pascal's theorem, and he ran into a wording problem. He wanted the general form of a rule about determinants, and the general form does not start with a square. It starts with a rectangle. He needed a name for the rectangle, so he made one up, mid-sentence, and capitalized it the way you do when you are coining a word rather than using one: "This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns" (S307, printed page 150). Then he went straight back to talking about determinants, because in 1850 the determinant was the interesting object and the rectangle was only its container.
A year later he said out loud what he had meant. At page 247 of the same collected volume: "I have in previous papers defined a 'Matrix' as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent" (S135, p. 247). Latin matrix is "womb, breeding female," from mater, "mother," and the core sense the word carries in English from the late fourteenth century is "that which encloses or gives origin to something" (S306, matrix). Sylvester picked it deliberately. The array gives birth to determinants, so the array is the mother.
Why this matters in your course. Course section 3.2 opens with the definition of a matrix, and the definition is Sylvester's. His own glossary entry reads "Matrix. A square or rectangular arrangement of terms in lines and columns" (S135, scan leaf 601), and he is already comfortable with non-square arrays and already counts rows before columns, exactly as your shape notation does. Appendix B.1 entry 19 is his rectangle with a capital letter on it.
The receipt. Sylvester, S307, printed page 150 (scan leaf 168), read in the 1904 Cambridge collected edition on the Internet Archive. The item's own leaf-to-page map gives leaf 168 as printed page 150, which is exactly the locator Miller cites (S302, m.html). The womb sentence is S135, p. 247. Latin from S306, matrix. (S307; REFERENCE | S302, S306)
So what? This is the single best-documented coinage in your entire course, and it is the only first-use claim in this chapter that somebody in this book confirmed by reading the mathematician's own text rather than a reference book about it. Everything else in section 9.6 rests on Cajori's testimony or Miller's, honestly labeled. One out of a hundred and thirty-six. That ratio is the real subject of section 9.7.
The factorial fight, 1808 to about 1860
Christian Kramp of Strasbourg published Elemens d'arithmetique universelle at Cologne in 1808, with a section headed "Notations," and in it he wrote a sentence that decided what a million exam papers would look like: "Je me sers de la notation tres simple n! pour designer le produit," I use the very simple notation n! to denote the product (S301, vol. 2, sect. 448, p. 72, Cajori quoting Kramp). That is the whole justification. It is simple. He needed a compact mark for a product that kept turning up, and the exclamation point was a sort a printer already had in the case and nobody else was using.
In the same year Adrien-Marie Legendre introduced a rival, the capital Greek Gamma, continued it in his integral calculus of 1811, and set it up so that Gamma(n+1) stands for n-factorial (S301, vol. 2, sect. 448, p. 73). Both notations survive. You will meet Kramp's in this course and Legendre's if you go on to analysis, which is unusual: most notation fights end with one corpse.
Kramp did not win quickly. In 1816 J. B. Durrande used n! in Gergonne's Annales and wrote, in print, "There is ground for surprise that a notation so simple and consequently so useful has not yet been universally adopted" (S301, vol. 2, sect. 449, p. 75, Cajori quoting Durrande). Eight years after Kramp, a working mathematician is complaining that nobody has picked it up.
Meanwhile England had its own sign and its own candidate. In 1827 Thomas Jarrett, who had just taken his BA at St Catherine's College, Cambridge, suggested a bar-and-corner symbol, a printer's rule bent at a right angle (S301, vol. 2, sect. 449, p. 74). Cajori is blunt about how well it did: "for a quarter of a century the notation was neglected." The Reverend Harvey Goodwin used it freely in 1846 in an article printed in 1849, and "the symbol made no substantial headway in England until it was adopted by Todhunter about 1860, and was used in his popular texts" (S301, vol. 2, sect. 449, p. 74). Henry Warburton of Cambridge tried !n! in 1847 (S301, vol. 2, sect. 449, p. 73). Nobody uses either now.
And here is the detail that makes the whole thing human. Cajori records that "some texts in the English language suggest the reading 'n-admiration' (the exclamation point [!] being a note of admiration), but most texts prefer 'factorial n,' or 'n-factorial'" (S301, vol. 2, sect. 449, p. 75). In the printing trade the exclamation point was called the note of admiration, which is where that reading comes. German readers said "n-Fakultat" (S301, vol. 2, sect. 449, p. 75).
Why this matters in your course. Course section 2.1 hands you n! as though it fell out of the sky, and Appendix B.1 entry 11 tells you to read it "n factorial." Both the mark and the reading are decisions, and both had rivals. Note also that the noun came after the sign: Kramp's notation is 1808 and the English noun "factorial" is dated 1816 (S306, factorial). We had the symbol for eight years before we had the word.
The receipt. Every quotation in this scene is Cajori's, read through the Internet Archive's search-inside index against the Wellcome Collection scan of volume 2, with printed pages derived from that item's own leaf-to-page map at a constant offset verified at eleven separate leaves (S301, access note). Kramp's sentence, sect. 448, p. 72. Durrande's complaint and the "note of admiration" line, sect. 449, p. 75. Jarrett, Goodwin, and Todhunter, sect. 449, p. 74. Warburton, sect. 449, p. 73. Legendre, sect. 448, p. 73. (S301, for what Cajori says; the 1808 book itself was not opened, see 9.7)
So what? Two things. First, notation is a popularity contest with no referee, and Cajori's volume 2 is the closest thing to a referee's notebook this book has found. Second, check the arithmetic on your own sources. Cajori writes "a quarter of a century" for the neglect of Jarrett's sign and then dates the revival to 1846, which is nineteen years after 1827, not twenty-five (verify/ch09_output.txt, section 2). Cajori is the standard authority on the history of notation and his own paragraph does not quite add up. Print the dates, not the phrase.
A bracket from Vienna, and a year that is wrong everywhere
You write perhaps a hundred times in a combinatorics unit. Cajori dates it: "The notation which has become the more common was introduced in 1827 by von Ettingshausen," footnoted to Vorlesungen uber hohere Mathematik, vol. 1 (Vienna, 1827), p. 38 (S301, vol. 2, sect. 439, p. 63). Raabe was using it by 1851 (S301, same page).
You will find 1826 printed in a great many places, including the brief that started this book. It is wrong, and this book settled it. Somebody here opened the Internet Archive scan of the book, read the title page, and read the preface. The title page of volume 1 says Vorlesungen uber die hohere Mathematik, Erster Band, Wien, 1827. The preface is signed off "Wien, im Sommer 1827." Volume 2 of the same first edition is also dated 1827, so the tidy reconciliation, that some catalogs date the set 1826 to 1827 and both camps could be right about different volumes, does not apply either. Page 38 of volume 1 is about the binomial theorem, which is consistent with Cajori's page citation (S387).
The book is a set of sixty lectures given at the University of Vienna (S387, cataloguer's note). So the symbol has a place and a season: a Vienna lecture course, written up in the summer of 1827.
Why this matters in your course. Course section 2.1 and Appendix B.1 entry 13 both use the bracket. The symbol is 199 years old in 2026 (verify/ch09_output.txt, section 3), it is nineteen years younger than Kramp's exclamation point, and the object it names, the binomial coefficient, was being computed in China, India, and the Islamic world centuries before Europe had a mark for it. Notation and mathematics are on different clocks.
The receipt. Cajori's sentence and footnote, S301, vol. 2, sect. 439, p. 63. The title page and the preface dateline, read in the scan, S387. The settlement is recorded in and is binding on this book. (S301, S387)
So what? Because this is what a one-year error looks like when you chase it. The difference between 1826 and 1827 changes nothing mathematically and everything about whether you can be trusted. And the job is still not finished: the OCR of an 1827 German scan does not render a two-line bracket, so nobody in this book has ever seen the mark on page 38. We have proved the year and not the symbol (S387). If this book ever reproduces that page, a human being has to look at the image first.
Unit 1 is younger than the aeroplane
Open your book at Course section 1.2 and read the four words in the heading: union, intersection, complement, universal set. Every one of them is about a hundred years old.
- "Intersection" in the set-theoretic sense is recorded in Webster's New International Dictionary of 1909 (S302, i.html).
- "Disjoint" is Cassius J. Keyser in Science in 1909: "two classes are disjoint if neither includes a term of the other" (S302, d.html).
- "The universal class" is Whitehead and Russell, Principia Mathematica vol. I, p. 30, 1910 (S302, u.html).
- "Union" is James Pierpont, Lectures on the Theory of Functions of Real Variables, vol. 2, p. 22, 1912. Before that people wrote "sum" (S302, u.html).
- "Set complementary to" is E. W. Chittenden in the Transactions of the AMS, 1914 (S302, c.html).
- "Empty set" is J. E. McAtee in the American Journal of Mathematics, 1919, used without explanation, and for years afterwards "null set" was much more common (S302, e.html).
- "Set theory," as an English phrase, is Orrin Frink in the Annals of Mathematics in 1926 (S302, s.html).
In 2026 those are 117, 117, 116, 114, 112, 107, and 100 years old, and the whole group is spread over just seventeen years (verify/ch09_output.txt, section 4). The word "set" itself is a translation and not a coinage: Cantor wrote Menge, English translators wrote "aggregate," and E. H. Moore had to announce in 1901 that "It is convenient to use set as the equivalent of Menge and ensemble." Jourdain was still translating Cantor's Menge as "aggregate" in 1915 (S302, s.html).
Why this matters in your course. Unit 1 is the unit students assume is oldest, because it feels the most basic. It is the newest vocabulary in the book. The mathematics underneath is older, since Cantor's Menge is fixed by 1883 and Peano's epsilon is 1889, but the English words on your page were settled inside a single seventeen-year window, from 1909 to 1926.
The receipt. All seven first uses from Miller's letter pages, read in full for these entries (S302, c.html, d.html, e.html, i.html, s.html, u.html). Moore's sentence and Jourdain's "aggregate," S302, s.html. Ages computed in verify/ch09.py, section 4, all checks passing. (S302)
So what? Because "since ancient times" is doing a lot of quiet damage. A subject presented as handed down from antiquity is a subject nobody feels allowed to argue with. A subject whose basic vocabulary was still being negotiated in 1919 is one you could imagine contributing to. Both descriptions are of the same syllabus, and only one of them is accurate.
Eigenvalue: a word half-translated, and the word that lost
In 1904, in "Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen," David Hilbert wrote Eigenwert and Eigenfunktion (S302, e.html; the printed volume is S184). German eigen means "own, proper, characteristic," and Wert means "value," so an Eigenwert is a thing's own value. English then did something strange: it translated the second half and kept the first. That is why you write "eigenvalue" and not "proper value," and why the word looks like nothing else in your book.
There was competition. Cauchy had l'equation caracteristique in 1840 (S302, e.html). Sylvester, in 1883, offered "latent roots of a matrix, latent in a somewhat similar sense as vapour may be said to be latent in water" (S302, e.html), a metaphor borrowed straight from the physics of latent heat, from Latin latere, "to lie hidden" (S306, latent). A. C. Aitken was still writing "latent roots and latent vectors" in 1937 (S302, e.html). Paul Halmos, who preferred "proper value," surveyed the wreckage in 1958: "almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature." In 1967 he gave up in print: "eigenvalues have won it" (S302, e.html). Hilbert to Halmos's surrender is sixty-three years (verify/ch09_output.txt, section 5).
Why this matters in your course. Course section 3.7 and Appendix B.1 entry 27 give you lambda and the word eigenvalue with no explanation of either. Now you have the explanation for the word. You do not have one for lambda: no source this book reached dates the choice of that letter, and the row in section 9.6 says so and stays tagged as unverified.
The receipt. Miller's eigen entry, read in full, for Hilbert 1904, Courant and Hilbert's Eigenvektor in 1924, Cauchy 1840, Sylvester 1883, Aitken 1937, and both Halmos quotations (S302, e.html). The printed Hilbert volume, with the word Eigenwert on the page and a chapter heading built on it, is S184 and is cleared as IMG-042 and IMG-043. (S302; PRIMARY | S184 for the page)
So what? One warning worth carrying. Miller reports that Hilbert's Eigenwerte are the reciprocals of what we would now call the characteristic or latent roots (S302, e.html). The man who coined the word did not mean quite what you mean by it. That is the ordinary condition of mathematical vocabulary, not an exception.
Stochastic: a word for aiming at something you cannot hit
Greek stokhos is the target, specifically the pillar an archer shoots, and stokhazesthai means to aim, and by extension to guess (S306, stochastic). The word reached English by the 1660s meaning "pertaining to conjecture" (S302, s.html; S306). Then in 1713 Jacob Bernoulli's posthumous Ars Conjectandi defined a whole subject with it: "Ars Conjectandi sive Stochastice nobis definitur ars metiendi quam fieri potest exactissime probabilitates rerum," the art of conjecturing, or stochastics, is defined by us as the art of measuring the probabilities of things as exactly as possible (S302, s.html, quoting Bernoulli through Chuprov).
Then the word went to sleep for two centuries. It comes back in 1917, in German statistics, when Ladislaus von Bortkiewicz proposed in Die Iterationen that the probability-based study of empirical multiplicities "moege als Stochastik bezeichnet werden," should be called stochastics (S302, s.html). A. A. Chuprov put it into English in 1923, explaining that he used "stochastical" as a synonym for "based on the theory of probability," and citing Bernoulli's page 213 and Bortkiewicz by name (S302, s.html). Kolmogorov writes "processo stocastico" in Italian in 1932; Doob writes "stochastic processes" in English in 1934 (S302, s.html). Bernoulli to Bortkiewicz is 204 years (verify/ch09_output.txt, section 7).
Why this matters in your course. Unit 4 calls a matrix row-stochastic and calls the whole apparatus a stochastic process, and the metaphor inside the word is a good one to keep. You aim at the target. You do not hit the same spot twice. The distribution of where the arrows land is the object of study, and Bernoulli's phrase, the art of measuring probabilities as exactly as possible, already contains the admission that exactness is the goal and not the result.
The receipt. The Greek from S306, stochastic, which also gives the Indo-European root that produced "sting" and "stag." Bernoulli's Latin from S302, s.html. Bortkiewicz and Chuprov from the same page. The 1713 title page and the page carrying the definition are cleared as IMG-006 and IMG-007 (S006). (S302, S306)
So what? There is an honest hole here, and it is instructive. This book tried to confirm Bernoulli's sentence at page 213 of the 1713 Basel printing by searching the Internet Archive scan's own text index, twice, for "Stochastice" and "Stochastic," and got nothing (S302, section 10). That is an optical character recognition failure on early modern Latin type, not evidence that the sentence is absent, and the note says so rather than quietly upgrading or dropping the claim. The chain here runs Bernoulli to Chuprov to Miller to you, and one link in it has never been opened.
Two of the biggest words in mathematics are the same man
Around 825, Abu Ja'far Muhammad ibn Musa al-Khwarizmi, whom the sources place at Khwarazm and then Baghdad, wrote a book whose title is usually given as al-mukhtasar fi hisab al-jabr wa al-muqabala, "the compendium on calculation by restoring and balancing" (S306, algebra). Al-jabr means the restoring, the reunion of broken parts, and in the book it names the operation of moving a subtracted quantity to the other side of an equation (S302, a.html). Europe took the word out of the title. It reached English in the 1550s, and for a while in fifteenth and sixteenth century English "algebra" also meant bone-setting, which is the same metaphor doing honest work: you are putting the broken parts back together (S306, algebra).
Europe also took his name. Al-Khwarizmi means "the man from Khwarazm," and it went through Medieval Latin as algorismus, then Old French algorisme, then Middle English algorism, and arrived as "algorithm" in the 1690s (S306, algorithm). Leibniz was already using the word in something close to the modern sense in 1684, in "Nova Methodus pro maximis et minimis," writing of "this rule, known as an algorithm, so to speak, of this calculus" (S302, a.html).
One book, one man, two of the largest words in the subject. And a third word in the same family arrives by the same route: Sanskrit sunya, empty, became Arabic sifr, empty, which gave English both "cipher" and, through Medieval Latin zephirum and Italian, "zero" (S306, zero and cipher). Cipher and zero are the same word twice.
Why this matters in your course. Appendix A is called Algebra Refreshers, Course section 3.6 teaches an elimination that every textbook calls an algorithm, and Unit 1 spends a week on a set with nothing in it. The names of all three came into European mathematics through Arabic, and two of them are one ninth-century author.
The receipt. S306 for algebra, algorithm, zero, and cipher, each entry read in full. S302, a.html, for al-jabr as the transposing operation and for Leibniz's 1684 use. (S302, S306)
So what? Two honest limits. The date is Miller's "c. 825" and this book has not opened anything of al-Khwarizmi's, so every claim in this scene is a dictionary's claim, and etymonline is itself a compilation from other dictionaries rather than an independent research instrument (S306, section 10). And the arithmetic that follows from an approximate year is approximate: about 1,200 years from that book to your desk, which the script computes and flags as arithmetic on a "c." date (verify/ch09_output.txt, section 7).
Two symbols younger than they look, and only one of them documented
Start with the bar in P(A|B). Harold Jeffreys, a Cambridge geophysicist writing about how evidence changes belief, used a vertical stroke for conditioning in Scientific Inference in 1931, writing P(p | q) (S303). Two years later Kolmogorov fixed P(A) itself in the Grundbegriffe (S303). William Feller made the stroke standard by using Pr{A | B} throughout his 1950 textbook, and later editions moved to P{A} (S303). So the two halves of the most-used symbol in Unit 2 arrived within two years of each other, from two people who disagreed about what probability even is, and the notation you think is ancient is, in the researcher's own phrase, younger than the aeroplane (S303, section 9).
Now the empty set. Miller records the symbol's first appearance in N. Bourbaki, Elements de mathematique (Paris, 1939), at page 4, in the phrase "la partie vide de E" (S054). Bourbaki was not a person. It was a collective pseudonym, working in Paris, and Weil calls it a group (S054). In 1992, in his autobiography The Apprenticeship of a Mathematician, Andre Weil claimed the glyph as his own: "The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar" (S054, quoting Weil, 1992, p. 114). He also records the group's feeling that "it was high time to fix these notations once and for all." So it is not a zero, and it is not a Greek phi. It is a Norwegian and Danish letter.
Why this matters in your course. Appendix B.1 entries 3 and 14 are these two symbols. Course section 1.1 draws the empty set on page one of the course, and Course section 2.3 lives inside the conditional bar. Both are twentieth century decisions, and one of them was made by a committee that did not use its own members' names.
The receipt. Jeffreys, Kolmogorov, and Feller from Miller's probability and statistics symbol page, read in full for these entries (S303). Bourbaki 1939, p. 4, and the Weil quotation from Miller's set theory and logic symbol page (S054). (S303, S054)
So what? Compare the evidence behind the two halves of this scene, because they are not the same. Jeffreys's bar is in a printed book of 1931 that anybody can check. The Norwegian story is one man's memory, written down in 1992 about a decision taken in the 1930s, with no contemporary document behind it, and Miller's own page is where this book read it rather than Weil's book. The note file says so in as many words: "Weil's claim is Weil's own, made in 1992 about a decision of the 1930s, and it is the only account. It is credible but uncorroborated by a contemporary document" (S054, section 10). Bourbaki's internal minutes, held in the Archives Bourbaki, would settle it. Nobody in this book has seen them. Believe the story if you like, and know exactly what you are believing.
Open your course book at Course section 1.2 and read the four words in the heading: union, intersection, complement, universal set. How old are they?
Show the answer▸
About a hundred years, all four. "Intersection" in this sense is recorded in Webster's New International Dictionary of 1909 (S302, i.html). "Union" is 1912, "complement" is 1914, "empty set" is 1919, and "set theory" reaches English in 1926 (S302, u.html and c.html; S055).
Unit 1's entire vocabulary is younger than the aeroplane. The Wright brothers flew in 1903.
That is worth carrying into the exam. The words look ancient because they are set in a serif typeface inside a textbook, and that is a fact about typography. The subject is one long lifetime old, and the people who chose these words had reasons you can read.
The master timeline: every date, in one place
583 dated events across ten eras, merged from 266 sources. Where sources disagree about a date, the disagreement is shown rather than resolved by picking a favorite. Read a row as a claim with a provenance, not as a fact from nowhere.
- Place any person or result in this book on a single shared timeline.
- Read a dated row together with its sources, and say how well attested it is.
- Spot a disputed date and say what the disagreement is about.
- Compare what was happening in different parts of the world in the same century.
- See how far apart the mathematics and the word for it usually are.
Part II
How to read the timeline
Every dated event from every source in this book, merged into one sequence. 583 entries, running from 480 BCE to 2025.
How to read it. Every row is a claim with a provenance, not a fact from nowhere. The Sources column names the documents it rests on, and each one links to its full reference in Appendix J.
A row carrying the Disputed label is not settled: the date, the attribution, or both. The reason is stated underneath the event, and Appendix F gives every position in full. 22 of these 583 entries carry that label, and 21 of this book's 48 logged disagreements are still open, so you will meet it often. That is the honest state of the evidence, not a gap in the research.
A word about the dates themselves. A great many are approximate, and the table says so rather than smoothing it over. "c." means circa, roughly. A range means the sources disagree or the event took years. Russian dates before 1918 are Old Style, with the New Style equivalent in brackets where a source gives one.
Timeline
Before 500: counting, chance, and the first systems
7 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 480 BCE to 221 BCE | Warring States period, during which counting rod numerals are in use | not named | China | S126 |
| c. 300 BCE | Babylonian clay tablets carry problems that amount to simultaneous linear equations | not named | Babylonia (Iraq) | S122 |
| 213 BCE | The Qin book burning that later compilers say they were repairing | the Qin government | Qin China | S123 |
| 200 BCE to 50 CE Disputed | The Nine Chapters on the Mathematical Art compiled; chapter 8 poses eighteen linear systems and states the rules for negative numbers Dispute: When the Nine Chapters was compiled. Unresolved and italicized. records that the only reasoned account available is MacTutor's units argument, and instructs that the range be printed and not a number. The row carries the widest range any source defends, 200 BCE to 50 CE. | anonymous compilers, later Zhang Cang | Han China | S120, S122, S123, S124, S126 |
| 2nd c. BCE to 1st c. CE | The Natyasastra applies the same combinatorial machinery to drama and music | Bharata | India | S002 |
| 2nd c. BCE | The Chandahsastra treats Sanskrit meter as a mathematical theory of combinations | Pingala | India | S002, S011 |
| 263 CE | Liu Hui's commentary on the Nine Chapters, the one firm date in the Chinese story | Liu Hui | Wei China | S123, S124, S125 |
Timeline
500 to 1400: India, the Islamic world, and China
26 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 6th c. CE | Brhat Samhita chapter 77 counts perfume blends drawn four at a time from sixteen substances | Varahamihira | India | S010 |
| c. 640 CE | Li Chunfeng corrects and expands Liu Hui's notes on the Nine Chapters | Li Chunfeng | Tang China | S123 |
| 7th c. CE | The matra recursion for meter, of Fibonacci type; meru prastara and suci prastara distinguished | Virahanka | India | S002 |
| c. 750 CE | The recursion for counting metrical patterns is replaced by a closed multiplicative formula | Sridhara | India | S002 |
| c. 825 | Al-Khwarizmi's al-jabr w'al-muqabalah, the book that gives algebra its name | al-Khwarizmi | Baghdad | S302 |
| c. 840 CE | An Arabic manuscript carries two full 8 by 8 knight's tours, one of them reentrant | Ali C. Mani and al-Adli ar-Rumi | Abbasid Baghdad | S431 |
| 9th c. CE Disputed | Ganita-sara-sangraha rule 218 states what we write as n choose r Dispute: Mahavira's century. Unresolved and italicized. Neither source argues its dating; they may not even be talking about the same man. | Mahavira | southern India | S009, S002 |
| c. 900 CE | A half-board knight's tour written into the Kavyalankara as a verse | Rudrata | Kashmir | S431 |
| 10th c. CE | Halayudha's commentary describes building the triangular array, each cell the sum of the two above | Halayudha | India | S002, S011 |
| 953 to c. 1029 | Life of al-Karaji, the man al-Samaw'al credits with the construction of the coefficient table | al-Karaji | Baghdad and al-Karaj | S003 |
| c. 1050 | The array of binomial coefficients now called the Jia Xian triangle | Jia Xian | Song dynasty China | S014 |
| 11th c. | The pattern of binomial coefficients studied | Omar Khayyam | Persia | S013 |
| 1069 | The commentary on Rudrata through which the tour is known | Nami of Guzerat | Gujarat | S431 |
| c. 1100 CE | Kedara standardizes the prastara procedures in verse | Kedara | India | S002 |
| c. 1100 Disputed | Miller dates the binomial theorem in Persia and China to about this year Dispute: When the binomial theorem appears in Persia and China. Unresolved and italicized. The row is kept because Miller's date is in print and is widely repeated, and the reader needs to see it against the earlier dates. | not named | Persia and China | S011 |
| c. 1130 | Birth of al-Samaw'al ibn Yahya al-Maghribi | al-Samaw'al | Baghdad | S003 |
| c. 1150 | Al-Bahir fi al-Jabr prints the coefficient table to its twelfth row and argues by an inductive descent | al-Samaw'al | Baghdad | S003 |
| 1228 | Death of Ibn Mun'im, whose Fiqh al-hisab gives nineteen pages to combinatorics | Ibn Mun'im al-Abdari | Marrakesh | S015, S016 |
| 1261 | Xiangjie jiuzhang suanfa absorbs about two thirds of Jia Xian's problems | Yang Hui | Qiantang, Zhejiang | S014 |
| 1288 | Birth of Levi ben Gershon | Levi ben Gershon | Provence, France | S004 |
| 1303 | Siyuan yujian prints the array, already labeled the Old Method | Zhu Shijie | China | S013, S017 |
| 1321 | Maaseh Hoshev completed in the spring: sixty-eight theorems with Euclid-style proofs | Levi ben Gershon | Provence, France | S004 |
| c. 1323 Disputed | The two laws later called De Morgan's are said to occur explicitly in the Summa Totius Logicae Dispute: Whether De Morgan's laws are Ockham's, c. 1323. Unresolved and italicized. The Latin was not read in this book, so the c. 1323 row stands as a reported claim, not a fact. | William of Ockham | Oxford and Munich | S055 |
| 1344 | Death of Levi ben Gershon | Levi ben Gershon | Provence, France | S004 |
| 1356 | Ganitakaumudi finished, Saka 1278, with chapter 13, Ankapasa, on combinatorics | Narayana Pandita | India | S020 |
| 1380 | The problem of points appears in Italian manuscripts | unknown | Italy | S072, S075 |
Timeline
1400 to 1650: print, gamblers, and the algebraic turn
23 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| c. 1400 | Madhava's series for sine, cosine and arctangent | Madhava | Sangamagrama, India | S284 |
| 1408 | The Yongle dadian encyclopaedia copies Yang Hui's text and so preserves it | imperial compilers | China | S014 |
| 1494 | Pacioli's Summa prints the problem of points and divides the stake 5 to 3, which is wrong | Luca Pacioli | Italy | S072, S075, S093 |
| 1499 | Birth of Niccolo Fontana, later called Tartaglia | Tartaglia | Brescia | S019 |
| 1501 | Birth of Girolamo Cardano on 24 September | Girolamo Cardano | Pavia | S094 |
| 1512 | A sabre cleaves his jaws and palate in the French sack of the city | Tartaglia | Brescia | S019 |
| c. 1525 | Cardano's rules for the single-die problem, on Ore's dating | Girolamo Cardano | Italy | S075 |
| 1544 | Arithmetica Integra uses the array for extracting roots, not for counting | Michael Stifel | Nuremberg | S018 |
| 1545 | Ars Magna gives the regula de modo, which is Cramer's rule for two equations | Girolamo Cardano | Italy | S122 |
| c. 1550 | Jyesthadeva's Yukti-Bhasa, written in Malayalam | Jyesthadeva | India | S284 |
| 1555 | De Censura Veri displays an Aristotelian syllogism with triangles | Juan Luis Vives | Low Countries | S051 |
| 1557 Disputed | Death of Tartaglia on 13 December, on Britannica's dating Dispute: Tartaglia's death year. Unresolved and italicized. Britannica's full date and age is the stronger form, but neither source cites a document, so the year is not settled. | Tartaglia | Venice | S019, S011 |
| 1560 | Execution of Giambatista Cardano, the mathematician's son | Giambatista Cardano | Italy | S094 |
| 1563 Disputed | The Liber de ludo aleae probably completed, on MacTutor's hedged dating Dispute: When Cardano wrote the Liber de ludo aleae. Unresolved and italicized. conflict 1 says to print no composition date without one of those two, so the row is kept only as MacTutor's hedged claim. | Girolamo Cardano | Milan | S094, S070, S073 |
| 1570 | Billingsley's English Euclid puts discrete and lemma into English mathematics | Henry Billingsley | London | S302 |
| 1570 | Opus novum de proportionibus uses the arithmetic triangle | Girolamo Cardano | Italy | S072 |
| 1576 | Death of Cardano on 21 September | Girolamo Cardano | Rome | S094 |
| 1603 | The first of the continuous weekly Bills of Mortality, 29 December, O.S. | the Company of Parish Clerks | London | S077 |
| 1607 to 1684 | Life of Antoine Gombaud, chevalier de Mere | Antoine Gombaud, chevalier de Mere | France | S072, S075 |
| 1620 Disputed | Sopra le scoperte dei dadi written, on Gorroochurn's dating Dispute: When Galileo wrote Sopra le scoperte dei dadi. Unresolved and italicized. Do not print 1620 on Gorroochurn's authority alone. | Galileo Galilei | Florence | S072, S071 |
| 1625 | The bills of mortality add parish-level breakdowns | the Company of Parish Clerks | London | S077, S078 |
| 1629 | The bills add causes of death and the split between male and female | the Company of Parish Clerks | London | S077 |
| 1646 | Viete uses magnitudines scalares, a different sense of scalar | Francois Viete | France | S176, S302 |
Timeline
1650 to 1800: the problems get their shape
76 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1654 | The Traite du triangle arithmetique composed | Blaise Pascal | Paris | S011 |
| 1654 | Pascal to Fermat, 24 August: the three-player problem, 17, 5 and 5 out of 27 | Blaise Pascal | Paris | S074 |
| 1654 | Pascal to Fermat, 29 July: the 32 pistoles and the odds of 671 to 625 | Blaise Pascal | Paris and Toulouse | S070, S074, S075, S302 |
| 1654 | Pascal's definitive religious conversion on 23 November, four weeks after the last mathematical letter | Blaise Pascal | Paris | S075 |
| 1654 | The Pascal to Fermat correspondence that founds the subject | Blaise Pascal and Pierre de Fermat | Paris and Toulouse | S070, S074, S075 |
| 1656 | Wallis writes per modum inductionis | John Wallis | Oxford | S302 |
| 1656 to 1657 | Composition of Van Rekeningh in Spelen van Geluck | Christiaan Huygens | The Hague | S076 |
| 1657 | De ratiociniis in ludo aleae printed in van Schooten's Exercitationum mathematicarum by J. Elsevir; expectatio enters the Latin | Christiaan Huygens | Leiden | S005, S076, S302 |
| 1660 | The last two Pascal to Fermat letters, in July and August | Blaise Pascal and Pierre de Fermat | Paris and Toulouse | S074 |
| 1661 | Universalia Euclidea puts circles into print for testing the validity of a syllogism | Johann Christoph Sturm | Germany | S051 |
| 1662 | Stochastic enters English meaning pertaining to conjecture, then does nothing in mathematics for two centuries | not named | England | S302, S306 |
| 1662 | Natural and Political Observations... upon the Bills of Mortality published | John Graunt | London | S077, S078 |
| 1662 | The Port Royal Logic, the model for the title Ars Conjectandi, puts probability into its modern sense | Antoine Arnauld and Pierre Nicole | Paris | S080, S302 |
| 1663 | The Liber de ludo aleae printed in the first volume of Cardano's collected works | Girolamo Cardano | not stated in the source | S070, S094 |
| 1665 | The Traite du triangle arithmetique printed by G. Desprez, 177 pages | Blaise Pascal | Paris | S001 |
| 1666 | Dissertatio de arte combinatoria, the first European book whose whole subject is combinations | Gottfried Wilhelm Leibniz | Leipzig | S008 |
| 1667 | Birth of Abraham de Moivre on 26 May | Abraham de Moivre | Vitry-le-Francois, Champagne | S082 |
| 1669 to 1670 | Newton writes the elimination note he would rather not have seen published | Isaac Newton | England | S120, S121 |
| 1673 Disputed | Miller dates a Wallis title carrying the word Combinations to this year; the book is usually dated 1685 Dispute: The year of a Wallis title carrying the word Combinations. Unresolved and italicized. conflict 9 flags this so that the book does not print 1673 without a check. | John Wallis | Oxford | S011, S302 |
| 1678 | Thomas Strode defines permutation in English | Thomas Strode | England | S011, S302 |
| 1682 to 1739 | Life of Nicholas Saunderson, the blind Lucasian Professor | Nicholas Saunderson | Cambridge | S094 |
| 1683 | Seki's method for fukudai problems: expand the array, pull common factors from rows and columns, count n factorial terms | Seki Takakazu | Edo (now Tokyo) | S122, S127, S128 |
| 1685 | The Edict of Fontainebleau revokes the Edict of Nantes | Louis XIV | France | S082 |
| c. 1686 | An eighteen page manuscript, De Formae Logicae Comprobatione per Linearum Ductus, draws circles and ellipses for the valid syllogisms | Gottfried Wilhelm Leibniz | Hanover | S051 |
| 1687 | De Moivre and his brother are admitted to the Savoy Church on 28 August, O.S. | Abraham de Moivre | London | S082 |
| 1688 | Claimed release of de Moivre from the Prieure de Saint-Martin on 27 April | Abraham de Moivre | Paris | S082 |
| c. 1689 | Bernoulli completes the proof of the law of large numbers and publishes nothing | Jacob Bernoulli | Basel | S079 |
| 1691 | Weise, Nucleus Logicae, first edition | Christian Weise | not stated in the source | S051 |
| 1693 | Leibniz writes the vanishing condition for three equations to the Marquis de l'Hopital, on 28 April | Gottfried Wilhelm Leibniz | Hanover | S129 |
| 1701 to 1702 | The window in which Thomas Bayes was probably born, July 1701 to April 1702 | Thomas Bayes | England | S084, S094 |
| c. 1702 | Bernoulli has settled on the word Stochastic | Jacob Bernoulli | Basel | S079 |
| 1705 | Death of Jacob Bernoulli, aged 50, with the manuscript unpublished | Jacob Bernoulli | Basel | S005, S079, S080 |
| 1708 | Montmort's Essay d'analyse sur les jeux de hazard prints Table de M. Pascal pour les combinaisons and is spurred by a review of Bernoulli's manuscript | Pierre Remond de Montmort | France | S011, S080 |
| 1708 | Death of Seki Takakazu on 5 December | Seki Takakazu | Edo (now Tokyo) | S128 |
| 1709 | Nicolaus I's dissertation lifts Jacob's work almost literally | Nicolaus I Bernoulli | Basel | S005, S079 |
| 1711 | De Moivre publishes De Mensura Sortis in Latin; Saunderson becomes the fourth Lucasian Professor | Abraham de Moivre and Nicholas Saunderson | London and Cambridge | S082, S094 |
| 1713 | Ars Conjectandi appears in August, eight years after its author's death; it prints stochastice, a priori and a posteriori | Jacob Bernoulli | Basel | S005, S006, S080, S302 |
| 1713 | Nicholas Bernoulli poses the St Petersburg problem to Montmort | Nicolaus I Bernoulli | Basel and Paris | S072 |
| 1718 | The Doctrine of Chances, first edition, printed by W. Pearson and dedicated to Isaac Newton; the word event in the probability sense | Abraham de Moivre | London | S081, S055, S302 |
| 1720 Disputed | Bayes enters the University of Edinburgh, on Bellhouse's dating Dispute: When Bayes entered the University of Edinburgh. Unresolved and italicized. Bellhouse is the archival historian and is the better bet, but he does not quote the roll entry, so the year is flagged. | Thomas Bayes | Edinburgh | S084, S094 |
| 1730 | Triangulum Arithmeticum PASCALIANUM in Miscellanea analytica | Abraham de Moivre | England | S011 |
| 1730s | Maclaurin's work on exterminating unknown quantities | Colin Maclaurin | Edinburgh | S122 |
| 1733 to 1734 | Bayes becomes minister at Tunbridge Wells | Thomas Bayes | Tunbridge Wells | S084, S094 |
| 1733 | The Approximatio pamphlet printed for a few friends | Abraham de Moivre | London | S082 |
| 1735 | Birth of Alexandre-Theophile Vandermonde | Alexandre-Theophile Vandermonde | Paris | S430 |
| 1735 | E053 read to the Academy on 26 August, in a calendar neither catalog states | Leonhard Euler | St Petersburg | S421, S422 |
| 1736 | Euler writes to Marinoni about the Konigsberg bridges, letter OO1468, on 13 March | Leonhard Euler | St Petersburg to Vienna | S421, S422 |
| 1736 | The letter to Carl Ehler: this bears little relationship to mathematics | Leonhard Euler | St Petersburg to Danzig | S422 |
| 1738 | The second edition of the Doctrine of Chances defines independent and dependent events in English | Abraham de Moivre | London | S081, S082, S088, S302 |
| 1741 | E053 printed, Commentarii volume 8, pp. 128 to 140, in the volume dated 1736 | Leonhard Euler | St Petersburg | S420, S421 |
| 1742 | Bayes elected Fellow of the Royal Society on 4 November, O.S. | Thomas Bayes | London | S084 |
| 1748 Disputed | A Treatise of Algebra published two years after its author's death, with a chapter on exterminating unknown quantities Dispute: What the 1748 first edition of Maclaurin's Treatise of Algebra contained. Unresolved and italicized. The S132 note records the claim as contested and partly unverified, because the exterminating chapter was read in a later edition. CH-04 section 4.10 item 3 carries the same open question as the Maclaurin against Cramer priority. | Colin Maclaurin | Edinburgh | S122, S132 |
| 1750 | Cramer's Introduction a l'analyse des lignes courbes algebriques states the general n by n rule at printed pp. 656 to 659, with no proof | Gabriel Cramer | Geneva | S129, S130, S131 |
| 1751 | The product 1.2.3....m written as a capital M | Leonhard Euler | not stated in the source | S301 |
| 1754 | Death of Abraham de Moivre on 27 November | Abraham de Moivre | London | S082 |
| 1755 | Euler introduces the capital sigma for summation | Leonhard Euler | St Petersburg | S305 |
| 1759 | E309 on the knight's tour, published 1766; not opened in this book | Leonhard Euler | St Petersburg | S431 |
| 1760 to 1762 | The 234 letters to Princess Charlotte Ludovica Luisa of Anhalt-Dessau are written; the logic letters take about three weeks | Leonhard Euler | St Petersburg and Berlin | S051, S052 |
| 1761 | Death of Thomas Bayes on 7 April, suddenly, reported in two London newspapers of the week | Thomas Bayes | Tunbridge Wells | S084, S094 |
| 1763 | Bayes's Essay read to the Royal Society on 23 December | Richard Price and Thomas Bayes | London | S083, S085 |
| 1763 | Price's covering letter dated 10 November | Richard Price | Newington Green | S083 |
| 1764 | Bezout, Recherches sur le degre des equations resultantes | Etienne Bezout | France | S122, S129 |
| 1764 | The Essay printed in Philosophical Transactions volume 53, pp. 370 to 418 | Thomas Bayes | London | S085, S094 |
| 1768 Disputed | Lettres a une princesse d'Allemagne published in three volumes and becomes a bestseller Dispute: When Euler's Lettres a une princesse d'Allemagne was published. Unresolved and italicized. The start year is agreed; the end year is not, and the difference matters for who saw the logic letters when. | Leonhard Euler | not stated in the source | S051, S052, S058 |
| 1771 | Memoire sur l'elimination read to the Academy on 12 January | Alexandre-Theophile Vandermonde | Paris | S129, S430 |
| 1771 | Remarques sur des problemes de situation, on the knight's tour; Vandermonde elected to the Academie | Alexandre-Theophile Vandermonde | Paris | S430 |
| 1772 | The Academy volume carrying Vandermonde at pages 516 to 532 and Laplace's expansion by minors at pages 267 to 376 | Alexandre-Theophile Vandermonde and Pierre-Simon Laplace | Paris | S129 |
| 1773 | Lagrange's mechanics paper, in which a determinant is read as a volume | Joseph-Louis Lagrange | not stated in the source | S122 |
| 1774 | Laplace's formal statement of the classical definition of probability, on Gorroochurn's dating | Pierre-Simon Laplace | Paris | S072 |
| 1779 | E530 read on 8 March: can 36 officers be paraded in a 6 by 6 square? | Leonhard Euler | St Petersburg | S421 |
| 1782 | The Latin square conjecture, as Bose and Shrikhande date it | Leonhard Euler | St Petersburg | S442 |
| 1789 | Birth of Augustin Louis Cauchy | Augustin Louis Cauchy | Paris | S176 |
| 1795 | The first English translation of Euler's letters appears | Henry Hunter | London | S051 |
| 1796 | Birth of Irenee-Jules Bienayme on 28 August | Irenee-Jules Bienayme | Paris | S094 |
| 1796 | The Principles of Algebra prints set of numbers | William Frend | England | S055, S302 |
| 1799 | A small Greek pi used for n factorial in a theory of equations | Paolo Ruffini | not stated in the source | S301 |
Timeline
1800 to 1850: elimination, determinants, and a new word
65 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1801 | Disquisitiones Arithmeticae, article 153, coins determinans for bb minus ac, which is not the determinant of a matrix | Carl Friedrich Gauss | Leipzig | S133, S134, S302 |
| 1802 | Hunter's second English edition in two volumes, the copy read for this book; Letter CIII is Of Syllogisms, and their different Forms | Henry Hunter | London | S049 |
| 1803 to 1809 | The Pallas observations: six equations in six unknowns | Carl Friedrich Gauss | Gottingen | S122, S180 |
| 1806 | Birth of Augustus De Morgan on 27 June | Augustus De Morgan | Madurai, India | S063 |
| 1808 | The capital Gamma introduced for the factorial in the same year as Kramp's n! | Adrien-Marie Legendre | not stated in the source | S301 |
| 1808 | The symbol n factorial printed for the first time | Christian Kramp | Cologne | S301 |
| 1809 | Theoria motus, page 214: the method is eliminatio vulgaris, common elimination | Carl Friedrich Gauss | Gottingen | S120, S121, S134, S302 |
| 1809 | Birth of Hermann Gunther Grassmann on 15 April | Hermann Grassmann | Stettin (now Szczecin) | S172, S181 |
| 1810 or 1811 Disputed | Disquisitio de elementis ellipticis Palladis, the memoir carrying Gauss's elimination Dispute: The year of Gauss's Pallas memoir, 1810 or 1811. Unresolved and italicized. The row prints both years rather than choosing. | Carl Friedrich Gauss | Gottingen | S120, S134, S302 |
| 1812 | Cauchy's determinant memoir addressed to the Institut on 30 November, with Binet presenting on the same occasion; the modern sense of determinant | Augustin-Louis Cauchy and Jacques Binet | Paris | S122, S129, S134, S180, S302 |
| 1812 | First edition of Theorie analytique des probabilites; the only copy reached in this book is the 1820 third edition, registered as a pointer | Pierre-Simon Laplace | Paris | S087 |
| 1814 | The Essai philosophique sur les probabilites states the Sixth Principle, which is Bayes' rule with the law of total probability in the denominator | Pierre-Simon Laplace | Paris | S086, S302 |
| 1814 | Birth of James Joseph Sylvester | James Joseph Sylvester | London | S170, S176 |
| 1815 | The memoir printed, Journal de l'Ecole Polytechnique, cahier 17, tome X, pp. 29 to 112 | Augustin-Louis Cauchy | Paris | S129, S134, S302 |
| 1815 | Birth of George Boole | George Boole | Lincoln, England | S060 |
| 1816 | Durrande uses n! in Gergonne's Annales and complains that nobody else has | J. B. Durrande | Nismes (Nimes) | S301 |
| 1818 | Combinatorial analysis appears in English in Nicholson's Essays on the Combinatorial Analysis | Peter Nicholson | England | S011, S302 |
| 1821 | Birth of Pafnuty Chebyshev on 16 May, in a calendar the source does not label | Pafnuty Chebyshev | Okatovo | S094 |
| 1824 | Birth of Gustav Robert Kirchhoff on 12 March, in the town of the bridges | Gustav Kirchhoff | Konigsberg (now Kaliningrad) | S429, S445 |
| 1826 Disputed | Tableau for the coefficient array; the eigenvalues found; every real symmetric matrix diagonalized Dispute: Cauchy's eigenvalue work, 1826 or 1829. Unresolved and both rows italicized. These are probably two papers in one body of work, but that is an inference and the timeline does not print inferences as facts. | Augustin-Louis Cauchy | Paris | S122, S180 |
| 1827 | A rival factorial sign, a bar and a corner, proposed by a new graduate of St Catherine's College | Thomas Jarrett | Cambridge | S301 |
| 1827 | The two-line bracket for a binomial coefficient printed in Vorlesungen uber die hohere Mathematik, volume 1 | Andreas von Ettingshausen | Vienna | S301, S387 |
| 1829 | Guerry's courbes circulaires, the first known polar-area diagrams | Andre-Michel Guerry | France | S267 |
| 1829 Disputed | The work on symmetric determinants with which the word secular is associated Dispute: Cauchy's eigenvalue work, 1826 or 1829. Unresolved and both rows italicized. These are probably two papers in one body of work, but that is an inference and the timeline does not print inferences as facts. | Augustin-Louis Cauchy | Paris | S176, S302 |
| 1830 | Lubbock and Drinkwater-Bethune, On Probability, use Bayes' theorem | John Lubbock and John Drinkwater-Bethune | London | S088 |
| 1832 | Birth of Mary Everest on 11 March | Mary Everest Boole | Wickwar, Gloucestershire | S061 |
| 1835 | Poisson writes la loi de grands nombres in the Comptes Rendus | Simeon Denis Poisson | Paris | S088, S302 |
| 1835 | Birth of Eugenio Beltrami on 16 November | Eugenio Beltrami | Cremona | S170, S201 |
| 1837 | Poisson coins the phrase on p. 7 of the Recherches sur la probabilite des jugements | Simeon Denis Poisson | Paris | S080 |
| 1838 | An Essay on Probabilities separates direct from inverse probability, and names mathematical expectation | Augustus De Morgan | London | S088, S302 |
| 1838 | De Morgan names mathematical induction in the Penny Cyclopedia | Augustus De Morgan | London | S063 |
| 1840 | L'equation caracteristique named in the Memoire sur l'integration des equations lineaires | Augustin Louis Cauchy | Paris | S176, S302 |
| 1841 | The first English paper on determinants, and the two vertical bars | Arthur Cayley | Cambridge | S122, S304 |
| 1841 | Three memoirs in volume 22 of Crelle's Journal turn determinants into common property | Carl Gustav Jacob Jacobi | Berlin | S122, S138 |
| 1842 | A printed edition of Yang Hui's text is issued | not named | China | S014 |
| 1842 | Menabrea's French original in the Bibliotheque Universelle de Geneve, No. 82, in October | Luigi Menabrea | Geneva | S260 |
| 1843 and 1845 | Cajori dates Cayley's double vertical lines for a matrix to these two papers | Arthur Cayley | England | S301 |
| 1843 | Lovelace's translation with Notes, Scientific Memoirs volume 3 | Ada Lovelace | London | S259, S260 |
| 1843 | The phrase la regle de Bayes is coined | Antoine Augustin Cournot | France | S088, S302 |
| 1843 | The quaternion relations conceived on the towpath on 16 October and cut into the stone of Brougham Bridge | William Rowan Hamilton | Dublin | S128, S139, S175 |
| 1843 | The quaternions paper read to the Royal Irish Academy on 13 November | William Rowan Hamilton | Dublin | S175 |
| 1844 | Linear substitutions treated as an algebra, with a multiplication that does not commute | Gotthold Eisenstein | Berlin | S122, S137 |
| 1844 | De Morgan's letter of 21 January to Lady Byron about Lovelace, archive reference LB 339 | Augustus De Morgan and Lady Byron | London | S258 |
| 1844 | Die lineale Ausdehnungslehre published by Otto Wigand, and ignored | Hermann Grassmann | Leipzig | S171, S181 |
| 1845 | The circuit laws announced | Gustav Kirchhoff | not stated in the source | S429 |
| 1845 | Bienayme's branching-process paper | Irenee-Jules Bienayme | Paris | S092 |
| 1845 Disputed | Birth of Georg Cantor on 3 March, in a calendar neither source states Dispute: Whether Cantor's birth date is Old Style or New Style. Unresolved and italicized. Russia used the Julian calendar until 1918, and this book's sourcing rule requires the label, so the year is flagged until somebody supplies it. | Georg Cantor | St Petersburg | S041, S062 |
| 1846 | First known English use of the nouns vector and scalar, the mathematical sense credited to Hamilton | William Rowan Hamilton | England | S202, S306 |
| 1846 | Jarrett's factorial sign revived after nineteen years of nobody using it | Harvey Goodwin | not stated in the source | S301 |
| 1846 | Chebyshev's Crelle paper on Poisson's law | Pafnuty Chebyshev | St Petersburg | S080, S092 |
| 1846 | Grassmann's prize announced by the Jablonowski Society on 1 July; his was the only entry submitted | Hermann Grassmann | Leipzig | S181 |
| 1846 | Scalar and vector coined, Philosophical Magazine xxix, 26 to 31, article 18, in July | William Rowan Hamilton | London | S174 |
| 1846 | Tensor and versor coined, Philosophical Magazine xxix, 326 to 328, article 19, in October | William Rowan Hamilton | London | S174, S302 |
| 1847 | n factorial written !n!, an idea that went nowhere | Henry Warburton | Cambridge | S301 |
| 1847 | Kirchhoff graduates and moves to Berlin; the spanning tree paper is dated to this year but was not opened here | Gustav Kirchhoff | Konigsberg to Berlin | S429 |
| 1847 | The word Topologie introduced in Vorstudien zur Topologie | Johann Benedict Listing | Gottingen | S302 |
| 1847 | Formal Logic, or, The Calculus of Inference, Necessary and Probable published by Taylor and Walton | Augustus De Morgan | London | S047 |
| 1847 | The Mathematical Analysis of Logic: 1 for the Universe, 1 minus x for the complement, and x squared equals x | George Boole | Cambridge and London | S045 |
| 1847 | Work begins on Paradoxien des Unendlichen | Bernard Bolzano | Villa Liboch near Melnik, Bohemia | S044 |
| 1847 | Kummer's report on the prize essay: commendably good material expressed in a deficient form | Ernst Kummer | Berlin | S173, S181 |
| 1848 | The treatise is finished in the summer, at the age of 67, in the last year of its author's life | Bernard Bolzano | Villa Liboch near Melnik, Bohemia | S044 |
| 1849 | Birth of Alfred Bray Kempe on 6 July | Alfred Bray Kempe | Kensington | S424 |
| 1849 | Boole appointed the first Professor of Mathematics at Queen's College Cork, taking the post in November | George Boole | Cork | S060 |
| 1849 | The universe of a proposition, or of a name introduced as a technical term, Trans. Camb. Phil. Soc. VIII, p. 380 | Augustus De Morgan | Cambridge | S055, S302 |
| 1849 | Birth of Ferdinand Georg Frobenius on 26 October | Georg Frobenius | Berlin-Charlottenburg | S187 |
Timeline
1850 to 1880: logic becomes algebra, and algebra gets a matrix
75 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1850 | Sylvester coins minor, in the homaloidal law paper, in November | James Joseph Sylvester | London | S134, S135 |
| 1850 | The word matrix coined, Philosophical Magazine, pp. 363 to 370 | James Joseph Sylvester | London | S134, S135, S178, S180, S307 |
| 1850 | Boole meets Mary Everest | George Boole and Mary Everest | Cork | S060, S061 |
| 1850 Disputed | De Morgan states the laws as the contrary of an aggregate is the compound of the contraries of the aggregants, Trans. Camb. Phil. Soc. 9, pp. 79 to 127 Dispute: The year of De Morgan's laws paper, 1850 or 1856. Unresolved and italicized. Cambridge Philosophical Society volumes were assembled from parts read on different dates, so both dates are probably right about different things, but nobody here has seen the head of the paper. | Augustus De Morgan | Cambridge | S055, S069 |
| 1851 | The metaphor spelled out: determinants engendered as from the womb of a common parent | James Joseph Sylvester | London | S135 |
| 1851 | Paradoxien des Unendlichen published posthumously by C. H. Reclam sen., edited from the manuscript remains | Frantisek Prihonsky | Leipzig | S044 |
| 1852 | Guthrie's coloring question reaches De Morgan on 23 October, and he writes to Hamilton the same day | Francis Guthrie and Augustus De Morgan | London | S423, S427 |
| 1852 | Hamilton declines on 26 October: your quaternion of color | William Rowan Hamilton | Dublin | S423 |
| 1853 | Cayley publishes the first matrix inverse | Arthur Cayley | England | S122 |
| 1853 | Lectures on Quaternions, 737 pages, with the special case of the Cayley-Hamilton theorem at p. 566 | William Rowan Hamilton | Dublin | S134, S177 |
| 1853 | The inequality now usually called Chebyshev's is derived in the Comptes Rendus | Irenee-Jules Bienayme | Paris | S080, S092, S094 |
| 1853 | Lectures on Quaternions uses set and theory of sets for n-tuples | William Rowan Hamilton | Dublin | S055, S302 |
| 1853 | Sur les clefs algebrique, Comptes Rendus, which Grassmann said took his ideas | Augustin Louis Cauchy | Paris | S179 |
| 1854 | Birth of Percy Alexander MacMahon on 26 September | P. A. MacMahon | Sliema, Malta | S012, S023 |
| 1854 | Boole writes of a random distribution of stars | George Boole | Cork | S302 |
| 1854 | Death of the child at 40 Broad Street on 2 September, the probable index case | not named | London | S269 |
| 1854 | Snow addresses the Board of Guardians on 7 September | John Snow | London | S269 |
| 1854 | Snow first exhibits a cholera spot map on 4 December, 87 days after the handle came off | John Snow | London | S269 |
| 1854 | The Broad Street outbreak, 31 August to 9 September: more than 500 dead in ten days | not named | London | S269 |
| 1854 | The Broad Street pump handle is removed on 8 September | the Board of Guardians | London | S269 |
| 1854 | An Investigation of the Laws of Thought published | George Boole | Cork | S046 |
| 1854 | A committee is set up on the Grassmann and Cauchy priority question, and never reports | the Paris Academy of Sciences | Paris | S179 |
| 1854 | Sylvester writes linear combination | James Joseph Sylvester | London | S176, S302 |
| 1855 | On the Mode of Communication of Cholera, second edition, not opened in this book | John Snow | London | S267, S269 |
| 1855 | Marriage of George Boole and Mary Everest on 11 September | George Boole and Mary Everest Boole | Wickwar, Gloucestershire | S060, S061 |
| 1856 | Birth of A. A. Markov on 2 June O.S. (14 June N.S.) | Andrei Markov | Ryazan | S211, S220, S271 |
| 1856 | The scanned volume 9 of the Transactions of the Cambridge Philosophical Society carries an 1856 imprint; the volume is registered as a pointer and De Morgan's paper itself was not reached in the OCR stream | Cambridge Philosophical Society | Cambridge | S069 |
| 1857 | A Memoir on the Theory of Matrices received by the Royal Society on 10 December | Arthur Cayley | London | S136 |
| 1857 | On the Theory of the analytical Forms called Trees | Arthur Cayley | Cambridge | S302 |
| 1857 | Death of Augustin Louis Cauchy | Augustin Louis Cauchy | Paris | S176, S202 |
| 1858 | A Memoir on the Theory of Matrices read on 14 January and published in Phil. Trans. volume 148, pp. 17 to 37; it defines inverse and transposed | Arthur Cayley | London | S136, S178, S180, S302 |
| 1858 to 1868 | Weierstrass's work on spectral theory | Karl Weierstrass | Berlin | S137 |
| 1858 | Notes on Matters Affecting the Health, Efficiency and Hospital Administration of the British Army | Florence Nightingale | London | S268 |
| 1858 | Birth of Giuseppe Peano on 27 August | Giuseppe Peano | Cuneo | S064, S183 |
| 1859 | A Contribution to the Sanitary History of the British Army | Florence Nightingale | London | S268 |
| c. 1860 | Todhunter puts the English factorial sign in his textbooks and it finally spreads | Isaac Todhunter | Cambridge | S301 |
| 1860 | Birth of Alicia Boole on 8 June | Alicia Boole Stott | Cork | S266 |
| 1861 | Birth of Percy John Heawood | Percy John Heawood | Newport, Shropshire | S425 |
| 1862 | The rewritten Ausdehnungslehre, 300 copies printed at Grassmann's own expense | Hermann Grassmann | Stettin (now Szczecin) | S172, S177 |
| 1863 | Grassmann's linguistics paper introducing what became Grassmann's Law; the citation is unverified in this book | Hermann Grassmann | Stettin (now Szczecin) | S203 |
| 1864 | De Morgan co-founds the London Mathematical Society and becomes its first president | Augustus De Morgan | London | S063 |
| 1864 | Death of George Boole on 8 December, aged 49 | George Boole | Cork | S060 |
| 1864 | About 600 copies of the 1844 Ausdehnungslehre pulped as waste paper | the publisher | Germany | S177 |
| 1865 | Todhunter's History of the Mathematical Theory of Probability published by Macmillan | Isaac Todhunter | Cambridge and London | S070 |
| 1865 | Death of William Rowan Hamilton | William Rowan Hamilton | Dublin | S177 |
| 1867 | Objection lodged to the word matrix, with block proposed instead | Charles Dodgson (Lewis Carroll) | Oxford | S134 |
| 1867 | Des valeurs moyennes printed in Liouville, 2nd series 12, 177 to 184, with no mention of Bienayme | Pafnuty Chebyshev | Paris | S091, S080 |
| 1867 | Tait's Elementary Treatise of Quaternions, and Thomson and Tait's Treatise on Natural Philosophy | Tait and Thomson | Edinburgh and Glasgow | S177 |
| 1868 to 1963 | Life of W. E. B. Du Bois, per the Library of Congress authority record | W. E. B. Du Bois | United States | S282 |
| 1870 | Traite des substitutions et des equations algebriques: the Jordan canonical form | Camille Jordan | Paris | S122, S180 |
| 1870 to 1871 | The Franco-Prussian War, in which General Bourbaki served | General Charles Denis Bourbaki | France | S276 |
| 1871 | Birth of Ernst Zermelo on 27 July | Ernst Zermelo | Berlin | S065 |
| 1871 | Death of Augustus De Morgan on 18 March | Augustus De Morgan | London | S063 |
| 1872 | The H-theorem proved on the principle that every collision is balanced by its exact reverse | Ludwig Boltzmann | Vienna | S230 |
| 1872 | Dedekind publishes Stetigkeit und irrationale Zahlen; he and Cantor meet and begin corresponding | Georg Cantor and Richard Dedekind | Switzerland | S059, S062 |
| 1873 | Cantor writes to Dedekind on 2 December unsure whether the reals are countable, and on 7 December with the proof that they are not | Georg Cantor | Halle | S053 |
| 1873 | Maxwell's Treatise on Electricity and Magnetism | James Clerk Maxwell | Cambridge | S177 |
| 1873 | Sulle funzioni bilineari, Giornale di Matematiche 11: the singular value decomposition, first publication | Eugenio Beltrami | Italy | S170 |
| 1874 to 1896 | Markov's course at St Petersburg University, from entering as a student to election as ordinary academician, O.S. | Andrei Markov | St Petersburg | S219 |
| 1874 | Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Crelle 77, pp. 258 to 262: two proofs, and the word diagonal nowhere in it | Georg Cantor | Halle an der Saale | S040 |
| 1874 | Memoire sur les formes bilineaires, J. Math. Pures Appl. (2) 19, 35 to 54: the same decomposition, independently | Camille Jordan | Paris | S170 |
| 1875 | An eighth bridge built, joining B to C, which makes the walk possible | the townspeople | Konigsberg (now Kaliningrad) | S422 |
| 1876 | Note sur le triangle arithmetique de Pascal et sur la serie de Lame | Edouard Lucas | France | S011 |
| 1876 | Birth of Tatiana Afanassjewa on 19 November | Tatiana Ehrenfest-Afanassjewa | Kiev (now Kyiv) | S222 |
| 1877 Disputed | Je le vois, mais je ne le crois pas is written to Dedekind, about a correspondence between a line and p-dimensional space Dispute: The date and setting of Je le vois, mais je ne le crois pas. Unresolved and italicized. MacTutor gives no archive, no printed edition and no exact day. The popular claim that the line is about the diagonal argument is separately wrong. | Georg Cantor | Halle | S062, S053 |
| 1877 | Death of Hermann Grassmann on 26 September | Hermann Grassmann | Stettin (now Szczecin) | S181 |
| 1878 Disputed | The general proof that a matrix satisfies its own equation, and the definition of rank; Hawkins dates the 63-page Crelle paper to 1877 Dispute: Frobenius on bilinear forms, 1877 or 1878. Unresolved and italicized. The row is placed at 1878 and names Hawkins's 1877. | Ferdinand Georg Frobenius | Berlin | S122, S137, S180 |
| 1878 | Chemistry and Algebra, Nature 17, p. 284, on 7 February: the word graph | James Joseph Sylvester | not stated in the source | S302 |
| 1878 | The four color problem put to the London Mathematical Society on 13 June | Arthur Cayley | London | S423 |
| c. 1878 | Hinton shows Alicia Boole the wooden cubes, at eighteen | Charles Howard Hinton and Alicia Boole | England | S266 |
| 1878 | Death of Bienayme on 19 October | Irenee-Jules Bienayme | Paris | S094 |
| 1878 | Cantor conjectures the continuum hypothesis | Georg Cantor | Halle | S065 |
| 1878 | Ein Beitrag zur Mannigfaltigkeitslehre, Crelle 84, delayed by Kronecker | Georg Cantor | Halle and Berlin | S043, S053 |
| 1879 | A four color proof announced in Nature on 17 July | Alfred Bray Kempe | London | S423 |
| 1879 | Kempe's proof printed in the American Journal of Mathematics; Cayley, On the coloring of maps | Alfred Bray Kempe and Arthur Cayley | Baltimore and London | S423, S424 |
Timeline
1880 to 1910: foundations, cardinals, and the first chains
89 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1880s | The decade in which Cayley's 1858 memoir began to be noticed abroad | Arthur Cayley | England | S137 |
| 1880 | Birth of Sergei Bernstein on 5 March, in a calendar the source does not label | Sergei Bernstein | Odessa | S094 |
| 1880 | On the Diagrammatic and Mechanical Representation of Propositions and Reasonings, Philosophical Magazine X, pp. 1 to 18, appears in July | John Venn | London | S051, S055, S302 |
| 1881 | Doolittle's elimination scheme at the US Coast and Geodetic Survey | Myrick Doolittle | United States | S120 |
| 1881 | Symbolic Logic published by Macmillan; chapter V offers a new scheme for complicated propositions | John Venn | London | S048, S302 |
| 1881 | Gibbs privately prints the first half of Elements of Vector Analysis | J. Willard Gibbs | New Haven | S177 |
| 1882 | Birth of Emmy Noether on 23 March | Emmy Noether | Erlangen | S278 |
| 1882 | Cantor first uses the German word Element, Math. Annalen XX, p. 114 | Georg Cantor | Halle | S055, S302 |
| 1883 | Cantor's first hard push at the continuum hypothesis; the Grundlagen prints Menge and Mengenlehre, and the French ensemble appears in Acta Mathematica | Georg Cantor | Halle | S053, S302 |
| 1883 | Heaviside begins introducing vectorial methods | Oliver Heaviside | England | S177 |
| 1883 | Latent roots coined in On the equation to the secular inequalities in the planetary theory | James Joseph Sylvester | Baltimore | S176, S178, S302 |
| 1884 | Iyer's English Brhat Samhita, in which the translator works the perfume count and finds the error | N. Chidambaram Iyer | India | S010 |
| 1884 | Cantor's first serious breakdown, in May, lasting somewhat over a month | Georg Cantor | Halle | S053, S062 |
| 1885 | Birth of Hermann Weyl; Heaviside's first unified presentation of vector methods | Hermann Weyl and Oliver Heaviside | Germany and England | S170, S177 |
| 1886 | The plain English phrase Pascal's triangle appears in Chrystal's Algebra | George Chrystal | Scotland | S011 |
| 1887 | Death of Kirchhoff on 17 October | Gustav Kirchhoff | Berlin | S429, S445 |
| 1887 | After Lorentz objects, a weaker cyclic balance condition, in the same journal | Ludwig Boltzmann | Vienna | S230 |
| 1887 | Dedekind's preface to the first edition of Was sind und was sollen die Zahlen?, dated 5 October | Richard Dedekind | Brunswick | S043 |
| 1888 | The same method published in the Annales de la Societe Scientifique de Bruxelles 12, 251 to 281 | Clasen | Brussels | S134, S142, S144 |
| 1888 | The third edition of Jordan's Handbuch, with a foreword dated May, writes the elimination into a surveying manual | Wilhelm Jordan | Germany | S134, S142, S144 |
| 1888 | Birth of Sydney Chapman | Sydney Chapman | Eccles near Manchester | S238 |
| 1888 Disputed | Calcolo geometrico is said by Cajori to introduce the union and intersection glyphs Dispute: What Peano's cup and cap meant in 1888. Unresolved and italicized. records that no open copy exists in reach and that Kennedy's translation is lending-restricted. Do not print Peano invented the union and intersection symbols in 1888 as a bare fact. | Giuseppe Peano | Turin | S054, S064, S050 |
| 1888 | Was sind und was sollen die Zahlen? prints Definition 64 and credits both Cantor 1878 and Bolzano 1851 | Richard Dedekind | not stated in the source | S043, S059 |
| 1888 | Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann, chapter 9: the vector space axioms; the preface is dated February | Giuseppe Peano | Turin | S179, S182, S183 |
| 1889 | Arithmetices principia, nova methodo exposita explains on p. x that Signum epsilon significat est | Giuseppe Peano | Turin | S050, S054 |
| 1889 | Sylvester's three items on the singular value decomposition, Messenger of Mathematics 19 and Comptes Rendus 108 | James Joseph Sylvester | England and Paris | S170 |
| 1890 | Kempe's proof broken, and five colors proved to suffice, in the same paper | Percy John Heawood | Durham | S423, S424, S425 |
| 1890 | Peano appointed professor of calculus at Turin in December, in Genocchi's chair | Giuseppe Peano | Turin | S064, S183 |
| 1890 | Vorlesungen uber die Algebra der Logik, volume 1, replaces the greater than and less than signs with the inclusion symbols | Ernst Schroder | not stated in the source | S054 |
| 1890 to 1894 | The vector analysis debate: 8 journals, 12 scientists, 38 publications | Tait, Gibbs, Heaviside and others | Britain and the United States | S177 |
| 1894 | Death of Chebyshev on 8 December, in a calendar the source does not label | Pafnuty Chebyshev | St Petersburg | S094 |
| 1895 | Death of Arthur Cayley on 26 January | Arthur Cayley | not stated in the source | S143 |
| c. 1895 | First contact between Alicia Boole Stott and P. H. Schoute | Alicia Boole Stott and P. H. Schoute | Groningen and England | S266 |
| 1895 | Karl Pearson coins binomial distribution | Karl Pearson | London | S088 |
| 1895 | Beitrage zur Begrundung der transfiniten Mengenlehre part one, Math. Annalen 46, prints aleph-null at p. 492 | Georg Cantor | Halle | S042, S054, S302 |
| 1895 | Cantor's letter of 30 April explains that the other alphabets were already over-used, which is why he chose the Hebrew aleph | Georg Cantor | Halle | S054 |
| 1896 | Frobenius learns of Cayley's memoir, adopts the word matrix, and hands back the credit | Ferdinand Georg Frobenius | Berlin | S122, S180 |
| 1896 | The same error in Kempe's proof found independently | Charles de la Vallee Poussin | Louvain | S423 |
| 1897 | Beitrage part two, Math. Annalen 49 | Georg Cantor | Halle | S041 |
| 1897 to 1902 Disputed | Zermelo's independent discovery of the paradox, as the Stanford Encyclopedia reports it Dispute: Whether Zermelo found the paradox before Russell. Unresolved and italicized. The span itself is only as good as the one source that gives it. | Ernst Zermelo | Gottingen | S056, S065 |
| 1897 | Death of James Joseph Sylvester | James Joseph Sylvester | England | S170 |
| 1899 | Markov calls a Nekrasov report an unsubstantiated declaration, O.S. | Andrei Markov | St Petersburg | S218 |
| 1899 | Cantor writes to Dedekind about the inconsistent multiplicity W | Georg Cantor | Halle | S053 |
| 1900 | Birth of Mary Cartwright on 17 December | Mary Cartwright | Aynho, Northamptonshire | S280 |
| 1900 to 1902 | Noether may study at Erlangen only unofficially, one of two women | Emmy Noether | Erlangen | S278 |
| 1900 | On certain series of sections of the regular four-dimensional hypersolids | Alicia Boole Stott | Groningen | S266 |
| 1900 | The Exposition Universelle, where the Du Bois data portraits were shown | W. E. B. Du Bois | Paris | S281, S282, S380, S381, S382, S383 |
| 1900 | Hilbert's problem list is presented and Russell meets Peano's notation at the International Congress | David Hilbert, Bertrand Russell and Giuseppe Peano | Paris | S064, S065 |
| 1900 | Death of Eugenio Beltrami on 18 February | Eugenio Beltrami | Rome | S201, S170, S204 |
| 1901 | Wilson uses bold type for vectors | Edwin Bidwell Wilson | New Haven | S304 |
| 1901 | Birth of Raj Chandra Bose on 19 June | R. C. Bose | Hoshangabad | S443 |
| 1901 | Detailed balance formulated for chemical kinetics | Rudolf Wegscheider | Vienna | S230, S231 |
| 1901 | Russell discovers the paradox in the spring; he later recalled June, spring, and May | Bertrand Russell | England | S056 |
| 1901 | Kelvin's thirty-eight years' war over quaternions remark | Lord Kelvin | Glasgow | S177 |
| 1902 | Afanassjewa arrives in Gottingen and meets Paul Ehrenfest | Tatiana Afanassjewa and Paul Ehrenfest | Gottingen | S222 |
| 1902 | Gorky's election to the Academy annulled on the tsar's orders, O.S. | Nicholas II and Maxim Gorky | St Petersburg | S210, S219 |
| 1902 | Philosophy and logic of the science of mass expressions of human activities, Matematicheskii Sbornik 23, 463 to 600, O.S. | Pavel Nekrasov | Moscow | S218, S271 |
| 1902 | Russell's letter of 16 June states that there is no class, as a whole, of those classes which, as wholes, are not members of themselves | Bertrand Russell and Gottlob Frege | Cambridge to Jena | S056 |
| 1903 | Birth of Frank Plumpton Ramsey on 22 February | Frank Plumpton Ramsey | Cambridge | S432 |
| 1903 | Birth of A. N. Kolmogorov on 25 April N.S. | A. N. Kolmogorov | Tambov | S237 |
| 1903 | Birth of Lars Onsager | Lars Onsager | Norway | S245 |
| 1903 | Markov refuses the honors awarded him, O.S. | Andrei Markov | St Petersburg | S219 |
| 1903 | The Principles of Mathematics carries the first English element and class in the modern senses | Bertrand Russell | Cambridge | S055, S302 |
| 1904 | The search for unavoidable sets in the four color problem begins | Paul Wernicke | not stated in the source | S423 |
| 1904 | German rules change on 24 October: women may matriculate on equal terms | not named | Germany | S278 |
| 1904 | Zermelo's well-ordering proof introduces the axiom of choice | Ernst Zermelo | Gottingen | S057, S065 |
| 1904 | Eigenwert and Eigenfunktion in the first Mitteilung, Nachrichten pp. 49 to 91 | David Hilbert | Gottingen | S176, S184, S302 |
| 1905 | The Problem of the Random Walk, Nature, 27 July | Karl Pearson | London | S302 |
| 1906 | Extension of the law of large numbers to dependent quantities, Izv. Fiz.-Matem. Obsch. Kazan Univ., 2nd ser., 15(4), 135 to 156, O.S. | Andrei Markov | Kazan | S211, S212, S302 |
| 1906 | Klein commissions the statistical mechanics encyclopedia article from both Ehrenfests | Felix Klein, Paul and Tatiana Ehrenfest | Gottingen | S223 |
| 1906 | Birth of Andre Weil on 6 May | Andre Weil | Paris | S277 |
| 1906 | Birth of Mary Celine Fasenmyer on 4 October | Mary Celine Fasenmyer | Crown, Pennsylvania | S253, S254 |
| 1906 to 1998 | Life of Andre Weil | Andre Weil | not stated in the source | S068 |
| 1906 | Birth of Olga Taussky on 30 August | Olga Taussky-Todd | Olmutz (now Olomouc) | S192 |
| 1906 | Birth of Wassily Leontief on 5 August, in a calendar this book has not verified | Wassily Leontief | St Petersburg | S199 |
| 1907 | Introduction to Higher Algebra prints the Hamilton-Cayley equation and singular matrix | Maxime Bocher | United States | S122, S134, S302 |
| 1907 | Investigation of a notable instance of dependent trials, Izv. Akad. Nauk 6th ser. 1(3), 61 to 80, O.S. | Andrei Markov | St Petersburg | S211 |
| 1907 | The Second Duma dissolved in June O.S.; Markov repudiates his membership of the electorate | Nicholas II and Andrei Markov | St Petersburg | S212, S219 |
| 1907 | The urn paper in Physikalische Zeitschrift 8, 311 to 314, issue of 6 May | Paul and Tatiana Ehrenfest | Leipzig | S221, S224 |
| 1907 | Mathematische Annalen 63, 433 to 476: the best rank-k approximation theorem | Erhard Schmidt | Germany | S170 |
| 1907 | Perron's theorem, proved in section 14 of Grundlagen fur eine Theorie des Jacobischen Kettenbruchalgorithmus, Math. Ann. 64, 11 to 76 | Oskar Perron | Germany | S185 |
| 1908 | Markov's chain limit theorems, Zapiski Akad. Nauk 7th ser. 22(9), O.S. | Andrei Markov | St Petersburg | S211 |
| 1908 | Hardy signs his letter on the equilibrium of genotype ratios on 5 April | G. H. Hardy | Trinity College, Cambridge | S270, S388 |
| 1908 | Hardy's letter appears in Science, N.S. 28(706), 49 to 50, on 10 July | G. H. Hardy | United States | S270, S388 |
| 1908 | Untersuchungen uber die Grundlagen der Mengenlehre I, Math. Annalen 65: seven axioms, and braces for sets at p. 263 | Ernst Zermelo | not stated in the source | S054, S057 |
| 1908 and 1909 | Uber Matrizen aus positiven Elementen, Sitzungsberichte Berlin | Georg Frobenius | Berlin | S185 |
| 1909 | Birth of Florence Nightingale David on 23 August | F. N. David | Ivington, England | S265 |
| 1909 | The strong law of large numbers, and countable additivity in probability | Emile Borel | Paris | S080, S089 |
| 1909 | Keyser prints disjoint, Science 30, 31 December, p. 956 | Cassius Jackson Keyser | not stated in the source | S055, S302 |
Timeline
1910 to 1935: axioms, spaces, and a settled probability
93 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1910 | Investigation of the general case of trials associated into a chain, Zapiski 8th ser. 25(3), O.S. | Andrei Markov | St Petersburg | S211 |
| 1910 | The newspaper item The Holy Synod and Tolstoy, Rech', p. 3, 8 November O.S. | not named | St Petersburg | S217 |
| 1910 | Principia Mathematica prints universal class and the symbol V | Alfred North Whitehead and Bertrand Russell | Cambridge | S055, S302 |
| 1910 | The Gibbs and Heaviside system has won | the physics profession | worldwide | S177 |
| 1911 | Birth of George Szekeres on 29 May | George Szekeres | Budapest | S434 |
| 1911 to 1913 | The Beilis affair, O.S. | Menahem Mendel Beilis | Kiev (now Kyiv) | S217 |
| 1912 | Kempe knighted, and made legal adviser to the Diocese of London | Alfred Bray Kempe | London | S423, S424 |
| 1912 | Lorentz invites Paul Ehrenfest to Leiden | Hendrik Lorentz and Paul Ehrenfest | Leiden | S222, S223 |
| 1912 | Markov asks the Most Holy Synod to excommunicate him in February O.S.; the Synod refuses | Andrei Markov | St Petersburg | S217 |
| 1912 | Markov publishes A rebuke to P. A. Nekrasov, O.S. | Andrei Markov | St Petersburg | S218 |
| 1912 | Union appears in English, Lectures on the Theory of Functions of Real Variables, volume 2, p. 22 | James Pierpont | not stated in the source | S055, S302 |
| 1912 | Uber Matrizen aus nicht negativen Elementen, Sitzungsberichte Berlin, 456 to 477: Perron's theorem extended | Georg Frobenius | Berlin | S185, S186 |
| 1912 | Weyl, Math. Annalen 71, 441 to 479: perturbation theory for singular values | Hermann Weyl | Germany | S170 |
| 1913 | Markov answers the Romanov tercentenary with a bicentenary of the law of large numbers, O.S. | Andrei Markov | St Petersburg | S219 |
| 1913 | The Eugene Onegin lecture to the physical-mathematical faculty on 23 January O.S. (5 February N.S.) | Andrei Markov | St Petersburg | S213, S214, S271 |
| 1913 | Autonne extends the singular value decomposition to complex matrices | Leon Autonne | France | S170 |
| 1914 | An honorary doctorate awarded to Alicia Boole Stott, and received in absentia | Alicia Boole Stott | Groningen | S266 |
| 1914 | Complement of a set appears in Trans. AMS 15, and Hausdorff writes die Summe | E. W. Chittenden and Felix Hausdorff | not stated in the source | S055, S302 |
| 1915 | Combinatory Analysis volume 1 gives the subject a name and a textbook | P. A. MacMahon | Cambridge | S007 |
| 1915 | Birth of Wolfgang Doeblin on 17 March | Wolfgang Doeblin | Berlin | S232, S233, S234 |
| 1915 | Polya reads Markov's French edition | George Polya | Zurich | S215 |
| 1915 | Hilbert and Klein invite Noether to Gottingen | David Hilbert, Felix Klein and Emmy Noether | Gottingen | S278 |
| 1915 | Frechet takes measure beyond Euclidean space | Maurice Frechet | France | S089 |
| 1916 | Combinatory Analysis volume 2 | P. A. MacMahon | Cambridge | S012 |
| 1916 | Detailed balance applied to the emission and absorption of radiation | Albert Einstein | Berlin | S230, S231 |
| 1916 | Nekrasov's newspaper attack calling Markov a panphysicist, O.S. | Pavel Nekrasov | Moscow | S218 |
| 1916 to 1917 | Noether lectures at Gottingen under Hilbert's name | Emmy Noether | Gottingen | S278 |
| 1916 | Death of Mary Everest Boole on 17 May | Mary Everest Boole | Notting Hill, London | S061 |
| 1917 | Markov asks to be sent to the interior in September O.S. and teaches school mathematics unpaid | Andrei Markov | Zaraisk | S219 |
| 1917 | Bernstein's qualitative axioms for probability | Sergei Bernstein | Russia | S089, S094 |
| 1917 | Bortkiewicz prints Stochastik in Die Iterationen, p. 3 | Ladislaus von Bortkiewicz | Germany | S055, S302 |
| 1917 | Death of Georg Frobenius on 3 August | Georg Frobenius | Berlin | S187 |
| 1918 | Birth of N. G. de Bruijn on 9 July | N. G. de Bruijn | The Hague | S444 |
| 1918 | Death of Georg Cantor on 6 January | Georg Cantor | Halle | S062 |
| 1918 | The dimensional axiom added to Peano's list | Hermann Weyl | Germany | S173 |
| 1919 | Birth of David Blackwell on 24 April | David Blackwell | Centralia, Illinois | S250 |
| 1919 | Birth of Julia Bowman on 8 December | Julia Bowman Robinson | St Louis, Missouri | S279 |
| 1919 | Noether's habilitation granted, and she is appointed Privatdozent | Emmy Noether | Gottingen | S278 |
| 1919 | The year covered by Leontief's 44-sector input-output table | Wassily Leontief | United States | S283 |
| 1919 | Daniell's integral appears, and von Mises's collective, in the Grundlagen der Wahrscheinlichkeitsrechnung with the Merkmalraum | Percy Daniell and Richard von Mises | Berlin | S089, S055, S302 |
| 1919 | Empty set appears in English without explanation, Amer. J. Math. 41, p. 237 | J. E. McAtee | not stated in the source | S055, S302 |
| 1919 | Birth of James Hardy Wilkinson on 27 September | James Hardy Wilkinson | Strood, Kent | S195 |
| 1920 | Birth of Vera Nikolaevna Kublanovskaya on 21 November | Vera Kublanovskaya | Krokino or Krokhono, Vologda Oblast | S193, S194 |
| 1920 | The fully axiomatic approach set out in a doctoral dissertation | Stefan Banach | Lwow (now Lviv) | S179 |
| 1921 | Birth of Alfred Renyi on 30 March | Alfred Renyi | Budapest | S438 |
| 1921 | Markov writes on 5 March that he cannot attend Academy meetings for lack of footwear | Andrei Markov | Petrograd (now St Petersburg) | S212 |
| 1921 | Fisher's technical use of likelihood, Metron 1, 3 to 32 | R. A. Fisher | England | S088, S302 |
| 1922 | Death of Alfred Bray Kempe on 21 April | Alfred Bray Kempe | London | S424 |
| 1922 | Maps with at most 25 regions proved four colorable | Philip Franklin | not stated in the source | S423 |
| 1922 | Death of A. A. Markov on 20 July, of sepsis | Andrei Markov | Petrograd (now St Petersburg) | S211, S212, S220, S271 |
| 1922 | Death of Camille Jordan | Camille Jordan | Paris | S134, S144 |
| 1922 | Fundamenta Mathematicae 3: sets of elements of which I will postulate certain properties | Stefan Banach | Poland | S173 |
| c. 1923 | The Husson prank lecture at which Bourbaki's theorem was announced | Raoul Husson | Paris | S275 |
| 1923 | Steinhaus axiomatizes the binary case; Antoni Lomnicki proposes a density definition | Hugo Steinhaus and Antoni Lomnicki | Lwow | S089 |
| 1924 | Death of P. A. Nekrasov | Pavel Nekrasov | Moscow | S210, S218 |
| 1924 | Eigenvektor printed in Methoden der Mathematischen Physik | Courant and Hilbert | Germany | S176, S302 |
| 1926 Disputed | Claimed first printing of the phrase Markov chain, Math. Annalen 97, 1 to 59 Dispute: Who first printed the phrase Markov chain. Unresolved and italicized,. Dictionary.com's 1940 to 1945 is contradicted by Miller's own page citation and is discarded. Do not print a bare the phrase was first used in 1926. | Sergei Bernstein | Berlin | S212, S216, S244, S393 |
| 1926 | Set theory appears in English in the Annals of Mathematics | Orrin Frink | not stated in the source | S055, S302 |
| 1927 | Birth of Arianna Wright on 15 September | Arianna Wright Rosenbluth | Houston, Texas | S257 |
| 1927 | Slutsky's general abstract sketch | Evgeny Slutsky | Moscow | S089 |
| 1927 | The earliest English eigenvalue, in a letter to Nature dated 23 July | Arthur Stanley Eddington | England | S176, S202 |
| 1928 | On a problem of formal logic read to the London Mathematical Society on 13 December | Frank Plumpton Ramsey | London | S432 |
| 1928 | The International Congress of Mathematicians at which Markov's theory crossed to the West | many | Bologna | S215 |
| 1929 | Death of MacMahon on 25 December | P. A. MacMahon | Bognor Regis | S012 |
| 1929 | Hill's cipher paper printed in the June to July American Mathematical Monthly 36(6), pp. 306 to 312 | Lester S. Hill | New York | S140 |
| 1929 | The Cayley-Hamilton theorem printed under that name for the first time | Herbert Westren Turnbull | St Andrews | S134 |
| 1929 | The Message Protector patent filed by Weisner and Hill on 14 February | Louis Weisner and Lester S. Hill | New York | S141 |
| 1929 | Chain-Links, in the collection Everything is Different: five intermediaries | Frigyes Karinthy | Budapest | S439 |
| 1929 | Sur les chaines de Markoff, C. R. Acad. Sci. URSS, A, no. 9, 203 to 208 | Vsevolod Romanovsky | Moscow | S216, S302 |
| 1929 | Kolmogorov's general abstract sketch | A. N. Kolmogorov | Moscow | S089 |
| 1930 | Death of Frank Ramsey on 19 January, aged 26, after surgery for jaundice | Frank Plumpton Ramsey | London | S432 |
| 1930 | Ramsey's paper published in the Proceedings of the London Mathematical Society | Frank Plumpton Ramsey | London | S432 |
| 1931 | The reciprocal relations papers, tied by their author to detailed balancing | Lars Onsager | New Haven | S231, S230 |
| 1931 | Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Annalen 104, 415 to 458: the Chapman-Kolmogorov equation | A. N. Kolmogorov | Berlin | S216, S236, S246 |
| c. 1931 | F. N. David becomes Karl Pearson's research student | F. N. David and Karl Pearson | London | S265 |
| 1931 | The vertical stroke written as P(p given q) in Scientific Inference | Harold Jeffreys | Cambridge | S088, S303 |
| 1931 to 1932 | Taussky at Gottingen editing Hilbert's collected works | Olga Taussky-Todd | Gottingen | S192 |
| 1931 | Uber die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR, 749 to 754: the disk theorem | Semyon Gersgorin | USSR | S191 |
| 1932 | The Message Protector patent granted on 16 February, US 1,845,947 | Louis Weisner and Lester S. Hill | United States | S141 |
| 1932 | Doeblin expelled from the School of Political Science | Wolfgang Doeblin | Berlin | S232 |
| 1932 | Death of Giuseppe Peano on 20 April | Giuseppe Peano | Turin | S064 |
| 1933 | MacDuffee uses a superscript T for the transpose and I for the identity | Cyrus Colton MacDuffee | United States | S304 |
| 1933 | Alfred Doblin flees Berlin at the Reichstag fire | Alfred Doblin | Berlin | S234 |
| 1933 | Paul Ehrenfest kills his son Vassily and himself on 25 September | Paul Ehrenfest | Leiden | S222 |
| 1933 | Noether dismissed from Gottingen by the Nazis in April, without pension | Emmy Noether | Gottingen | S278 |
| 1933 | Noether sails for Bryn Mawr College in October | Emmy Noether | Gottingen to Bryn Mawr | S278 |
| 1933 | Grundbegriffe der Wahrscheinlichkeitsrechnung, 62 pages, defines elementary events as elements of a set and random events as its subsets, and prints the symbol P(A) | A. N. Kolmogorov | Moscow | S066, S089, S090, S088, S303 |
| 1933 | Neyman and Pearson coin sample space, Phil. Trans. A 231, 289 to 337 | Jerzy Neyman and Egon Pearson | London | S055 |
| 1934 | Wedderburn uses I and O for the identity and zero matrices | Joseph Wedderburn | United States | S304 |
| 1934 to 1939 | Doeblin studies mathematics in Paris | Wolfgang Doeblin | Paris | S232 |
| 1934 | Khintchine, Korrelationstheorie stationarer stochastischer Prozesse, Math. Ann. 109, 604 to 615, puts Markov process into German; Doob puts stochastic process into English | Aleksandr Khintchine and Joseph Doob | Germany and United States | S216, S302 |
| late 1934 | The collaborators adopt the pseudonym Bourbaki | the Bourbaki group | Paris | S275 |
| 1934 | The first Bourbaki meeting, Monday 10 December: six mathematicians invent an author | Cartan, Chevalley, Delsarte, Dieudonne, de Possel, Weil | Cafe Capoulade, Boulevard Saint-Michel, Paris | S276 |
Timeline
1935 to 1960: war, computers, and the machine age
70 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1935 | A combinatorial problem in geometry, Compositio Math. 2, 463 to 470, answering Esther Klein's question and naming her in the paper | Paul Erdos and George Szekeres | Budapest | S433 |
| 1935 | Death of Emmy Noether on 14 April | Emmy Noether | Bryn Mawr, Pennsylvania | S278 |
| 1936 | Theorie der endlichen und unendlichen Graphen, the first systematic study | Denes Konig | Leipzig | S302 |
| 1936 | Frechet's Traite, the standard matrix treatment of Markov chains | Maurice Frechet | Paris | S221 |
| 1936 | Zur Theorie der Markoffschen Ketten, Math. Ann. 112, 155 to 160, on the EuDML catalog record; the scan itself could not be opened in this book | A. N. Kolmogorov | Berlin | S246 |
| 1936 | The stained-glass Archimedean lampshade given to Coxeter as he leaves for Toronto | Alicia Boole Stott | England | S266 |
| 1936 | Psychometrika 1, 211 to 218, later named after its authors by mistake | Carl Eckart and Gale Young | United States | S170, S176 |
| 1937 | George Szekeres marries Esther Klein on 13 June | George Szekeres and Esther Klein | Budapest | S434 |
| 1937 | Frechet's address praising Kolmogorov; Cramer adopts P(A) in English | Maurice Frechet and Harald Cramer | Geneva | S089, S088, S303 |
| 1938 | Doeblin's doctorate under Frechet, and the start of his military service | Wolfgang Doeblin | Paris | S233, S234 |
| 1938 | Doob, Trans. AMS 44, p. 102, puts Markov process into English | Joseph Doob | United States | S216 |
| 1938 | The phrase Markov chain first attested in English, Amer. Math. Monthly 45, p. 410 | unnamed | United States | S216, S244, S302 |
| 1938, 1939 and 1941 | Blackwell's AB, AM and PhD | David Blackwell | University of Illinois | S250, S252 |
| 1939 | Szekeres leaves for a job as a leather chemist, and stays until 1948 | George Szekeres | Budapest to Shanghai | S434 |
| 1939 | Weil arrested in November with Bourbaki calling cards in his pocket, and released on 12 December | Andre Weil | Finland | S277 |
| 1939 | Elements de mathematique, p. 4, prints the slashed-O glyph in the phrase la partie vide de E | Nicolas Bourbaki | Paris | S054 |
| 1940 | Combinatorics used as a noun by F. W. Levi | F. W. Levi | not stated in the source | S011 |
| 1940 | Doeblin burns his papers and kills himself on 21 June, aged 25 | Wolfgang Doeblin | Housseras, Vosges | S232, S233, S234 |
| 1940 | Sealed envelope 11668 deposited at the Academie des Sciences on 26 February | Wolfgang Doeblin | Paris | S232, S234 |
| 1940 | Death of Alicia Boole Stott on 17 December | Alicia Boole Stott | Highgate, Middlesex | S266 |
| 1941 to 1942 | Blackwell's Rosenwald Fellowship at the Institute for Advanced Study | David Blackwell | Princeton | S250 |
| c. 1941 to 1942 | Estimated date of Turing's HW 25/37 manuscript | Alan Turing | Bletchley Park | S261 |
| 1942 | Blackwell applies to all 105 Black colleges in the country | David Blackwell | United States | S252 |
| 1943 to 1946 | Aerodynamic flutter at the National Physical Laboratory, and a 6 by 6 matrix | Olga Taussky-Todd | Teddington | S191, S192 |
| 1944 | The Bureau of Labor Statistics calculates its first comprehensive employment forecast for the War Production Board | the Bureau of Labor Statistics | United States | S283 |
| 1945 | The phrase De Morgan's Laws appears in an index to the Journal of Symbolic Logic | not named | not stated in the source | S055 |
| 1945 to 1948 | Turing at the National Physical Laboratory, working on the Pilot ACE | Alan Turing | Teddington | S189 |
| 1946 | A combinatorial problem: de Bruijn sequences | N. G. de Bruijn | Amsterdam | S444 |
| 1946 | Ulam's solitaire problem, the origin of the Monte Carlo method | Stanislaw Ulam | Los Alamos | S216, S228, S302 |
| 1946 Disputed | Fasenmyer's PhD conferred in June Dispute: The year of Fasenmyer's doctorate. Unresolved and italicized. The likely reconciliation is a 1945 dissertation with a June 1946 degree, but that is inference, so the year stays flagged. | Mary Celine Fasenmyer | University of Michigan | S254, S255 |
| 1946 | Wilkinson becomes Turing's assistant on the ACE project in May | James Hardy Wilkinson | Teddington | S195 |
| 1947 | Some remarks on the theory of graphs, Bull. AMS 53, 292 to 294, three pages, no coloring exhibited | Paul Erdos | Syracuse University | S435 |
| 1947 | Kac puts a number on the reversibility paradox, Amer. Math. Monthly 54, 369 to 391 | Mark Kac | Ithaca | S221 |
| 1947 | Cartwright is the first woman mathematician elected to the Royal Society | Mary Cartwright | London | S280 |
| 1947 | Fasenmyer, Some generalized hypergeometric polynomials, Bull. AMS 53 | Mary Celine Fasenmyer | United States | S253, S254 |
| 1947 | Numerical inverting of matrices of high order presented on 5 September, received by the Bulletin editors on 1 October, and published in November, BAMS 53(11), 1021 to 1099: the LU factorization introduced | John von Neumann and Herman Goldstine | United States | S188, S189, S120 |
| 1947 | Taussky and John Todd move to the National Bureau of Standards | Olga Taussky-Todd and John Todd | United States | S191, S192 |
| 1948 | A Mathematical Theory of Communication, Bell System Technical Journal 27, 379 to 423 and 623 to 656, with English generated from a Markoff process by hand | Claude Shannon | Murray Hill | S225 |
| 1948 | Input-output work folded into Project SCOOP, funded by the US Air Force | the Bureau of Labor Statistics | United States | S283 |
| 1948 | Kublanovskaya takes her Leningrad State University degree; Turing leaves the National Physical Laboratory | Vera Kublanovskaya and Alan Turing | Leningrad (now St Petersburg) and Teddington | S189, S194, S195 |
| 1948 | Rounding-off errors in matrix processes, QJMAM 1, 287 to 308, names LU and the condition number; received 4 November 1947 | Alan Turing | Teddington | S189, S390, S120 |
| 1949 | Metropolis and Ulam, The Monte Carlo Method, JASA 44, 335 to 341 | Nicholas Metropolis and Stanislaw Ulam | Los Alamos | S216, S302 |
| 1949 | Arianna Wright's PhD in physics under J. H. Van Vleck | Arianna Wright Rosenbluth | Harvard | S257 |
| 1949 | Bourbaki's application for individual AMS membership refused: the AMS Secretary draws a two-column table showing the biographical details contradict each other | John Kline and the Bourbaki group | United States | S275 |
| 1950 | Renyi becomes Director of the new Institute of Applied Mathematics | Alfred Renyi | Budapest | S438 |
| 1950 | Feller makes Pr{A given B} standard and uses sample space abstractly, crediting von Mises; Morrison's English Grundbegriffe appears from Chelsea | William Feller and Nathan Morrison | Princeton and New York | S088, S090, S055, S066, S303 |
| c. 1950 | The word Bayesian enters circulation; Fisher uses it | R. A. Fisher | not stated in the source | S088 |
| 1950 | The Pilot ACE becomes operational in May | the National Physical Laboratory team | Teddington | S195 |
| 1951 | Taussky organizes the first matrix theory conference at the National Bureau of Standards | Olga Taussky-Todd | United States | S191 |
| 1952 | The ILLIAC completed at the University of Illinois, the first computer owned entirely by an educational institution; the first hydrogen bomb | the University of Illinois | Urbana | S227, S241, S272 |
| 1952 | Conjugate gradients, Journal of Research of the NBS 49(6), 409 to 436, in December | Magnus Hestenes and Eduard Stiefel | United States | S190 |
| 1953 | The English phrase Gaussian elimination appears in print | George Forsythe | United States | S120 |
| 1953 | Equation of State Calculations by Fast Computing Machines received on 6 March and published in June, J. Chem. Phys. 21(6), 1087 to 1092 | Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller | Los Alamos | S226, S228, S256, S257 |
| 1953 | Death of Ernst Zermelo on 21 May | Ernst Zermelo | Freiburg im Breisgau | S065 |
| 1954 | Emeliakh's archival study of the Markov excommunication case | L. I. Emeliakh | USSR | S217 |
| 1954 | Blackwell and Girshick, Theory of Games and Statistical Decisions; the ICM at Amsterdam; the Berkeley appointment | David Blackwell | Amsterdam and Berkeley | S250 |
| 1955 | Death of Percy John Heawood | Percy John Heawood | Durham | S425 |
| 1955 | Kublanovskaya's candidate's degree; Taussky's matrix theory course at New York University | Vera Kublanovskaya and Olga Taussky-Todd | Leningrad (now St Petersburg) and New York | S191, S194 |
| 1956 | The complete four-movement ILLIAC Suite, in November | Lejaren Hiller and Leonard Isaacson | Urbana | S241, S272 |
| 1956 | The first three movements of the ILLIAC Suite premiered in the Illini Union on 9 August | Lejaren Hiller and Leonard Isaacson | Urbana | S241, S242, S273 |
| 1958 | On random graphs I received on 19 November | Paul Erdos and Alfred Renyi | Budapest | S436 |
| 1958 to 1959 | Xenakis composes Analogique A and Analogique B | Iannis Xenakis | Paris | S274 |
| 1958 | Finite Dimensional Vector Spaces, p. 102, lists the rival adjectives for eigenvalue | Paul Halmos | United States | S176, S302 |
| 1958 | Rutishauser's LR algorithm paper | Heinz Rutishauser | Zurich | S193 |
| 1959 | Front page of the Sunday New York Times on 26 April: Euler's spoilers | Bose, Shrikhande and Parker | New York | S443 |
| 1959 | On random graphs I published, Publ. Math. Debrecen 6 | Paul Erdos and Alfred Renyi | Debrecen | S436 |
| 1959 | On the evolution of random graphs received on 28 December | Paul Erdos and Alfred Renyi | Budapest | S437 |
| 1959 | The Latin squares paper received by the Transactions of the AMS on 10 April | R. C. Bose and S. S. Shrikhande | University of North Carolina | S442 |
| 1959 | Hiller and Isaacson, Experimental Music, McGraw-Hill | Lejaren Hiller and Leonard Isaacson | New York | S241, S242, S273 |
| 1959 | Francis submits QR Part 1 to The Computer Journal on 29 October, written in assembly on a Pegasus | John G. F. Francis | England | S193 |
Timeline
1960 to now: networks, search, and machine learning
59 entries.
| When | What happened | Who | Where | Sources |
|---|---|---|---|---|
| 1960 | On the evolution of random graphs published; the Bose and Shrikhande Transactions paper; BCH error-correcting codes | Erdos, Renyi, Bose, Ray-Chaudhuri and Hocquenghem | Budapest and the University of North Carolina | S437, S442, S443 |
| 1960 | Kublanovskaya submits her Doklady paper on 5 July, independently | Vera Kublanovskaya | Leningrad (now St Petersburg) | S193, S194 |
| 1961 | Kublanovskaya, Doklady Akad. Nauk SSSR 136, 26 to 28; Francis moves to Ferranti | Vera Kublanovskaya and John G. F. Francis | Leningrad (now St Petersburg) and England | S193 |
| 1962 | F. N. David, Games, Gods and Gambling | F. N. David | England | S265 |
| 1964 | Death of Tatiana Ehrenfest-Afanassjewa on 14 April | Tatiana Ehrenfest-Afanassjewa | Leiden | S222 |
| 1964 | Cartwright receives the Sylvester Medal, the first woman to do so | Mary Cartwright | London | S280 |
| 1965 | Scherr applies continuous-time Markov chains to time-sharing systems | Allan Scherr | Massachusetts | S243 |
| 1965 | Blackwell elected to the National Academy of Sciences, its first Black member | David Blackwell | United States | S250, S251, S252 |
| 1965 | Wilkinson, The Algebraic Eigenvalue Problem; Golub and Kahan make the singular value decomposition computable without forming the square | Wilkinson, Golub and Kahan | Teddington and the United States | S195, S200 |
| 1966 | Baum and Petrie on probabilistic functions of finite state Markov chains, Ann. Math. Statist. 37(6), 1554 to 1563 | Leonard Baum and Ted Petrie | Princeton | S239 |
| 1967 | Golub and Businger, Stanford Technical Report CS73; Halmos concedes that eigenvalues have won it | Golub, Businger and Halmos | Stanford | S176, S200, S302 |
| 1968 | Onsager's Nobel Prize in Chemistry | Lars Onsager | Stockholm | S245 |
| 1968 | Death of Sergei Bernstein on 26 October | Sergei Bernstein | Moscow | S094 |
| 1969 | The method of discharging introduced | Heinrich Heesch | Germany | S423 |
| 1970 | Death of Alfred Renyi on 1 February, aged 48 | Alfred Renyi | Budapest | S438 |
| 1970 | Death of Sydney Chapman | Sydney Chapman | Boulder, Colorado | S238 |
| 1970 | Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika 57, 97 to 109, then ignored for twenty years | W. Keith Hastings | Toronto | S227, S228 |
| 1970 | Matiyasevich establishes the J. R. Hypothesis, completing the solution of Hilbert's tenth problem | Yuri Matiyasevich | Leningrad (now St Petersburg) | S279 |
| 1970 | Wilkinson wins the Turing Award and gives the von Neumann Lecture; Golub and Reinsch, Numer. Math. 14, 403 to 420 | Wilkinson, Golub and Reinsch | United States | S195, S200 |
| 1971 | Taussky promoted to full professor, the first woman at Caltech at that rank | Olga Taussky-Todd | Pasadena | S192 |
| 1973 | The economics Nobel for the development of the input-output method | Wassily Leontief | Stockholm | S199 |
| 1975 | Julia Robinson elected to the National Academy of Sciences, the first woman in the mathematical section | Julia Robinson | United States | S279 |
| 1976 | Every planar map is four colorable received by the Bulletin of the AMS on 26 July, and published in September, Bull. AMS 82(5), 711 to 712 | Kenneth Appel and Wolfgang Haken | University of Illinois | S423, S426, S427, S428 |
| 1976 | Death of Lars Onsager | Lars Onsager | United States | S245 |
| 1977 | Appel and Haken's full papers appear in the Illinois Journal of Mathematics 21 | Kenneth Appel and Wolfgang Haken | University of Illinois | S423 |
| 1978 | Zeilberger recognizes Fasenmyer's method | Doron Zeilberger | United States | S253 |
| 1979 | Four objections to computer proof: not a priori, not certain, not surveyable, not checkable | Thomas Tymoczko | Smith College | S428 |
| 1980 | Ferguson's Institute for Defense Analyses lectures, The Blue Book | Jack Ferguson | United States | S239 |
| 1984 | Geman and Geman introduce the Gibbs sampler | Stuart and Donald Geman | United States | S227 |
| 1985 | Death of Julia Robinson on 30 July | Julia Robinson | United States | S279 |
| 1986 | Death of James Hardy Wilkinson on 5 October | James Hardy Wilkinson | Teddington | S195 |
| 1987 | Death of Raj Chandra Bose on 31 October | R. C. Bose | Fort Collins, Colorado | S443 |
| 1987 | Death of A. N. Kolmogorov on 20 October | A. N. Kolmogorov | Moscow | S237 |
| 1990 | Gelfand and Smith make Markov chain Monte Carlo ordinary Bayesian practice | Alan Gelfand and Adrian Smith | United States | S227 |
| 1992 | The Apprenticeship of a Mathematician, p. 114: the empty set symbol came from the Norwegian alphabet | Andre Weil | not stated in the source | S054 |
| 1995 | The four color proof revised down to 633 configurations | Robertson, Sanders, Seymour and Thomas | not stated in the source | S427 |
| 1995 | The Erdos Number Project begins on 25 May | Jerrold Grossman | Oakland University | S264 |
| 1995 | Death of Olga Taussky-Todd on 7 October | Olga Taussky-Todd | Pasadena | S192 |
| 1996 | Death of Paul Erdos on 20 September, aged 83 | Paul Erdos | Warsaw | S263 |
| 1996 | Death of Sister Mary Celine Fasenmyer on 27 December | Mary Celine Fasenmyer | Erie, Pennsylvania | S253, S254 |
| 1998 | Collective dynamics of small-world networks, Nature 393, 440 to 442, 4 June | Duncan Watts and Steven Strogatz | Cornell | S441 |
| 1998 | Death of Andre Weil on 6 August | Andre Weil | Princeton, New Jersey | S277 |
| 1998 | Death of Mary Cartwright on 3 April | Mary Cartwright | Cambridge | S280 |
| 1998 | The Anatomy of a Large-Scale Hypertextual Web Search Engine: PageRank corresponds to the principal eigenvector of the normalized link matrix of the web | Sergey Brin and Larry Page | Stanford | S198, S243 |
| 1999 | Death of Wassily Leontief on 5 February | Wassily Leontief | New York | S199 |
| 2000 | Envelope 11668 opened in May, and the memoir printed in full in the Comptes rendus in December | the Academie des Sciences and Bernard Bru | Paris | S232, S233, S234 |
| 2000 | Map-making and myth-making in Broad Street publishes Snow's real chronology in The Lancet | Brody, Rip, Vinten-Johansen, Paneth, and Rachman | London | S269 |
| 2002 | Six Degrees: Urban Myth?, Psychology Today, March: 3 of 60 letters arrived in Milgram's pilot | Judith Kleinfeld | University of Alaska Fairbanks | S440 |
| 2003 | Marshall Rosenbluth tells a fiftieth anniversary conference who did the work on the 1953 paper | Marshall Rosenbluth | United States | S256 |
| 2005 | George and Esther Szekeres die on 28 August, within an hour of each other | George and Esther Szekeres | Adelaide | S434 |
| 2005 | The four color theorem formalized in Coq | Georges Gonthier | Cambridge, England | S427 |
| 2007 | Gene Golub visits John Francis at his home on 7 August, forty six years after the papers | Gene Golub and John G. F. Francis | England | S193 |
| 2008 | The Statistical Science footnote that takes the polar area diagram away from Nightingale | Michael Friendly | not stated in the source | S267 |
| 2010 | Death of David Blackwell on 8 July | David Blackwell | Berkeley, California | S250 |
| 2012 | Death of N. G. de Bruijn on 17 February | N. G. de Bruijn | Nuenen | S444 |
| 2012 | Death of Vera Nikolaevna Kublanovskaya on 21 February | Vera Kublanovskaya | St Petersburg | S194 |
| 2018 | Floros reports that between 4 and 8 per cent of adolescents who gamble show significant gambling-related problems | Georgios Floros | not stated in the source | S389 |
| 2020 | Death of Arianna Wright Rosenbluth on 28 December | Arianna Wright Rosenbluth | United States | S257 |
| 2025 | The current Erdos Number Project data version, August 2025, gives 514 Erdos co-authors | Jerrold Grossman | Oakland University | S264 |
Reference: the registers everything else is built on
Eleven appendices and a glossary. Nobody reads this part front to back, and it is not written to be read that way. Everything in the story links into it, and every entry links back out to where it is used.
- Look up any person, word, glyph or source used anywhere in this book.
- Check a worked calculation for yourself, since every one was re-run in code.
- See exactly where two sources disagree, and what document would settle it.
- Find where a story belongs in your Discrete Math and Linear Algebra course.
- Tell the difference between a story that checks out and one that does not.
- Find a hook into another subject, with the anchor fact and its source.
Appendix A
People register
306 people, sorted by birth year, each with life dates, the name in its own script where one exists, how to say it, where they were born and where they worked, and the chapters they appear in. Life dates in italics are not settled, which is true of 204 of them.
Pronunciation is a respelling you can say out loud, not a phonetic transcription. 100 of the 306 are sourced to a named dictionary entry or a published romanization. The rest are marked not sourced: they are careful approximations, not evidence, and they are labeled so rather than hidden. 1 person has no respelling at all, because nothing consulted states one. An empty cell means nobody here knows, not that nobody looked.
No portrait of any person in this book is cleared for reuse. Not one. The largest open collection of mathematicians' portraits refuses to warrant its images for publication, so this book prints no faces rather than printing faces it has no right to. Where you would expect a portrait, you get a name, a place, and a date, which is the part the evidence supports.
| Name | Lived | Say it | Born in / worked in | Chapters | Sources |
|---|---|---|---|---|---|
| al-Karaji الكرجي | 953 to 1029 Disputed | al-kuh-RAH-jee not sourced | al-Karaj region / Baghdad | 2Appears in Chapter 2 5 times, and in the timeline under 500 to 1400 twice. | S003 |
| al-Samaw'al al-Maghribi السموأل المغربي | 1130 Not known | as-suh-MOW-al not sourced | Baghdad / Baghdad, Maragha | 2Appears in Chapter 2, and in the timeline under 500 to 1400. | S003 |
| Zhu Shijie 朱世傑 | 1260 to 1320 Disputed | joo shr-jyeh not sourced | China / China | 2Appears in Chapter 2 twice, and in the timeline under 500 to 1400. | S013, S017 |
| William of Ockham Guillelmus de Ockham | 1287 to 1347 Disputed | ˈɒk əm OCK-uhm | Ockham, Surrey / Oxford, Munich | S055 | |
| Levi ben Gershon (Gersonides) לוי בן גרשון | 1288 to 1344 | LAY-vee ben GAIR-shon not sourced | Provence / Provence | 2Named in the registers, not in the story or the timeline. | S004 |
| Madhava of Sangamagrama Malayalam script NOT sourced | 1350 to 1425 | MAH-dhuh-vuh not sourced | near Cochin, Kerala, India / Kerala | S284 | |
| Michael Stifel Michael Stifelius | 1487 to 1567 | MIKH-ah-el SHTEE-fel not sourced | Germany / Nuremberg, Jena | 2 | S018 |
| Juan Luis Vives Joannes Ludovicus Vives | 1493 to 1540 | hwahn loo-EES VEE-vays not sourced | Valencia / Low Countries | S051 | |
| Niccolo Fontana "Tartaglia" Nicolo Tartaglia | 1499 to 1557 Disputed | tar-TAH-lyah not sourced | Brescia / Venice | 2 | S011, S019 |
| Girolamo Cardano Hieronymus Cardanus | 1501 to 1576 | jee-ROH-lah-moh kar-DAH-noh not sourced | Pavia, Duchy of Milan / Milan, Bologna, Rome | 3 | S018, S072, S075, S093, S094, S122 |
| Giulio Pace Julius Pacius | 1550 to 1635 | JOOL-yoh PAH-chay not sourced | Vicenza / Italy, France | S051 | |
| Galileo Galilei Galileo Galilei | 1564 to 1642 | ˌgæl əˈleɪ oʊ, -ˈli oʊ; MW ˌga-lə-ˈlē-(ˌ)ō, -ˈlā- gal-uh-LAY-oh | Pisa (not sourced here) / Padua, Florence | 3 | S070, S071, S072 |
| Pierre de Fermat Pierre de Fermat | 1601 to 1665 | fer-ˈmä fer-MAH | France / Toulouse | 3 | S070, S074, S075, S094 |
| Antoine Gombaud, chevalier de Mere Gombaud / Gombauld | 1607 to 1684 | duh MAIR not sourced | France / court of Louis XIV | 3 | S070, S072, S074, S075, S093 |
| John Graunt | 1620 to 1674 | GRAWNT not sourced | London / London | 3Appears in Chapter 3 7 times, and in the timeline under 1650 to 1800. | S077, S078, S079 |
| Christiaan Huygens Christiaan Huygens, van Zuylichem | 1629 to 1695 | ˈhaɪ gənz, ˈhɔɪ-; British ˈhœixəns HY-genz (English), HOY-khens (Dutch) | The Hague / The Hague, Paris, London | 3 | S005, S076, S081, S094 |
| Christian Weise Christian Weise | 1642 to 1708 | KRIS-tee-ahn VY-zuh not sourced | Germany / Zittau | S051 | |
| Seki Takakazu (Seki Kowa) 関孝和 | 1642 to 1708 | SEH-kee tah-kah-KAH-zoo not sourced | Fujioka, Kozuke (now Gunma) / Edo (now Tokyo) | 4Named in the registers, not in the story or the timeline. | S122, S127, S128 |
| Gottfried Wilhelm Leibniz Leibnitz (Muir's spelling) | 1646 to 1716 | "ˈlīb-nəts" (Eng.), "ˈlīp-nits" (Ger.) LYBE-nits | Leipzig / Leipzig, Hanover | 2, 4, 7 | S005, S008, S051, S079, S122, S129, S143, S420 |
| Jacob Bernoulli Jacobus Bernoulli | 1654 to 1705 | bərˈnu li, bɛrˈnu li ber-NOO-lee | Basel / Basel | 3, 9 | S005, S079, S080, S091, S260 |
| Abraham de Moivre Abraham de Moivre | 1667 to 1754 | də ˈmwɑv, ˈmwɑ vrə, ˈmɔɪ vər duh MWAHV, or duh MWAH-vruh | Vitry, Champagne / England | 2, 3 | S011, S081, S082, S094 |
| Johann Bernoulli Johannes Bernoulli | 1667 to 1748 | bərˈnu li ber-NOO-lee | Basel / Basel, Groningen | S079, S094 | |
| Nicholas Saunderson | 1682 to 1739 | SAWN-der-son not sourced | Thurlstone, Yorkshire / Cambridge | In the timeline 1650 to 1800 4 times. | S094 |
| Nicolaus I Bernoulli | 1687 to 1759 | bərˈnu li ber-NOO-lee | Basel / Basel | 3In the timeline 1650 to 1800 twice. | S005, S006, S079, S080, S081 |
| Colin Maclaurin | 1698 to 1746 | məkˈlɔr ɪn, məˈklɔr- muh-KLOR-in | Scotland / Edinburgh | 4Appears in Chapter 4 4 times, and in the timeline under 1650 to 1800 5 times. | S122, S132 |
| Daniel Bernoulli | 1700 to 1782 | bərˈnu li ber-NOO-lee | Groningen / Basel, St Petersburg | Named in the registers, not in the story or the timeline. | S072, S094 |
| Leonhard Euler Leonhardus Eulerus | 1707 to 1783 | "ˈɔɪ lər" (Amer.), "ˈɔɪlər" (Brit.) OY-ler | Basel (not sourced here) / St Petersburg, Berlin | 7, 9 | S007, S049, S051, S058, S121, S305, S420, S421, S445 |
| Alexandre-Theophile Vandermonde | 1735 to 1796 | van-der-MOND not sourced | Paris / Paris | 4, 7Appears in Chapter 4 7 times and Chapter 7 twice, and in the timeline under 1650 to 1800 7 times. | S122, S129, S430 |
| Pierre-Simon Laplace Pierre-Simon Laplace | 1749 to 1827 | lə-ˈpläs; laˈplas luh-PLAHSS | Normandy / Paris | 3, 4 | S007, S086, S087, S092, S094, S122, S129 |
| Carl Friedrich Gauss | 1777 to 1855 | gaʊs; MW ˈgau̇s GOWSS | Brunswick / Gottingen | 4Appears in Chapter 4 14 times, and in the timeline under 1800 to 1850 6 times. | S120, S121, S122, S133, S180 |
| Justus Gunther Grassmann Justus Günther Grassmann | 1779 to 1852 | ˈgrɑs mən, ˈgrɑsˌmɑn GRAHSS-mun | Prussia / Stettin | 5 | S172, S181, S204 |
| Bernard Bolzano Bernard Bolzano | 1781 to 1848 | boʊltˈsɑ noʊ (US); bolˈtsano (UK) bolt-SAH-no | Prague / Prague, Villa Liboch | 1 | S043, S044, S068 |
| Charles Joseph Minard | 1781 to 1870 | mee-NAR not sourced | France / Paris | Named in the registers, not in the story or the timeline. | S267 |
| Simeon-Denis Poisson Simeon-Denis Poisson | 1781 to 1840 | pwasɔ̃ pwah-SOHN | France / Paris | 3 | S080, S088 |
| Augustin-Louis Cauchy | 1789 to 1857 | "koʊˈʃi" koh-SHEE | Paris / Paris | 4, 5Appears in Chapter 4 9 times, Chapter 5 9 times and Chapter 9 twice, and in the timeline under 1650 to 1800 twice, 1800 to 1850 8 times and 1850 to 1880 4 times. | S122, S129, S143, S176, S179, S180, S202, S204 |
| Andreas von Ettingshausen | 1796 to 1878 | AN-dray-ahs fon ET-tings-how-zen not sourced | Vienna | 9Appears in Chapter 9 twice, and in the timeline under 1800 to 1850. | S301, S387 |
| Irenee-Jules Bienayme Irenee-Jules Bienayme | 1796 to 1878 | byen-eh-MAY not sourced | Paris / Paris | 3 | S080, S092, S094 |
| Carl Gustav Jacob Jacobi | 1804 to 1851 | "jɑˈkoʊ bi", "dʒəˈkoʊ bi" yah-KOH-bee (German), juh-KOH-bee (English) | Potsdam / Konigsberg, Berlin | 4Appears in Chapter 4 twice, Chapter 5 twice and Chapter 7 twice, and in the timeline under 1800 to 1850. | S122, S129, S138, S143, S180, S185 |
| William Rowan Hamilton | 1805 to 1865 | ˈhæm əl tən HAM-il-ton | Dublin / Dublin, Dunsink Observatory | 4, 5, 7Appears in Chapter 4 twice, Chapter 5 18 times and Chapter 7 6 times, and in the timeline under 1800 to 1850 6 times and 1850 to 1880 7 times. | S122, S128, S139, S174, S175, S177, S302, S306, S423 |
| Augustus De Morgan Augustus De Morgan | 1806 to 1871 | dɪ ˈmɔr gən di MOR-gun | Madurai, India / University College London | 1, 7 | S047, S055, S060, S063, S069, S088, S258, S423 |
| Johann Benedict Listing | 1808 to 1882 | LIST-ing not sourced | Frankfurt / Gottingen | 7Appears in Chapter 7 twice, and in the timeline under 1800 to 1850. | S302 |
| Hermann Gunther Grassmann Hermann Günther Grassmann | 1809 to 1877 | ˈgrɑs mən, ˈgrɑsˌmɑn GRAHSS-mun (also GRAHSS-mahn) | Stettin, Prussia (now Szczecin, Poland) / Stettin | 5 | S171, S177, S179, S181, S182, S203, S204 |
| James Joseph Sylvester | 1814 to 1897 | "sɪlˈvɛs tər" sil-VESS-ter | London / London, Virginia, Baltimore, Oxford | 4, 5, 7, 9Appears in Chapter 4 9 times, Chapter 5 7 times, Chapter 7 8 times and Chapter 9 7 times, and in the timeline under 1800 to 1850 twice, 1850 to 1880 7 times, 1880 to 1910 5 times and 1960 to now. | S007, S023, S122, S128, S135, S143, S170, S176, S178, S180, S301, S302, S307 |
| George Boole George Boole | 1815 to 1864 | bul (US); buːl (UK) BOOL | Lincoln / Lincoln; Queen's College Cork | 1 | S045, S046, S060, S068, S135, S266 |
| Karl Weierstrass Karl Theodor Weierstrass | 1815 to 1897 | ˈvaɪ ərˌʃtrɑs VY-er-shtrahss | Ostenfelde (not sourced here) / Berlin | S062, S068, S122, S137, S187 | |
| Isaac Todhunter | 1820 to 1884 | TOD-hun-ter not sourced | England / Cambridge | 3, 9Appears in Chapter 3 5 times and Chapter 9 5 times, and in the timeline under 1850 to 1880 4 times. | S301 |
| Arthur Cayley | 1821 to 1895 | "ˈkeɪ li" KAY-lee | Richmond, Surrey / London, Cambridge | 4, 7, 9Appears in Chapter 4 10 times and Chapter 7 5 times, and in the timeline under 1800 to 1850 3 times, 1850 to 1880 8 times and 1880 to 1910 5 times. | S007, S023, S122, S136, S143, S178, S180, S301, S304, S423 |
| Pafnuty Lvovich Chebyshev Пафнутий Львович Чебышёв | 1821 to 1894 | tʃə bəˈʃɔf chuh-buh-SHAWF | Okatovo, Kaluga region / St Petersburg | 3Appears in Chapter 3 5 times, and in the timeline under 1800 to 1850 4 times, 1850 to 1880 twice and 1880 to 1910 twice. | S080, S094, S219, S220 |
| Leopold Kronecker Leopold Kronecker | 1823 to 1891 | ˈkroʊ nɛk ər KROH-neck-er | Liegnitz (not sourced here) / Berlin | 1 | S053, S062, S068, S122, S137, S187 |
| Gustav Robert Kirchhoff Gustav Robert Kirchhoff | 1824 to 1887 | "ˈkirx hɔf" (Amer.), "ˈkɪrçhɔf" (Brit.) KEERKH-hawf | Konigsberg, Prussia / Breslau, Heidelberg, Berlin | 7 | S429, S445 |
| Peter Guthrie Tait | 1831 to 1901 | TAYT not sourced | Scotland / Edinburgh | 5Appears in Chapter 5 6 times, and in the timeline under 1850 to 1880 3 times and 1880 to 1910. | S177 |
| Richard Dedekind Richard Dedekind | 1831 to 1916 | ˈdeɪ də kɪnt (US); ˈdedəˌkɪnt (UK) DAY-duh-kint | Braunschweig / Braunschweig, Gottingen | 1 | S043, S053, S059, S062, S068 |
| Mary Everest Boole Mary Everest Boole | 1832 to 1916 | surname bul MAIR-ee EV-rist bool | Wickwar, Gloucestershire / London | 1 | S060, S061, S068, S266 |
| John Venn John Venn | 1834 to 1923 | vɛn VEN | Hull (not sourced here) / Gonville and Caius College, Cambridge | 1 | S048, S051, S058, S068 |
| Eugenio Beltrami Eugenio Beltrami | 1835 to 1900 | bɛlˈtrɑ mi bel-TRAH-mee | Cremona, Lombardy / Bologna, Pisa, Rome, Pavia | 5 | S170, S201, S204 |
| Marie Ennemond Camille Jordan | 1838 to 1922 | ʒɔrˈdɑ̃ zhor-DAHN | France / Paris | 4, 5Appears in Chapter 4 5 times and Chapter 5 3 times, and in the timeline under 1850 to 1880 3 times, 1880 to 1910 twice and 1910 to 1935 twice. | S122, S134, S144, S170, S180 |
| Josiah Willard Gibbs | 1839 to 1903 | ˈdʒoʊsiə ˈwɪlərd (forenames) GIBZ (forenames joh-SY-uh WIL-erd) | New Haven, Connecticut / Yale, New Haven | 5Appears in Chapter 5 10 times and Chapter 6, and in the timeline under 1880 to 1910 3 times, 1910 to 1935 and 1960 to now. | S177, S204 |
| Wilhelm Jordan (geodesist) | 1842 to 1899 | VIL-helm YOR-dahn not sourced | Germany / Karlsruhe, Hanover | 4Named in the registers, not in the story or the timeline. | S134, S142, S144 |
| Ludwig Boltzmann | 1844 to 1906 | ˈbɔltsˌmɑn, ˈboʊlts mən (US); ˈbɔltsman (UK); forename ˈluːtvɪç LOOT-vikh BAWLTS-mahn | Vienna, Austria / Vienna; Graz; Leipzig | 6Appears in Chapter 6 4 times, and in the timeline under 1850 to 1880 and 1880 to 1910. | S221, S230, S302 |
| Thomas Muir | 1844 to 1934 | MYOOR not sourced | Scotland / Glasgow, Cape Colony | Appears in Chapter 4 9 times. | S129 |
| Georg Cantor Georg Cantor | 1845 to 1918 | ˈkæn tər, -tɔr (surname) GAY-org KAN-ter | St Petersburg / Halle | 1, 9 | S040, S041, S042, S043, S053, S054, S059, S062, S068, S302 |
| Gottlob Frege Gottlob Frege | 1848 to 1925 | ˈfreɪ gə (US); ˈfreːɡə (UK) FRAY-guh | Wismar (not sourced here) / Jena | 1 | S056, S068 |
| Alfred Bray Kempe | 1849 to 1922 | KEMP not sourced | Kensington, London / London | 7Appears in Chapter 7 10 times, and in the timeline under 1800 to 1850 twice, 1850 to 1880 3 times, 1880 to 1910 twice and 1910 to 1935 4 times. | S423, S424 |
| Ferdinand Georg Frobenius Ferdinand Georg Frobenius | 1849 to 1917 | froh-BAY-nee-oos not sourced | Berlin-Charlottenburg / Zurich, Berlin | 4, 5 | S122, S128, S137, S180, S185, S186, S187, S204, S302 |
| Oliver Heaviside | 1850 to 1925 | ˈhɛv iˌsaɪd HEV-ee-side | England / England | 5Appears in Chapter 5 7 times, and in the timeline under 1880 to 1910 5 times and 1910 to 1935. | S177, S202, S204 |
| George Chrystal | 1851 to 1911 | KRIS-tal not sourced | Scotland / Edinburgh | 2Appears in Chapter 2 3 times and Chapter 7, and in the timeline under 1880 to 1910 twice. | S011 |
| Pavel Alekseevich Nekrasov Павел Алексеевич Некрасов | 1853 to 1924 | nɪˈkrasəf nih-KRASS-uff | Russia (Ryazan seminary) / Moscow University | 6Appears in Chapter 6 11 times, and in the timeline under 1880 to 1910 twice and 1910 to 1935 5 times. | S210, S211, S212, S218, S271, S385 |
| Percy Alexander MacMahon | 1854 to 1929 | mak-MAHN not sourced | Sliema, Malta / Woolwich, London, Cambridge | 2In the timeline 1850 to 1880 twice and 1910 to 1935 4 times. | S007, S012, S023 |
| Andrei Andreevich Markov Андрей Андреевич Марков | 1856 to 1922 Disputed | ˈmar.kəf (Russian); ˈmɑr kɔf, ˈmɑːkɒf (English) MAR-kuff (Russian), mahr-KAWF (English) | Ryazan, Russia / St Petersburg University; Imperial Academy of Sciences; Zaraisk 1917 | 6Appears in Chapter 5, Chapter 6 54 times and Chapter 8 6 times, and in the timeline under 1850 to 1880 4 times, 1880 to 1910 10 times, 1910 to 1935 20 times, 1935 to 1960 4 times and 1960 to now 4 times. | S210, S211, S212, S213, S214, S217, S219, S220, S225, S244, S245, S271, S302 |
| Giuseppe Peano Giuseppe Peano | 1858 to 1932 | piˈɑ noʊ, pɛˈɑ nɔ pay-AH-noh (English pee-AH-noh) | Cuneo, Piemonte / Turin | 1, 5, 9 | S050, S054, S064, S068, S173, S179, S182, S183, S204, S394 |
| Alicia Boole Stott | 1860 to 1940 | uh-LISS-ee-uh BOOL STOT not sourced | Cork, Ireland / England; corresponded with Groningen | 8Appears in Chapter 8 3 times, and in the timeline under 1850 to 1880, 1880 to 1910 3 times, 1910 to 1935 twice and 1935 to 1960 3 times. | S266 |
| Percy John Heawood | 1861 to 1955 | HEE-wood not sourced | Newport, Shropshire / Durham | 7Appears in Chapter 7 8 times, and in the timeline under 1850 to 1880 twice, 1880 to 1910 and 1935 to 1960 twice. | S423, S425 |
| David Hilbert David Hilbert | 1862 to 1943 | ˈhɪl bərt HIL-bert (German HIL-bairt) | Konigsberg, Prussia / Gottingen | 5, 9 | S065, S176, S179, S184, S192, S202, S204, S222, S278, S302 |
| Paul Gustav Stackel Stäckel | 1862 to 1919 | SHTEK-'l not sourced | Germany / Heidelberg | S138 | |
| Ladislaus Josephowitsch Bortkiewicz Bortkiewicz | 1868 to 1931 | bort-KYEH-vitch not sourced | Saint Petersburg, Berlin | 9 | S302 |
| William Edward Burghardt Du Bois | 1868 to 1963 | du ˈbɔɪs doo-BOYSS | USA (Great Barrington, MA, NOT verified here) / Atlanta University; Paris 1900 | 8Appears in Chapter 8 4 times, and in the timeline under 1850 to 1880 twice and 1880 to 1910 twice. | S281, S282, S331, S380, S382 |
| Ernst Zermelo Ernst Friedrich Ferdinand Zermelo | 1871 to 1953 | tser-MAY-loh not sourced | Berlin / Gottingen, Zurich, Freiburg | 1, 9 | S054, S056, S057, S059, S065, S221 |
| Felix Edouard Emile Borel Emile Borel | 1871 to 1956 | bɔˈrɛl baw-REL | Saint-Affrique (not sourced here) / Paris | 3 | S080, S089, S094 |
| Erhard Schmidt Erhard Schmidt | 1876 to 1959 | ʃmɪt (surname) AIR-hart SHMIT | Dorpat / Gottingen, Berlin | 5 | S170, S179 |
| Tatiana Alexeyevna Ehrenfest-Afanassjewa Татьяна Алексеевна Афанасьева | 1876 to 1964 | AIR-un-fest ah-fah-NAH-syeh-vuh not sourced | Kiev (now Kyiv, Ukraine) / Gottingen; St Petersburg; Leiden | 6In the timeline 1850 to 1880 and 1960 to now twice. | S222, S223 |
| Rene Maurice Frechet Maurice Frechet | 1878 to 1973 | freɪˈʃɛ fray-SHAY | Maligny (not sourced here) / Paris, Strasbourg | 3 | S089, S090, S094, S215, S233, S236 |
| Sergei Natanovich Bernstein Сергей Натанович Бернштейн | 1880 to 1968 | BERN-shtyne not sourced | Odessa, Russian Empire / Kharkov, Paris, Moscow | 3, 6In the timeline 1880 to 1910 twice, 1910 to 1935 3 times and 1960 to now twice. | S089, S094, S212 |
| Emmy Amalie Noether | 1882 to 1935 | ˈnʌ tər NUH-ter | Erlangen, Bavaria, Germany / Gottingen; Bryn Mawr | 8Appears in Chapter 8 4 times, and in the timeline under 1880 to 1910 4 times, 1910 to 1935 10 times and 1935 to 1960 twice. | S059, S191, S278 |
| Richard von Mises Richard von Mises | 1883 to 1953 | RIKH-art fon MEE-zuhs not sourced | Lemberg (not sourced here) / Berlin, Istanbul, Harvard | S055, S066, S067, S089, S215 | |
| Hermann Weyl Hermann Weyl | 1885 to 1955 | vaɪl VILE (rhymes with "mile") | Germany / Gottingen, Zurich, Princeton | 5 | S170, S173 |
| Sydney Chapman | 1888 to 1970 | CHAP-mun not sourced | Eccles, near Manchester, England / Cambridge; Oxford; Boulder, Colorado; Alaska | 6Appears in Chapter 6 4 times, and in the timeline under 1880 to 1910 twice and 1960 to now twice. | S238 |
| Dame Mary Lucy Cartwright | 1900 to 1998 | KART-rite not sourced | Aynho, Northamptonshire, England / Girton College, Cambridge | In the timeline 1880 to 1910 twice, 1935 to 1960 twice and 1960 to now 4 times. | S280 |
| Raj Chandra Bose राज चंद्र बोस | 1901 to 1987 | boʊs (surname) rahj CHUN-druh BOHSS | Hoshangabad, Madhya Pradesh / Calcutta, Chapel Hill, Fort Collins | 7Appears in Chapter 7 5 times, and in the timeline under 1650 to 1800, 1880 to 1910 twice, 1935 to 1960 twice and 1960 to now 4 times. | S442, S443 |
| Andrei Andreevich Markov Jr. Андрей Андреевич Марков (мл.) | 1903 to 1979 | ˈmar.kəf (Russian); ˈmɑr kɔf (English) MAR-kuff (Russian), mahr-KAWF (English) | Russia / Moscow | Named in the registers, not in the story or the timeline. | S212 |
| Andrei Nikolaevich Kolmogorov Андрей Николаевич Колмогоров | 1903 to 1987 | /ˌkɒlmɒˈɡɔːrɒf/ kol-muh-GAW-rof | Tambov (not sourced here) / Moscow State University | 3, 6, 9Appears in Chapter 3 8 times, Chapter 6 6 times and Chapter 9 3 times, and in the timeline under 1880 to 1910 twice, 1910 to 1935 4 times, 1935 to 1960 twice and 1960 to now twice. | S066, S089, S090, S094, S215, S236, S237, S245, S303 |
| Frank Plumpton Ramsey | 1903 to 1930 | ˈræm zi (US); ˈræmzɪ (UK) RAM-zee | Cambridge, England / Cambridge | 7Appears in Chapter 7 4 times, and in the timeline under 1880 to 1910 twice and 1910 to 1935 5 times. | S432, S435 |
| Lars Onsager n/a (Norwegian) | 1903 to 1976 | /ˈɒn sɑ gər/, /ˈɔn-/ ON-sah-ger (also AWN-) | Norway / Yale University | 6 | S230, S231, S245 |
| Andre Weil André Weil | 1906 to 1998 | veɪ (US); vail (UK) ahn-DRAY vay | Paris, France / Strasbourg; Aligarh; Sao Paulo; Chicago; Princeton | 1, 8, 9 | S054, S068, S275, S276, S277 |
| Olga Taussky-Todd | 1906 to 1995 | TOW-skee TOD ("TOW" as in "how") not sourced | Olmutz (now Olomouc, Czech Republic) / Vienna, Gottingen, Teddington, Washington, Pasadena | 5, 8In the timeline 1880 to 1910, 1910 to 1935, 1935 to 1960 4 times and 1960 to now 3 times. | S192, S204 |
| Sister Mary Celine Fasenmyer | 1906 to 1996 | FAY-zen-my-er not sourced | Crown, Pennsylvania, USA / Mercyhurst College, Erie, PA | 8Appears in Chapter 8 twice, and in the timeline under 1880 to 1910 twice, 1935 to 1960 5 times and 1960 to now 3 times. | S253, S254, S255 |
| Wassily Leontief Василий Леонтьев | 1906 to 1999 | liˈɒn tiˌɛf lee-ON-tee-ef | St Petersburg / Harvard, New York University | 5Appears in Chapter 5 and Chapter 8, and in the timeline under 1880 to 1910 twice, 1910 to 1935 twice and 1960 to now 3 times. | S185, S199, S204, S283 |
| Florence Nightingale David | 1909 Not known | ˈdeɪ vɪd DAY-vid | Ivington, England / UCL; UC Berkeley; UC Riverside | 8Appears in Chapter 5 twice, Chapter 8 4 times and Chapter 9, and in the timeline under 1880 to 1910 4 times, 1910 to 1935 5 times, 1935 to 1960 4 times and 1960 to now 5 times. | S071, S265 |
| Simone Adolphine Weil Simone Weil | 1909 to 1943 | veɪ see-MOHN vay | Paris, France / Paris; London | 8 | S068, S277 |
| George Szekeres Szekeres Gyorgy | 1911 to 2005 | ˈsɛ.kɛ.rɛʃ SEH-keh-resh | Budapest / Shanghai, Adelaide | 7 | S433, S434 |
| Paul Erdos Hungarian: Erdos Pal (double acute accent on the o) | 1913 to 1996 | ˈer-ˌdərsh AIR-dursh | Budapest (NOT verified this session) / itinerant; Hungary, USA, Israel | 7, 8 | S245, S263, S264, S433, S434, S435, S436, S437, S445 |
| Wolfgang Doeblin (Vincent Doblin) Wolfgang Döblin | 1915 to 1940 | ˈdœ blin DUR-bleen (German oe, the vowel of French "oeuf") | Berlin, Germany / Paris; French army, Vosges | 6 | S233, S234 |
| Nicolaas Govert de Bruijn | 1918 to 2012 | duh BROWN not sourced | The Hague / Eindhoven, Nuenen | 7Appears in Chapter 7 6 times, and in the timeline under 1910 to 1935 twice, 1935 to 1960 twice and 1960 to now twice. | S444 |
| David Harold Blackwell | 1919 to 2010 | ˈblæk wəl, -ˌwɛl BLACK-wel | Centralia, Illinois, USA / Howard University; UC Berkeley | 8Appears in Chapter 8 6 times, and in the timeline under 1910 to 1935 twice, 1935 to 1960 8 times and 1960 to now 4 times. | S250, S251, S252 |
| James Hardy Wilkinson | 1919 to 1986 | WIL-kin-sun not sourced | Strood, Kent / Teddington | 5Appears in Chapter 5 4 times, and in the timeline under 1910 to 1935 twice, 1935 to 1960 twice and 1960 to now 6 times. | S189, S195, S196, S204 |
| Julia Bowman Robinson | 1919 to 1985 | ROB-in-sun not sourced | St Louis, Missouri, USA / UC Berkeley | In the timeline 1910 to 1935 and 1960 to now 4 times. | S279 |
| Vera Nikolaevna Kublanovskaya Вера Николаевна Кублановская | 1920 to 2012 | koob-luh-NOFF-skuh-yuh not sourced | Krokino (also Krokhono), Vologda Oblast / Leningrad / St Petersburg | 5, 8Appears in Chapter 5 4 times and Chapter 8 3 times, and in the timeline under 1910 to 1935 twice, 1935 to 1960 4 times and 1960 to now 6 times. | S193, S194, S204 |
| Alfred Renyi Renyi Alfred | 1921 to 1970 | RAYN-yee not sourced | Budapest / Budapest | 7 | S436, S437, S438 |
| Iannis Xenakis Greek: Iannis Xenakis (Greek script NOT sourced) | 1922 to 2001 | /ksɛˈnakis/ or /zɛˈnɑːkɪs/ zeh-NAH-kiss | Braila, Romania / Paris | 8 | S274 |
| Lejaren Arthur Hiller Jr | 1924 Not known | LEDGE-uh-rin HILL-er not sourced | USA / University of Illinois; SUNY Buffalo | 8Named in the registers, not in the story or the timeline. | S241, S242, S272, S273, S274 |
| Arianna Wright Rosenbluth | 1927 to 2020 | ah-ree-AH-nuh ROH-zen-blooth not sourced | Houston, Texas, USA / Los Alamos National Laboratory | 6, 8Appears in Chapter 8 twice, and in the timeline under 1910 to 1935, 1935 to 1960 and 1960 to now twice. | S226, S256, S257 |
| Doron Zeilberger Hebrew form not sourced | 1950 Not known | DOR-on ZYLE-ber-ger not sourced | Rutgers University | S253, S254, S255 | |
| Adam Makkai Makkai Adam | dates unknown Not known | AH-dahm MAH-koy not sourced | Budapest / Chicago | S439 | |
| Adrien-Marie Legendre | dates unknown Not known | ləˈʒɑn dər, -ˈʒɑnd, ləˈʒɑ̃ drə luh-ZHAHN-der | 9Appears in Chapter 4 and Chapter 9 4 times, and in the timeline under 1800 to 1850. | S121, S301 | |
| Alan Baker (philosopher) | dates unknown Not known | BAY-ker not sourced | Swarthmore College | Named in the registers, not in the story or the timeline. | S428 |
| Alan Mathison Turing | dates unknown Not known | ˈtʊər ɪŋ TOOR-ing | London / Teddington, Manchester | 5, 8Appears in Chapter 5 9 times and Chapter 8, and in the timeline under 1935 to 1960 8 times and 1960 to now. | S120, S189, S195, S196, S261, S262, S390 |
| Aleksandr Aleksandrovich Chuprov Александр Александрович Чупров | dates unknown Not known | choo-PROFF not sourced | Russia / St Petersburg | Appears in Chapter 9 4 times. | S220, S302 |
| Alexander Craig Aitken | dates unknown Not known | AY-ken not sourced | New Zealand / Edinburgh | Appears in Chapter 9 twice. | S122, S302 |
| Alfred Doblin Alfred Döblin | dates unknown Not known | ˈdœ blin DUR-bleen (German oe, the vowel of French "oeuf") | Stettin, Germany / Berlin; exile in France and the USA | S232, S233, S234 | |
| Andre-Louis Cholesky | dates unknown Not known | shoh-LESS-kee not sourced | France / France | Named in the registers, not in the story or the timeline. | S120 |
| Andre-Michel Guerry | dates unknown Not known | zhe-REE not sourced | France / Paris | 8Appears in Chapter 8, and in the timeline under 1800 to 1850 twice. | S267 |
| Anna Johnson Pell Wheeler | dates unknown Not known | WHEE-ler not sourced | USA / Bryn Mawr College | Named in the registers, not in the story or the timeline. | S278 |
| Antoine Augustin Cournot | dates unknown Not known | koor-NOH not sourced | France / Paris | 3Appears in Chapter 3 7 times, and in the timeline under 1800 to 1850. | S088 |
| August Ferdinand Mobius August Ferdinand Möbius | dates unknown Not known | ˈmœ bi əs, ˈmeɪ-, ˈmoʊ- MER-bee-oos | Saxony / Leipzig | S172, S177 | |
| Augusta Ada King, Countess of Lovelace | dates unknown Not known | ˈlʌvˌleɪs LUV-layss | London, England / London | 8Appears in Chapter 8 4 times, and in the timeline under 1800 to 1850 3 times. | S063, S258, S260 |
| Augusta H. (Mici) Teller Hungarian: Harkanyi Auguszta | dates unknown Not known | MEE-tsee TELL-er not sourced | Hungary / Los Alamos | 6 | S226, S256 |
| Bertrand Russell Bertrand Russell | dates unknown Not known | ˈrʌs əl RUSS-uhl | Wales / Cambridge | S056, S064 | |
| Bharata भरत | dates unknown Not known | BHUH-ruh-tuh not sourced | India / India | In the timeline Before 500. | S002 |
| Blaise Pascal Blaise Pascal | dates unknown Not known | "pa-ˈskal", "pä-ˈskäl" pa-SKAL | Clermont (not sourced here) / Paris | 2, 3 | S001, S070, S074, S075 |
| Bohuslav Hostinsky Bohuslav Hostinsky (Czech) | dates unknown Not known | HOS-tin-skee not sourced | Bohemia / Brno | S215 | |
| Camille Flye Sainte-Marie | dates unknown Not known | fly sant mah-REE not sourced | France / France | Named in the registers, not in the story or the timeline. | S121, S122, S129, S134, S144, S301, S444 |
| Carl Leonhard Gottlieb Ehler | dates unknown Not known | AY-ler not sourced | Danzig | 7Appears in Chapter 7 twice, and in the timeline under 1650 to 1800. | S422 |
| Cassius Jackson Keyser | dates unknown Not known | KASH-us KY-zer not sourced | United States / Columbia | 9Appears in Chapter 9, and in the timeline under 1880 to 1910 twice. | S055, S302 |
| Charles Denis Sauter Bourbaki (General) French; MacTutor prints "Charles Soter Bourbaki" | dates unknown Not known | boor-buh-KEE not sourced | France / France | S276 | |
| Charles Jean de la Vallee Poussin | dates unknown Not known | duh lah vah-LAY poo-SAN not sourced | Belgium / Louvain | Appears in Chapter 7, and in the timeline under 1880 to 1910. | S423 |
| Charles Sanders Peirce | dates unknown Not known | pɜrs, pɪərs PURSS | USA / USA | Named in the registers, not in the story or the timeline. | S058, S064, S423 |
| Charlotte Ludovica Luisa of Anhalt-Dessau | dates unknown Not known | not established | Anhalt-Dessau / Prussia | Appears in Chapter 1, and in the timeline under 1650 to 1800. | S051 |
| Christian Genest | dates unknown Not known | zhuh-NEH not sourced | Canada / McGill University | Appears in Chapter 3. | S081, S082, S094 |
| Christian Kramp | dates unknown Not known | KRAHMP not sourced | Cologne, Strasbourg | 9Appears in Chapter 9 17 times, and in the timeline under 1800 to 1850 twice. | S301 |
| Christian Reinsch Christian Reinsch | dates unknown Not known | RINE-sh not sourced | Germany / Munich | S200 | |
| Clasen (initials contested: "J. B." or "B.-I.") | dates unknown Not known | KLAH-sen not sourced | Brussels | Named in the registers, not in the story or the timeline. | S134, S142, S144 |
| Claude Chevalley | dates unknown Not known | shuh-vah-LAY not sourced | France / France; USA | Appears in Chapter 8, and in the timeline under 1910 to 1935. | S276 |
| Claude Elwood Shannon | dates unknown Not known | ˈʃæn ən SHAN-un | USA / Bell Telephone Laboratories; MIT | 6Appears in Chapter 6 9 times, and in the timeline under 1935 to 1960. | S215, S225 |
| Cuthbert Edmund Cullis | dates unknown Not known | KUTH-bert KUL-is not sourced | Cambridge, Calcutta | Named in the registers, not in the story or the timeline. | S301 |
| Cyrus Colton MacDuffee | dates unknown Not known | mak-DUFF-ee not sourced | United States / Wisconsin | In the timeline 1910 to 1935 twice. | S304 |
| David R. Bellhouse | dates unknown Not known | BEL-howss not sourced | University of Western Ontario | Appears in Chapter 3 twice, and in the timeline under 1650 to 1800 twice. | S268 |
| Denes Konig Konig Denes | dates unknown Not known | DAY-nesh KUR-nig not sourced | Budapest / Budapest | 7 | S122, S129, S134, S138, S232, S420, S421, S422, S429, S445 |
| Duncan J. Watts | dates unknown Not known | wɒts WOTS | Australia / Cornell, Columbia | 7Appears in Chapter 7 5 times, and in the timeline under 1960 to now. | S440, S441 |
| Edouard Lucas | dates unknown Not known | ay-DWAR loo-KAH not sourced | France / France | 2Appears in Chapter 2 twice, and in the timeline under 1850 to 1880. | S011 |
| Eduard Stiefel Eduard Stiefel | dates unknown Not known | SHTEE-fel not sourced | Switzerland / ETH Zurich | S190 | |
| Edward Teller | dates unknown Not known | TELL-er not sourced | Budapest, Hungary / Los Alamos; University of Chicago | Appears in Chapter 6. | S226, S256 |
| Edward Wilson Chittenden | dates unknown Not known | CHIT-en-den not sourced | United States / Iowa | Appears in Chapter 9, and in the timeline under 1910 to 1935. | S055, S302 |
| Edwin Bidwell Wilson | dates unknown Not known | ˈwɪl sən WIL-son | United States / Yale, Harvard | 9In the timeline 1880 to 1910 twice. | S304 |
| Egon S. Pearson Egon Sharpe Pearson | dates unknown Not known | EE-gon PEER-suhn not sourced | England / University College London | S055 | |
| Ernest Tilden Parker | dates unknown Not known | ˈpɑr kər PAR-ker | USA / Illinois | 7Appears in Chapter 7. | S443 |
| Ernst Eduard Kummer Ernst Eduard Kummer | dates unknown Not known | ˈkʊm ər KOOM-er | Prussia / Berlin | S062, S173, S181, S187 | |
| Ernst Schroder Ernst Schröder | dates unknown Not known | ernst SHRUH-der not sourced | Germany / Karlsruhe | S054, S064 | |
| Esther Klein Szekeres Klein Eszter | 2005 Not known | ˈsɛ.kɛ.rɛʃ (Szekeres) KLYNE SEH-keh-resh | Budapest / Shanghai, Adelaide | 7 | S434 |
| Etienne Bezout | dates unknown Not known | bay-ZOO not sourced | France / Paris | Appears in Chapter 4 3 times, and in the timeline under 1650 to 1800 twice. | S122, S129 |
| Eugene Seneta | dates unknown Not known | suh-NET-uh not sourced | University of Sydney | Appears in Chapter 3 and Chapter 6 3 times. | S385, S392 |
| Evgeny Slutsky Евгений Слуцкий | dates unknown Not known | SLOOT-skee not sourced | Russian Empire / Kiev, Moscow | In the timeline 1910 to 1935 twice. | S089 |
| Florian Cajori | dates unknown Not known | ka-JOR-ee not sourced | Colorado Springs, Berkeley | 9Appears in Chapter 1 and Chapter 9 13 times, and in the timeline under 1800 to 1850 and 1880 to 1910. | S300 |
| Francis Guthrie | dates unknown Not known | AHD respelling (gŭthrē) GUTH-ree | England / London, South Africa | 7Appears in Chapter 7 twice, and in the timeline under 1850 to 1880. | S423 |
| Frans van Schooten Frans van Schooten | dates unknown Not known | frahns vahn SKHOH-ten not sourced | Leiden / Leiden | S076 | |
| Frantisek Prihonsky František Přihonský | dates unknown Not known | FRAHN-tyi-shek PRZHI-hon-skee not sourced | Bohemia / Bohemia | S044 | |
| Frederick Guthrie | dates unknown Not known | AHD respelling (gŭthrē) GUTH-ree | England / London | Named in the registers, not in the story or the timeline. | S423 |
| Friedrich Wilhelm Bessel | dates unknown Not known | ˈbɛs əl BESS-'l | Germany / Konigsberg | Named in the registers, not in the story or the timeline. | S134 |
| Frigyes Karinthy Karinthy Frigyes | dates unknown Not known | FRIH-jesh KAH-rin-tee not sourced | Budapest / Budapest | 7 | S439 |
| Gabriel Andrew Dirac | dates unknown Not known | dɪˈræk dih-RAK | Hungary and Britain / Aarhus | Appears in Chapter 5 twice and Chapter 7 twice. | S425 |
| Gabriel Cramer | dates unknown Not known | ˈkreɪ mərz (in "Cramer's rule") KRAY-mer (English) | Geneva / Geneva | 4Appears in Chapter 4 10 times, and in the timeline under 1400 to 1650, 1650 to 1800 3 times and 1935 to 1960 twice. | S122, S129, S130, S131 |
| Gene H. Golub | dates unknown Not known | GOH-lub not sourced | USA / Stanford | 5Appears in Chapter 5 8 times and Chapter 8, and in the timeline under 1960 to now 8 times. | S193, S200 |
| George Forsythe | dates unknown Not known | FOR-syth not sourced | United States / Stanford | Appears in Chapter 4 4 times and Chapter 5, and in the timeline under 1935 to 1960. | S120, S195 |
| George Udny Yule | dates unknown Not known | jul (US); juːl (UK) YOOL | Scotland / London, Cambridge | Named in the registers, not in the story or the timeline. | S270 |
| Georges Gonthier | dates unknown Not known | ZHORZH gon-tee-AY not sourced | France / Cambridge, England | 7Appears in Chapter 7 4 times, and in the timeline under 1960 to now. | S427 |
| Giovanni Jacopo Marinoni | dates unknown Not known | mah-ree-NOH-nee not sourced | Italy / Vienna | 7Appears in Chapter 7 twice, and in the timeline under 1650 to 1800. | S421, S422 |
| Glenn Shafer | dates unknown Not known | SHAY-fer not sourced | United States / Rutgers University | Appears in Chapter 3 twice and Chapter 6 3 times. | S089, S090, S092, S217, S218, S236, S237, S384 |
| Godfrey Harold Hardy | dates unknown Not known | ˈhɑr di HAR-dee | England / Trinity College, Cambridge | 8Appears in Chapter 5 and Chapter 8, and in the timeline under 1880 to 1910 4 times, 1910 to 1935 twice, 1935 to 1960 and 1960 to now twice. | S270, S388 |
| Gotthold Eisenstein | dates unknown Not known | ˈaɪ zənˌʃtaɪn, ˈaɪ zənˌstaɪn EYE-zen-shtyne | Berlin / Berlin | Appears in Chapter 4, and in the timeline under 1800 to 1850. | S122, S137 |
| Guillaume de l'Hopital, Marquis l'Hospital (older spelling) | dates unknown Not known | loh-pee-TAL not sourced | France / Paris | S079, S129 | |
| Guo Shuchun 郭書春 | dates unknown Not known | gwoh SHOO-chwun not sourced | China / Chinese Academy of Sciences, Beijing | Appears in Chapter 4 twice. | S125 |
| Halayudha हलायुध | dates unknown Not known | huh-LAH-yoo-dhuh not sourced | India / India | 2Appears in Chapter 2 9 times, and in the timeline under 500 to 1400 twice. | S002 |
| Harold James Ruthven Murray | dates unknown Not known | MUR-ee not sourced | England / Oxford | Appears in Chapter 7 twice, and in the timeline under 1935 to 1960. | S431 |
| Harold Jeffreys | dates unknown Not known | ˈdʒɛf riz JEF-reez | England / Cambridge | 9Appears in Chapter 3 twice and Chapter 9 3 times, and in the timeline under 1910 to 1935. | S088, S303 |
| Harvey Goodwin | dates unknown Not known | GOOD-win not sourced | England / Caius College, Cambridge | 9Appears in Chapter 9 5 times, and in the timeline under 1800 to 1850. | S189, S301 |
| Hayashi Tsuruichi 林鶴一 | dates unknown Not known | hah-YAH-shee tsoo-roo-EE-chee not sourced | Japan / Japan | Named in the registers, not in the story or the timeline. | S127 |
| Heinrich Heesch | dates unknown Not known | HYNE-rikh HAYSH not sourced | Germany / Hanover | 7Appears in Chapter 7, and in the timeline under 1960 to now. | S423 |
| Heinz Rutishauser Heinz Rutishauser | dates unknown Not known | ROO-tee-how-zer not sourced | Switzerland / ETH Zurich | S193 | |
| Henri Cartan | dates unknown Not known | on-REE kar-TAHN not sourced | France / Strasbourg; Paris | Appears in Chapter 8, and in the timeline under 1910 to 1935. | S276, S277 |
| Henry Billingsley | dates unknown Not known | BIL-ingz-lee not sourced | England / London | In the timeline 1400 to 1650 twice. | S302 |
| Henry Lewis Rietz | dates unknown Not known | REETS not sourced | United States / Iowa | Named in the registers, not in the story or the timeline. | S303 |
| Henry Thomas Colebrooke | dates unknown Not known | KOHL-brook not sourced | England / Calcutta, London | Named in the registers, not in the story or the timeline. | S021 |
| Henry Warburton | dates unknown Not known | WOR-ber-ton not sourced | England / Cambridge | 9Appears in Chapter 9 3 times, and in the timeline under 1800 to 1850. | S301 |
| Henry Whitehead (Reverend) | dates unknown Not known | ˈʰwaɪtˌhɛd, ˈwaɪt- WITE-hed | England / St Luke's, Berwick Street, London | 8Named in the registers, not in the story or the timeline. | S269 |
| Herman H. Goldstine | dates unknown Not known | GOLD-steen not sourced | USA / Princeton | 5Appears in Chapter 5 4 times, and in the timeline under 1935 to 1960. | S120, S188, S189 |
| Hilda Geiringer Hilda Geiringer | dates unknown Not known | HIL-duh GY-ring-er not sourced | Vienna / Berlin, USA | S067 | |
| Hugo Steinhaus Hugo Steinhaus | dates unknown Not known | SHTYNE-hows not sourced | Galicia / Lwow, Wroclaw | S089 | |
| Ibn Mun'im al-Abdari ابن منعم العبدري | 1228 Not known | ib-n moon-EEM not sourced | Denia, Andalusia / Marrakesh | 2Appears in Chapter 2, and in the timeline under 500 to 1400. | S015, S016 |
| Ibn al-Banna al-Marrakushi ابن البنّاء المراكشي | dates unknown Not known | ib-n al-BAN-nah not sourced | Marrakesh / Marrakesh | 2Appears in Chapter 2. | S016 |
| Ignaz Klein | dates unknown Not known | IG-nahts KLYNE not sourced | Hungary / Budapest | Appears in Chapter 6, Chapter 7 8 times and Chapter 8 5 times, and in the timeline under 1880 to 1910 twice, 1910 to 1935 twice and 1935 to 1960 3 times. | S434 |
| J. B. Durrande | dates unknown Not known | doo-RAHND not sourced | France / Gergonne's Annales, Nismes | 9Appears in Chapter 9 6 times, and in the timeline under 1800 to 1850 twice. | S301 |
| Jacques Philippe Marie Binet | dates unknown Not known | bɪˈneɪ, biˈnɛ bih-NAY | France / Paris | 4Appears in Chapter 4 5 times, and in the timeline under 1800 to 1850 twice. | S122, S129 |
| James Pierpont | dates unknown Not known | PEER-pont not sourced | United States / Yale | 9Appears in Chapter 9, and in the timeline under 1910 to 1935. | S055, S302 |
| Jean Delsarte | dates unknown Not known | del-SART not sourced | France / Nancy | Appears in Chapter 8, and in the timeline under 1910 to 1935. | S276 |
| Jean Dieudonne | dates unknown Not known | dyuh-doh-NAY not sourced | France / Nancy; Nice | Appears in Chapter 5 and Chapter 8, and in the timeline under 1910 to 1935. | S276 |
| Jerzy Neyman Jerzy Neyman | dates unknown Not known | YAIR-zhee NAY-mahn not sourced | Bendery (not sourced here) / London, Berkeley | S055, S067, S250, S265 | |
| Jia Xian 賈憲 | dates unknown Not known | jyah shyen not sourced | China / Song China | 2Appears in Chapter 2 9 times, and in the timeline under 500 to 1400 3 times. | S013, S014, S017 |
| Johann Christian Lange Johann Christian Lange | dates unknown Not known | YO-hahn KRIS-tee-ahn LAHNG-uh not sourced | Germany / Giessen | S051 | |
| Johann Christoph Sturm Johann Christoph Sturm | dates unknown Not known | YO-hahn KRIS-toff SHTOORM not sourced | Germany / Altdorf | 1 | S051 |
| John Canton | dates unknown Not known | KAN-ton not sourced | England / London | Named in the registers, not in the story or the timeline. | S083 |
| John G. F. Francis | dates unknown Not known | FRAN-sis not sourced | England / NRDC, Ferranti | 5, 8Appears in Chapter 5 5 times, Chapter 7 twice and Chapter 8 5 times, and in the timeline under 1850 to 1880, 1935 to 1960 twice and 1960 to now 4 times. | S193 |
| John Koch | dates unknown Not known | KOKE not sourced | USA / Urbana, Illinois | 7Appears in Chapter 7. | S423, S426 |
| John Snow | dates unknown Not known | snoʊ SNOH | England / London | 8Appears in Chapter 8 twice, and in the timeline under 1850 to 1880 5 times and 1960 to now. | S267, S269, S330, S333 |
| John von Neumann Neumann Janos | dates unknown Not known | vɒn ˈnɔɪ mɑn, -mən von NOY-mahn | Budapest / Princeton | 5 | S120, S188, S189, S228 |
| Joseph Henry Maclagan Wedderburn | dates unknown Not known | WED-er-burn not sourced | Scotland / Princeton | In the timeline 1910 to 1935 twice. | S304 |
| Joseph-Louis Lagrange | dates unknown Not known | ləˈɡreɪndʒ, laˈɡrɑ̃ʒ luh-GRAYNJ (English), la-GRAHNZH (French) | Turin / Berlin, Paris | Appears in Chapter 4 and Chapter 5, and in the timeline under 1650 to 1800 twice. | S122, S305 |
| Judith S. Kleinfeld | dates unknown Not known | KLYNE-feld not sourced | USA / Fairbanks, Alaska | 7Appears in Chapter 7 3 times, and in the timeline under 1960 to now. | S440 |
| Judith Stupanus | dates unknown Not known | YOO-dit shtoo-PAH-nus not sourced | Basel / Basel; Jacob Bernoulli's widow | Appears in Chapter 3 twice. | S079 |
| Jyesthadeva Malayalam script NOT sourced | dates unknown Not known | jyesh-tuh-DAY-vuh not sourced | Kerala, India / Kerala | S284 | |
| Kedara Bhatta केदार | dates unknown Not known | KAY-dah-ruh not sourced | India / India | Named in the registers, not in the story or the timeline. | S002 |
| Kenneth Ira Appel | dates unknown Not known | AP-pel not sourced | USA / Urbana, Illinois | 7Appears in Chapter 7 3 times, and in the timeline under 1960 to now 3 times. | S423, S426 |
| Lam Lay Yong 藍麗蓉 | dates unknown Not known | lahm lay yong not sourced | Singapore / National University of Singapore | Appears in Chapter 4 twice. | S126 |
| Lawrence (Larry) Page | dates unknown Not known | PAYJ not sourced | USA / Stanford, Google | Appears in Chapter 5 twice and Chapter 9, and in the timeline under 1960 to now. | S197, S198 |
| Leonard E. Baum | dates unknown Not known | BAWM not sourced | USA / Institute for Defense Analyses, Princeton | 6In the timeline 1960 to now twice. | S239, S243 |
| Leonard Eugene Dickson | dates unknown Not known | DIK-son not sourced | United States / Chicago | Named in the registers, not in the story or the timeline. | S302 |
| Leonard Isaacson | dates unknown Not known | EYE-zuk-sun not sourced | USA / University of Illinois | In the timeline 1935 to 1960 4 times. | S241, S242, S272, S273, S274 |
| Lester S. Hill | dates unknown Not known | HILL not sourced | United States / Hunter College, New York | 4Appears in Chapter 4 11 times and Chapter 8, and in the timeline under 1910 to 1935 6 times and 1935 to 1960. | S140, S141 |
| Li Chunfeng 李淳風 | dates unknown Not known | lee chwun-FUNG not sourced | China / Tang China | In the timeline 500 to 1400 twice. | S123 |
| Liu Hui 劉徽 | dates unknown Not known | lyoh HWAY not sourced | China / Wei state, China | 4Appears in Chapter 4 3 times, and in the timeline under Before 500 twice and 500 to 1400. | S120, S123, S124, S125 |
| Louis Weisner | dates unknown Not known | WYZE-ner not sourced | United States / Hunter College, New York | 4Appears in Chapter 4 6 times, and in the timeline under 1910 to 1935 3 times. | S141 |
| Luca Pacioli Fra Luca Pacioli | dates unknown Not known | LOO-kah pah-CHOH-lee not sourced | Italy / Italy | S072, S093 | |
| Ludwig Stickelberger | dates unknown Not known | STIK-el-bair-ger not sourced | Switzerland / Freiburg | Named in the registers, not in the story or the timeline. | S128 |
| Luigi Federico Menabrea Italian | dates unknown Not known | meh-nah-BRAY-ah not sourced | Chambery (NOT verified) / Turin | 8 | S259, S260 |
| M. Rangacharya | dates unknown Not known | ran-gah-CHAR-yuh not sourced | India / Madras | Appears in Chapter 2 twice. | S009 |
| Magnus R. Hestenes | dates unknown Not known | HESS-tuh-neez not sourced | USA / NBS, UCLA | Appears in Chapter 5, and in the timeline under 1935 to 1960. | S190 |
| Mahavira (Mahaviracarya) महावीराचार्य | dates unknown Not known | məˌhɑˈvɪər ə muh-hah-VEER-uh | India / southern India | 2Named in the registers, not in the story or the timeline. | S009 |
| Mark Kac Marek Kac | dates unknown Not known | KATS not sourced | Krzemieniec, Poland (now Ukraine) / Cornell; Rockefeller University | 6 | S221 |
| Marko Petkovsek Marko Petkovsek (Slovene, hacek on s and c) | dates unknown Not known | PET-kov-shek not sourced | Slovenia / University of Ljubljana | S254, S255 | |
| Marshall N. Rosenbluth | dates unknown Not known | ROH-zun-blooth not sourced | USA / Los Alamos; UC San Diego | 6Appears in Chapter 6. | S226, S256 |
| Martin C. Kohli | dates unknown Not known | KOH-lee not sourced | USA / U.S. Bureau of Labor Statistics | Named in the registers, not in the story or the timeline. | S283 |
| Martin Mattmuller Martin Mattmuller | dates unknown Not known | MAT-mue-ler not sourced | Switzerland / Bernoulli-Euler Zentrum, Basel | S079, S080 | |
| Maxime Bocher | dates unknown Not known | mak-SEEM BOH-ker not sourced | United States / Harvard | Appears in Chapter 4, and in the timeline under 1880 to 1910. | S122, S134, S176, S304 |
| Myrick Hascall Doolittle | dates unknown Not known | DOO-lit-'l not sourced | United States / US Coast Survey, Washington | In the timeline 1880 to 1910 twice. | S120 |
| N. Chidambaram Iyer | dates unknown Not known | chih-DUM-buh-rum EYE-yer not sourced | India / India | Appears in Chapter 2 3 times, and in the timeline under 1880 to 1910 twice. | S010 |
| Nami of Guzerat | dates unknown Not known | NAH-mee not sourced | Gujarat / Gujarat | Appears in Chapter 7, and in the timeline under 500 to 1400. | S431 |
| Narayana Pandita नारायण पण्डित | dates unknown Not known | nah-RAH-yuh-nuh PUN-dih-tuh not sourced | India / India | 2Appears in Chapter 2, and in the timeline under 500 to 1400. | S020 |
| Nathan Morrison | dates unknown Not known | MOR-i-son not sourced | United States / translator of the Grundbegriffe | Appears in Chapter 3, and in the timeline under 1935 to 1960 twice. | S066, S090 |
| Nicholas Metropolis Greek surname, Latin script | dates unknown Not known | muh-TROP-uh-liss not sourced | Chicago, USA / Los Alamos Scientific Laboratory | 6 | S216, S226, S256, S302 |
| Nicolas Bourbaki (collective pseudonym) | dates unknown Not known | /ˈbɔːbəkɪ/ BOOR-buh-kee | Poldavia (fictional) / Paris; "Nancago" | 8Named in the registers, not in the story or the timeline. | S054, S275 |
| Nicolas Chuquet Nicolas Chuquet | dates unknown Not known | shoo-KAY not sourced | France / Lyon | ||
| Omar Khayyam عمر خیام | dates unknown Not known | kaɪˈjɑm, -ˈjæm (US); kaɪˈɑːm (UK) oh-MAR ky-YAHM | Persia / Persia | 2Appears in Chapter 2, and in the timeline under 500 to 1400. | S013 |
| Oskar Perron Oskar Perron | dates unknown Not known | peh-ROHN not sourced | Germany / Munich | 5 | S185, S186, S204 |
| Oswald Veblen | dates unknown Not known | ˈvɛb lən VEB-lun | USA / Princeton | Named in the registers, not in the story or the timeline. | S423 |
| Oystein Ore Oystein Ore | dates unknown Not known | OY-styne OR-uh not sourced | Norway / Yale University | S073, S093 | |
| Paul Ehrenfest | 1933 Not known | AIR-un-fest not sourced | Vienna, Austria / St Petersburg 1907-1912; Leiden | 6Appears in Chapter 6 10 times and Chapter 8, and in the timeline under 1880 to 1910 4 times and 1910 to 1935 4 times. | S221, S222, S223, S224 |
| Paul Richard Halmos | dates unknown Not known | HAL-mohs not sourced | Hungary / United States | 5, 9Appears in Chapter 5 twice and Chapter 9 3 times, and in the timeline under 1935 to 1960 and 1960 to now twice. | S176, S302 |
| Paul Turan Turan Pal | dates unknown Not known | TOO-rahn not sourced | Budapest / Budapest | S438 | |
| Paul Wernicke | dates unknown Not known | VAIR-nih-keh not sourced | Appears in Chapter 7, and in the timeline under 1880 to 1910. | S423, S442 | |
| Philip E. B. Jourdain Philip Edward Bertrand Jourdain | dates unknown Not known | JOR-dayn not sourced | England / England | S041 | |
| Philip Franklin | dates unknown Not known | FRANK-lin not sourced | USA / MIT | In the timeline 1910 to 1935. | S423 |
| Pierre Remond de Montmort | dates unknown Not known | mon-MOR not sourced | France / France | 2Appears in Chapter 2 3 times, and in the timeline under 1650 to 1800 3 times. | S011, S072, S079, S080, S081 |
| Pieter Hendrik Schoute Dutch | dates unknown Not known | SKHOW-tuh not sourced | Netherlands / University of Groningen | S266 | |
| Pingala पिङ्गल | dates unknown Not known | PING-guh-luh not sourced | India / India | 2Appears in Chapter 2 7 times and Chapter 7, and in the timeline under Before 500. | S002, S011 |
| Prakash Gorroochurn | dates unknown Not known | gor-oo-CHURN not sourced | Mauritius (not sourced here) / Columbia University | Appears in Chapter 3 twice, and in the timeline under 1400 to 1650 twice and 1650 to 1800. | S071, S072, S073, S086, S093 |
| Raoul Husson | dates unknown Not known | rah-OOL oo-SOHN not sourced | France / Ecole Normale Superieure, Paris | Appears in Chapter 8, and in the timeline under 1910 to 1935 twice. | S276 |
| Reginald Crundall Punnett | dates unknown Not known | PUN-et not sourced | England / Cambridge | Named in the registers, not in the story or the timeline. | S270 |
| Rene de Possel | dates unknown Not known | duh poh-SELL not sourced | France / Clermont-Ferrand | Appears in Chapter 8, and in the timeline under 1910 to 1935. | S276 |
| Richard J. Pulskamp | dates unknown Not known | PULSS-kamp not sourced | United States / Xavier University | Appears in Chapter 3. | S091, S092 |
| Richard Price | dates unknown Not known | PRYSS not sourced | Wales / Newington Green, London | 3Appears in Chapter 3 4 times, and in the timeline under 1650 to 1800 3 times. | S083, S084, S085, S094 |
| Richard Rado | dates unknown Not known | RAH-doh not sourced | Germany / Reading, England | Named in the registers, not in the story or the timeline. | S433 |
| Ronald Aylmer Fisher | dates unknown Not known | ˈfɪʃ ər FISH-er | London / Rothamsted, Cambridge | Named in the registers, not in the story or the timeline. | S023, S055, S088, S265 |
| Rudrata रुद्रट | dates unknown Not known | ROOD-ruh-tuh not sourced | Kashmir / Kashmir | 7Appears in Chapter 7 twice, and in the timeline under 500 to 1400 twice. | S431 |
| Semyon Aranovich Gersgorin (Gershgorin) Семён Аранович Гершгорин | dates unknown Not known | GERSH-gaw-rin not sourced | Russian Empire / USSR | 5Named in the registers, not in the story or the timeline. | S191 |
| Sergei Aksakov | dates unknown Not known | ahk-SAH-koff not sourced | Russia / Russia | Appears in Chapter 6 twice. | S213, S271 |
| Sergey Brin Сергей Брин | dates unknown Not known | SUR-gay BRIN not sourced | Moscow / Stanford, Google | Appears in Chapter 5 twice, and in the timeline under 1960 to now. | S197, S198 |
| Sharadchandra Shankar Shrikhande शरदचंद्र शंकर श्रीखंडे | dates unknown Not known | shuh-rud-CHUN-druh shree-KUN-day not sourced | India / India, Chapel Hill | 7Appears in Chapter 7 3 times, and in the timeline under 1650 to 1800, 1935 to 1960 twice and 1960 to now. | S442, S443 |
| Sridhara श्रीधर | dates unknown Not known | SHREE-duh-ruh not sourced | India / India | 2In the timeline 500 to 1400. | S002 |
| Stanislaw Marcin Ulam | dates unknown Not known | STAN-iss-wahf OO-lahm not sourced | Lwow / Los Alamos | In the timeline 1935 to 1960 4 times. | S089, S216, S226, S228, S302 |
| Stanley Milgram | dates unknown Not known | MIL-gram not sourced | New York / Harvard, Yale, CUNY | 7Appears in Chapter 7 7 times, and in the timeline under 1960 to now. | S440 |
| Stefan Banach Stefan Banach | dates unknown Not known | BAH-nahkh not sourced | Krakow / Lwow | 5 | S173, S179, S390 |
| Stephen M. Stigler | dates unknown Not known | STIG-ler not sourced | United States / University of Chicago | Appears in Chapter 3 twice. | S384, S391 |
| Steven H. Strogatz | dates unknown Not known | STROH-gats not sourced | USA / Cornell | 7Appears in Chapter 7 5 times, and in the timeline under 1960 to now. | S441 |
| Thomas Bayes | 1761 Not known | beɪz BAYZ | London / Tunbridge Wells | 3Appears in Chapter 3 16 times, and in the timeline under 1650 to 1800 14 times and 1800 to 1850 3 times. | S083, S084, S085, S094 |
| Thomas Digges | dates unknown Not known | DIGZ not sourced | England / England | Named in the registers, not in the story or the timeline. | S302 |
| Thomas Hawkins | dates unknown Not known | HAW-kinz not sourced | United States / Boston University | Appears in Chapter 4 and Chapter 5, and in the timeline under 1850 to 1880 twice. | S137 |
| Thomas Jarrett | dates unknown Not known | JARR-et not sourced | England / St Catherine's College, Cambridge | 9Appears in Chapter 9 10 times, and in the timeline under 1800 to 1850 twice. | S301 |
| Thomas Strode | dates unknown Not known | STROHD not sourced | England / England | In the timeline 1650 to 1800 twice. | S011, S302 |
| Thomas Tymoczko | dates unknown Not known | tih-MOTCH-koh not sourced | USA / Smith College | 7Appears in Chapter 7 3 times, and in the timeline under 1960 to now. | S428 |
| Varahamihira वराहमिहिर | dates unknown Not known | vuh-RAH-huh-MIH-hih-ruh not sourced | India / India | 2Appears in Chapter 2 3 times, and in the timeline under 500 to 1400. | S010 |
| Vera Sanford | dates unknown Not known | SAN-ford not sourced | United States / Western Reserve University | Appears in Chapter 3. | S074 |
| Virahanka विरहाङ्क | dates unknown Not known | vih-ruh-HAHN-kuh not sourced | India / India | 2In the timeline 500 to 1400. | S002 |
| Vladimir Vovk | dates unknown Not known | VOAFK not sourced | Russia / Royal Holloway, London | Appears in Chapter 3 twice and Chapter 6 3 times. | S092 |
| Vsevolod Ivanovich Romanovsky Всеволод Иванович Романовский | dates unknown Not known | ruh-mah-NOFF-skee not sourced | Russian Empire / Tashkent | 6In the timeline 1910 to 1935. | S216 |
| W. Keith Hastings | dates unknown Not known | ˈheɪ stɪŋz HAY-stingz | Canada / University of Toronto; Memorial University of Newfoundland | 6Appears in Chapter 6 twice, and in the timeline under 1960 to now. | S227, S228 |
| William (Velvel) Kahan | dates unknown Not known | KAY-hun not sourced | Canada / Toronto, Berkeley | Appears in Chapter 5, and in the timeline under 1960 to now twice. | S200 |
| William Allen Whitworth | dates unknown Not known | ˈʰwɪtˌwɜrθ, ˈwɪt- WIT-werth | England / Cambridge, Liverpool | 9Named in the registers, not in the story or the timeline. | S301, S303 |
| William Andrew Rogers | dates unknown Not known | ROJ-erz not sourced | USA / Atlanta University | Appears in Chapter 8. | S281 |
| William Feller Vilibald Feller | dates unknown Not known | FELL-er not sourced | Zagreb (not sourced here) / London, Princeton | 9 | S055, S088, S233, S303 |
| William Frend | dates unknown Not known | FREND not sourced | England / Cambridge, London | In the timeline 1650 to 1800. | S259 |
| William Petty | dates unknown Not known | PET-ee not sourced | England / London, Ireland | Named in the registers, not in the story or the timeline. | S077 |
| Woldemar Voigt | dates unknown Not known | VOHL-de-mar FOHKT not sourced | Germany / Gottingen | Named in the registers, not in the story or the timeline. | S302 |
| Wolfgang Haken | dates unknown Not known | VOLF-gang HAH-ken not sourced | Germany / Urbana, Illinois | 7Appears in Chapter 7 3 times, and in the timeline under 1960 to now 3 times. | S423, S426 |
| Yang Hui 楊輝 | dates unknown Not known | yahng hway not sourced | Qiantang, Zhejiang / Southern Song China | 2Appears in Chapter 2 5 times, and in the timeline under 500 to 1400, 1400 to 1650 and 1800 to 1850. | S011, S013, S014, S022, S128 |
| Yoshio Mikami 三上義夫 | dates unknown Not known | mee-KAH-mee not sourced | Japan / Japan | Appears in Chapter 4 3 times. | S022, S127 |
| Yuri Vladimirovich Matiyasevich | dates unknown Not known | mah-tee-yah-SAY-vich not sourced | Russia / Leningrad / St Petersburg | In the timeline 1960 to now twice. | S279 |
| Zhang Cang 張蒼 | dates unknown Not known | jahng TSAHNG not sourced | China / Han China | Appears in Chapter 4, and in the timeline under Before 500. | S123 |
| al-Adli ar-Rumi العدلي الرومي | dates unknown Not known | al-AD-lee ar-ROO-mee not sourced | Abbasid Baghdad | 7Appears in Chapter 7, and in the timeline under 500 to 1400. | S431 |
| al-Khwarizmi Abu Ja'far Muhammad ibn Musa al-Khwarizmi | dates unknown Not known | ˈxwɑr ɪzˌmi al-KHWAR-iz-mee | Khwarazm / Baghdad | 9 | S306 |
Appendix B
Symbols, words, and where they came from
142 entries: 107 words and 35 glyphs, with where each came from and what it meant before it meant what it means now. 108 of the 142 carry a first-use year. The rest do not, because nothing consulted establishes one, and a plausible year would be worse than a blank.
Read the literal meanings in one sitting and a pattern appears. A startling number of these words are about wombs, dice, boards, threads and doors: physical things a person could point at. The abstractions came later and kept the old names. The other pattern is how recent the vocabulary is. Most of Unit 1's words entered English between 1909 and 1926, which is to say inside one lifetime.
| Term or glyph | Kind | First use | Came from | What it literally meant | Sources |
|---|---|---|---|---|---|
| Aleph-null | symbol | 1895 Georg Cantor | the first letter of the Hebrew alphabet | cardinality of a countable set | S054 |
| Arrow over a vector | symbol | not established | an arrow | handwritten vector mark | S304 |
| Binomial coefficient bracket | symbol | 1827 Andreas von Ettingshausen | a large parenthesis holding two stacked numerals | n choose k | S301 |
| Bold face, for vectors and matrices | symbol | 1901 Edwin Bidwell Wilson | Clarendon, a heavy slab-serif printing type | v, A | S304 |
| Bold pi, for a stationary distribution | symbol | not established | the sixteenth letter of the Greek alphabet, already carrying the circle constant | the stationary distribution vector | |
| Braces, { and } | symbol | 1908 Ernst Zermelo | printers' braces, used for bracketing lines of verse and music | set-builder and roster notation | S054 |
| Complement notation | symbol | not established | letters and diacritics | A with a superscript c, a prime, or a bar | S302, S054 |
| Determinant bars, a single vertical line each side | symbol | 1841 Arthur Cayley | the vertical rule, a printer's sort | det A written with bars | S304 |
| Double subscript, a with i and j | symbol | not established | subscript numerals | matrix entry | S304 |
| Double vertical bars, around an array | symbol | 1843 Arthur Cayley | two vertical rules | matrix delimiter | S301, S304 |
| Empty set glyph, a slashed O | symbol | 1939 Nicolas Bourbaki (Andre Weil claims personal responsibility) | a letter of the Norwegian and Danish alphabet | the empty set | S054 |
| Exclamation point, n! | symbol | 1808 Christian Kramp | the printers' "note of admiration" | factorial | S301 |
| Factorial, !n! | symbol | 1847 Henry Warburton | two exclamation points | an obsolete English notation | S301 |
| Factorial, bar and corner | symbol | 1827 Thomas Jarrett | a printer's rule bent at a right angle | the English rival to n! | S301 |
| Factorial, capital Gamma | symbol | 1808 Adrien-Marie Legendre | the third letter of the Greek alphabet | the factorial function extended | S301 |
| Factorial, capital M | symbol | 1751 Leonhard Euler | a capital letter | Euler's first contraction | S301 |
| Factorial, capital Pi | symbol | not established Carl Friedrich Gauss | the sixteenth letter of the Greek alphabet | n-factorial as Pi(n) | S301 |
| I, for the identity matrix | symbol | 1933 J. H. M. Wedderburn and C. C. MacDuffee | a capital letter | I | S304 |
| Inclusion signs | symbol | 1890 Ernst Schroder | modified less-than and greater-than signs | subset and superset | S054 |
| Lambda, for eigenvalues | symbol | not established | the eleventh letter of the Greek alphabet | the eigenvalue symbol | |
| Membership epsilon | symbol | 1889 Giuseppe Peano | the Greek letter epsilon | is an element of | S054 |
| nCr, nVr, nPr | symbol | 1886 W. A. Whitworth and contemporaries | letter-and-subscript compounds | combinations, variations, permutations | S301 |
| Norm, double bars | symbol | not established | two vertical rules | the length of a vector | |
| nPr alone | symbol | 1869 Harvey Goodwin | as above | permutations | S301 |
| P(A) | symbol | 1933 Andrei Nikolaevich Kolmogorov | a capital letter | probability of an event | S303 |
| Parentheses, round, around an array | symbol | 1909 Maxime Bocher and others | printers' parentheses | matrix delimiter | S304, S301 |
| Pr{A} | symbol | 1950 William Feller | a two-letter abbreviation | probability of an event | S303 |
| Script E | symbol | 1901 William Allen Whitworth | a capital letter in a script face | expectation | S303 |
| Sigma, capital | symbol | 1755 Leonhard Euler | the eighteenth letter of the Greek alphabet, S for summa | summation | S305 |
| Subfactorial sign | symbol | 1878 William Allen Whitworth | an inverted version of the Jarrett factorial sign | subfactorial n | S301 |
| Superscript T | symbol | 1933 C. C. MacDuffee | a capital letter set as a superscript | transpose | S304 |
| Triple vertical lines | symbol | not established W. A. Whitworth | three vertical rules | around a matrix | S301 |
| Union and intersection, cup and cap | symbol | 1888 Giuseppe Peano | rounded brackets rotated | union, intersection | S054 |
| Vertical bar, in P(A given B) | symbol | 1931 Harold Jeffreys | the vertical rule | conditional probability | S303 |
| Vertical bars, for cardinality | symbol | not established | the vertical rule | the size of a set | S054 |
| Absorbing (state) | word | not established | Latin, via Old French | absorbere, "to suck away," from ab "away" plus sorbere "suck in" | S306, S302 |
| Aleatory | word | not established | Latin | aleatorius, "pertaining to a dice player," from alea, "a die, a game with dice"; literally "depending on the throw of a die" | S306 |
| Algebra | word | 825 Abu Ja'far Muhammad ibn Musa al-Khwarizmi | Arabic | al-jabr, "the restoring" or "reunion of broken parts," from the title al-mukhtasar fi hisab al-jabr wa al-muqabala, "the compendium on calculation by restoring and balancing" | S302, S306 |
| Algorithm | word | 1684 Gottfried Wilhelm Leibniz | Arabic proper name, through Medieval Latin | algorismus, a mangled Latin transliteration of al-Khwarizmi, "the man from Khwarazm" | S302, S306 |
| Aperiodic | word | not established | Greek plus Latin prefix | a- "not" plus periodos, "a going round" | S302 |
| Arithmetic | word | not established Greek mathematicians | Greek | arithmetike (tekhne), "the counting art," from arithmos, "number" | S302, S306 |
| Axiom | word | 1485 not named | Greek | axioma, "that which is thought worthy or fit," from axios, "worthy" | S302, S306 |
| Basis | word | 1879 F. G. Frobenius and L. Stickelberger | Greek | basis, "a going, a step; a stand, a base," from bainein, "to go, walk, step" | S302, S306 |
| Bayes' rule | word | 1843 Antoine Augustin Cournot | English proper name plus Latin regula, "a straight stick, a rule" | "the rule of Bayes" | S302 |
| Binomial coefficient | word | 1733 not named | Latin | bi- "two" plus nomen "name, term"; coefficiens, "working together" | S302 |
| Cardinal (number) | word | 1538 Glareanus | Latin | cardinalis, "pertaining to a hinge," from cardo, "door hinge, pole of the sky"; hence "principal, the one the rest turns on" | S302, S306 |
| Cardinality | word | 1935 not named | Latin, via cardinal | "the condition of being cardinal" | S306 |
| Chance | word | 1778 not named | Latin, through Old French | Vulgar Latin cadentia, "that which falls out," from cadere, "to fall"; a dice metaphor | S306 |
| Characteristic equation | word | 1840 Augustin-Louis Cauchy | Greek kharakter, "an engraved mark, a stamp" | "the equation that stamps the thing" | S302 |
| Cipher | word | 1390 not named | Arabic, through Old French | sifr, "zero," literally "empty, nothing," itself rendering Sanskrit sunya-, "empty" | S306 |
| Combination | word | 1654 Blaise Pascal | Latin | combinare, from com- "together" plus bini "two by two"; literally "to yoke together two at a time" | S302, S306 |
| Combinatorics | word | 1666 Gottfried Wilhelm Leibniz | Latin, via Leibniz's coinage | ars combinatoria, "the combining art" | S302, S008 |
| Complement | word | 1914 E. W. Chittenden | Latin | complementum, "that which fills up," from complere, "to fill up" | S302, S306 |
| Conditional probability | word | not established | Latin | condicere, "to speak together, to agree upon"; a condition is a thing agreed in advance | S302 |
| Corollary | word | not established | Latin | corollarium, "money paid for a garland," from corolla, "a small garland"; hence "a tip, something extra" | S306, S302 |
| Detailed balance | word | not established | English plus Latin | balance from Latin (libra) bilanx, "a scale having two pans" | S302, S306 |
| Determinant | word | 1801 Carl Friedrich Gauss | Latin | determinare, "to mark the end or boundary of," from terminus, "an end, a limit"; the determining number | S302 |
| Determinant (modern sense) | word | 1815 Augustin-Louis Cauchy | as above | as above | S302 |
| Diagonal | word | not established Heron of Alexandria | Greek | diagonios, from dia "across, through" plus gonia "angle, corner"; "from angle to angle" | S302, S306 |
| Dimension | word | not established | Latin | dimensionem, "a measuring," from dimetiri, "to measure out" | S306, S302 |
| Discrete | word | 1570 Henry Billingsley | Latin | discretus, "separated," past participle of discernere, "to separate, to distinguish" | S302, S306 |
| Disjoint | word | 1909 Cassius J. Keyser | Latin, through Old French | disiungere, "to unyoke," from dis- apart plus iungere, "to join"; literally "unyoked" | S302, S306 |
| Eigenvalue | word | 1904 David Hilbert | German plus English, a hybrid | German eigen, "own, proper, characteristic," plus Wert, "value"; "its own value" | S302 |
| Eigenvector | word | 1924 Richard Courant and David Hilbert | German plus English | as above, plus Latin vector, "carrier" | S302 |
| Element | word | 1882 Georg Cantor | Latin | elementum, "a rudiment, a first principle, matter in its most basic form," rendering Greek stoikheion; ultimate origin unknown | S302, S306 |
| Empty set | word | 1919 J. E. McAtee | Old English plus Latin | aemettig, "at leisure, unoccupied," from aemetta, "leisure" | S302, S306 |
| Ensemble (French for set) | word | 1883 Georg Cantor, in translation | Latin, through French | insimul, "at the same time, together" | S302 |
| Ergodic | word | 1884 Ludwig Boltzmann | Greek | Boltzmann's coinage Ergode, from ergon, "work," plus hodos, "way" | S302 |
| Event | word | 1718 Abraham De Moivre | Latin | eventus, "an outcome, an occurrence," from evenire, "to come out," from ex- "out" plus venire "to come" | S302, S306 |
| Expectation | word | 1657 Frans van Schooten, translating Christiaan Huygens | Latin | expectatio, "an awaiting," from expectare, "to await, look out for" | S302 |
| Experiment | word | not established | Latin | experimentum, "a trial, a test," from experiri, "to try" | S306, S302 |
| Factorial | word | 1816 not named for the English word | Latin, through French and English | from factor, "a doer, a maker," from facere, "to do, to make"; the numbers that make the product | S306, S301 |
| Gaussian elimination (the English name in print) | word | 1953 George E. Forsythe for the name, not the method | German proper name plus Latin | eliminare, "to turn out of doors," from ex "out" plus limen "threshold" | S120 |
| Gaussian elimination (the method Gauss described) | word | 1811 Carl Friedrich Gauss for the method, not the name | German proper name plus Latin | eliminare, "to turn out of doors," from ex "out" plus limen "threshold" | S302 |
| Geometry | word | not established Greek mathematicians | Greek | geometria, from ge, "earth, land," plus -metria, "a measuring of"; "land measurement" | S302, S306 |
| Graph | word | 1878 James Joseph Sylvester | Greek, through the English compound "graphic formula" | graphein, "to write, to draw, to scratch" | S302, S306 |
| Identity matrix | word | 1908 Leonard Eugene Dickson | Latin | identitas, "sameness, oneness," from idem, "the same" | S302, S306 |
| Independence (of events) | word | 1738 Abraham De Moivre | Latin | in- "not" plus dependere, "to hang from" | S302, S306 |
| Induction (mathematical) | word | 1656 John Wallis | Latin | inductio, "a leading," Cicero's rendering of Greek epagoge, "leading to" | S302, S306 |
| Insieme (Italian for set) | word | not established | Latin | in simul, "in one and the same time, together" | |
| Intersection | word | 1909 not named | Latin | intersecare, "to cut asunder," from inter- "between" plus secare "to cut" | S302, S306 |
| Inverse (matrix) | word | 1858 Arthur Cayley | Latin | inversus, "turned about, turned upside down," from invertere | S302, S306 |
| Irreducible | word | 1856 Arthur Cayley | Latin | in- "not" plus reducere, "to lead back" | S302 |
| Latent root | word | 1883 James Joseph Sylvester | Latin | latere, "to hide, to lie hidden"; a root that is hiding inside the matrix | S302, S306 |
| Lemma | word | 1570 Henry Billingsley | Greek | lemma, "something received or taken," from a root meaning "to seize, to take" | S302, S306 |
| Likelihood | word | 1921 Ronald Aylmer Fisher | Old English plus suffix | likely plus -hood; the earliest sense is "resemblance, similarity" | S302, S306 |
| Linear | word | not established | Latin | linearis, from linea, "a string, a line," itself from linum, flax | S306 |
| Linear algebra (Benjamin Peirce's sense, a finite dimensional algebra over a field) | word | 1875 Benjamin Peirce | Latin plus Arabic | "algebra of lines" | S176 |
| Linear algebra (the phrase in English) | word | 1870 or 1881 not named | Latin plus Arabic | "algebra of lines" | S176, S302 |
| Linear combination | word | 1854 James Joseph Sylvester | Latin | as above | S302 |
| Markov chain | word | 1929 Vsevolod Romanovsky for the phrase, A. A. Markov for the object | Russian proper name plus Latin catena, "a chain" | "the chain of Markov" | S302 |
| Mathematics | word | not established Greek mathematicians | Greek | mathematike tekhne, "the learning art," from manthanein, "to learn" | S302, S306 |
| Matrix | word | 1850 James Joseph Sylvester | Latin | matrix, "womb, breeding female," from mater, "mother"; "the thing that gives birth to something" | S307, S302, S306 |
| Menge (Bolzano's sense, a totality whose arrangement is a matter of indifference) | word | 1851 Bernard Bolzano | German | Menge, "a quantity, a crowd, a multitude" | S044 |
| Menge (von Staudt's technical use, then Cantor's settled term) | word | 1856 Karl von Staudt, then Georg Cantor | German | Menge, "a quantity, a crowd, a multitude" | S302 |
| Monte Carlo | word | 1949 Nicholas Metropolis and Stanislaw Ulam | Monegasque place name | the casino quarter of Monaco, "Charles's mountain," after Charles III of Monaco | S302 |
| Odds | word | 1560 not named | English, from odd | things that "do not come out even"; an inequality | S302, S306 |
| Orthogonal | word | 1571 Thomas Digges | Greek | orthogonios, from orthos, "straight, right," plus gonia, "angle" | S302, S306 |
| Outcome | word | 1788 | English, Scottish | "that which comes out," from the verbal phrase out plus come | S306, S302 |
| Partition | word | not established | Latin | partitionem, "a sharing, a division, a distribution," from partire, "to divide," from pars, "a part" | S306, S302 |
| Permutation | word | 1678 Thomas Strode | Latin | permutationem, from, "thoroughly," plus mutare, "to change"; "a thorough change, an exchange" | S302, S306 |
| Pivot | word | 1924 E. T. Whittaker and G. Robinson | French, of uncertain origin | "the pin on which a wheel turns" | S302, S306 |
| Posterior | word | 1713 Jacob Bernoulli | Latin | posterior, "later, coming after," comparative of posterus, from post, "after" | S302, S306 |
| Prior | word | 1713 Jacob Bernoulli | Latin | prior, "former, previous, first," comparative of Old Latin pri, "before" | S302, S306 |
| Probability | word | 1662 Antoine Arnauld and Pierre Nicole | Latin | probabilitatem, from probabilis, "credible, provable," from probare, "to test, to prove" | S302, S306 |
| Proof | word | not established | Latin, through Old French | Late Latin proba, back-formed from probare, "to test, to try, to prove" | S306, S302 |
| Random | word | 1854 George Boole | Old French | randon, "a rush, disorder, force," from randir, "to run fast"; the English adverb originally meant "at great speed" | S302, S306 |
| Random walk | word | 1905 Karl Pearson | English compound | see random | S302 |
| Rank (of a matrix) | word | 1879 F. G. Frobenius | Frankish or Germanic, through Old French | renc, ranc, from Proto-Germanic hringaz, "a circle, a ring"; the English sense "a row, a line, a series" | S302, S306 |
| Recurrent | word | not established | Latin | recurrere, "to run back, to hasten back, to return" | S306, S302 |
| Reversible | word | not established | Latin, through French | reverse plus -ible, from revertere, "to turn back" | S306 |
| Row echelon | word | not established | French | echelon, "a rung of a ladder," from eschiele, "ladder," from Late Latin scala, "stair" | S306, S302 |
| Sample space | word | 1933 not settled to one person | Latin, through Old French | sample from exemplum, "a specimen taken out," from eximere, "to take out"; space from spatium | S302 |
| Scalar (Hamilton's modern sense, the real part of a quaternion, hence a real number) | word | 1846 William Rowan Hamilton | Latin | scalaris, "of or pertaining to a ladder," from scalae, "ladder, steps" | S174, S306 |
| Scalar (Viete's magnitudines scalares) | word | 1646 Francois Viete, then William Rowan Hamilton | Latin | scalaris, "of or pertaining to a ladder," from scalae, "ladder, steps" | S302, S306 |
| Secular equation | word | 1829 Augustin-Louis Cauchy | Latin | saecularis, "of an age, of a generation," from saeculum, "an age"; in astronomy, the changes that take centuries | S302, S306 |
| Set | word | 1796 William Frend, then E. H. Moore | Old English and Old French, a tangle | the OED records "set" for a collection of things from the 17th century; the technical sense is a translation of German Menge | S302 |
| Singular (matrix) | word | 1907 Maxime Bocher | Latin | singularis, "alone of its kind," from singulus, "one to each" | S302 |
| Span | word | not established | Old English | span, a hand measurement of about nine inches, from a root meaning "to draw, to stretch" | S306, S302 |
| Spur (German for trace) | word | 1922 Hermann Weyl, through H. L. Brose | German | Spur, "a track, a footprint, a trail" | S302 |
| State | word | 1913 | Latin | status, "a standing, a condition," from stare, "to stand" | S306 |
| Stationary distribution | word | 1934 Aleksandr Khinchin | Latin | stationarius, "of a military post," from statio, "a standing still" | S302 |
| Steady state | word | not established | English | plain compound | |
| Stochastic | word | 1713 Jacob Bernoulli, then Ladislaus von Bortkiewicz | Greek | stokhastikos, "able to guess, conjecturing," from stokhazesthai, "to aim, to guess," from stokhos, "the target, the pillar an archer shoots at" | S302, S306 |
| Stochastic matrix | word | not established | as above plus Latin matrix | "an aiming array" | |
| Subset | word | not established | Latin prefix plus English | sub-, "under," plus set | S302 |
| Tensor | word | 1846 William Rowan Hamilton | Latin | tensor, "a stretcher," from tendere, "to stretch"; originally an anatomical term for a muscle that tightens | S302, S306 |
| Tensor (modern sense) | word | 1898 Woldemar Voigt | as above | as above | S302 |
| Theorem | word | 1551 Robert Recorde | Greek | theorema, "that which is looked, a spectacle," from theorein, "to look, to behold" | S302, S306 |
| Trace (of a matrix) | word | 1922 H. L. Brose, translating Hermann Weyl | Latin, through Old French | tractus, "a track, a course," from trahere, "to pull, to draw"; a trace is a mark left by passage | S302, S306 |
| Transient | word | not established | Latin | transire, "to go across," from trans, "across," plus ire, "to go" | S306, S302 |
| Transition | word | not established | Latin | transitionem, "a going across," from transire | S306, S302 |
| Transpose | word | 1858 Arthur Cayley | Latin, through Old French | transponere, "to place across, to set over," from trans plus ponere, "to put" | S302, S306 |
| Union | word | 1912 James Pierpont | Latin | unionem, from unus, "one"; "a making into one" | S302, S306 |
| Universal set | word | 1910 A. N. Whitehead and Bertrand Russell | Latin | universum, "turned into one," from unus plus vertere, "to turn" | S302, S306 |
| Universe (of discourse) | word | 1849 Augustus De Morgan | Latin | as above | S302 |
| Vector (Hamilton's directed-magnitude sense, the imaginary part of a quaternion) | word | 1846 William Rowan Hamilton | Latin | as above | S174, S302 |
| Vector (the astronomical radius vector) | word | 1704 John Harris | Latin | vector, "a carrier, one who conveys," from vehere, "to carry, to convey" | S302 |
| Venn diagram | word | 1880 John Venn | English proper name plus Greek | diagramma, "that which is marked out by lines" | S302 |
| Versor (Hamilton's coinage in the quaternion papers) | word | 1846 William Rowan Hamilton | Latin | versor, "a turner," from vertere, "to turn" | S174 |
| Versor (the definition in Elements of Quaternions) | word | 1866 William Rowan Hamilton | Latin | as above | S302 |
| Zero | word | 1600 not named | Arabic, through Italian and Medieval Latin | sifr, "empty, nothing," through Medieval Latin zephirum, rendering Sanskrit sunya-m, "empty place, naught" | S306, S300 |
Appendix C
Pronunciation quick reference
Every name in this book that anything can be said about: 305 of 306. The other 1 is absent because nothing consulted states how to say it, and a guess would be worse than a gap.
100 of these are sourced to a named dictionary entry or a published romanization. The other 205 are approximations built from standard romanization rules, and every one of them says so in its own row. Do not quote an approximated respelling as evidence about a language.
Say them out loud, and say them wrong rather than not at all. A great deal of this book happened in Russian, Sanskrit, Arabic, Chinese, German and Italian. A student who says a name imperfectly is doing better than a student who avoids the sentence.
| Name | Say it | IPA | Evidence | Worked in |
|---|---|---|---|---|
| Abraham de Moivre | duh MWAHV, or duh MWAH-vruh | də ˈmwɑv, ˈmwɑ vrə, ˈmɔɪ vər | sourced | England |
| Adam Makkai | AH-dahm MAH-koy | not recorded | approximated, not sourced | Chicago |
| Adrien-Marie Legendre | luh-ZHAHN-der | ləˈʒɑn dər, -ˈʒɑnd, ləˈʒɑ̃ drə | sourced | |
| al-Adli ar-Rumi | al-AD-lee ar-ROO-mee | not recorded | approximated, not sourced | Abbasid Baghdad |
| al-Karaji | al-kuh-RAH-jee | not recorded | approximated, not sourced | Baghdad |
| al-Khwarizmi | al-KHWAR-iz-mee | ˈxwɑr ɪzˌmi | sourced | Baghdad |
| al-Samaw'al al-Maghribi | as-suh-MOW-al | not recorded | approximated, not sourced | Baghdad, Maragha |
| Alan Baker (philosopher) | BAY-ker | not recorded | approximated, not sourced | Swarthmore College |
| Alan Mathison Turing | TOOR-ing | ˈtʊər ɪŋ | sourced | Teddington, Manchester |
| Aleksandr Aleksandrovich Chuprov | choo-PROFF | not recorded | approximated, not sourced | St Petersburg |
| Alexander Craig Aitken | AY-ken | not recorded | approximated, not sourced | Edinburgh |
| Alexandre-Theophile Vandermonde | van-der-MOND | not recorded | approximated, not sourced | Paris |
| Alfred Bray Kempe | KEMP | not recorded | approximated, not sourced | London |
| Alfred Doblin | DUR-bleen (German oe, the vowel of French "oeuf") | ˈdœ blin | sourced | Berlin; exile in France and the USA |
| Alfred Renyi | RAYN-yee | not recorded | approximated, not sourced | Budapest |
| Alicia Boole Stott | uh-LISS-ee-uh BOOL STOT | not recorded | approximated, not sourced | England; corresponded with Groningen |
| Andre Weil | ahn-DRAY vay | veɪ (US); vail (UK) | sourced | Strasbourg; Aligarh; Sao Paulo; Chicago; Princeton |
| Andre-Louis Cholesky | shoh-LESS-kee | not recorded | approximated, not sourced | France |
| Andre-Michel Guerry | zhe-REE | not recorded | approximated, not sourced | Paris |
| Andreas von Ettingshausen | AN-dray-ahs fon ET-tings-how-zen | not recorded | approximated, not sourced | Vienna |
| Andrei Andreevich Markov | MAR-kuff (Russian), mahr-KAWF (English) | ˈmar.kəf (Russian); ˈmɑr kɔf, ˈmɑːkɒf (English) | sourced | St Petersburg University; Imperial Academy of Sciences; Zaraisk 1917 |
| Andrei Andreevich Markov Jr. | MAR-kuff (Russian), mahr-KAWF (English) | ˈmar.kəf (Russian); ˈmɑr kɔf (English) | sourced | Moscow |
| Andrei Nikolaevich Kolmogorov | kol-muh-GAW-rof | /ˌkɒlmɒˈɡɔːrɒf/ | sourced | Moscow State University |
| Anna Johnson Pell Wheeler | WHEE-ler | not recorded | approximated, not sourced | Bryn Mawr College |
| Antoine Augustin Cournot | koor-NOH | not recorded | approximated, not sourced | Paris |
| Antoine Gombaud, chevalier de Mere | duh MAIR | not recorded | approximated, not sourced | court of Louis XIV |
| Arianna Wright Rosenbluth | ah-ree-AH-nuh ROH-zen-blooth | not recorded | approximated, not sourced | Los Alamos National Laboratory |
| Arthur Cayley | KAY-lee | "ˈkeɪ li" | sourced | London, Cambridge |
| August Ferdinand Mobius | MER-bee-oos | ˈmœ bi əs, ˈmeɪ-, ˈmoʊ- | sourced | Leipzig |
| Augusta Ada King, Countess of Lovelace | LUV-layss | ˈlʌvˌleɪs | sourced | London |
| Augusta H. (Mici) Teller | MEE-tsee TELL-er | not recorded | approximated, not sourced | Los Alamos |
| Augustin-Louis Cauchy | koh-SHEE | "koʊˈʃi" | sourced | Paris |
| Augustus De Morgan | di MOR-gun | dɪ ˈmɔr gən | sourced | University College London |
| Bernard Bolzano | bolt-SAH-no | boʊltˈsɑ noʊ (US); bolˈtsano (UK) | sourced | Prague, Villa Liboch |
| Bertrand Russell | RUSS-uhl | ˈrʌs əl | sourced | Cambridge |
| Bharata | BHUH-ruh-tuh | not recorded | approximated, not sourced | India |
| Blaise Pascal | pa-SKAL | "pa-ˈskal", "pä-ˈskäl" | sourced | Paris |
| Bohuslav Hostinsky | HOS-tin-skee | not recorded | approximated, not sourced | Brno |
| Camille Flye Sainte-Marie | fly sant mah-REE | not recorded | approximated, not sourced | France |
| Carl Friedrich Gauss | GOWSS | gaʊs; MW ˈgau̇s | sourced | Gottingen |
| Carl Gustav Jacob Jacobi | yah-KOH-bee (German), juh-KOH-bee (English) | "jɑˈkoʊ bi", "dʒəˈkoʊ bi" | sourced | Konigsberg, Berlin |
| Carl Leonhard Gottlieb Ehler | AY-ler | not recorded | approximated, not sourced | Danzig |
| Cassius Jackson Keyser | KASH-us KY-zer | not recorded | approximated, not sourced | Columbia |
| Charles Denis Sauter Bourbaki (General) | boor-buh-KEE | not recorded | approximated, not sourced | France |
| Charles Jean de la Vallee Poussin | duh lah vah-LAY poo-SAN | not recorded | approximated, not sourced | Louvain |
| Charles Joseph Minard | mee-NAR | not recorded | approximated, not sourced | Paris |
| Charles Sanders Peirce | PURSS | pɜrs, pɪərs | sourced | USA |
| Christiaan Huygens | HY-genz (English), HOY-khens (Dutch) | ˈhaɪ gənz, ˈhɔɪ-; British ˈhœixəns | sourced | The Hague, Paris, London |
| Christian Genest | zhuh-NEH | not recorded | approximated, not sourced | McGill University |
| Christian Kramp | KRAHMP | not recorded | approximated, not sourced | Cologne, Strasbourg |
| Christian Reinsch | RINE-sh | not recorded | approximated, not sourced | Munich |
| Christian Weise | KRIS-tee-ahn VY-zuh | not recorded | approximated, not sourced | Zittau |
| Clasen (initials contested: "J. B." or "B.-I.") | KLAH-sen | not recorded | approximated, not sourced | Brussels |
| Claude Chevalley | shuh-vah-LAY | not recorded | approximated, not sourced | France; USA |
| Claude Elwood Shannon | SHAN-un | ˈʃæn ən | sourced | Bell Telephone Laboratories; MIT |
| Colin Maclaurin | muh-KLOR-in | məkˈlɔr ɪn, məˈklɔr- | sourced | Edinburgh |
| Cuthbert Edmund Cullis | KUTH-bert KUL-is | not recorded | approximated, not sourced | Cambridge, Calcutta |
| Cyrus Colton MacDuffee | mak-DUFF-ee | not recorded | approximated, not sourced | Wisconsin |
| Dame Mary Lucy Cartwright | KART-rite | not recorded | approximated, not sourced | Girton College, Cambridge |
| Daniel Bernoulli | ber-NOO-lee | bərˈnu li | sourced | Basel, St Petersburg |
| David Harold Blackwell | BLACK-wel | ˈblæk wəl, -ˌwɛl | sourced | Howard University; UC Berkeley |
| David Hilbert | HIL-bert (German HIL-bairt) | ˈhɪl bərt | sourced | Gottingen |
| David R. Bellhouse | BEL-howss | not recorded | approximated, not sourced | University of Western Ontario |
| Denes Konig | DAY-nesh KUR-nig | not recorded | approximated, not sourced | Budapest |
| Doron Zeilberger | DOR-on ZYLE-ber-ger | not recorded | approximated, not sourced | Rutgers University |
| Duncan J. Watts | WOTS | wɒts | sourced | Cornell, Columbia |
| Edouard Lucas | ay-DWAR loo-KAH | not recorded | approximated, not sourced | France |
| Eduard Stiefel | SHTEE-fel | not recorded | approximated, not sourced | ETH Zurich |
| Edward Teller | TELL-er | not recorded | approximated, not sourced | Los Alamos; University of Chicago |
| Edward Wilson Chittenden | CHIT-en-den | not recorded | approximated, not sourced | Iowa |
| Edwin Bidwell Wilson | WIL-son | ˈwɪl sən | sourced | Yale, Harvard |
| Egon S. Pearson | EE-gon PEER-suhn | not recorded | approximated, not sourced | University College London |
| Emmy Amalie Noether | NUH-ter | ˈnʌ tər | sourced | Gottingen; Bryn Mawr |
| Erhard Schmidt | AIR-hart SHMIT | ʃmɪt (surname) | sourced | Gottingen, Berlin |
| Ernest Tilden Parker | PAR-ker | ˈpɑr kər | sourced | Illinois |
| Ernst Eduard Kummer | KOOM-er | ˈkʊm ər | sourced | Berlin |
| Ernst Schroder | ernst SHRUH-der | not recorded | approximated, not sourced | Karlsruhe |
| Ernst Zermelo | tser-MAY-loh | not recorded | approximated, not sourced | Gottingen, Zurich, Freiburg |
| Esther Klein Szekeres | KLYNE SEH-keh-resh | ˈsɛ.kɛ.rɛʃ (Szekeres) | sourced | Shanghai, Adelaide |
| Etienne Bezout | bay-ZOO | not recorded | approximated, not sourced | Paris |
| Eugene Seneta | suh-NET-uh | not recorded | approximated, not sourced | University of Sydney |
| Eugenio Beltrami | bel-TRAH-mee | bɛlˈtrɑ mi | sourced | Bologna, Pisa, Rome, Pavia |
| Evgeny Slutsky | SLOOT-skee | not recorded | approximated, not sourced | Kiev, Moscow |
| Felix Edouard Emile Borel | baw-REL | bɔˈrɛl | sourced | Paris |
| Ferdinand Georg Frobenius | froh-BAY-nee-oos | not recorded | approximated, not sourced | Zurich, Berlin |
| Florence Nightingale David | DAY-vid | ˈdeɪ vɪd | sourced | UCL; UC Berkeley; UC Riverside |
| Florian Cajori | ka-JOR-ee | not recorded | approximated, not sourced | Colorado Springs, Berkeley |
| Francis Guthrie | GUTH-ree | AHD respelling (gŭthrē) | sourced | London, South Africa |
| Frank Plumpton Ramsey | RAM-zee | ˈræm zi (US); ˈræmzɪ (UK) | sourced | Cambridge |
| Frans van Schooten | frahns vahn SKHOH-ten | not recorded | approximated, not sourced | Leiden |
| Frantisek Prihonsky | FRAHN-tyi-shek PRZHI-hon-skee | not recorded | approximated, not sourced | Bohemia |
| Frederick Guthrie | GUTH-ree | AHD respelling (gŭthrē) | sourced | London |
| Friedrich Wilhelm Bessel | BESS-'l | ˈbɛs əl | sourced | Konigsberg |
| Frigyes Karinthy | FRIH-jesh KAH-rin-tee | not recorded | approximated, not sourced | Budapest |
| Gabriel Andrew Dirac | dih-RAK | dɪˈræk | sourced | Aarhus |
| Gabriel Cramer | KRAY-mer (English) | ˈkreɪ mərz (in "Cramer's rule") | sourced | Geneva |
| Galileo Galilei | gal-uh-LAY-oh | ˌgæl əˈleɪ oʊ, -ˈli oʊ; MW ˌga-lə-ˈlē-(ˌ)ō, -ˈlā- | sourced | Padua, Florence |
| Gene H. Golub | GOH-lub | not recorded | approximated, not sourced | Stanford |
| Georg Cantor | GAY-org KAN-ter | ˈkæn tər, -tɔr (surname) | sourced | Halle |
| George Boole | BOOL | bul (US); buːl (UK) | sourced | Lincoln; Queen's College Cork |
| George Chrystal | KRIS-tal | not recorded | approximated, not sourced | Edinburgh |
| George Forsythe | FOR-syth | not recorded | approximated, not sourced | Stanford |
| George Szekeres | SEH-keh-resh | ˈsɛ.kɛ.rɛʃ | sourced | Shanghai, Adelaide |
| George Udny Yule | YOOL | jul (US); juːl (UK) | sourced | London, Cambridge |
| Georges Gonthier | ZHORZH gon-tee-AY | not recorded | approximated, not sourced | Cambridge, England |
| Giovanni Jacopo Marinoni | mah-ree-NOH-nee | not recorded | approximated, not sourced | Vienna |
| Girolamo Cardano | jee-ROH-lah-moh kar-DAH-noh | not recorded | approximated, not sourced | Milan, Bologna, Rome |
| Giulio Pace | JOOL-yoh PAH-chay | not recorded | approximated, not sourced | Italy, France |
| Giuseppe Peano | pay-AH-noh (English pee-AH-noh) | piˈɑ noʊ, pɛˈɑ nɔ | sourced | Turin |
| Glenn Shafer | SHAY-fer | not recorded | approximated, not sourced | Rutgers University |
| Godfrey Harold Hardy | HAR-dee | ˈhɑr di | sourced | Trinity College, Cambridge |
| Gottfried Wilhelm Leibniz | LYBE-nits | "ˈlīb-nəts" (Eng.), "ˈlīp-nits" (Ger.) | sourced | Leipzig, Hanover |
| Gotthold Eisenstein | EYE-zen-shtyne | ˈaɪ zənˌʃtaɪn, ˈaɪ zənˌstaɪn | sourced | Berlin |
| Gottlob Frege | FRAY-guh | ˈfreɪ gə (US); ˈfreːɡə (UK) | sourced | Jena |
| Guillaume de l'Hopital, Marquis | loh-pee-TAL | not recorded | approximated, not sourced | Paris |
| Guo Shuchun | gwoh SHOO-chwun | not recorded | approximated, not sourced | Chinese Academy of Sciences, Beijing |
| Gustav Robert Kirchhoff | KEERKH-hawf | "ˈkirx hɔf" (Amer.), "ˈkɪrçhɔf" (Brit.) | sourced | Breslau, Heidelberg, Berlin |
| Halayudha | huh-LAH-yoo-dhuh | not recorded | approximated, not sourced | India |
| Harold James Ruthven Murray | MUR-ee | not recorded | approximated, not sourced | Oxford |
| Harold Jeffreys | JEF-reez | ˈdʒɛf riz | sourced | Cambridge |
| Harvey Goodwin | GOOD-win | not recorded | approximated, not sourced | Caius College, Cambridge |
| Hayashi Tsuruichi | hah-YAH-shee tsoo-roo-EE-chee | not recorded | approximated, not sourced | Japan |
| Heinrich Heesch | HYNE-rikh HAYSH | not recorded | approximated, not sourced | Hanover |
| Heinz Rutishauser | ROO-tee-how-zer | not recorded | approximated, not sourced | ETH Zurich |
| Henri Cartan | on-REE kar-TAHN | not recorded | approximated, not sourced | Strasbourg; Paris |
| Henry Billingsley | BIL-ingz-lee | not recorded | approximated, not sourced | London |
| Henry Lewis Rietz | REETS | not recorded | approximated, not sourced | Iowa |
| Henry Thomas Colebrooke | KOHL-brook | not recorded | approximated, not sourced | Calcutta, London |
| Henry Warburton | WOR-ber-ton | not recorded | approximated, not sourced | Cambridge |
| Henry Whitehead (Reverend) | WITE-hed | ˈʰwaɪtˌhɛd, ˈwaɪt- | sourced | St Luke's, Berwick Street, London |
| Herman H. Goldstine | GOLD-steen | not recorded | approximated, not sourced | Princeton |
| Hermann Gunther Grassmann | GRAHSS-mun (also GRAHSS-mahn) | ˈgrɑs mən, ˈgrɑsˌmɑn | sourced | Stettin |
| Hermann Weyl | VILE (rhymes with "mile") | vaɪl | sourced | Gottingen, Zurich, Princeton |
| Hilda Geiringer | HIL-duh GY-ring-er | not recorded | approximated, not sourced | Berlin, USA |
| Hugo Steinhaus | SHTYNE-hows | not recorded | approximated, not sourced | Lwow, Wroclaw |
| Iannis Xenakis | zeh-NAH-kiss | /ksɛˈnakis/ or /zɛˈnɑːkɪs/ | sourced | Paris |
| Ibn al-Banna al-Marrakushi | ib-n al-BAN-nah | not recorded | approximated, not sourced | Marrakesh |
| Ibn Mun'im al-Abdari | ib-n moon-EEM | not recorded | approximated, not sourced | Marrakesh |
| Ignaz Klein | IG-nahts KLYNE | not recorded | approximated, not sourced | Budapest |
| Irenee-Jules Bienayme | byen-eh-MAY | not recorded | approximated, not sourced | Paris |
| Isaac Todhunter | TOD-hun-ter | not recorded | approximated, not sourced | Cambridge |
| J. B. Durrande | doo-RAHND | not recorded | approximated, not sourced | Gergonne's Annales, Nismes |
| Jacob Bernoulli | ber-NOO-lee | bərˈnu li, bɛrˈnu li | sourced | Basel |
| Jacques Philippe Marie Binet | bih-NAY | bɪˈneɪ, biˈnɛ | sourced | Paris |
| James Hardy Wilkinson | WIL-kin-sun | not recorded | approximated, not sourced | Teddington |
| James Joseph Sylvester | sil-VESS-ter | "sɪlˈvɛs tər" | sourced | London, Virginia, Baltimore, Oxford |
| James Pierpont | PEER-pont | not recorded | approximated, not sourced | Yale |
| Jean Delsarte | del-SART | not recorded | approximated, not sourced | Nancy |
| Jean Dieudonne | dyuh-doh-NAY | not recorded | approximated, not sourced | Nancy; Nice |
| Jerzy Neyman | YAIR-zhee NAY-mahn | not recorded | approximated, not sourced | London, Berkeley |
| Jia Xian | jyah shyen | not recorded | approximated, not sourced | Song China |
| Johann Benedict Listing | LIST-ing | not recorded | approximated, not sourced | Gottingen |
| Johann Bernoulli | ber-NOO-lee | bərˈnu li | sourced | Basel, Groningen |
| Johann Christian Lange | YO-hahn KRIS-tee-ahn LAHNG-uh | not recorded | approximated, not sourced | Giessen |
| Johann Christoph Sturm | YO-hahn KRIS-toff SHTOORM | not recorded | approximated, not sourced | Altdorf |
| John Canton | KAN-ton | not recorded | approximated, not sourced | London |
| John G. F. Francis | FRAN-sis | not recorded | approximated, not sourced | NRDC, Ferranti |
| John Graunt | GRAWNT | not recorded | approximated, not sourced | London |
| John Koch | KOKE | not recorded | approximated, not sourced | Urbana, Illinois |
| John Snow | SNOH | snoʊ | sourced | London |
| John Venn | VEN | vɛn | sourced | Gonville and Caius College, Cambridge |
| John von Neumann | von NOY-mahn | vɒn ˈnɔɪ mɑn, -mən | sourced | Princeton |
| Joseph Henry Maclagan Wedderburn | WED-er-burn | not recorded | approximated, not sourced | Princeton |
| Joseph-Louis Lagrange | luh-GRAYNJ (English), la-GRAHNZH (French) | ləˈɡreɪndʒ, laˈɡrɑ̃ʒ | sourced | Berlin, Paris |
| Josiah Willard Gibbs | GIBZ (forenames joh-SY-uh WIL-erd) | ˈdʒoʊsiə ˈwɪlərd (forenames) | sourced | Yale, New Haven |
| Juan Luis Vives | hwahn loo-EES VEE-vays | not recorded | approximated, not sourced | Low Countries |
| Judith S. Kleinfeld | KLYNE-feld | not recorded | approximated, not sourced | Fairbanks, Alaska |
| Judith Stupanus | YOO-dit shtoo-PAH-nus | not recorded | approximated, not sourced | Basel; Jacob Bernoulli's widow |
| Julia Bowman Robinson | ROB-in-sun | not recorded | approximated, not sourced | UC Berkeley |
| Justus Gunther Grassmann | GRAHSS-mun | ˈgrɑs mən, ˈgrɑsˌmɑn | sourced | Stettin |
| Jyesthadeva | jyesh-tuh-DAY-vuh | not recorded | approximated, not sourced | Kerala |
| Karl Weierstrass | VY-er-shtrahss | ˈvaɪ ərˌʃtrɑs | sourced | Berlin |
| Kedara Bhatta | KAY-dah-ruh | not recorded | approximated, not sourced | India |
| Kenneth Ira Appel | AP-pel | not recorded | approximated, not sourced | Urbana, Illinois |
| Ladislaus Josephowitsch Bortkiewicz | bort-KYEH-vitch | not recorded | approximated, not sourced | Saint Petersburg, Berlin |
| Lam Lay Yong | lahm lay yong | not recorded | approximated, not sourced | National University of Singapore |
| Lars Onsager | ON-sah-ger (also AWN-) | /ˈɒn sɑ gər/, /ˈɔn-/ | sourced | Yale University |
| Lawrence (Larry) Page | PAYJ | not recorded | approximated, not sourced | Stanford, Google |
| Lejaren Arthur Hiller Jr | LEDGE-uh-rin HILL-er | not recorded | approximated, not sourced | University of Illinois; SUNY Buffalo |
| Leonard E. Baum | BAWM | not recorded | approximated, not sourced | Institute for Defense Analyses, Princeton |
| Leonard Eugene Dickson | DIK-son | not recorded | approximated, not sourced | Chicago |
| Leonard Isaacson | EYE-zuk-sun | not recorded | approximated, not sourced | University of Illinois |
| Leonhard Euler | OY-ler | "ˈɔɪ lər" (Amer.), "ˈɔɪlər" (Brit.) | sourced | St Petersburg, Berlin |
| Leopold Kronecker | KROH-neck-er | ˈkroʊ nɛk ər | sourced | Berlin |
| Lester S. Hill | HILL | not recorded | approximated, not sourced | Hunter College, New York |
| Levi ben Gershon (Gersonides) | LAY-vee ben GAIR-shon | not recorded | approximated, not sourced | Provence |
| Li Chunfeng | lee chwun-FUNG | not recorded | approximated, not sourced | Tang China |
| Liu Hui | lyoh HWAY | not recorded | approximated, not sourced | Wei state, China |
| Louis Weisner | WYZE-ner | not recorded | approximated, not sourced | Hunter College, New York |
| Luca Pacioli | LOO-kah pah-CHOH-lee | not recorded | approximated, not sourced | Italy |
| Ludwig Boltzmann | LOOT-vikh BAWLTS-mahn | ˈbɔltsˌmɑn, ˈboʊlts mən (US); ˈbɔltsman (UK); forename ˈluːtvɪç | sourced | Vienna; Graz; Leipzig |
| Ludwig Stickelberger | STIK-el-bair-ger | not recorded | approximated, not sourced | Freiburg |
| Luigi Federico Menabrea | meh-nah-BRAY-ah | not recorded | approximated, not sourced | Turin |
| M. Rangacharya | ran-gah-CHAR-yuh | not recorded | approximated, not sourced | Madras |
| Madhava of Sangamagrama | MAH-dhuh-vuh | not recorded | approximated, not sourced | Kerala |
| Magnus R. Hestenes | HESS-tuh-neez | not recorded | approximated, not sourced | NBS, UCLA |
| Mahavira (Mahaviracarya) | muh-hah-VEER-uh | məˌhɑˈvɪər ə | sourced | southern India |
| Marie Ennemond Camille Jordan | zhor-DAHN | ʒɔrˈdɑ̃ | sourced | Paris |
| Mark Kac | KATS | not recorded | approximated, not sourced | Cornell; Rockefeller University |
| Marko Petkovsek | PET-kov-shek | not recorded | approximated, not sourced | University of Ljubljana |
| Marshall N. Rosenbluth | ROH-zun-blooth | not recorded | approximated, not sourced | Los Alamos; UC San Diego |
| Martin C. Kohli | KOH-lee | not recorded | approximated, not sourced | U.S. Bureau of Labor Statistics |
| Martin Mattmuller | MAT-mue-ler | not recorded | approximated, not sourced | Bernoulli-Euler Zentrum, Basel |
| Mary Everest Boole | MAIR-ee EV-rist bool | surname bul | sourced | London |
| Maxime Bocher | mak-SEEM BOH-ker | not recorded | approximated, not sourced | Harvard |
| Michael Stifel | MIKH-ah-el SHTEE-fel | not recorded | approximated, not sourced | Nuremberg, Jena |
| Myrick Hascall Doolittle | DOO-lit-'l | not recorded | approximated, not sourced | US Coast Survey, Washington |
| N. Chidambaram Iyer | chih-DUM-buh-rum EYE-yer | not recorded | approximated, not sourced | India |
| Nami of Guzerat | NAH-mee | not recorded | approximated, not sourced | Gujarat |
| Narayana Pandita | nah-RAH-yuh-nuh PUN-dih-tuh | not recorded | approximated, not sourced | India |
| Nathan Morrison | MOR-i-son | not recorded | approximated, not sourced | translator of the Grundbegriffe |
| Niccolo Fontana "Tartaglia" | tar-TAH-lyah | not recorded | approximated, not sourced | Venice |
| Nicholas Metropolis | muh-TROP-uh-liss | not recorded | approximated, not sourced | Los Alamos Scientific Laboratory |
| Nicholas Saunderson | SAWN-der-son | not recorded | approximated, not sourced | Cambridge |
| Nicolaas Govert de Bruijn | duh BROWN | not recorded | approximated, not sourced | Eindhoven, Nuenen |
| Nicolas Bourbaki (collective pseudonym) | BOOR-buh-kee | /ˈbɔːbəkɪ/ | sourced | Paris; "Nancago" |
| Nicolas Chuquet | shoo-KAY | not recorded | approximated, not sourced | Lyon |
| Nicolaus I Bernoulli | ber-NOO-lee | bərˈnu li | sourced | Basel |
| Olga Taussky-Todd | TOW-skee TOD ("TOW" as in "how") | not recorded | approximated, not sourced | Vienna, Gottingen, Teddington, Washington, Pasadena |
| Oliver Heaviside | HEV-ee-side | ˈhɛv iˌsaɪd | sourced | England |
| Omar Khayyam | oh-MAR ky-YAHM | kaɪˈjɑm, -ˈjæm (US); kaɪˈɑːm (UK) | sourced | Persia |
| Oskar Perron | peh-ROHN | not recorded | approximated, not sourced | Munich |
| Oswald Veblen | VEB-lun | ˈvɛb lən | sourced | Princeton |
| Oystein Ore | OY-styne OR-uh | not recorded | approximated, not sourced | Yale University |
| Pafnuty Lvovich Chebyshev | chuh-buh-SHAWF | tʃə bəˈʃɔf | sourced | St Petersburg |
| Paul Ehrenfest | AIR-un-fest | not recorded | approximated, not sourced | St Petersburg 1907-1912; Leiden |
| Paul Erdos | AIR-dursh | ˈer-ˌdərsh | sourced | itinerant; Hungary, USA, Israel |
| Paul Gustav Stackel | SHTEK-'l | not recorded | approximated, not sourced | Heidelberg |
| Paul Richard Halmos | HAL-mohs | not recorded | approximated, not sourced | United States |
| Paul Turan | TOO-rahn | not recorded | approximated, not sourced | Budapest |
| Paul Wernicke | VAIR-nih-keh | not recorded | approximated, not sourced | |
| Pavel Alekseevich Nekrasov | nih-KRASS-uff | nɪˈkrasəf | sourced | Moscow University |
| Percy Alexander MacMahon | mak-MAHN | not recorded | approximated, not sourced | Woolwich, London, Cambridge |
| Percy John Heawood | HEE-wood | not recorded | approximated, not sourced | Durham |
| Peter Guthrie Tait | TAYT | not recorded | approximated, not sourced | Edinburgh |
| Philip E. B. Jourdain | JOR-dayn | not recorded | approximated, not sourced | England |
| Philip Franklin | FRANK-lin | not recorded | approximated, not sourced | MIT |
| Pierre de Fermat | fer-MAH | fer-ˈmä | sourced | Toulouse |
| Pierre Remond de Montmort | mon-MOR | not recorded | approximated, not sourced | France |
| Pierre-Simon Laplace | luh-PLAHSS | lə-ˈpläs; laˈplas | sourced | Paris |
| Pieter Hendrik Schoute | SKHOW-tuh | not recorded | approximated, not sourced | University of Groningen |
| Pingala | PING-guh-luh | not recorded | approximated, not sourced | India |
| Prakash Gorroochurn | gor-oo-CHURN | not recorded | approximated, not sourced | Columbia University |
| Raj Chandra Bose | rahj CHUN-druh BOHSS | boʊs (surname) | sourced | Calcutta, Chapel Hill, Fort Collins |
| Raoul Husson | rah-OOL oo-SOHN | not recorded | approximated, not sourced | Ecole Normale Superieure, Paris |
| Reginald Crundall Punnett | PUN-et | not recorded | approximated, not sourced | Cambridge |
| Rene de Possel | duh poh-SELL | not recorded | approximated, not sourced | Clermont-Ferrand |
| Rene Maurice Frechet | fray-SHAY | freɪˈʃɛ | sourced | Paris, Strasbourg |
| Richard Dedekind | DAY-duh-kint | ˈdeɪ də kɪnt (US); ˈdedəˌkɪnt (UK) | sourced | Braunschweig, Gottingen |
| Richard J. Pulskamp | PULSS-kamp | not recorded | approximated, not sourced | Xavier University |
| Richard Price | PRYSS | not recorded | approximated, not sourced | Newington Green, London |
| Richard Rado | RAH-doh | not recorded | approximated, not sourced | Reading, England |
| Richard von Mises | RIKH-art fon MEE-zuhs | not recorded | approximated, not sourced | Berlin, Istanbul, Harvard |
| Ronald Aylmer Fisher | FISH-er | ˈfɪʃ ər | sourced | Rothamsted, Cambridge |
| Rudrata | ROOD-ruh-tuh | not recorded | approximated, not sourced | Kashmir |
| Seki Takakazu (Seki Kowa) | SEH-kee tah-kah-KAH-zoo | not recorded | approximated, not sourced | Edo (now Tokyo) |
| Semyon Aranovich Gersgorin (Gershgorin) | GERSH-gaw-rin | not recorded | approximated, not sourced | USSR |
| Sergei Aksakov | ahk-SAH-koff | not recorded | approximated, not sourced | Russia |
| Sergei Natanovich Bernstein | BERN-shtyne | not recorded | approximated, not sourced | Kharkov, Paris, Moscow |
| Sergey Brin | SUR-gay BRIN | not recorded | approximated, not sourced | Stanford, Google |
| Sharadchandra Shankar Shrikhande | shuh-rud-CHUN-druh shree-KUN-day | not recorded | approximated, not sourced | India, Chapel Hill |
| Simeon-Denis Poisson | pwah-SOHN | pwasɔ̃ | sourced | Paris |
| Simone Adolphine Weil | see-MOHN vay | veɪ | sourced | Paris; London |
| Sister Mary Celine Fasenmyer | FAY-zen-my-er | not recorded | approximated, not sourced | Mercyhurst College, Erie, PA |
| Sridhara | SHREE-duh-ruh | not recorded | approximated, not sourced | India |
| Stanislaw Marcin Ulam | STAN-iss-wahf OO-lahm | not recorded | approximated, not sourced | Los Alamos |
| Stanley Milgram | MIL-gram | not recorded | approximated, not sourced | Harvard, Yale, CUNY |
| Stefan Banach | BAH-nahkh | not recorded | approximated, not sourced | Lwow |
| Stephen M. Stigler | STIG-ler | not recorded | approximated, not sourced | University of Chicago |
| Steven H. Strogatz | STROH-gats | not recorded | approximated, not sourced | Cornell |
| Sydney Chapman | CHAP-mun | not recorded | approximated, not sourced | Cambridge; Oxford; Boulder, Colorado; Alaska |
| Tatiana Alexeyevna Ehrenfest-Afanassjewa | AIR-un-fest ah-fah-NAH-syeh-vuh | not recorded | approximated, not sourced | Gottingen; St Petersburg; Leiden |
| Thomas Bayes | BAYZ | beɪz | sourced | Tunbridge Wells |
| Thomas Digges | DIGZ | not recorded | approximated, not sourced | England |
| Thomas Hawkins | HAW-kinz | not recorded | approximated, not sourced | Boston University |
| Thomas Jarrett | JARR-et | not recorded | approximated, not sourced | St Catherine's College, Cambridge |
| Thomas Muir | MYOOR | not recorded | approximated, not sourced | Glasgow, Cape Colony |
| Thomas Strode | STROHD | not recorded | approximated, not sourced | England |
| Thomas Tymoczko | tih-MOTCH-koh | not recorded | approximated, not sourced | Smith College |
| Varahamihira | vuh-RAH-huh-MIH-hih-ruh | not recorded | approximated, not sourced | India |
| Vera Nikolaevna Kublanovskaya | koob-luh-NOFF-skuh-yuh | not recorded | approximated, not sourced | Leningrad / St Petersburg |
| Vera Sanford | SAN-ford | not recorded | approximated, not sourced | Western Reserve University |
| Virahanka | vih-ruh-HAHN-kuh | not recorded | approximated, not sourced | India |
| Vladimir Vovk | VOAFK | not recorded | approximated, not sourced | Royal Holloway, London |
| Vsevolod Ivanovich Romanovsky | ruh-mah-NOFF-skee | not recorded | approximated, not sourced | Tashkent |
| W. Keith Hastings | HAY-stingz | ˈheɪ stɪŋz | sourced | University of Toronto; Memorial University of Newfoundland |
| Wassily Leontief | lee-ON-tee-ef | liˈɒn tiˌɛf | sourced | Harvard, New York University |
| Wilhelm Jordan (geodesist) | VIL-helm YOR-dahn | not recorded | approximated, not sourced | Karlsruhe, Hanover |
| William (Velvel) Kahan | KAY-hun | not recorded | approximated, not sourced | Toronto, Berkeley |
| William Allen Whitworth | WIT-werth | ˈʰwɪtˌwɜrθ, ˈwɪt- | sourced | Cambridge, Liverpool |
| William Andrew Rogers | ROJ-erz | not recorded | approximated, not sourced | Atlanta University |
| William Edward Burghardt Du Bois | doo-BOYSS | du ˈbɔɪs | sourced | Atlanta University; Paris 1900 |
| William Feller | FELL-er | not recorded | approximated, not sourced | London, Princeton |
| William Frend | FREND | not recorded | approximated, not sourced | Cambridge, London |
| William of Ockham | OCK-uhm | ˈɒk əm | sourced | Oxford, Munich |
| William Petty | PET-ee | not recorded | approximated, not sourced | London, Ireland |
| William Rowan Hamilton | HAM-il-ton | ˈhæm əl tən | sourced | Dublin, Dunsink Observatory |
| Woldemar Voigt | VOHL-de-mar FOHKT | not recorded | approximated, not sourced | Gottingen |
| Wolfgang Doeblin (Vincent Doblin) | DUR-bleen (German oe, the vowel of French "oeuf") | ˈdœ blin | sourced | Paris; French army, Vosges |
| Wolfgang Haken | VOLF-gang HAH-ken | not recorded | approximated, not sourced | Urbana, Illinois |
| Yang Hui | yahng hway | not recorded | approximated, not sourced | Southern Song China |
| Yoshio Mikami | mee-KAH-mee | not recorded | approximated, not sourced | Japan |
| Yuri Vladimirovich Matiyasevich | mah-tee-yah-SAY-vich | not recorded | approximated, not sourced | Leningrad / St Petersburg |
| Zhang Cang | jahng TSAHNG | not recorded | approximated, not sourced | Han China |
| Zhu Shijie | joo shr-jyeh | not recorded | approximated, not sourced | China |
Appendix D
The doors out of the math room
54 places where this history walks into another subject, grouped by that subject so a teacher co-planning with a colleague can find their own lesson fast. 11 subjects are represented.
This is not decoration. The mathematics in your course arrived through history, English, biology, chemistry, music, economics, philosophy and public health. In several cases it arrived that way because somebody in one of those rooms needed it and no mathematician had bothered. Markov counted Pushkin's letters. Hardy wrote a page of algebra for biologists and apologized for intruding. Snow argued about a water pump. Du Bois and five students drew 72 items by hand for a world's fair. Each row below is an anchor fact with its source, so a colleague can check it before they teach it.
History
11 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Bolzano had been removed from his university post before he wrote Paradoxien des Unendlichen, and Zermelo renounced his Freiburg chair in 1935 in opposition to the Hitler regime and was reinstated in 1946. Ask what happens to ideas when the people who have them are pushed out of institutions | 1 | S044, S065 |
| The chapter's scenes are set in Song China, Abbasid Baghdad, Almohad Marrakesh, Jewish Provence, Renaissance Brescia, and Reformation Nuremberg. Six pins in a map and one date under each is a lesson on simultaneous invention | 2 | S002, S003, S013, S015, S018 |
| London started counting its dead because of plague, and the counting was done by sworn women called Searchers who inspected the bodies. Wednesday compilation, Thursday publication, four shillings a year. Pair it with the Edict of Fontainebleau of 1685, which made de Moivre a refugee | 3 | S077, S082 |
| The Nine Chapters sits directly after the Qin book burning of 213 BCE, and Seki worked inside the closed Tokugawa state, in a mathematical culture that kept its methods secret. Both facts are about politics shaping what gets written down and what survives | 4 | S123, S128 |
| Three names in Unit 4 are attached to the wrong people or to only half the people: the Ehrenfest urn has two authors, the Metropolis algorithm was coded by Arianna Rosenbluth, and Chapman-Kolmogorov carries the name of a geophysicist whose own biography does not mention it | 6 | S224, S256, S238 |
| Konigsberg was flattened in 1944 to 1945 and renamed Kaliningrad in 1946. Two of the founding ideas of graph theory came out of that town, a century apart, and neither man thought he was doing graph theory | 7 | S422, S445 |
| Hook 1. Library of Congress LOT 11931, "72 items: pen, ink, wash and photographs; on 56 poster boards 71 x 56 cm", rights line "No known restrictions on publication" | 8 | S282, S380, S381, S383 |
| Hook 2. Nightingale did not invent the polar area diagram. Guerry drew courbes circulaires in 1829 to show seasonal wind direction. What she did was aim a statistical graph at people who would never read a table | 8 | S267, S268 |
| Hook 4. TNA HW 25/37, section 1.5 "The Factor Principle" and section 1.6 "Decibanage", with the unit named "in honour of the famous town of Banbury". Crown copyright has expired, so the whole document is printable | 8 | S261, S262 |
| Hook 5. "In 1948, with the Cold War growing chillier, Wood had the Bureau's input-output work included in an interagency project, funded by the Air Force, known as Project SCOOP." Funding soared after 1950 and stopped in 1953 | 8 | S283, S199 |
| Hook 12. Andre Weil arrested in Finland in November 1939 carrying calling cards reading "Nicolas Bourbaki, member of the Royal Academy of Poldavia", released 12 December 1939, and imprisoned at Rouen where he proved the Riemann hypothesis for curves over finite fields. His sister was Simone Weil | 8 | S277 |
English
7 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Frege's reply to Russell, conceding in print that "my law V... is false", is a short text worth reading in an essay-writing class as a model of intellectual honesty | 1 | S056 |
| Pingala's whole subject is scansion. A Sanskrit meter is a string of light and heavy syllables, which is a binary string, so "how many metres of eight syllables are there" and "how many bytes are there" are the same question. Shah calls the related power-of-two device "the closest Indian mathematicians came to inventing binary numbers" | 2 | S002 |
| Three dead metaphors: matrix is a womb, eliminate is throwing somebody out of the door across the threshold, and Maclaurin's exterminate is driving them past the boundary marker. Ask a class to find the metaphor still living inside a word they use every day | 4 | S306, S132 |
| Markov's data is the opening of Eugene Onegin, and his second study used 100,000 letters of Aksakov. Any class can repeat the experiment in its own language, and the two conditional probabilities will come out different | 6 | S213, S214 |
| The most famous idea in network science is a bar bet in a Hungarian short story of 1929, and it is worth reading aloud. Karinthy dates his own story by giving the population of the Earth as 1.5 billion | 7 | S439 |
| Hook 7. Markov tallied 8,638 vowels and 11,362 consonants in the first 20,000 letters, by hand. Independence predicts about 3,731 vowel-vowel pairs and he counted 1,104 | 8 | S271 |
| Five dead metaphors in one lesson: a matrix is a womb, a pivot is the pin a wheel turns on, an echelon is a rung of a ladder, a cardinal number is named after a door hinge, and latent roots are latent the way heat is latent in water | 9 | S306, S302 |
Biology
1 hook.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Hook 9. Hardy's letter, dated 5 April 1908 from Trinity College, Cambridge, printed in Science, New Series, 28(706), 49 to 50, opening "I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge". He wrote the three genotype counts as p: 2q: r, squared a sum, and showed the proportions lock after one generation. Worked in full at Chapter 8, section 8.7.2 | 8 | S270, S388 |
Chemistry
3 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Varahamihira's problem is a blending problem: choose four ingredients from sixteen and vary the strengths. It is the same structure as a recipe, a paint mix, or a fragrance pyramid | 2 | S010 |
| The word "graph" is borrowed from a Kekule structural diagram, in the same sentence in which Sylvester coins "chemicograph". Put a benzene ring and a state diagram on the board side by side and ask what is the same | 7 | S302 |
| George Szekeres trained as a chemical engineer, worked six years as an analytical chemist in Budapest, and escaped to Shanghai in 1939 to work as a leather chemist, all while doing the mathematics he is remembered for | 7 | S434 |
Art and design
4 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Hook 1. One Du Bois chart curls a bar into a spiral because the numbers grew too fast for a straight line to fit the board, and in no chart does any measure related to his subject trend down. That is not an accident, it is an argument | 8 | S281, S282 |
| Hook 2. Nightingale's chart carries its own title, "Diagram of the Causes of Mortality in the Army in the East", and her own category word, "zymotic". The wedge she made biggest was infection, not battle | 8 | S267, S268 |
| Bold face for vectors exists because a printer had Clarendon in the case in 1901 and a pencil cannot make it, which is why students invented the arrow and the underline. Notation is constrained by manufacturing | 9 | S304 |
| The exclamation point became the factorial sign partly because it was a sort nobody else was competing for, and in the printing trade it was called the note of admiration | 9 | S301 |
Computer science
7 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Hill's cipher is modular arithmetic plus a matrix inverse, and it is the first job matrices ever did outside pure mathematics that a student can run by hand | 4 | S140 |
| An n-gram model of order k is a Markov chain on the space of k-grams, exactly. A transformer is not, in any useful sense. Researchers deliberately train transformers on data generated by known Markov chains to check whether the network recovers the chain | 6 | S225, S240 |
| De Bruijn's 1946 sequences are built by taking an Euler circuit through a small graph, so Euler's 1736 bookkeeping runs inside modern position encoders and sequence assembly | 7 | S444 |
| AUTOMATH, de Bruijn's language, is "so designed that it is incorrect to state a theorem without first 'constructing' a proof of the theorem", which is a direct ancestor of the Coq system Gonthier used in 2005 | 7 | S444, S427 |
| Hook 4. Turing needed a unit for evidence, invented one, and named it after a town in Oxfordshire where the tally sheets were printed. Because logarithms turn multiplication into addition, you add decibans up until you have enough to act | 8 | S261, S262 |
| Hook 6. Hill, "Cryptography in an Algebraic Alphabet", 1929: "We say that T is a 'normal' transformation if its determinant is primary." US Patent 1,845,947, filed 14 February 1929, granted 16 February 1932, with Louis Weisner as first-named inventor | 8 | S140, S141 |
| Hook 11. The Erdos Number Project, running since 25 May 1995, gives 514 people at distance 1 and 13,782 at distance 2 as of August 2025. Computing the distances is breadth-first search done by hand on real people | 8 | S264, S263 |
Economics
2 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Leontief's input-output method, the 1973 economics Nobel, is a matrix inverse applied to an entire national economy: solve and you know how much everything must produce | 5 | S199 |
| Hook 5. The 1947 table had 450 industrial sectors, and a 450 by 450 elimination needs about 30 million multiplications, which is why the Air Force cared | 8 | S283, S199 |
Music
2 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Movement four of the ILLIAC Suite, premiered in a student lounge on 9 August 1956, used "Markov chains (zero and first order) for interval and harmony selection" | 6 | S241, S242 |
| Hook 8. Four undergraduates played a string quartet written by a five-ton computer on 9 August 1956 in the Wedgewood Lounge of the Illini Union. Two years later Xenakis defined eight "screens" and an 8 by 8 matrix of probabilities for moving between them, printed as Table 2 of a peer reviewed journal | 8 | S273, S272, S274 |
Philosophy
5 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Venn was a lecturer in the Moral Sciences, not in mathematics, so the Venn diagram arrives out of a philosophy course | 1 | S048 |
| De Moivre closed the 1756 preface with a design argument: the doctrine that finds chance where chance really is can prove "that where Uniformity, Order and Constancy reside, there also reside Choice and Design". Price's offprint title turns a gambling paper into an argument about knowledge | 3 | S081, S085 |
| Nekrasov argued that the law of large numbers requires independence, that social statistics obey the law, and therefore that human choices are free. Markov attacked the weakest link with a counterexample | 6 | S212, S218 |
| Tymoczko's four objections to computer proof, from 1979, are the four objections a class will raise unprompted: not a priori, not certain, not surveyable, and not checkable by human mathematicians | 7 | S428 |
| Hook 12. Students who have met Simone Weil in a philosophy lesson can be shown that her brother was in a Paris cafe in December 1934 inventing a fictional mathematician whose calling cards got him arrested in 1939 | 8 | S277 |
Health
1 hook.
| The anchor fact | Chapter | Sources |
|---|---|---|
| Hook 10. Floros (2018), a peer reviewed review: "between 4% and 8 % of adolescent gamblers are experiencing significant gambling-related problems", with reported problem prevalence of 2.1% to 2.6% in North America, 0.2% to 4.4% in Oceania, and 0.2% to 12.3% in Europe, and "the proportion of pathological gambling among adolescents in the US could be more than three times that of adults". The European range is as much a statement about different screening instruments as about different behavior. The arithmetic that goes with it is the house edge: every bet on a roulette layout has the same expected value and it is negative, exactly of the stake on a European wheel and on an American one, worked in full at Chapter 8, section 8.7.1 | 8 | S389 |
Also present, and not on the ten-subject list
11 hooks.
| The anchor fact | Chapter | Sources |
|---|---|---|
| The same German sentence of Cantor's reads "aggregate" in Jourdain's 1915 English and "set" in a 2024 translation. Klyve makes the point harder: comparing editions shows translators and editors altered Euler's text, so a quotation from an English edition is not automatically a quotation from Euler. Hook: give a language class the two English versions of one sentence and ask which one is Cantor | 1 | S041, S042, S052 |
| Italy says triangolo di Tartaglia, China says Yang Hui's triangle, and India said meru-prastara. What a country names a thing is a curriculum decision | 2 | S011, S013 |
| "Stochastic" is Greek for aiming at a target, taken from Plato's Philebus 55e; "la regle de Bayes" is French, coined in 1843 by a man who was not naming his own work | 3 | S079, S088 |
| Eigenwert is a transparent compound to a German speaker, own-value. English took the Wert and left the eigen untranslated, and Merriam-Webster labels the result a "partial translation". Ask a German class what other half-translated compounds they can find in English science | 5 | S202 |
| Grassmann's Law is a real rule of sound change in Sanskrit and Greek, still being refined in Language in 1966. The man who wrote the founding book of linear algebra got his honorary doctorate for the linguistics | 5 | S203, S181 |
| Algebra and algorithm are one ninth-century author twice, one from his book's title and one from his name. Zero and cipher are the same Arabic word, itself rendering Sanskrit sunya, empty | 9 | S302, S306 |
| The vector analysis war of 1890 to 1894 ran across eight journals with twelve named participants, and the winning side won on convenience for electrical engineering, not on mathematical elegance. Pair it with any modern argument about a standard | 5 | S177 |
| Detailed balance arrived in Unit 4 from Boltzmann's 1872 H-theorem by way of Wegscheider's chemical kinetics in 1901 and Einstein's radiation theory in 1916 | 6 | S230, S231 |
| The Ehrenfest urn is a thermodynamics argument you can run with counters on a table, and the journal's own contents page for 6 May 1907 names both authors, "P. u. T. Ehrenfest" | 6 | S221, S224 |
| Hook 3. Brody et al. in The Lancet, 2000: the board met on 7 September 1854, the handle came off on 8 September, the first spot map was exhibited on 4 December, and "The map did not give rise to the insight, but rather it tended to confirm theories already held". The person who found the index case was a curate called Henry Whitehead, who set out to prove Snow wrong | 8 | S269 |
| Konigsberg on the Pregel, the island called der Kneiphof, seven bridges in 1735, an eighth added in 1875 that makes the walk possible, the town flattened in 1944 to 1945, renamed in 1946 | 7 | S420, S422, S445 |
Appendix E
Every number in this book, re-derived and checked in code
1,157 checks across 22 scripts, and 0 failures. Not checked by rereading them. Recomputed, in code, from the same starting values the original author used, and compared against the value that author printed.
This is the appendix that lets you distrust the rest of the book productively. If Markov's four numbers do not multiply out, if Galileo's 216 outcomes do not add up, if a claimed eigenvalue is not an eigenvalue, it shows up here rather than in your head three weeks later.
Where a widely repeated modern retelling is wrong, that is printed too. One example runs through Chapter 6: a well known magazine account of Markov's Onegin count gives 15,069 mixed pairs, and 20,000 letters contain 19,999 overlapping pairs, not 20,000. The difference is one pair and it is printed rather than smoothed over.
The scripts are the source and this page is their output, not a transcription of it. Change a number in the book without changing the mathematics and the build fails.
One thing you will see inside the raw output. Some checks count how many rows of a register carry a particular label: [PRIMARY] means the claim was read in an original document, [SCHOLARLY] in a peer reviewed source, [REFERENCE] in a reference work, [CONTESTED] where sources disagree, and [UNVERIFIED] where nothing consulted confirmed it. Those labels belong to the research files, not to this book's prose, and they appear here only because this is what the script printed.
| Script | Checks passed | Checks failed |
|---|---|---|
verify/book_numbers.py | 30 | 0 |
verify/ch01.py | 36 | 0 |
verify/ch03.py | 27 | 0 |
verify/ch04.py | 82 | 0 |
verify/ch05.py | 122 | 0 |
verify/ch06.py | 144 | 0 |
verify/ch08.py | 87 | 0 |
verify/ch09.py | 94 | 0 |
verify/claims.py | 5 | 0 |
verify/domainA.py | 11 | 0 |
verify/domainC.py | 70 | 0 |
verify/domainD.py | 34 | 0 |
verify/domainE.py | 51 | 0 |
verify/domainF.py | 98 | 0 |
verify/domainG.py | 47 | 0 |
verify/domainJ.py | 124 | 0 |
verify/figures.py | 20 | 0 |
verify/gap_closures.py | 16 | 0 |
verify/registers.py | 15 | 0 |
verify/tags.py | 5 | 0 |
verify/timeline.py | 37 | 0 |
verify/translit.py | 2 | 0 |
E.1 The mathematics of Chapter 1: The words are younger than you are
Three short pieces, each tied to a document named above. Every number below is produced by verify/ch01.py, output in verify/ch01_output.txt, which runs 36 checks and reports checks failed: 0. (a) Boole's one equation, 1847
Boole's whole system rests on the claim that logic obeys a law ordinary quantity does not: performing a selection twice gives the same result as performing it once. In symbols, from p. 16 of the 1847 pamphlet, "xx = x, or x squared equals x" (S045). Solve it as ordinary algebra and see what survives.
Nothing else works. The script searched every integer from to and every rational with denominator up to 100, and found exactly two solutions. Quoting the output:
[PASS] S-045 integer solutions of x^2 = x in the range -10000 to 10000
computed = [0, 1]
claimed = [0, 1]That is Boole's point in one line. If the symbols of logic obey , then the only numbers logic needs are 0 and 1, nothing and everything. The complement follows immediately: if the Universe is 1, then "not x" is , which is p. 15 of the same pamphlet. (b) De Morgan's laws, checked on every case
De Morgan's 1850 sentence, as Miller reports it, is "The contrary of an aggregate is the compound of the contraries of the aggregants" (S055, d.html). In your book's notation, with the universal set:
Take , , .
and for the second law,
One example proves nothing, so the script did all of them. A 4 element universe has subsets, so there are ordered pairs . Both laws hold in all 256:
[PASS] S-055 De Morgan law one: (A union B) complement = A complement intersect B complement, all 256 pairs
computed = True
claimed = True(c) Dedekind's Definition 64, 1888
"A system S is said to be infinite when it is similar to a proper part of itself; in the contrary case S is said to be a finite system" (S043, p. 31). "Similar" means matched one to one. Take the counting numbers and the doubling map.
Two things have to be true for Dedekind's definition to bite. The map must be one to one, and its image must be a proper part, meaning something is left out. Both check:
[PASS] S-043 the map n -> 2n is injective on the first 5000 naturals
computed = 5000
claimed = 5000and separately, 1 is a counting number that is not of the form , so the even numbers are a proper part. Meanwhile, below any finite cutoff the evens look like exactly half: 50 of the first 100. That mismatch, half by counting and equal by matching, is the thing Bolzano called a paradox in 1851 and Dedekind turned into the definition in 1888. (d) One number from Venn's own survey
Venn consulted 60 logic treatises and found 34 already using diagrams (S051, p. 112).
So 56.67 percent of the logic books on Venn's own shelf already had diagrams in them, and 26 did not. The script confirms computed = 56.67.
What verify/ch01.py printed
==========================================================================
SECTION 1. Boole's index law, x squared equals x (S-045 p. 16; S-046 p. 6)
==========================================================================
[PASS] S-045 integer solutions of x^2 = x in the range -10000 to 10000
computed = [0, 1]
claimed = [0, 1]
source = Boole, The Mathematical Analysis of Logic (1847), p. 16 of the Gutenberg PDF
[PASS] S-045 rational solutions of x^2 = x with denominator up to 100
computed = ['0', '1']
claimed = ['0', '1']
source = Boole, 1847, p. 16; the point of the law is that logic keeps only 0 and 1
[PASS] S-045 the factorisation x^2 - x = x(x - 1) holds for every integer tested
computed = True
claimed = True
source = algebraic identity used in the worked example of section 1.7
==========================================================================
SECTION 2. De Morgan's laws, checked exhaustively (S-055 d.html; S-069)
==========================================================================
[PASS] number of subsets of a 4 element universal set
computed = 16
claimed = 16
source = derived: 2^4 = 16, the frame for the exhaustive check below
[PASS] number of ordered pairs (A, B) tested
computed = 256
claimed = 256
source = derived: 16 x 16 = 256
[PASS] S-055 De Morgan law one: (A union B) complement = A complement intersect B complement, all 256 pairs
computed = True
claimed = True
source = De Morgan, Trans. Camb. Phil. Soc. 9 (1850), as reported by Miller, S-055 d.html
[PASS] S-055 De Morgan law two: (A intersect B) complement = A complement union B complement, all 256 pairs
computed = True
claimed = True
source = De Morgan, 1850, as reported by Miller, S-055 d.html
[PASS] worked instance: A = {1,2}, B = {2,3}, U = {1,2,3,4}; (A union B) complement
computed = [4]
claimed = [4]
source = worked example in chapter section 1.7
[PASS] worked instance: A complement intersect B complement
computed = [4]
claimed = [4]
source = worked example in chapter section 1.7
[PASS] worked instance: (A intersect B) complement
computed = [1, 3, 4]
claimed = [1, 3, 4]
source = worked example in chapter section 1.7
[PASS] worked instance: A complement union B complement
computed = [1, 3, 4]
claimed = [1, 3, 4]
source = worked example in chapter section 1.7
==========================================================================
SECTION 3. Dedekind's Definition 64, tested on the even numbers (S-043 p. 31)
==========================================================================
[PASS] S-043 the map n -> 2n is injective on the first 5000 naturals
computed = 5000
claimed = 5000
source = Dedekind, Was sind und was sollen die Zahlen? (1888), Definition 64
[PASS] S-043 every image 2n is itself a natural number (the map lands inside the set)
computed = True
claimed = True
source = Dedekind, 1888, Definition 64
[PASS] S-043 the image is a PROPER part: 1 is a natural number and is not of the form 2n
computed = False
claimed = False
source = Dedekind, 1888, Definition 64: 'similar to a proper part of itself'
[PASS] S-043 how many of the first 5000 naturals the map misses
computed = 2500
claimed = 2500
source = derived: exactly the odd numbers are left out, and there are still as many evens as naturals
[PASS] the finite illusion: count of evens at or below 100 versus count of naturals at or below 100
computed = (50, 100)
claimed = (50, 100)
source = derived: the finite count is exactly half, which is why the definition looks wrong before it looks right
==========================================================================
SECTION 4. Venn's own survey of the logic literature (S-051 p. 112)
==========================================================================
[PASS] S-051 books Venn consulted that already used diagrams
computed = 34
claimed = 34
source = Bennett, Origins of the Venn Diagram (2015), p. 112
[PASS] S-051 books Venn consulted in total
computed = 60
claimed = 60
source = Bennett, 2015, p. 112
[PASS] S-051 books with no diagrams: 60 minus 34
computed = 26
claimed = 26
source = derived from Bennett, 2015, p. 112
[PASS] S-051 34 of 60 as a fraction in lowest terms
computed = 17/30
claimed = 17/30
source = derived from Bennett, 2015, p. 112
[PASS] S-051 34 of 60 as a percentage, to two decimal places
computed = 56.67
claimed = 56.67
source = derived from Bennett, 2015, p. 112
==========================================================================
SECTION 5. Gaps between the mathematics and the words for it (S-055, S-054)
==========================================================================
[PASS] S-055 years from De Morgan's death (1871) to the first printed phrase 'De Morgan's Laws' (1945)
computed = 74
claimed = 74
source = Miller, S-055 d.html: JSL index, 1945; MacTutor, S-063: died 18 March 1871
[PASS] S-055 years from Cantor's Beitrage (1895) to the English word 'union' (1912)
computed = 17
claimed = 17
source = Miller, S-055 u.html (Pierpont 1912) and S-054 (Beitrage 1895)
[PASS] S-055 years from Bolzano's posthumous book (1851) to the English phrase 'set theory' (1926)
computed = 75
claimed = 75
source = Miller, S-055 s.html (Frink 1926); S-044 (Bolzano 1851)
[PASS] S-051 years from Sturm's printed circles (1661) to Venn's paper (1880)
computed = 219
claimed = 219
source = Bennett, S-051, pp. 108 and 112
[PASS] S-051 years from Leibniz's manuscript (c. 1686) to its first printing (1903), matching Bennett's 'over 200 years'
computed = 217
claimed = 217
source = Bennett, S-051, pp. 108-109
[PASS] S-040 and S-055 years from Cantor's first uncountability paper (1874) to the English 'empty set' (1919)
computed = 45
claimed = 45
source = S-040 (Crelle 77, 1874); Miller, S-055 e.html (McAtee 1919)
==========================================================================
SECTION 6. Ages and intervals stated in the chapter
==========================================================================
[PASS] S-044 Bolzano's age when he finished the treatise in summer 1848, taking 1781 as the birth year
computed = 67
claimed = 67
source = S-044 foreword gives the age as 67; S-068 gives life dates 1781 to 1848
[PASS] S-060 Boole's age at death on 8 December 1864, born 1815
computed = 49
claimed = 49
source = MacTutor, S-060: 'died on 8 December 1864, aged 49'
[PASS] S-060 Boole's age when appointed first Professor of Mathematics at Cork in 1849
computed = 34
claimed = 34
source = MacTutor, S-060
[PASS] S-061 Mary Everest Boole's age when widowed (born 11 March 1832, widowed 8 December 1864)
computed = 32
claimed = 32
source = MacTutor, S-061: 'a widow at 32'
[PASS] S-063 De Morgan's age when appointed at London University in 1827, born 27 June 1806
computed = 21
claimed = 21
source = MacTutor, S-063: 'in 1827, aged 21'
[PASS] S-053 days between Cantor's two December 1873 letters to Dedekind (2 December to 7 December)
computed = 5
claimed = 5
source = Dauben, S-053
[PASS] S-062 years Cantor lived after the 1874 paper (1874 to 6 January 1918)
computed = 44
claimed = 44
source = MacTutor, S-062
==========================================================================
SECTION 7. Zermelo's system (S-057, S-065)
==========================================================================
[PASS] S-057 number of axioms in Zermelo's 1908 system
computed = 7
claimed = 7
source = SEP, Zermelo's Axiomatization of Set Theory, S-057; MacTutor, S-065
[PASS] S-057 'Every set M possesses at least one subset M_0 that is not an element of M', checked on all 16 subsets of {1,2,3,4}
computed = True
claimed = True
source = SEP, S-057, quoting Zermelo 1908
==========================================================================
SUMMARY
==========================================================================
checks run : 36
checks passed : 36
checks failed : 0
Every number printed in CH-01-sets-and-infinity.md sections 1.2 to 1.7
is produced by this script.E.2 The mathematics of Chapter 2: The triangle with five names, and the one thing Pascal added
Everything in this section is checked by verify/domainA.py, and its output is in verify/domainA_output.txt. Worked example 1. Varahamihira's perfumes, and exactly where his number goes wrong
The problem, as the sixth century text sets it. Sixteen aromatic substances are named in verses 13 and 14. Choose four of them. Give the four chosen substances the strengths one part, two parts, three parts, and four parts, in some order. How many different perfumes? (S010)
Step 1: choose the four substances. Order does not matter yet, so this is a combination.
The verifier confirms it:
[PASS] S-010 number of ways to choose 4 perfume substances from 16
computed = 1820
claimed = 1820
source = Brhat Samhita ch. 77, vv. 13 to 17, via Iyer 1884Step 2: hand out the four proportions. Four distinct strengths, four chosen substances, one strength each. That is a permutation of four objects.
Step 3: multiply, by the multiplication principle.
[PASS] S-010 corrected total perfumes, C(16,4) x 4!
computed = 43680
claimed = 43680
source = Iyer's translator's note after v. 21Step 4: now look at the printed figure. Verse 17 says: "The process gives us 174720 (4000+70000+100000+720) different varieties of perfumes." (S010, ch. 77 v. 17, trans. Iyer 1884) Two things are worth checking separately, and this is the part students should do themselves.
First, is the text's own addition right? Add the four pieces it offers:
[PASS] S-010 the text's own addition 4000 + 70000 + 100000 + 720
computed = 174720
claimed = 174720
source = Brhat Samhita ch. 77 v. 17, as printedSo the addition is fine. The sum of the parts really is the total that is printed. The mistake is upstream of the addition.
Second, what multiplier does 174720 imply?
[PASS] S-010 the figure printed in the text, C(16,4) x 96
computed = 174720
claimed = 174720
source = Brhat Samhita ch. 77 v. 17: '174720 (4000+70000+100000+720)'
[PASS] S-010 ratio between the printed figure and the corrected one
computed = 4
claimed = 4
source = derived: the error is a factor of 4 in the arrangement count, 96 versus 24Where the printed figure goes wrong, in one sentence. The number of substances is right, the choice count is right, and the addition of 4000, 70000, 100000, and 720 is right; the error is in the arrangement count, where the text uses 96 arrangements for each chosen set of four when there are only , so every perfume has been counted exactly four times over. That is Iyer's finding: he says Varahamihira "counted 96 arrangements for each set of four substances where the correct count is 24," and gives the corrected total as 43680, or 28392 once the restrictions on which substances may be mixed are applied. (S010, translator's note)
One caution before you call it a mistake. "Error" is Iyer's judgment, not the author's admission. If repeated substances were allowed, or if the four proportions could be assigned in a way that the translation does not capture, 96 might be a different convention rather than a slip. Nobody in this book has read a second translation, so the honest label is: the arithmetic as translated does not work, and the reason it does not work is a factor of 4.
A bonus from the same chapter. Verse 10 gives a nine-substance recipe with the parts 1 through 9, which the translation associates with 362880 varieties. That is , and it checks:
[PASS] S-010 arrangements of the nine-substance recipe, 9!
computed = 362880
claimed = 362880
source = Brhat Samhita ch. 77 v. 10 and note, via Iyer 1884Worked example 2. The additive rule, from Halayudha's mountain and al-Karaji's board
The rule, as two of our sources state it. Halayudha: write one square cell at the top, then two cells below it extending half way on both sides, and so on, with each cell the sum of the two above. (S002) Al-Karaji, quoted by al-Samaw'al: "in order to achieve that, you place on a board one and one below it". (S003, PDF p. 17) Neither of them wrote it as an equation. In modern notation it is
Build the array up to the row that Shah prints. Start with a single 1, and make every new entry the sum of the two entries above it, with 1 at each end.
| Row | Entries |
|---|---|
| 0 | 1 |
| 1 | 1, 1 |
| 2 | 1, 2, 1 |
| 3 | 1, 3, 3, 1 |
| 4 | 1, 4, 6, 4, 1 |
| 5 | 1, 5, 10, 10, 5, 1 |
| 6 | 1, 6, 15, 20, 15, 6, 1 |
Take one entry and check it against the closed formula, so you can see the two descriptions are the same object:
[PASS] S-002 sixth row of the meru-prastara as printed by Shah
computed = [1, 6, 15, 20, 15, 6, 1]
claimed = [1, 6, 15, 20, 15, 6, 1]
source = Shah, A History of Pingala's Combinatorics, Halayudha's constructionNow add a row up. This is the property Pascal states as a consequence, that the sum of the cells of each base is double the base before it. (S001, and see IMG-005)
Al-Samaw'al's table runs to the twelfth row, so his deepest row has thirteen entries and they sum to :
[PASS] S-003 twelfth row of al-Samaw'al's table has 13 entries summing to 2^12
computed = (13, 4096)
claimed = (13, 4096)
source = Bajri, Hannah and Montelle 2015, PDF p. 1 and Table 2 p. 18That doubling is the same fact as: add one element to a set and you double the number of subsets, which is Course section 1.4 of your course arriving in Course section 2.1 wearing a different hat.
And what Pascal proved by induction. His Douziesme Consequence is a ratio between two neighboring cells in the same base, which in modern symbols says
Check it on row 6 with : the ratio is , and is . The verifier tests the identity for every up to 19:
[PASS] S-001 Pascal's Douziesme Consequence ratio identity for n up to 19
computed = True
claimed = True
source = Pascal, Traite du triangle arithmetique, 1665, OCR p. 7Notice what the induction is doing. The identity has, as Pascal says, "an infinity of cases." He does not check them. He checks the second base, shows that any base passing the test forces the next base to pass it, and lets the two lemmas do the rest of the work forever. That machine is the thing Pascal added to a table that four other civilizations already had.
The 1303 Chinese table gives us one more row to check, since it tabulates the coefficients up to the eighth power:
[PASS] S-011 eighth row of the 1303 Chinese table
computed = [1, 8, 28, 56, 70, 56, 28, 8, 1]
claimed = [1, 8, 28, 56, 70, 56, 28, 8, 1]
source = Miller, Earliest Known Uses, entry PASCAL'S TRIANGLEE.3 The mathematics of Chapter 3: Chance: from a gambler's notebook to five axioms
Four worked examples, one per course section. Every line below is checked by code. Sections 1 to 16 of verify/domainC.py produce verify/domainC_output.txt (70 checks run, 70 passed, 0 failed). The arithmetic that appears only in this chapter is checked by verify/ch03.py, whose output is verify/ch03_output.txt (27 checks run, 27 passed, 0 failed). 3.7.1 Galileo's three dice: why 10 beats 9 (supports 2.2)
Three dice, each with six faces. Count ordered outcomes, because a die does not know it is interchangeable with the other two.
Both 9 and 10 can be made from exactly six unordered combinations, which is why the gamblers' claim looked absurd. The rescue is Galileo's rule that a combination of three equal numbers has 1 ordering, two equal and one different has 3, and three all different has 6.
| Total 9 | Orderings | Total 10 | Orderings |
|---|---|---|---|
| 1 + 2 + 6 | 6 | 1 + 3 + 6 | 6 |
| 1 + 3 + 5 | 6 | 1 + 4 + 5 | 6 |
| 1 + 4 + 4 | 3 | 2 + 2 + 6 | 3 |
| 2 + 2 + 5 | 3 | 2 + 3 + 5 | 6 |
| 2 + 3 + 4 | 6 | 2 + 4 + 4 | 3 |
| 3 + 3 + 3 | 1 | 3 + 3 + 4 | 3 |
| Total | 25 | Total | 27 |
From verify/domainC_output.txt, Section 1:
[OK ] Total ordered outcomes with three dice
computed = 216
claimed = 216
[OK ] Ordered ways to make 10
computed = 27
claimed = 27
[OK ] Ordered ways to make 9
computed = 25
claimed = 25
[OK ] Unordered triples summing to 9
computed = 6
claimed = 6
note: this is why 9 and 10 look equal: both have 6 partitionsFrom verify/ch03_output.txt, Sections 1 and 2:
[OK ] Ordering multiplicities for 10, in the chapter's row order
computed = [6, 6, 3, 6, 3, 3]
claimed = [6, 6, 3, 6, 3, 3]
note: 136, 145, 226, 235, 244, 334
[OK ] The gap, as a reduced fraction
computed = 1/108
claimed = 1/1083.7.2 The de Mere calculation, and how small the gap really is (supports 2.5)
Two dice thrown together give 36 equally likely outcomes, one of which is a double six. Assume the throws are independent, which is de Moivre's 1738 word for it, and use the complement rule from 2.2.
So 24 throws is a losing bet and 25 is a winning one, and the whole difference between them is 0.014128, which verify/book_numbers.py section 2 recomputes. From verify/domainC_output.txt, Section 2:
[OK ] P(double six in 24), Ore's .4914 and Gorroochurn's 0.491
computed = 0.491404
claimed = 0.491400 (tolerance 0.0001)
[OK ] Smallest n with P(double six in n) > 1/2
computed = 25
claimed = 25
[OK ] Old Gambler's Rule prediction for two dice
computed = 24
claimed = 24
note: the rule predicts 24; the true answer is 25And from verify/ch03_output.txt, Section 3:
[OK ] Difference between the two
computed = 0.01412767010747491
claimed = 0.014128 (tolerance 1e-06)
note: one and four tenths of a percentage point3.7.3 Bayes' rule the way Laplace stated it (supports 2.4)
Here is the Sixth Principle again, in Truscott and Emory's English: "The probability of the existence of any one of these causes is then a fraction whose numerator is the probability of the event resulting from this cause and whose denominator is the sum of the similar probabilities relative to all the causes" (S086). Numerator: one cause. Denominator: every cause, summed. That sum is the law of total probability, and it is the reason 2.4 teaches the two rules together.
Take a disease that one person in a hundred has, a test that catches 99 percent of the people who have it, and a test that correctly clears 95 percent of the people who do not. Write for having the disease and for a positive test. Laplace's denominator first.
Now his numerator over his denominator.
The same computation as a head count of ten thousand people, which is the version to use on a class that mistrusts fractions:
From verify/domainC_output.txt, Section 9:
[OK ] P(+) by the law of total probability
computed = 297/5000
claimed = 297/5000
[OK ] P(D|+) by Bayes' rule
computed = 1/6
claimed = 1/6
[OK ] Partition check: P(D|+) + P(not D|+) = 1
computed = 1
claimed = 1From verify/ch03_output.txt, Section 4:
[OK ] All positives
computed = 594
claimed = 594
[OK ] Share of positives who are ill, as a fraction
computed = 1/6
claimed = 1/6
[OK ] False positives outnumber true positives five to one
computed = 5
claimed = 5A test that is 99 percent sensitive, applied to a rare disease, gives five false alarms for every real case. The denominator is where the surprise lives, which is why Laplace spent a whole clause on it. 3.7.4 Pairwise independence is not mutual independence (supports 2.5)
De Moivre's 1738 definition covers two events. Three events are harder, and the standard counterexample is short. Take four equally likely outcomes, written as three-digit labels: 112, 121, 211, and 222. Let be "the first digit is 1", "the second digit is 1", "the third digit is 1".
Every pair is independent and the triple is not, because no outcome has three 1s. From verify/domainC_output.txt, Section 12:
[OK ] Mutual independence FAILS: P(ABC) is not P(A)P(B)P(C)
computed = True
claimed = True
note: P(ABC) = 0 but the product is 1/8
[OK ] A and B are NOT independent in this second example
computed = True
claimed = True
note: so the triple product alone does not imply pairwise independenceThat second line matters: the script also builds an eight-point example where the triple product holds and yet and are dependent. Neither condition implies the other, which is why 2.5 defines independence for a whole family and not just pair by pair. This example is usually attributed to Sergei Bernstein, and that attribution is here: the mathematics is checked from first principles, the credit is not (see 3.10, item 7).
What verify/ch03.py printed
------------------------------------------------------------------------------
SECTION 1. Galileo's six combinations for 9 and for 10, broken out one by one.
------------------------------------------------------------------------------
Total 9: unordered combinations and the orderings each contributes
1 + 2 + 6 = 9 -> 6 orderings
1 + 3 + 5 = 9 -> 6 orderings
1 + 4 + 4 = 9 -> 3 orderings
2 + 2 + 5 = 9 -> 3 orderings
2 + 3 + 4 = 9 -> 6 orderings
3 + 3 + 3 = 9 -> 1 orderings
combinations = 6, orderings = 25
Total 10: unordered combinations and the orderings each contributes
1 + 3 + 6 = 10 -> 6 orderings
1 + 4 + 5 = 10 -> 6 orderings
2 + 2 + 6 = 10 -> 3 orderings
2 + 3 + 5 = 10 -> 6 orderings
2 + 4 + 4 = 10 -> 3 orderings
3 + 3 + 4 = 10 -> 3 orderings
combinations = 6, orderings = 27
[OK ] Number of unordered combinations making 9
computed = 6
claimed = 6
[OK ] Number of unordered combinations making 10
computed = 6
claimed = 6
[OK ] Ordering multiplicities for 9, in the chapter's row order
computed = [6, 6, 3, 3, 6, 1]
claimed = [6, 6, 3, 3, 6, 1]
note: 126, 135, 144, 225, 234, 333
[OK ] Ordering multiplicities for 10, in the chapter's row order
computed = [6, 6, 3, 6, 3, 3]
claimed = [6, 6, 3, 6, 3, 3]
note: 136, 145, 226, 235, 244, 334
[OK ] Orderings summing to 9
computed = 25
claimed = 25
[OK ] Orderings summing to 10
computed = 27
claimed = 27
[OK ] Brute-force ordered count for 9
computed = 25
claimed = 25
[OK ] Brute-force ordered count for 10
computed = 27
claimed = 27
------------------------------------------------------------------------------
SECTION 2. How big is the gap between 10 and 9?
------------------------------------------------------------------------------
P(10) = 27/216 = 1/8 = 0.125000
P(9) = 25/216 = 25/216 = 0.115741
P(10) - P(9) = 1/108 = 0.009259
[OK ] P(total is 10) as a reduced fraction
computed = 1/8
claimed = 1/8
[OK ] P(total is 9) as a reduced fraction
computed = 25/216
claimed = 25/216
[OK ] The gap, as a reduced fraction
computed = 1/108
claimed = 1/108
[OK ] The gap, as a decimal
computed = 0.009259259259259259
claimed = 0.009259 (tolerance 1e-06)
note: about nine throws in a thousand
[OK ] Ratio of the two counts
computed = 27/25
claimed = 27/25
note: 27 to 25 is 1.08, an eight percent edge
[OK ] Ratio as a decimal
computed = 1.08
claimed = 1.08 (tolerance 1e-09)
------------------------------------------------------------------------------
SECTION 3. The size of the gap Ore says nobody could have felt.
------------------------------------------------------------------------------
P(at least one double six in 24 throws) = 0.491404
P(at least one double six in 25 throws) = 0.505532
difference = 0.014128
[OK ] P at 24 throws
computed = 0.49140387613090325
claimed = 0.491404 (tolerance 1e-06)
note: Ore prints .4914
[OK ] P at 25 throws
computed = 0.5055315462383781
claimed = 0.505532 (tolerance 1e-06)
note: Ore prints .5055
[OK ] Difference between the two
computed = 0.01412767010747491
claimed = 0.014128 (tolerance 1e-06)
note: one and four tenths of a percentage point
[OK ] How far 24 throws falls short of an even bet
computed = 0.00859612386909674
claimed = 0.008596 (tolerance 1e-06)
------------------------------------------------------------------------------
SECTION 4. The Laplace-style Bayes computation, done on 10000 people.
------------------------------------------------------------------------------
people = 10000
have the disease = 100
do not have it = 9900
of those who have it, test + = 99
of those who do not, test + = 495
everyone who tests + = 594
share of positives who are ill = 99/594 = 1/6
[OK ] People who have the disease
computed = 100
claimed = 100
[OK ] People who do not
computed = 9900
claimed = 9900
[OK ] True positives
computed = 99
claimed = 99
[OK ] False positives
computed = 495
claimed = 495
[OK ] All positives
computed = 594
claimed = 594
[OK ] Share of positives who are ill, as a fraction
computed = 1/6
claimed = 1/6
[OK ] Share of positives who are ill, as a decimal
computed = 0.16666666666666666
claimed = 0.166667 (tolerance 1e-06)
[OK ] Head count agrees with the fraction arithmetic in verify_domainC.py Section 9
computed = 297/5000
claimed = 297/5000
note: 594/10000 = 297/5000, which is P(+) by the law of total probability
[OK ] False positives outnumber true positives five to one
computed = 5
claimed = 5
------------------------------------------------------------------------------
SUMMARY
------------------------------------------------------------------------------
checks run : 27
checks passed : 27
checks failed : 0
Every other number quoted in CH-03 is checked in qa/verify_domainC.py and
quoted in the chapter from qa/verify_domainC_output.txt.E.4 The mathematics of Chapter 4: The algorithm came first, the word came last
Three solutions of one problem. The problem is the paddy problem of chapter 8 of the Nine Chapters, in the numbers Schwartz quotes from the Shen, Crossley, and Lun translation (S124):
Here , , and are the yields in dou of one bundle of top, medium, and low grade paddy. 4.7.1 On the board, the way a Han clerk did it
The board writes each equation as a column, filled from right to left, so the first equation sits on the right (S124). The rule is: multiply a whole column through by a number, then subtract the right column from it as many times as possible (S122). Every entry stays a whole number until the last division, which is the whole point of doing it this way with sticks.
The left column now says that 36 low-grade bundles yield 99 dou, and the rest is back substitution:
From verify/domainD_output.txt, section 1:
step 1a middle x 3 -> [6, 9, 3, 102]
step 1b minus 2 x right -> [0, 5, 1, 24]
step 2a left x 3 -> [3, 6, 9, 78]
step 2b minus 1 x right -> [0, 4, 8, 39]
step 3a left x 5 -> [0, 20, 40, 195]
step 3b minus 4 x new middle -> [0, 0, 36, 99] x = 37/4 = 9 1/4 dou, y = 17/4 = 4 1/4 dou, z = 11/4 = 2 3/4 dou Check by substitution:
equation 1 : 39 (target 39) OK
equation 2 : 34 (target 34) OK
equation 3 : 26 (target 26) OKStandard translations report the Nine Chapters' own answer in exactly those mixed numbers. That claim is tagged as unverified in the notes, because nobody in this book could open the Shen, Crossley, and Lun translation to read the answer as printed (S124). 4.7.2 The same system, the way Appendix A.4 does it
Appendix A.4 works in rows, left to right, and allows the legal moves: swap two equations, scale an equation, and add a multiple of one equation to another. The arithmetic is the same arithmetic. The only difference is that fractions arrive early instead of late, because we divide as we go instead of scaling up to clear.
The multipliers, written out so you can see where each one comes from:
And the back substitution, with the equals signs lined up:
Same three numbers. From verify/ch04_output.txt, section 1:
R2 - (2/3) R1 [ 0 5/3 1/3 | 8 ]
R3 - (1/3) R1 [ 0 4/3 8/3 | 13 ] R3a - (4/5) R2a [ 0 0 12/5 | 33/5 ] back substitution:
z = (33/5) / (12/5) = 11/4
y = (8 - (1/3)(11/4)) / (5/3) = 17/4
x = (39 - 2(17/4) - 11/4) / 3 = 37/4Two things to notice. First, the board's last column read and the row method's last row reads . Those are the same equation scaled differently, and the script confirms that and are both . Second, the Chinese method never leaves the whole numbers until the final division, which matters enormously when your calculator is a pile of sticks and matters not at all when it is a computer. That is the entire practical difference between the two columns of this page, across two thousand years.
Wilhelm Jordan's contribution, in 1888, was to keep going. Instead of stopping at the triangle and back substituting, scale each pivot to 1 and clear upwards as well, so the left block becomes the identity and the answers are simply sitting in the last column:
From verify/ch04_output.txt, section 2:
scale row 3 by 1/(12/5) [ 0 0 1 | 11/4 ]
row 2 minus (its z) x row 3 [ 0 1 0 | 17/4 ]
row 1 minus (its z) x row 3 [ 1 2/3 0 | 145/12 ]
row 1 minus (its y) x row 2 [ 1 0 0 | 37/4 ]Jordan wanted this because his survey crews were adjusting angle measurements by least squares and needed the answers without a second pass (S142, S144). 4.7.3 The same system by Cramer's rule, 1750
Cramer's rule builds one determinant for the denominator and one for each unknown. Expand along the first row, which is Laplace's 1772 method (S122, S129):
Now replace one column at a time by the right-hand side:
From verify/domainD_output.txt, section 2:
common denominator det(A) = 12
numerator for x = 111 -> x = 111/12 = 37/4
numerator for y = 51 -> y = 51/12 = 17/4
numerator for z = 33 -> z = 33/12 = 11/4
Cramer's rule reproduces the fangcheng answer exactly.Cramer's own sign rule, the derangement count, is checked in the same place, and gives three positive terms and three negative for :
Cramer's derangement (inversion) count for n = 3, and the sign it gives:
exponents 123 -> 0 derangement(s) -> sign +1
exponents 132 -> 1 derangement(s) -> sign -1
exponents 213 -> 1 derangement(s) -> sign -1
exponents 231 -> 2 derangement(s) -> sign +1
exponents 312 -> 2 derangement(s) -> sign +1
exponents 321 -> 3 derangement(s) -> sign -1 3 terms positive, 3 negative, and the count of terms is 3! = 6, which is Cramer's 'autant de termes qu'il y a
de divers arrangements de n choses differentes'.
For n = 4 the count is 4! = 24, for n = 5 it is 5! = 120.Those factorials are also Seki's count, from 1683: he "knew that the number of terms in the expansion of a determinant of the nth order was n!" (S127, p. 138). Two men, on opposite sides of the world, counting the same terms sixty-seven years apart (verify/ch04.py, section 5). 4.7.4 One extra, because Course section 3.3 needs it
Cayley's warning is one line of arithmetic. Take
The products differ, and the determinants do not notice. That is why "matrices are not in general convertible" (S136, p. 17) is a statement about matrices and not about determinants. Checked in verify/ch04.py section 6 and in verify/domainD.py section 13.
Every number printed in 4.7 is recomputed in verify/ch04.py, whose last line reads:
TOTAL: 82 passed, 0 failedand everything drawn from the domain script is confirmed by the last line of verify/domainD_output.txt:
ALL ASSERTIONS PASSEDWhat verify/ch04.py printed
========================================================================
1. Appendix A.4 method: rows, left to right, exact fractions
========================================================================
System:
3x + 2y + 1z = 39
2x + 3y + 1z = 34
1x + 2y + 3z = 26
Augmented rows at the start:
R1 [ 3 2 1 | 39 ]
R2 [ 2 3 1 | 34 ]
R3 [ 1 2 3 | 26 ]
multiplier for R2 : m21 = 2/3 = 2/3
multiplier for R3 : m31 = 1/3 = 1/3
R2 - (2/3) R1 [ 0 5/3 1/3 | 8 ]
R3 - (1/3) R1 [ 0 4/3 8/3 | 13 ]
[PASS] R2a leading entry is 0 computed = 0 claimed = 0
[PASS] R2a y coefficient computed = 5/3 claimed = 5/3
[PASS] R2a z coefficient computed = 1/3 claimed = 1/3
[PASS] R2a right side computed = 8 claimed = 8
[PASS] R3a leading entry is 0 computed = 0 claimed = 0
[PASS] R3a y coefficient computed = 4/3 claimed = 4/3
[PASS] R3a z coefficient computed = 8/3 claimed = 8/3
[PASS] R3a right side computed = 13 claimed = 13
multiplier for the second pass : m32 = (4/3)/(5/3) = 4/5
R3a - (4/5) R2a [ 0 0 12/5 | 33/5 ]
[PASS] R3b y coefficient is 0 computed = 0 claimed = 0
[PASS] R3b z coefficient computed = 12/5 claimed = 12/5
[PASS] R3b right side computed = 33/5 claimed = 33/5
back substitution:
z = (33/5) / (12/5) = 11/4
y = (8 - (1/3)(11/4)) / (5/3) = 17/4
x = (39 - 2(17/4) - 11/4) / 3 = 37/4
[PASS] x computed = 37/4 claimed = 37/4
[PASS] y computed = 17/4 claimed = 17/4
[PASS] z computed = 11/4 claimed = 11/4
[PASS] x as a mixed number, 9 and 1/4 computed = 37/4 claimed = 37/4
[PASS] y as a mixed number, 4 and 1/4 computed = 17/4 claimed = 17/4
[PASS] z as a mixed number, 2 and 3/4 computed = 11/4 claimed = 11/4
substitution check, all three original equations:
[PASS] equation 1 computed = 39 claimed = 39
[PASS] equation 2 computed = 34 claimed = 34
[PASS] equation 3 computed = 26 claimed = 26
intermediate right-hand sides printed in N.7 : 39, 8, 13, 33/5
[PASS] 39 stays put computed = 39 claimed = 39
[PASS] 34 - (2/3)(39) = 8 computed = 8 claimed = 8
[PASS] 26 - (1/3)(39) = 13 computed = 13 claimed = 13
[PASS] 13 - (4/5)(8) = 33/5 computed = 33/5 claimed = 33/5
========================================================================
2. Wilhelm Jordan's continuation: all the way to the identity
========================================================================
Jordan's 1888 handbook does not stop at the triangle. It clears upwards
as well, so no back substitution is needed. Same three rows, continued.
scale row 1 by 1/(3) [ 1 2/3 1/3 | 13 ]
scale row 2 by 1/(5/3) [ 0 1 1/5 | 24/5 ]
scale row 3 by 1/(12/5) [ 0 0 1 | 11/4 ]
row 2 minus (its z) x row 3 [ 0 1 0 | 17/4 ]
row 1 minus (its z) x row 3 [ 1 2/3 0 | 145/12 ]
row 1 minus (its y) x row 2 [ 1 0 0 | 37/4 ]
The left block is now the identity matrix and the right column is the answer.
[PASS] row 1 left block is the identity row computed = [Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)] claimed = [Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)]
[PASS] row 2 left block is the identity row computed = [Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)] claimed = [Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)]
[PASS] row 3 left block is the identity row computed = [Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)] claimed = [Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)]
[PASS] reduced row 1 gives x computed = 37/4 claimed = 37/4
[PASS] reduced row 2 gives y computed = 17/4 claimed = 17/4
[PASS] reduced row 3 gives z computed = 11/4 claimed = 11/4
========================================================================
3. Cramer's rule on the same system, every minor printed
========================================================================
det(A) = 3(3*3 - 1*2) - 2(2*3 - 1*1) + 1(2*2 - 3*1)
= 3(7) - 2(5) + 1(1)
= 21 - 10 + 1 = 12
[PASS] 2 by 2 minor for the 3 computed = 7 claimed = 7
[PASS] 2 by 2 minor for the 2 computed = 5 claimed = 5
[PASS] 2 by 2 minor for the 1 computed = 1 claimed = 1
[PASS] det(A) computed = 12 claimed = 12
numerator for x = 111
[PASS] value of x from Cramer computed = 37/4 claimed = 37/4
numerator for y = 51
[PASS] value of y from Cramer computed = 17/4 claimed = 17/4
numerator for z = 33
[PASS] value of z from Cramer computed = 11/4 claimed = 11/4
[PASS] numerator for x computed = 111 claimed = 111
[PASS] numerator for y computed = 51 claimed = 51
[PASS] numerator for z computed = 33 claimed = 33
x = 111/12 = 37/4, y = 51/12 = 17/4, z = 33/12 = 11/4
[PASS] 111/12 reduces to 37/4 computed = 37/4 claimed = 37/4
[PASS] 51/12 reduces to 17/4 computed = 17/4 claimed = 17/4
[PASS] 33/12 reduces to 11/4 computed = 11/4 claimed = 11/4
The board's last column read 36z = 99. Cramer's z reads 33/12.
[PASS] 99/36 equals 33/12 computed = 11/4 claimed = 11/4
========================================================================
4. Seki's count: an n-th order expansion has n! terms
========================================================================
n = 3 -> n! = 6
n = 4 -> n! = 24
n = 5 -> n! = 120
[PASS] 3! computed = 6 claimed = 6
[PASS] 4! computed = 24 claimed = 24
[PASS] 5! computed = 120 claimed = 120
[PASS] 5! computed computed = 120 claimed = 120
========================================================================
5. Every date interval and age stated in the chapter
========================================================================
[PASS] Leibniz's letter 1693 to Gerhardt's printing 1850 computed = 157 claimed = 157
[PASS] Seki 1683 to Leibniz 1693 computed = 10 claimed = 10
[PASS] Gauss's 'eliminatio vulgaris' 1809 to Forsythe 1953 computed = 144 claimed = 144
[PASS] Gauss coins 'determinans' 1801 to Cauchy printed 1815 computed = 14 claimed = 14
[PASS] Cauchy read 1812 to Cauchy printed 1815 computed = 3 claimed = 3
[PASS] Cayley's memoir 1858 to Turnbull's naming 1929 computed = 71 claimed = 71
[PASS] Cayley 1858 to Frobenius 1878 computed = 20 claimed = 20
[PASS] Cayley 1858 to Frobenius adopting 'matrix' 1896 computed = 38 claimed = 38
[PASS] Jordan's 2nd edition 1877 to his 3rd edition 1888 computed = 11 claimed = 11
[PASS] Sylvester's refused degree 1837 to Johns Hopkins 1877 computed = 40 claimed = 40
[PASS] Seki 1683 to Cramer's printed rule 1750 computed = 67 claimed = 67
[PASS] 200 BC to Gauss's Theoria motus, 1809 computed = 2008 claimed = 2008
[PASS] 200 BC to Sylvester's coinage, 1850 computed = 2049 claimed = 2049
[PASS] 200 BC to Forsythe, 1953 computed = 2152 claimed = 2152
[PASS] 100 BC (Lam's early bound) to 1809 computed = 1908 claimed = 1908
So 'about two thousand years' is honest for every date in the defended
range, and no single number is being smuggled in.
[PASS] age when he took the Virginia chair, 1841 (autumn) computed = 27 claimed = 27
[PASS] age in the spring of 1842 at Virginia computed = 27 claimed = 27
[PASS] age when he reached Johns Hopkins, 1877 (before 3 Sept) computed = 62 claimed = 62
[PASS] age at the Savilian chair, 1883 (before 3 Sept) computed = 68 claimed = 68
[PASS] age at death, 15 March 1897 computed = 82 claimed = 82
[PASS] 19 March 1842 plus three days computed = 672492 claimed = 672492
Hill and Weisner file : 1929-02-14
earliest day the June-July 1929 issue could carry a June date : 1929-06-01
gap in days : 107
[PASS] days from the filing to 1 June 1929 computed = 107 claimed = 107
[PASS] the gap is more than three months computed = True claimed = True
[PASS] the gap is less than five months computed = True claimed = True
[PASS] filing to grant, in days computed = 1097 claimed = 1097
[PASS] filing to grant, in whole years computed = 3 claimed = 3
The digest's phrase 'five months before the issue appeared' and the
S-141 note's phrase 'five days before that issue was even in print'
cannot both be right. The documented facts are the two dates. The
chapter prints the dates and the computed gap, and logs the clash at 4.10.
========================================================================
6. Noncommutativity, the fact Cayley printed in 1858
========================================================================
PQ = [[Fraction(2, 1), Fraction(1, 1)], [Fraction(1, 1), Fraction(1, 1)]]
QP = [[Fraction(1, 1), Fraction(1, 1)], [Fraction(1, 1), Fraction(2, 1)]]
[PASS] PQ is not QP computed = True claimed = True
[PASS] PQ top left computed = 2 claimed = 2
[PASS] QP top left computed = 1 claimed = 1
[PASS] det(PQ) computed = 1 claimed = 1
[PASS] det(QP) computed = 1 claimed = 1
[PASS] the two determinants agree even though the products do not computed = True claimed = True
========================================================================
7. The 2 by 2 inverse the course teaches in 3.5, on the paddy problem
========================================================================
[PASS] ad - bc for [[3, 2], [2, 3]] computed = 5 claimed = 5
M^{-1} = [[3/5, -2/5], [-2/5, 3/5]]
[PASS] M M^{-1} is the identity computed = [[Fraction(1, 1), Fraction(0, 1)], [Fraction(0, 1), Fraction(1, 1)]] claimed = [[Fraction(1, 1), Fraction(0, 1)], [Fraction(0, 1), Fraction(1, 1)]]
========================================================================
TOTAL: 82 passed, 0 failed
========================================================================E.5 The mathematics of Chapter 5: The planets, a half-translated word, and the machine age
Three worked examples, each tied to a named person in this chapter. Everything here is recomputed in verify/ch05.py, output in verify/ch05_output.txt, and the first, second, and third are also confirmed independently in verify/domainE.py, output in verify/domainE_output.txt. 5.7.1 An eigenvalue by hand, in exact fractions
This is the method Course section 3.7 teaches: form , set its determinant to zero, solve for , and then solve for each root. Take a two state transition matrix, which is a matrix you will meet again in Unit 4:
The characteristic polynomial, worked vertically:
Notice that the coefficients are the trace and the determinant, which is a free check on your algebra. Now the roots:
The discriminant is a perfect square, so the roots are rational and no decimals are needed anywhere. Now the eigenvectors, one row of one equation each:
From verify/ch05_output.txt, section 1:
[PASS] discriminant = 9/25 computed = 9/25 claimed = 9/25
[PASS] lambda_1 = 1 computed = 1 claimed = 1
[PASS] lambda_2 = 2/5 computed = 2/5 claimed = 2/5
[PASS] lambda_1 + lambda_2 = trace computed = 7/5 claimed = 7/5
[PASS] lambda_1 * lambda_2 = det computed = 2/5 claimed = 2/5and the two eigenvector checks:
[PASS] A (1, 1) = 1 * (1, 1) computed = [Fraction(1, 1), Fraction(1, 1)] claimed = [Fraction(1, 1), Fraction(1, 1)]
[PASS] A (1, -5) = (2/5) * (1, -5) computed = [Fraction(2, 5), Fraction(-2, 1)] claimed = [Fraction(2, 5), Fraction(-2, 1)]Two things carry forward. The eigenvalue 1 is not a coincidence: the rows of sum to 1, so leaves the vector alone. And the left eigenvector for , normalized so its entries sum to 1, is the stationary distribution , which is the whole of Course section 4.3 in one line. The domain script confirms it:
[PASS] stationary pi = (5/6, 1/6) satisfies pi P = pi: got 5/6, want 5/6The same script also works Cauchy's case, the symmetric matrix , whose characteristic polynomial factors as , and confirms the two things Cauchy proved in the 1820s: the eigenvalues are real and the eigenvectors are perpendicular.
[PASS] eigenvalues are real (max |imag| of general solver): got 0.0, want 0.0
[PASS] eigenvectors are orthogonal (dot product 0): got 0.0, want 0.05.7.2 The PageRank vector, exactly
This is Bryan and Leise's four page web (S197). Column of the matrix says where page 's links go: if page has outgoing links and one of them points to page , then . Page 1 links to pages 2, 3, and 4. Page 2 links to 3 and 4. Page 3 links only to page 1. Page 4 links to 1 and 3.
Every column sums to 1, which is exactly why 1 is an eigenvalue. The claim is that the eigenvector for , normalized so the entries sum to 1, is . Check it row by row, in exact fractions:
Four lines, and every one comes back to where it started. And , so the entries sum to exactly 1 and the vector is a probability distribution. The ranking is page 1, then page 3, then page 4, then page 2. Page 2 comes last although page 1 links to it, because page 1 splits its vote three ways and nobody else votes for page 2 at all.
Now the way a computer would do it, which is the way Brin and Page's own paper says they did it, with "a simple iterative algorithm" (S198). Start from a uniform guess and multiply by over and over. From verify/ch05_output.txt, section 2, computed in exact rationals and printed to six places:
step page 1 page 2 page 3 page 4
0 0.250000 0.250000 0.250000 0.250000
1 0.375000 0.083333 0.333333 0.208333
2 0.437500 0.125000 0.270833 0.166667
3 0.354167 0.145833 0.291667 0.208333
5 0.390625 0.131944 0.286458 0.190972
10 0.387587 0.128858 0.290244 0.193311
20 0.387096 0.129032 0.290323 0.193549
40 0.387097 0.129032 0.290323 0.193548
exact 0.387097 0.129032 0.290323 0.193548Step 1 is exact too: . By step 20 the answer is right to five decimal places, and the domain script confirms convergence to the exact vector:
[PASS] A x = x exactly, in rationals: got [Fraction(12, 31), Fraction(4, 31), Fraction(9, 31), Fraction(6, 31)], want [Fraction(12, 31), Fraction(4, 31), Fraction(9, 31), Fraction(6, 31)]
[PASS] power iteration converges to (12/31, 4/31, 9/31, 6/31): got 6.439015987069752e-12, want 0.0Bryan and Leise's real question is the one at the end of the article: "At present the web contains at least eight billion pages, how does one compute an eigenvector for an eight billion by eight billion matrix?" (S197, written in 2006). The answer is the table above. You never form the characteristic polynomial. You multiply, and you wait.
Google's actual matrix is damped, where has every entry and (S197). The domain script builds it and confirms two things Perron and Frobenius guarantee: the ranking stays strictly positive, and the second largest eigenvalue in size is well under 1, which is what makes the iteration converge and how fast.
damped Google matrix, m = 0.15: PageRank = [0.368151 0.141809 0.287962 0.202078]
[PASS] damped PageRank is strictly positive: got True, want True
second largest |eigenvalue| of M = 0.464749 (should be <= 1 - m = 0.85)5.7.3 The best rank one picture of a matrix
This is Schmidt's 1907 theorem, in the smallest case that shows anything. Take
and ask: of all the matrices of rank 1, which one is closest to ? "Closest" means smallest Frobenius norm of the difference, which is just the square root of the sum of the squares of the four entry-by-entry errors. The recipe is: eigen-decompose , take the square roots of its eigenvalues as the singular values, and build the rank one piece from the top one.
Two free checks: , which is the trace, and , which is the determinant. Now the vectors:
and the answer is the outer product:
Both columns of are the same, which is what rank 1 looks like. And the error is exactly the singular value you threw away:
From verify/ch05_output.txt, section 3:
u_1 = B v_1 / sigma_1 = (0.316228, 0.948683) = (1, 3) / sqrt(10)
B_1 = sigma_1 u_1 v_1^T = [[1.500000, 1.500000], [4.500000, 4.500000]]
||B - B_1||_F = 2.236068, sigma_2 = 2.236068
[PASS] Frobenius error of the best rank one equals sigma_2 computed = 2.2360679775 claimed = 2.2360679775
||B||_F^2 = 9 + 0 + 16 + 25 = 50, and sigma_1^2 + sigma_2^2 = 45 + 5 = 50That last line is the compression idea in one equation. The total squared size of is 50, split as 45 plus 5 between the two singular values, so the rank one piece carries , which is 90 per cent of it, using half the numbers. Replace 2 by 2 with a 1000 by 1000 photograph and you have image compression.
Schmidt's theorem says no rank one matrix does better. The domain script does not take that on trust: it tried 200,000 random rank one matrices and none of them beat .
best Frobenius error found by 200000 random rank-1 probes: 2.236068
[PASS] no random rank-1 matrix beats sigma_2: got True, want True5.7.4 Two more, in one line each
Perron's 1907 lemma, on the smallest positive matrix worth looking, : the characteristic polynomial is , the Perron root is , the other eigenvalue is negative and smaller in size, and the Perron vector is strictly positive, exactly as MacCluer states the theorem (S185).
[PASS] Perron root = (5 + sqrt(33))/2: got 5.372281323269014, want 5.372281323269014
[PASS] the other eigenvalue is strictly smaller in modulus: got True, want True
Perron vector (normalised to sum 1): [0.313859 0.686141]Leontief's model, with a two sector technology matrix and a final demand of : solve and the economy has to produce exactly 200 and 300.
det(I - A) = 0.8*0.9 - (-0.3)(-0.4) = 0.72 - 0.12 = 0.60
x = (I - A)^-1 d = [200. 300.]
[PASS] A x + d = x (the accounting identity closes): got 0.0, want 0.0
spectral radius of A = 0.500000 (must be < 1 for a nonnegative solution)Both of those are from verify/domainE_output.txt, sections 5 and 6, whose last line reads:
checks run: 51 passed: 51 failed: 0
ALL CHECKS PASSEDand everything new in 5.7 and in the prose of this chapter is confirmed by the last line of verify/ch05_output.txt:
TOTAL: 122 passed, 0 failedWhat verify/ch05.py printed
==============================================================================
1. The 2 by 2 characteristic polynomial, exact rationals
==============================================================================
A = [[9/10, 1/10], [1/2, 1/2]]
trace(A) = 9/10 + 1/2 = 7/5
det(A) = (9/10)(1/2) - (1/10)(1/2) = 2/5
[PASS] trace = 7/5 computed = 7/5 claimed = 7/5
[PASS] det = 2/5 computed = 2/5 claimed = 2/5
det(A - lambda I) = (9/10 - lambda)(1/2 - lambda) - (1/10)(1/2)
= lambda^2 - (7/5) lambda + (2/5)
discriminant = (7/5)^2 - 4(2/5) = 9/25
[PASS] discriminant = 9/25 computed = 9/25 claimed = 9/25
[PASS] discriminant is (3/5)^2 computed = 9/25 claimed = 9/25
lambda = ((7/5) +/- (3/5)) / 2
[PASS] lambda_1 = 1 computed = 1 claimed = 1
[PASS] lambda_2 = 2/5 computed = 2/5 claimed = 2/5
[PASS] lambda_1 + lambda_2 = trace computed = 7/5 claimed = 7/5
[PASS] lambda_1 * lambda_2 = det computed = 2/5 claimed = 2/5
(A - 1 I) row 1: (9/10 - 1) v1 + (1/10) v2 = 0
-(1/10) v1 + (1/10) v2 = 0 -> v1 = v2
[PASS] A (1, 1) = 1 * (1, 1) computed = [Fraction(1, 1), Fraction(1, 1)] claimed = [Fraction(1, 1), Fraction(1, 1)]
(A - (2/5) I) row 1: (9/10 - 2/5) v1 + (1/10) v2 = 0
(1/2) v1 + (1/10) v2 = 0 -> v2 = -5 v1
[PASS] A (1, -5) = (2/5) * (1, -5) computed = [Fraction(2, 5), Fraction(-2, 1)] claimed = [Fraction(2, 5), Fraction(-2, 1)]
[PASS] pi A = pi for pi = (5/6, 1/6) computed = [Fraction(5, 6), Fraction(1, 6)] claimed = [Fraction(5, 6), Fraction(1, 6)]
[PASS] pi entries sum to 1 computed = 1 claimed = 1
==============================================================================
2. PageRank on the Bryan and Leise four page web, exact
==============================================================================
A =
[0, 0, 1, 1/2]
[1/3, 0, 0, 0]
[1/3, 1/2, 0, 1/2]
[1/3, 1/2, 0, 0]
(Ax)_1 = 12/31 x_1 = 12/31
(Ax)_2 = 4/31 x_2 = 4/31
(Ax)_3 = 9/31 x_3 = 9/31
(Ax)_4 = 6/31 x_4 = 6/31
[PASS] A x = x exactly computed = [Fraction(12, 31), Fraction(4, 31), Fraction(9, 31), Fraction(6, 31)] claimed = [Fraction(12, 31), Fraction(4, 31), Fraction(9, 31), Fraction(6, 31)]
[PASS] numerators sum to the denominator: 12 + 4 + 9 + 6 computed = 31 claimed = 31
[PASS] components sum to 1 computed = 1 claimed = 1
[PASS] column 1 of A sums to 1 computed = 1 claimed = 1
[PASS] column 2 of A sums to 1 computed = 1 claimed = 1
[PASS] column 3 of A sums to 1 computed = 1 claimed = 1
[PASS] column 4 of A sums to 1 computed = 1 claimed = 1
power iteration from x0 = (1/4, 1/4, 1/4, 1/4), exact rationals,
printed to six decimal places:
step page 1 page 2 page 3 page 4
0 0.250000 0.250000 0.250000 0.250000
1 0.375000 0.083333 0.333333 0.208333
2 0.437500 0.125000 0.270833 0.166667
3 0.354167 0.145833 0.291667 0.208333
5 0.390625 0.131944 0.286458 0.190972
10 0.387587 0.128858 0.290244 0.193311
20 0.387096 0.129032 0.290323 0.193549
40 0.387097 0.129032 0.290323 0.193548
exact 0.387097 0.129032 0.290323 0.193548
[PASS] power iteration step 40 matches component 1 computed = 0.3870967742 claimed = 0.3870967742
[PASS] power iteration step 40 matches component 2 computed = 0.1290322581 claimed = 0.1290322581
[PASS] power iteration step 40 matches component 3 computed = 0.2903225806 claimed = 0.2903225806
[PASS] power iteration step 40 matches component 4 computed = 0.1935483871 claimed = 0.1935483871
[PASS] step 1 page 1 = 3/8 computed = 3/8 claimed = 3/8
[PASS] step 1 page 2 = 1/12 computed = 1/12 claimed = 1/12
[PASS] step 1 page 3 = 1/3 computed = 1/3 claimed = 1/3
[PASS] step 1 page 4 = 5/24 computed = 5/24 claimed = 5/24
[PASS] 12/31 to three decimals computed = 0.3870000000 claimed = 0.3870000000
[PASS] 4/31 to three decimals computed = 0.1290000000 claimed = 0.1290000000
[PASS] 9/31 to three decimals computed = 0.2900000000 claimed = 0.2900000000
[PASS] 6/31 to three decimals computed = 0.1940000000 claimed = 0.1940000000
[PASS] Bryan and Leise m plus Brin and Page d = 1 computed = 1 claimed = 1
==============================================================================
3. The rank one SVD approximation, built from scratch
==============================================================================
B = [[3, 0], [4, 5]]
B^T B = [[25, 20], [20, 25]]
[PASS] B^T B = [[25, 20], [20, 25]] computed = [[Fraction(25, 1), Fraction(20, 1)], [Fraction(20, 1), Fraction(25, 1)]] claimed = [[Fraction(25, 1), Fraction(20, 1)], [Fraction(20, 1), Fraction(25, 1)]]
det(B^T B - s I) = (25 - s)^2 - 400 = s^2 - 50 s + 225 = (s - 45)(s - 5)
[PASS] trace(B^T B) = 50 computed = 50 claimed = 50
[PASS] det(B^T B) = 225 computed = 225 claimed = 225
[PASS] 45 + 5 = 50 computed = 50 claimed = 50
[PASS] 45 * 5 = 225 computed = 225 claimed = 225
sigma_1 = sqrt(45) = 3 sqrt(5) = 6.708204
sigma_2 = sqrt(5) = 2.236068
[PASS] sigma_1 = 3 sqrt(5) computed = 6.7082039325 claimed = 6.7082039325
[PASS] sigma_1 * sigma_2 = |det B| = 15 computed = 15.0000000000 claimed = 15.0000000000
[PASS] |det B| = 15 computed = 15 claimed = 15
(B^T B - 45 I) row 1: (25 - 45) v1 + 20 v2 = 0 -> v1 = v2
so v_1 = (1, 1) / sqrt(2)
B v_1 = (2.121320, 6.363961), and its length is 6.708204 = sigma_1
[PASS] ||B v_1|| = sigma_1 computed = 6.7082039325 claimed = 6.7082039325
u_1 = B v_1 / sigma_1 = (0.316228, 0.948683) = (1, 3) / sqrt(10)
[PASS] u_1 first entry = 1/sqrt(10) computed = 0.3162277660 claimed = 0.3162277660
[PASS] u_1 second entry = 3/sqrt(10) computed = 0.9486832981 claimed = 0.9486832981
[PASS] u_1 is a unit vector computed = 1.0000000000 claimed = 1.0000000000
B_1 = sigma_1 u_1 v_1^T = [[1.500000, 1.500000], [4.500000, 4.500000]]
[PASS] B_1[0][0] = 3/2 computed = 1.5000000000 claimed = 1.5000000000
[PASS] B_1[0][1] = 3/2 computed = 1.5000000000 claimed = 1.5000000000
[PASS] B_1[1][0] = 9/2 computed = 4.5000000000 claimed = 4.5000000000
[PASS] B_1[1][1] = 9/2 computed = 4.5000000000 claimed = 4.5000000000
||B - B_1||_F = 2.236068, sigma_2 = 2.236068
[PASS] Frobenius error of the best rank one equals sigma_2 computed = 2.2360679775 claimed = 2.2360679775
||B||_F^2 = 9 + 0 + 16 + 25 = 50, and sigma_1^2 + sigma_2^2 = 45 + 5 = 50
[PASS] ||B||_F^2 = 50 computed = 50 claimed = 50
[PASS] sigma_1^2 + sigma_2^2 = 50 computed = 50.0000000000 claimed = 50.0000000000
fraction of squared size kept by the rank one piece = 45/50 = 0.90
[PASS] 45/50 = 9/10 computed = 9/10 claimed = 9/10
==============================================================================
4. Dates, intervals, ages and spans stated in the chapter prose
==============================================================================
[PASS] Grassmann's own 23 years run 1844 to 1867 computed = 1867 claimed = 1867
[PASS] Grassmann turned 35 in 1844 computed = 35 claimed = 35
[PASS] Grassmann's age at death, 15 Apr 1809 to 26 Sep 1877 computed = 68 claimed = 68
[PASS] Kummer's 1847 report is 3 years after the 1844 book computed = 3 claimed = 3
[PASS] 1844 book to the 1862 rewrite computed = 18 claimed = 18
[PASS] 1864 pulping is 20 years after publication computed = 20 claimed = 20
[PASS] 1878 second printing, one year after Grassmann died computed = 1 claimed = 1
[PASS] 1844 to the 1878 second printing computed = 34 claimed = 34
[PASS] Langendoen 1966 is 103 years after the 1863 linguistics paper computed = 103 claimed = 103
[PASS] Broom Bridge to the RIA reading, in days computed = 28 claimed = 28
16 Oct 1843 to 5 Aug 1865 = 7964 days
[PASS] Broom Bridge to the 1865 letter, completed years computed = 21 claimed = 21
[PASS] Broom Bridge to the 1865 letter, in days computed = 7964 claimed = 7964
[PASS] July 1846 to October 1846, in months computed = 3 claimed = 3
[PASS] quaternions 1843 to the naming of vector and scalar 1846 computed = 3 claimed = 3
[PASS] Hamilton wrote 109 of about 150 quaternion papers before 1865 computed = 41 claimed = 41
[PASS] Peano 1888 to Weyl's dimensional axiom 1918 computed = 30 claimed = 30
[PASS] Peano 1888 to Banach's dissertation 1920 computed = 32 claimed = 32
[PASS] Peano 1888 to Banach's 1922 paper computed = 34 claimed = 34
[PASS] Grassmann 1844 to Peano 1888 computed = 44 claimed = 44
[PASS] Peano's age in 1888 computed = 30 claimed = 30
[PASS] the debate ran 1890 to 1894, inclusive calendar years computed = 5 claimed = 5
[PASS] 594 quaternion publications minus 217 Grassmannian computed = 377 claimed = 377
[PASS] Kelvin's thirty eight years' war, counted back from 1901 computed = 1863 claimed = 1863
[PASS] Gibbs's two privately printed halves, 1881 and 1884 computed = 3 claimed = 3
[PASS] Cauchy 1840 to Sylvester 1883 computed = 43 claimed = 43
[PASS] Sylvester's latent roots 1883 to Hilbert's Eigenwert 1904 computed = 21 claimed = 21
[PASS] Hilbert 1904 to Eddington's English eigenvalue 1927 computed = 23 claimed = 23
[PASS] Eddington 1927 to this book's compilation year 2026 computed = 99 claimed = 99
[PASS] Hamilton's vector 1846 to 2026 computed = 180 claimed = 180
[PASS] Halmos 1958 to Halmos conceding in 1967 computed = 9 claimed = 9
Hilbert's six Mitteilungen, printed pages: [43, 47, 32, 71, 42, 63]
[PASS] first Mitteilung, 1904, pp. 49-91 computed = 43 claimed = 43
[PASS] the six Mitteilungen total computed = 298 claimed = 298
[PASS] there are six Mitteilungen computed = 6 claimed = 6
[PASS] Perron 1907 to Frobenius 1912 computed = 5 claimed = 5
[PASS] Frobenius 1912 to Brin and Page 1998 computed = 86 claimed = 86
[PASS] Perron 1907 to Bryan and Leise 2006 computed = 99 claimed = 99
[PASS] Frobenius's Habilitation year at Berlin, doctorate 1870 to Zurich 1875 computed = 5 claimed = 5
[PASS] Frobenius's age at death, 26 Oct 1849 to 3 Aug 1917 computed = 67 claimed = 67
[PASS] von Neumann and Goldstine, presented to received, in days computed = 26 claimed = 26
[PASS] the 1947 paper runs pp. 1021 to 1099, so its length in pages computed = 79 claimed = 79
[PASS] von Neumann and Goldstine 1947 to Turing 1948 computed = 1 claimed = 1
[PASS] Turing's 1948 paper runs pp. 287 to 308, so its length in pages computed = 22 claimed = 22
[PASS] digit loss at order 15, 50, 150: the three figures computed = [8, 10, 12] claimed = [8, 10, 12]
[PASS] Taussky at the NPL, 1943 to 1946, inclusive calendar years computed = 4 claimed = 4
[PASS] Taussky turned 24 in 1930, the year of her doctorate computed = 24 claimed = 24
[PASS] Taussky turned 57 in 1963, the year she was granted tenure computed = 57 claimed = 57
[PASS] Taussky turned 65 in 1971, the year she became a full professor computed = 65 claimed = 65
[PASS] Taussky's first matrix theory conference 1951, four years after 1947 computed = 4 claimed = 4
[PASS] Gersgorin 1931 to Taussky's NPL work beginning in 1943 computed = 12 claimed = 12
[PASS] Wilkinson becomes Turing's assistant May 1946, Pilot ACE May 1950 computed = 4 claimed = 4
Francis submits 29 Oct 1959; Kublanovskaya submits 5 Jul 1960
[PASS] days between the two submissions computed = 250 claimed = 250
[PASS] Francis, first submission to resubmission, in days computed = 586 claimed = 586
[PASS] Rutishauser 1958 to Francis's first submission, in years computed = 1 claimed = 1
[PASS] October 1961 publication to Golub's visit on 7 Aug 2007 computed = 45 claimed = 45
[PASS] October 1961 publication to 7 Aug 2007, in whole months computed = 550 claimed = 550
[PASS] 550 months is forty five years and ten months computed = (45, 10) claimed = (45, 10)
[PASS] Kublanovskaya turned 40 in November of 1960 computed = 40 claimed = 40
[PASS] Kublanovskaya's age on 5 July 1960 computed = 39 claimed = 39
[PASS] Kublanovskaya, degree 1948 to candidate's degree 1955 computed = 7 claimed = 7
[PASS] QR is sixth on the top ten algorithms list computed = 6 claimed = 6
[PASS] Beltrami 1873 to Jordan 1874 computed = 1 claimed = 1
[PASS] Beltrami 1873 to Schmidt 1907 computed = 34 claimed = 34
[PASS] Schmidt 1907 to Eckart and Young 1936 computed = 29 claimed = 29
[PASS] Beltrami 1873 to Golub and Kahan 1965 computed = 92 claimed = 92
[PASS] Sylvester's three SVD items of 1889, after Beltrami computed = 16 claimed = 16
[PASS] Beltrami's age at the 1873 paper, born 16 Nov 1835 computed = 38 claimed = 38
[PASS] Beltrami leaves university 1856, returns to a post 1862 computed = 6 claimed = 6
[PASS] Leontief's Nobel 1973, Perron 1907 to that prize computed = 66 claimed = 66
[PASS] Brin and Page's 1997 crawl: pages, anchors, links computed = [24, 259, 322] claimed = [24, 259, 322]
[PASS] Leontief born 5 Aug 1906, died 5 Feb 1999, completed years computed = 92 claimed = 92
==============================================================================
TOTAL: 122 passed, 0 failed
==============================================================================E.6 The mathematics of Chapter 6: Built to win an argument
Four worked examples, one for each section of Unit 4, plus the Metropolis rule as a fifth. Every number here is recomputed in verify/ch06.py, output in verify/ch06_output.txt, and anything already checked in the domain pass is quoted from verify/domainF_output.txt rather than re-derived.
Throughout, the book's row convention is in force: states index the rows, rows sum to 1, and distributions are row vectors that multiply the matrix from the left. That matches Levin and Peres, who write (S235). 6.7.1 Powers of a two-state transition matrix, one step at a time (Course section 4.1)
Two states, fine and wet. From a fine day, three chances in four of another fine day. From a wet day, an even chance either way.
Squaring it is four dot products. Written out so you can see every one:
Keep going, always in exact fractions:
Both rows are converging on the same thing, and there is a closed form for it. With and second eigenvalue :
Checked for to , both rows, in verify/ch06_output.txt, section 1:
Q^2 = row 0 [ 11/16 5/16 ]
row 1 [ 5/8 3/8 ]
Q^3 = row 0 [ 43/64 21/64 ]
row 1 [ 21/32 11/32 ]
Q^4 = row 0 [ 171/256 85/256 ]
row 1 [ 85/128 43/128 ] [PASS] closed form (Q^10)_00 = 2/3 + (1/3)(1/4)^10 computed = 699051/1048576 claimed = 699051/1048576
[PASS] closed form (Q^10)_10 = (2/3)(1 - (1/4)^10) computed = 349525/524288 claimed = 349525/524288Two conditions have to hold for that convergence, and they do different jobs. Irreducible, meaning you can get from anywhere to anywhere, is what makes the answer unique. Aperiodic, meaning the chain does not march in lockstep, is what makes the powers settle. Levin and Peres split them exactly that way: "Let P be the transition matrix of an irreducible Markov chain. There exists a unique probability distribution pi satisfying pi = piP" (S235, Corollary 1.17), and convergence is the separate Theorem 4.9 (S235). Both counterexamples are one line each. The swap chain has the perfectly good stationary distribution and never converges, because . The identity chain has , , and all stationary, so uniqueness fails. From verify/domainF_output.txt:
[PASS] F-5.4 Sw^101 = Sw, not the limit matrix
[PASS] F-6.4 so is (1/2, 1/2): uniqueness fails without irreducibility6.7.2 The stationary distribution as a linear system, on Markov's own numbers (Course section 4.3)
Take state 0 to be a vowel and state 1 a consonant, and build the matrix from Markov's integer counts rather than from his rounded decimals. He had 8,638 vowels, 11,362 consonants, and 1,104 vowel-vowel pairs, so the consonant-then-vowel count is (S214; F-1.5).
Now solve with the equals signs aligned:
The first equation says . Substitute into the third:
That is exactly , the raw vowel frequency. It is not a coincidence and it is worth a minute in class: both conditional probabilities were built from the same counts, and the numerator 3767 is shared by and by , so it cancels and the chain fitted to the data reproduces the data's own vowel rate. From verify/ch06_output.txt, section 2:
[PASS] pi_0 = p0 / ((1 - p1) + p0) computed = 4319/10000 claimed = 4319/10000
[PASS] the fitted chain reproduces the raw vowel frequency exactly computed = 4319/10000 claimed = 4319/10000
[PASS] the shared factor that makes it work: numerator of p0 equals numerator of 1 - p1 computed = 3767 claimed = 3767Run the same solve on Markov's printed decimals and you get , which still rounds to his 0.432 and sits within 0.0001 of the exact answer. The second eigenvalue is , which rounds to Markov's own (S214). Because is that large, the chain forgets slowly: after 8 steps the rows of are still 0.003829 away from , and it takes 22 steps to get inside (F-3.6b, F-3.5).
Two more facts that belong in the same breath. The independence prediction: if letters were independent you would expect about 3,731 vowel-vowel pairs among 19,999, and Markov counted 1,104 (F-2.4). And PageRank, in von Hilgers and Langville's phrase, is "the stationary vector of an enormous Markov chain" (S243), which is this same calculation with billions of states instead of two. Chapter 5 owns the eigenvalue machinery that makes it work. 6.7.3 A detailed balance check, and one that fails (Course section 4.4)
It holds: the Ehrenfest urn with four balls. State is the number of balls in urn I. Pick a ball number at random and move that ball.
Detailed balance says for every pair. There are only four pairs to check, because everything else is zero on both sides:
Four comparisons, and is stationary. No system was solved. That is the whole practical value of Course section 4.4. From verify/ch06_output.txt, section 3:
pi_0 P_01 = 1/16 pi_1 P_10 = 1/16
pi_1 P_12 = 3/16 pi_2 P_21 = 3/16
pi_2 P_23 = 3/16 pi_3 P_32 = 3/16
pi_3 P_34 = 1/16 pi_4 P_43 = 1/16
[PASS] pi E = pi, so detailed balance really did the work computed = [Fraction(1, 16), Fraction(1, 4), Fraction(3, 8), Fraction(1, 4), Fraction(1, 16)] ...The expected return time to a state is . With four balls, the all-in-one-urn state returns every 16 steps. With ten balls it is steps, against steps for the balanced state (F-7.7 to F-7.9). Scale that ratio up to Kac's 20,000 balls and you have his answer to Loschmidt and Zermelo.
It fails: a three-state cycle. Stationary and reversible are not the same thing, and here is the smallest honest counterexample.
Every row sums to 1 and so does every column, so is doubly stochastic and is stationary. But Kolmogorov's criterion compares the two ways round a cycle:
Those differ by a factor of 8, so no distribution at all can satisfy detailed balance with this matrix. The chain has a preferred direction: it drifts round the cycle one way. Watch a film of it and you can tell whether the film is running backwards, which is exactly what "not reversible" means. From verify/domainF_output.txt, section F-8:
[PASS] F-8.5 Kolmogorov cycle product forwards 0->1->2->0
got = 8/27
[PASS] F-8.6 Kolmogorov cycle product backwards 0->2->1->0
got = 1/27
[PASS] F-8.7 the two products differ, so the chain is not reversible6.7.4 An absorbing chain, and how long it lasts (Course section 4.2)
Gambler's ruin, the oldest absorbing-chain problem in the book. You start with pounds, the target is 5, and each round is a fair coin. States 0 and 5 are absorbing, states 1 to 4 are transient.
The fundamental matrix answers the question "starting at , how many visits do I expect to make to before I am absorbed?" Inverting exactly gives
Total expected duration is the row sum, because every step is a visit to some transient state:
Those are exactly , which is the classical answer, and the absorption probabilities come out for reaching the target and for ruin.
| Start k | Expected steps | Probability of ruin | Probability of reaching 5 |
|---|---|---|---|
| 1 | 4 | ||
| 2 | 6 | ||
| 3 | 6 | ||
| 4 | 4 |
From verify/ch06_output.txt, section 5:
N = row 0 [ 8/5 6/5 4/5 2/5 ]
row 1 [ 6/5 12/5 8/5 4/5 ]
row 2 [ 4/5 8/5 12/5 6/5 ]
row 3 [ 2/5 4/5 6/5 8/5 ]
[PASS] expected steps from 2 pounds = row sum of N = k(5 - k) computed = 6 claimed = 6and the same durations and probabilities appear independently at F-9.2 to F-9.4 of verify/domainF_output.txt. 6.7.5 Metropolis, run backwards from the answer (Course section 4.4)
Everything above solves for given . Metropolis and the Rosenbluths did the opposite: name the you want, then build a that satisfies detailed balance with respect to it. Take the target , propose the other state with probability , and accept with probability .
Detailed balance holds by construction, so the target is stationary without any solving, and is already within of it (F-10.1 to F-10.6, and verify/ch06_output.txt section 6). The rule in the 1953 paper is the same rule with written as , which is what the Boltzmann distribution makes that ratio (S226, p. 1088).
Every number in 6.7 is recomputed in verify/ch06.py, whose last line reads:
TOTAL: 144 passed, 0 failedand everything drawn from the domain pass is confirmed by the last lines of verify/domainF_output.txt:
98 of 98 checks passed.
No failures.What verify/ch06.py printed
==========================================================================
1. The classroom two-state chain, powers computed entry by entry
==========================================================================
[PASS] Q row 0 sums to 1 computed = 1 claimed = 1
[PASS] Q row 1 sums to 1 computed = 1 claimed = 1
[PASS] (Q^2)_00 = 3/4*3/4 + 1/4*1/2 computed = 11/16 claimed = 11/16
[PASS] (Q^2)_01 = 3/4*1/4 + 1/4*1/2 computed = 5/16 claimed = 5/16
[PASS] (Q^2)_10 = 1/2*3/4 + 1/2*1/2 computed = 5/8 claimed = 5/8
[PASS] (Q^2)_11 = 1/2*1/4 + 1/2*1/2 computed = 3/8 claimed = 3/8
Q^2 = row 0 [ 11/16 5/16 ]
row 1 [ 5/8 3/8 ]
Q^3 = row 0 [ 43/64 21/64 ]
row 1 [ 21/32 11/32 ]
Q^4 = row 0 [ 171/256 85/256 ]
row 1 [ 85/128 43/128 ]
[PASS] Q^3 row 0 computed = [Fraction(43, 64), Fraction(21, 64)] claimed = [Fraction(43, 64), Fraction(21, 64)]
[PASS] Q^3 row 1 computed = [Fraction(21, 32), Fraction(11, 32)] claimed = [Fraction(21, 32), Fraction(11, 32)]
[PASS] Q^4 row 0 computed = [Fraction(171, 256), Fraction(85, 256)] claimed = [Fraction(171, 256), Fraction(85, 256)]
[PASS] Q^4 row 1 computed = [Fraction(85, 128), Fraction(43, 128)] claimed = [Fraction(85, 128), Fraction(43, 128)]
[PASS] closed form (Q^1)_00 = 2/3 + (1/3)(1/4)^1 computed = 3/4 claimed = 3/4
[PASS] closed form (Q^1)_10 = (2/3)(1 - (1/4)^1) computed = 1/2 claimed = 1/2
[PASS] closed form (Q^2)_00 = 2/3 + (1/3)(1/4)^2 computed = 11/16 claimed = 11/16
[PASS] closed form (Q^2)_10 = (2/3)(1 - (1/4)^2) computed = 5/8 claimed = 5/8
[PASS] closed form (Q^3)_00 = 2/3 + (1/3)(1/4)^3 computed = 43/64 claimed = 43/64
[PASS] closed form (Q^3)_10 = (2/3)(1 - (1/4)^3) computed = 21/32 claimed = 21/32
[PASS] closed form (Q^4)_00 = 2/3 + (1/3)(1/4)^4 computed = 171/256 claimed = 171/256
[PASS] closed form (Q^4)_10 = (2/3)(1 - (1/4)^4) computed = 85/128 claimed = 85/128
[PASS] closed form (Q^5)_00 = 2/3 + (1/3)(1/4)^5 computed = 683/1024 claimed = 683/1024
[PASS] closed form (Q^5)_10 = (2/3)(1 - (1/4)^5) computed = 341/512 claimed = 341/512
[PASS] closed form (Q^6)_00 = 2/3 + (1/3)(1/4)^6 computed = 2731/4096 claimed = 2731/4096
[PASS] closed form (Q^6)_10 = (2/3)(1 - (1/4)^6) computed = 1365/2048 claimed = 1365/2048
[PASS] closed form (Q^7)_00 = 2/3 + (1/3)(1/4)^7 computed = 10923/16384 claimed = 10923/16384
[PASS] closed form (Q^7)_10 = (2/3)(1 - (1/4)^7) computed = 5461/8192 claimed = 5461/8192
[PASS] closed form (Q^8)_00 = 2/3 + (1/3)(1/4)^8 computed = 43691/65536 claimed = 43691/65536
[PASS] closed form (Q^8)_10 = (2/3)(1 - (1/4)^8) computed = 21845/32768 claimed = 21845/32768
[PASS] closed form (Q^9)_00 = 2/3 + (1/3)(1/4)^9 computed = 174763/262144 claimed = 174763/262144
[PASS] closed form (Q^9)_10 = (2/3)(1 - (1/4)^9) computed = 87381/131072 claimed = 87381/131072
[PASS] closed form (Q^10)_00 = 2/3 + (1/3)(1/4)^10 computed = 699051/1048576 claimed = 699051/1048576
[PASS] closed form (Q^10)_10 = (2/3)(1 - (1/4)^10) computed = 349525/524288 claimed = 349525/524288
[PASS] pi Q = pi for pi = (2/3, 1/3) computed = [Fraction(2, 3), Fraction(1, 3)] claimed = [Fraction(2, 3), Fraction(1, 3)]
[PASS] trace Q = 1 + 1/4 computed = 5/4 claimed = 5/4
[PASS] det Q = 1 * 1/4 computed = 1/4 claimed = 1/4
[PASS] second eigenvalue = 1 - a - b with a = 1/4, b = 1/2 computed = 1/4 claimed = 1/4
[PASS] detailed balance pi_0 Q_01 = pi_1 Q_10 computed = 1/6 claimed = 1/6
[PASS] that common value is 1/6 computed = 1/6 claimed = 1/6
==========================================================================
2. Markov's Onegin chain from the INTEGER counts, exact fractions
==========================================================================
[PASS] vowels + consonants computed = 20000 claimed = 20000
[PASS] consonant-then-vowel pairs CV = V - VV computed = 7534 claimed = 7534
[PASS] p = 8638/20000 in lowest terms computed = 4319/10000 claimed = 4319/10000
[PASS] p1 = 1104/8638 in lowest terms computed = 552/4319 claimed = 552/4319
[PASS] p0 = 7534/11362 in lowest terms computed = 3767/5681 claimed = 3767/5681
[PASS] 1 - p1 computed = 3767/4319 claimed = 3767/4319
[PASS] 1 - p0 computed = 1914/5681 claimed = 1914/5681
p = 4319/10000 = 0.431900
p1 = 552/4319 = 0.127807
p0 = 3767/5681 = 0.663087
[PASS] p1 rounds to Markov's 0.128 computed = 0.128 claimed = 0.128
[PASS] p0 rounds to Markov's 0.663 computed = 0.663 claimed = 0.663
[PASS] Onegin P row 0 sums to 1 computed = 1 claimed = 1
[PASS] Onegin P row 1 sums to 1 computed = 1 claimed = 1
[PASS] pi_0 = p0 / ((1 - p1) + p0) computed = 4319/10000 claimed = 4319/10000
[PASS] pi_1 = (1 - p1) / ((1 - p1) + p0) computed = 5681/10000 claimed = 5681/10000
[PASS] pi sums to 1 computed = 1 claimed = 1
[PASS] pi P = pi, first component computed = 4319/10000 claimed = 4319/10000
[PASS] pi P = pi, second component computed = 5681/10000 claimed = 5681/10000
[PASS] the fitted chain reproduces the raw vowel frequency exactly computed = 4319/10000 claimed = 4319/10000
[PASS] the shared factor that makes it work: numerator of p0 equals numerator of 1 - p1 computed = 3767 claimed = 3767
[PASS] 4319 + 5681 = 10000 computed = 10000 claimed = 10000
[PASS] rounded-value pi_0 = 0.663 / (0.872 + 0.663) = 663/1535 computed = 663/1535 claimed = 663/1535
663/1535 = 0.431922
[PASS] and it still rounds to 0.432 computed = 0.432 claimed = 0.432
[PASS] the rounded answer differs from the exact one by less than 0.0001 computed = True claimed = True
second eigenvalue p1 - p0 = -13133761/24536239 = -0.535280
[PASS] second eigenvalue rounds to Markov's delta = -0.535 computed = -0.535 claimed = -0.535
[PASS] a run of 20,000 letters has 19,999 overlapping pairs computed = 19999 claimed = 19999
[PASS] mixed pairs on the 19,999 convention computed = 15068 claimed = 15068
[PASS] Hayes's printed 15,069 needs a 20,000-pair loop computed = 15069 claimed = 15069
[PASS] the two conventions differ by exactly one pair computed = 1 claimed = 1
[PASS] independence would predict about 3,731 vowel-vowel pairs computed = 3731 claimed = 3731
[PASS] Markov observed far fewer computed = True claimed = True
==========================================================================
3. The Ehrenfest urn, four balls: detailed balance pair by pair
==========================================================================
[PASS] Ehrenfest row 0 sums to 1 computed = 1 claimed = 1
[PASS] Ehrenfest row 1 sums to 1 computed = 1 claimed = 1
[PASS] Ehrenfest row 2 sums to 1 computed = 1 claimed = 1
[PASS] Ehrenfest row 3 sums to 1 computed = 1 claimed = 1
[PASS] Ehrenfest row 4 sums to 1 computed = 1 claimed = 1
[PASS] pi = (1, 4, 6, 4, 1)/16 computed = [Fraction(1, 16), Fraction(1, 4), Fraction(3, 8), Fraction(1, 4), Fraction(1, 16)] claimed = [Fraction(1, 16), Fraction(1, 4), Fraction(3, 8), Fraction(1, 4), Fraction(1, 16)]
[PASS] pi sums to 1 computed = 1 claimed = 1
pi_0 P_01 = 1/16 pi_1 P_10 = 1/16
[PASS] detailed balance on the edge 0-1 computed = 1/16 claimed = 1/16
pi_1 P_12 = 3/16 pi_2 P_21 = 3/16
[PASS] detailed balance on the edge 1-2 computed = 3/16 claimed = 3/16
pi_2 P_23 = 3/16 pi_3 P_32 = 3/16
[PASS] detailed balance on the edge 2-3 computed = 3/16 claimed = 3/16
pi_3 P_34 = 1/16 pi_4 P_43 = 1/16
[PASS] detailed balance on the edge 3-4 computed = 1/16 claimed = 1/16
[PASS] all non-adjacent pairs balance at zero computed = True claimed = True
[PASS] pi E = pi, so detailed balance really did the work computed = [Fraction(1, 16), Fraction(1, 4), Fraction(3, 8), Fraction(1, 4), Fraction(1, 16)] claimed = [Fraction(1, 16), Fraction(1, 4), Fraction(3, 8), Fraction(1, 4), Fraction(1, 16)]
[PASS] expected return time to the all-in-one-urn state is 1/pi_0 computed = 16 claimed = 16
[PASS] with ten balls that return time is 2^10 computed = 1024 claimed = 1024
[PASS] with ten balls the balanced state returns in 1024/252 computed = 256/63 claimed = 256/63
[PASS] 1024/252 as a decimal computed = 4.063492 claimed = 4.0635 (tol 0.0001)
==========================================================================
4. Kolmogorov's cycle criterion: stationary but not reversible
==========================================================================
[PASS] C is row-stochastic computed = True claimed = True
[PASS] C is column-stochastic too, so it is doubly stochastic computed = True claimed = True
[PASS] the uniform distribution is stationary computed = [Fraction(1, 3), Fraction(1, 3), Fraction(1, 3)] claimed = [Fraction(1, 3), Fraction(1, 3), Fraction(1, 3)]
[PASS] cycle product 0->1->2->0 computed = 8/27 claimed = 8/27
[PASS] cycle product 0->2->1->0 computed = 1/27 claimed = 1/27
[PASS] the two differ, so no reversible pi can exist computed = True claimed = True
[PASS] they differ by a factor of 8 computed = 8 claimed = 8
[PASS] detailed balance fails on the edge 0-1 computed = False claimed = False
==========================================================================
5. Absorbing chain: gambler's ruin, target 5, fundamental matrix
==========================================================================
N = row 0 [ 8/5 6/5 4/5 2/5 ]
row 1 [ 6/5 12/5 8/5 4/5 ]
row 2 [ 4/5 8/5 12/5 6/5 ]
row 3 [ 2/5 4/5 6/5 8/5 ]
[PASS] N matches 2 min(i,j)(5 - max(i,j))/5 entry for entry computed = [[Fraction(8, 5), Fraction(6, 5), Fraction(4, 5), Fraction(2, 5)], [Fraction(6, 5), Fraction(12, 5), Fraction(8, 5), Fraction(4, 5)], [Fraction(4, 5), Fraction(8, 5), Fraction(12, 5), Fraction(6, 5)], [Fraction(2, 5), Fraction(4, 5), Fraction(6, 5), Fraction(8, 5)]] claimed = [[Fraction(8, 5), Fraction(6, 5), Fraction(4, 5), Fraction(2, 5)], [Fraction(6, 5), Fraction(12, 5), Fraction(8, 5), Fraction(4, 5)], [Fraction(4, 5), Fraction(8, 5), Fraction(12, 5), Fraction(6, 5)], [Fraction(2, 5), Fraction(4, 5), Fraction(6, 5), Fraction(8, 5)]]
[PASS] N_11 = 8/5 computed = 8/5 claimed = 8/5
[PASS] N_22 = 12/5 computed = 12/5 claimed = 12/5
[PASS] expected steps from 1 pounds = row sum of N = k(5 - k) computed = 4 claimed = 4
start k = 1 expected duration = 4
[PASS] expected steps from 2 pounds = row sum of N = k(5 - k) computed = 6 claimed = 6
start k = 2 expected duration = 6
[PASS] expected steps from 3 pounds = row sum of N = k(5 - k) computed = 6 claimed = 6
start k = 3 expected duration = 6
[PASS] expected steps from 4 pounds = row sum of N = k(5 - k) computed = 4 claimed = 4
start k = 4 expected duration = 4
[PASS] probability of ruin from 1 is (5 - k)/5 computed = 4/5 claimed = 4/5
[PASS] probability of reaching 5 from 1 is k/5 computed = 1/5 claimed = 1/5
[PASS] probability of ruin from 2 is (5 - k)/5 computed = 3/5 claimed = 3/5
[PASS] probability of reaching 5 from 2 is k/5 computed = 2/5 claimed = 2/5
[PASS] probability of ruin from 3 is (5 - k)/5 computed = 2/5 claimed = 2/5
[PASS] probability of reaching 5 from 3 is k/5 computed = 3/5 claimed = 3/5
[PASS] probability of ruin from 4 is (5 - k)/5 computed = 1/5 claimed = 1/5
[PASS] probability of reaching 5 from 4 is k/5 computed = 4/5 claimed = 4/5
[PASS] each row of B sums to 1 computed = True claimed = True
==========================================================================
6. The Metropolis rule as a two-state chain
==========================================================================
M = row 0 [ 1/2 1/2 ]
row 1 [ 1/6 5/6 ]
[PASS] M_01 = 1/2 * min(1, 3) computed = 1/2 claimed = 1/2
[PASS] M_10 = 1/2 * min(1, 1/3) computed = 1/6 claimed = 1/6
[PASS] M is row-stochastic computed = True claimed = True
[PASS] detailed balance pi_0 M_01 = pi_1 M_10 computed = 1/8 claimed = 1/8
[PASS] that common value is 1/8 computed = 1/8 claimed = 1/8
[PASS] so the target is stationary computed = [Fraction(1, 4), Fraction(3, 4)] claimed = [Fraction(1, 4), Fraction(3, 4)]
[PASS] M^20 row 0, first entry, against the target 0.25 computed = 0.25 claimed = 0.25 (tol 1e-06)
==========================================================================
7. Dates, ages, intervals and anniversaries stated in the prose
==========================================================================
[PASS] Markov turned 50 in June 1906, the year of the chain paper computed = 50 claimed = 50
[PASS] he was still 49 in January 1906 computed = 49 claimed = 49
[PASS] he was 55, not 56, in February 1912 computed = 55 claimed = 55
[PASS] he was 56 at the Onegin lecture, 23 January 1913 computed = 56 claimed = 56
[PASS] he was 61 in September 1917 computed = 61 claimed = 61
[PASS] he was 64 when he wrote the footwear letter, 5 March 1921 computed = 64 claimed = 64
[PASS] he died at 66, on 20 July 1922 computed = 66 claimed = 66
[PASS] Doeblin was 25 when he died computed = 25 claimed = 25
[PASS] deposit to death, in days computed = 116 claimed = 116
[PASS] 116 days is not two months computed = True claimed = True
[PASS] deposit to the May 2000 opening, in whole years computed = 60 claimed = 60
[PASS] 1913 was 200 years after Ars Conjectandi, 1713 computed = 200 claimed = 200
[PASS] 1913 was 300 years after the Romanovs took power in 1613 computed = 300 claimed = 300
[PASS] Markov's chains, 1906, to the Onegin lecture, 1913 computed = 7 claimed = 7
[PASS] Markov died in 1922, sixteen years after the 1906 paper computed = 16 claimed = 16
[PASS] the English phrase 'Markov chain', 1938, arrives 32 years after 1906 computed = 32 claimed = 32
[PASS] and 16 years after Markov's death computed = 16 claimed = 16
[PASS] Bernstein 1926 would antedate Romanovsky 1929 by three years computed = 3 claimed = 3
[PASS] Hastings 1970 to Gelfand and Smith 1990 computed = 20 claimed = 20
[PASS] Boltzmann 1872 to Metropolis 1953 computed = 81 claimed = 81
[PASS] Onegin lecture 1913 to Shannon 1948 computed = 35 claimed = 35
[PASS] 2 June 1856 O.S. plus 12 days is 14 June N.S. computed = 14 claimed = 14
[PASS] 23 January 1913 O.S. plus 13 days is 5 February N.S. computed = 5 claimed = 5
2^20000 seconds = 10^6020.6 seconds = 10^6013.1 years
[PASS] 2^20000 seconds is about 10^6020.6 seconds computed = 6020.599913 claimed = 6020.6 (tol 0.1)
[PASS] which is about 10^6013 years computed = 6013.100809 claimed = 6013.1 (tol 0.2)
[PASS] Kac's printed '10^6000 years' is a round order of magnitude, not exact computed = True claimed = True
exact 1/pi for the balanced state = 177.25 seconds
[PASS] the balanced state returns in about 177 seconds computed = 177.247601 claimed = 177.2 (tol 0.5)
[PASS] which is 100 * sqrt(pi), not 100 / sqrt(pi) computed = 177.245385 claimed = 177.2 (tol 0.5)
[PASS] 100 / sqrt(pi) is about 56, so that form cannot be Kac's figure computed = True claimed = True
[PASS] Kac's 'about 175 seconds' matches the exact value to within 3 seconds computed = True claimed = True
[PASS] the universe, at about 10^10 years, is negligible beside it computed = True claimed = True
==========================================================================
TOTAL: 144 passed, 0 failed
==========================================================================E.7 The mathematics of Chapter 7: The picture, and the table underneath it
Everything below is verified in verify/domainJ.py, whose output is in verify/domainJ_output.txt. That file's last block reads:
checks run: 124
checks passed: 124
checks failed: 07.7.1 Euler's own argument, which uses no matrices at all
Konigsberg has four land masses and seven bridges: two bridges join A to B, two join A to C, and one each joins A to D, B to D, and C to D. Count the bridges at each land mass:
That last line is the handshake lemma, and it has to hold because every bridge has two ends. Now run Euler's rule, which is the one from S422: a land mass reached by an odd number k of bridges must appear (k+1)/2 times in a written route.
Nine will not fit into eight, so no route exists. From verify/domainJ_output.txt, section J-1:
A: (5+1)/2 = 3
B: (3+1)/2 = 2
C: (3+1)/2 = 2
D: (3+1)/2 = 2
[PASS] J-1.11 letters demanded by Euler's rule
got = 9
want = 9
[PASS] J-1.12 letters available in a 7-bridge route
got = 8
want = 8
[PASS] J-1.13 Euler's contradiction: 9 demanded > 8 available
got = True
want = TrueAnd the brute force confirmation, which Euler refused to do by hand and which took a machine no time at all:
[PASS] J-1.14 exhaustive search over all 7! x 4 = 20160 routes finds a trail?
got = False
want = False
[PASS] J-1.15 with the 1875 eighth bridge B-C, odd-degree regions are A and D
got = ['A', 'D']
want = ['A', 'D']
[PASS] J-1.16 with eight bridges an Eulerian trail exists
got = True
want = True
one such route: A -> B -> A -> C -> A -> D -> B -> C -> D7.7.2 The same picture as a matrix, and what squaring it does
Write the picture as a table. Put in row i and column j the number of bridges joining region i to region j:
Two things to notice before doing any arithmetic. The matrix is symmetric, because a bridge works in both directions. The row sums are 5, 3, 3, and 3, which are the degrees you counted in 7.7.1, so the handshake lemma is now a statement about the sum of all the entries: 14, which is 2 times 7 (J-2.1, J-2.2, J-2.3).
Now square it, by the rule from Course section 3.3, where entry (i, j) of the product is the row i of the first matrix dotted into column j of the second:
Take the entry for A to D and write it out longhand:
Four. And here is the point of the whole chapter: that 4 is not an abstraction. It is a count of walks, and you can name them. Euler numbered the bridges a to g in E053, but this book never reached the section where he assigns the letters, so the labels below are ours, not his.
Call the two bridges from A to B a and b, the two from A to C c and d, the bridge from A to D e, the bridge from B to D f, and the bridge from C to D g. Then the four two-bridge walks from the island A to the region D are:
- A to B by bridge a, then B to D by bridge f.
- A to B by bridge b, then B to D by bridge f.
- A to C by bridge c, then C to D by bridge g.
- A to C by bridge d, then C to D by bridge g.
That is all of them. There is no walk A to A to D, because the term is : there is no bridge from A back to A. There is no walk A to D to D either, for the same reason on the other side. Each product counts the ways to take one step from i to k and then one step from k to j, and summing over k sweeps up every two-step route. Matrix multiplication is route counting. It always was.
From verify/domainJ_output.txt, section J-2:
(A^2)[A][D] = A[A][A]*A[A][D] + A[A][B]*A[B][D] + A[A][C]*A[C][D] + A[A][D]*A[D][D]
= 0*1 + 2*1 + 2*1 + 1*0
= 0 + 2 + 2 + 0
= 4[PASS] J-2.8 full A^2
got = [[9, 1, 1, 4], [1, 5, 5, 2], [1, 5, 5, 2], [4, 2, 2, 3]]
want = [[9, 1, 1, 4], [1, 5, 5, 2], [1, 5, 5, 2], [4, 2, 2, 3]]
Step 4. Independent check by enumeration: count 2-bridge walks directly.
[PASS] J-2.9 A^2 equals the bridge-by-bridge count of 2-bridge walks
got = [[9, 1, 1, 4], [1, 5, 5, 2], [1, 5, 5, 2], [4, 2, 2, 3]]
want = [[9, 1, 1, 4], [1, 5, 5, 2], [1, 5, 5, 2], [4, 2, 2, 3]]The script counted the walks one bridge at a time, without multiplying any matrices, and got the same sixteen numbers.
Two more readings of the same object. The diagonal of is 9, 5, 5, 3, which is not the degree list, because Konigsberg is a multigraph and you can go out on bridge a and come back on bridge b. In a simple graph, where no pair of regions has two bridges, the only way home in two steps is out and back along the same edge, so the diagonal of is exactly the degree list. On the 4-cycle A to B to C to D to A, the diagonal comes out 2, 2, 2, 2 (J-2.10, J-2.11). And counts walks of length three, so the trace of counts closed three-step walks, which is six times the number of triangles: for K4 the trace is 24 and the triangle count is 4 (J-2.12, J-2.13). 7.7.3 Now put probabilities in the entries, and change nothing else
This is Course section 4.1. Three states, Sunny, Cloudy, and Rainy. Row i, column j holds the probability of going to j given that you are at i. Exact fractions throughout, because decimals hide what is going on:
Every row sums to exactly 1, which is what makes it row-stochastic (J-3.1). Square it, using precisely the operation from 7.7.2:
And read out the same entry position, sunny today to rainy the day after tomorrow:
Put the two sums side by side. In 7.7.2 the terms were , , , and , and the sum counted routes. Here the terms are , , and , and the sum adds up the probabilities of the same routes. Identical algebra. There are three routes from Sunny to Rainy in two days, through Sunny, through Cloudy, and through Rainy, and each product is the probability of one of them because the two steps are independent given the middle state. That is the whole of Course section 4.1: a transition matrix is a weighted adjacency matrix, and matrix powers are k-step forecasting because matrix powers were always route counting.
From verify/domainJ_output.txt, section J-3:
(P^2)[Sun][Rain] = P[Sun][Sun]*P[Sun][Rain]
+ P[Sun][Cloud]*P[Cloud][Rain]
+ P[Sun][Rain]*P[Rain][Rain]
= 3/5 * 1/10 + 3/10 * 1/5 + 1/10 * 3/10
= 3/50 + 3/50 + 3/100
= 6/100 + 6/100 + 3/100
= 15/100 = 3/20
[PASS] J-3.3 (P^2)[Sun][Rain] = 3/20
got = 3/20
want = 3/20 Compare with J-2: in the unweighted graph the same sum COUNTS two-step
routes; here it ADDS UP their probabilities. Same algebra, two readings.For Course section 4.3, the same matrix has an exact stationary distribution, and the script checks it as an equality of fractions rather than to some number of decimal places:
stationary distribution pi (exact): Sun=16/35, Cloud=13/35, Rain=6/35
[PASS] J-3.6 pi P = pi exactly
got = [Fraction(16, 35), Fraction(13, 35), Fraction(6, 35)]
want = [Fraction(16, 35), Fraction(13, 35), Fraction(6, 35)]7.7.4 The determinant version, for Course section 3.5
Take the Konigsberg picture again, but throw away the doubled bridges so that it is a simple graph on A, B, C, and D with five edges: AB, AC, AD, BD, and CD. Build the Laplacian , with the degrees on the diagonal and minus the adjacency entries off it:
Delete any one row and the matching column. Delete the last of each, and expand the 3 by 3 determinant that is left along its first row, which is the method Course section 3.5 teaches:
Eight. That is the number of spanning trees of the graph: the number of ways to choose a set of bridges that keeps all four regions connected and contains no loop. From verify/domainJ_output.txt, section J-4:
[PASS] J-4.6 spanning trees of Konigsberg simple version: brute force = Laplacian minor
got = 8
want = 8The script checks the same theorem against brute force on five other graphs, including K5 with its 125 spanning trees, and confirms Cayley's formula for the complete graphs K2 through K6, where K6 has 1296 (J-4.1 to J-4.5, J-4.C2 to J-4.C6). It does not matter which row and column you delete; every choice gives the same number. That is the second bridge from a picture to a matrix, and unlike the first it needs a determinant. The attribution of it to Kirchhoff in 1847 is unverified in this book (7.10, item 5); the mathematics is not.
E.8 The mathematics of Chapter 8: The people, and the twelve doors out of the math room
Three calculations. The first is the one this chapter has a duty to get right, because a book that tells teenagers about Cardano owes them the arithmetic that shows why the house wins. The second is Hardy's, from April 1908. The third is Turing's, from Bletchley Park. 8.7.1 Roulette: why every bet on the table is the same bet
A European wheel has 37 pockets: the numbers 1 to 36, plus a single green zero. A bet on one number pays 35 to 1, meaning that if it wins you keep your 1 unit stake and receive 35 more, so the net gain is +35 and the net loss otherwise is -1. Write out the expected value with the equals signs lined up:
Now try to do better by betting on red, which wins 18 ways out of 37 and pays 1 to 1:
And a dozen, 12 numbers paying 2 to 1, and a split, 2 numbers paying 17 to 1:
Every bet on the layout has the same expected value, and it is negative. That is not an accident of the numbers, it is the design: the payouts are set as though the zero were not there, and the zero is there.
An American wheel adds a second green pocket, 00, making 38, and does not change the payout:
Two things worth saying carefully. First, the American five-number bet on 0, 00, 1, 2, and 3 is the one bet on either wheel with a worse value for the player than the rest, at . Second, people say the second zero "doubles" the house edge, and it nearly does but not exactly:
| Wheel and bet | Pockets | Ways to win | Payout | Exact value | Value per unit staked | Player's value, % |
|---|---|---|---|---|---|---|
| European, one number | 37 | 1 | 35 to 1 | -0.0270 | -2.7027 | |
| European, red or black | 37 | 18 | 1 to 1 | -0.0270 | -2.7027 | |
| European, dozen | 37 | 12 | 2 to 1 | -0.0270 | -2.7027 | |
| European, split | 37 | 2 | 17 to 1 | -0.0270 | -2.7027 | |
| American, one number | 38 | 1 | 35 to 1 | -0.0526 | -5.2632 | |
| American, red or black | 38 | 18 | 1 to 1 | -0.0526 | -5.2632 | |
| American, five-number | 38 | 5 | 6 to 1 | -0.0789 | -7.8947 |
From verify/ch08_output.txt, sections 1 and 2:
[PASS] European straight-up, E = (1/37)(+35) + (36/37)(-1) computed = -1/37 claimed = -1/37
[PASS] European red or black, E = (18/37)(+1) + (19/37)(-1) computed = -1/37 claimed = -1/37
[PASS] all four European bets have the SAME expected value computed = 1 claimed = 1 [PASS] American five-number bet on 0, 00, 1, 2, 3, paying 6 to 1 computed = -3/38 claimed = -3/38
[PASS] ratio of the American edge to the European edge, exactly computed = 37/19 claimed = 37/19
[PASS] the ratio is NOT exactly 2 computed = False claimed = Falseand, independently, from verify/gap_closures_output.txt:
European roulette, straight-up, EV per 1 unit staked: -1/37 [expected -1/37: OK]
American roulette, five-number bet, EV per 1 unit staked: -3/38 [expected -3/38: OK]What the number means over an evening: bet one unit on each of 100 European spins and your expected loss is 2.7027 units; on an American wheel it is 5.2632. There is no bet, no pattern, and no system on the layout that changes it, because every line above came out the same. That is the whole lesson, and it is the reason this section exists. 8.7.2 Hardy, April 1908: square a sum and the population stops moving
Take a generation of parents in the proportions for the three genotypes AA, Aa, and aa. Hardy's one line of algebra is that the children come out as (S388, p. 49). Start deliberately away from equilibrium, with , , and , so that :
One generation of random mating and Hardy's condition holds exactly. Run it once more and nothing moves:
And the point Hardy was making, in one number: the frequency of the dominant allele A is , and it is in the parents, in the children, and in the grandchildren. It does not spread. That is why he wrote to Science at all: "In a word, there is not the slightest foundation for the idea that a dominant character should show a tendency to spread over a whole population, or that a recessive should tend to die out" (S270, closing sentence).
From verify/ch08_output.txt, section 4:
[PASS] Hardy's condition now holds exactly: q1^2 - p1*r1 computed = 0 claimed = 0
[PASS] the dominant allele does not spread: all three are equal computed = True claimed = True
[PASS] class of 640: Aa children computed = 300 claimed = 300and from verify/domainG_output.txt, section 3:
[PASS] non-equilibrium population reaches q^2 = p*r after exactly one generation
computed = -0.0 expected = 0.0 tol = 1e-09For a class, scale to whole people: out of 640 children the split is exactly 250 AA, 300 Aa, and 90 aa. 8.7.3 Turing's decibans: why you add instead of multiply
Turing defines the unit in section 1.6 of HW 25/37: "The logarithm of the factor, taken to the base 10 to the power 1/10, is called decibanage in favour of the theory" (S261). In modern notation the weight of evidence in decibans is of the likelihood ratio (S262). Because logarithms turn multiplication into addition, three separate clues can be added up rather than multiplied together. Take clues worth 4 to 1, 5 to 1, and 2 to 1:
Twenty decibans is exactly 100 to 1, and MacKay records that about 400 characters of two intercepted Enigma messages, compared letter by letter, was usually enough to get there (S262).
From verify/ch08_output.txt, section 5:
[PASS] the three added together, in decibans computed = 16.0206 claimed = 16.0206 (tol 0.0001)
[PASS] converting the total back to odds: 10^(16.0206/10) computed = 40.0 claimed = 40.0 (tol 0.0001)
[PASS] and multiplying the odds directly gives the same 40 computed = 40 claimed = 40Every number printed in 8.7 and in the prose of 8.3 is recomputed in verify/ch08.py, whose last line reads:
TOTAL: 87 passed, 0 failedand the Domain G figures this chapter quotes are confirmed by the last line of verify/domainG_output.txt:
47 checks, 47 passed, 0 failed.What verify/ch08.py printed
==============================================================================
CHAPTER 8 ARITHMETIC VERIFICATION
==============================================================================
--- 1. Roulette, European wheel: 37 pockets, one green zero
[PASS] European straight-up, E = (1/37)(+35) + (36/37)(-1) computed = -1/37 claimed = -1/37
[PASS] European red or black, E = (18/37)(+1) + (19/37)(-1) computed = -1/37 claimed = -1/37
[PASS] European dozen, E = (12/37)(+2) + (25/37)(-1) computed = -1/37 claimed = -1/37
[PASS] European split, E = (2/37)(+17) + (35/37)(-1) computed = -1/37 claimed = -1/37
[PASS] all four European bets have the SAME expected value computed = 1 claimed = 1
[PASS] European house edge as a percentage computed = 2.702703 claimed = 2.7027 (tol 0.0001)
--- 2. Roulette, American wheel: 38 pockets, a zero and a double zero
[PASS] American straight-up, E = (1/38)(+35) + (37/38)(-1) computed = -1/19 claimed = -1/19
[PASS] -2/38 in lowest terms computed = -1/19 claimed = -1/19
[PASS] American red or black, E = (18/38)(+1) + (20/38)(-1) computed = -1/19 claimed = -1/19
[PASS] American five-number bet on 0, 00, 1, 2, 3, paying 6 to 1 computed = -3/38 claimed = -3/38
[PASS] American house edge as a percentage computed = 5.263158 claimed = 5.2632 (tol 0.0001)
[PASS] American five-number bet edge as a percentage computed = 7.894737 claimed = 7.8947 (tol 0.0001)
[PASS] the five-number bet is the ONLY American bet with a different edge computed = True claimed = True
[PASS] ratio of the American edge to the European edge, exactly computed = 37/19 claimed = 37/19
[PASS] that ratio as a decimal: near two, but not two computed = 1.947368 claimed = 1.9474 (tol 0.0001)
[PASS] the ratio is NOT exactly 2 computed = False claimed = False
--- 3. What the edge costs over an evening
[PASS] expected loss on 100 one-unit European spins, in units computed = 2.702703 claimed = 2.7027 (tol 0.0001)
[PASS] expected loss on 100 one-unit American spins, in units computed = 5.263158 claimed = 5.2632 (tol 0.0001)
[PASS] P(one chosen number never comes up in 37 European spins) computed = 0.362851 claimed = 0.3629 (tol 0.0001)
[PASS] so P(it comes up at least once in 37 spins) computed = 0.637149 claimed = 0.6371 (tol 0.0001)
--- 4. Hardy 1908, worked from p : 2q : r = 1/2 : 1/4 : 1/4
[PASS] the starting generation sums to 1: p + 2q + r computed = 1 claimed = 1
[PASS] the starting generation is NOT already at equilibrium: q^2 - pr computed = -7/64 claimed = -7/64
[PASS] p1 = (p + q)^2 computed = 25/64 claimed = 25/64
[PASS] q1 = (p + q)(q + r) computed = 15/64 claimed = 15/64
[PASS] r1 = (q + r)^2 computed = 9/64 claimed = 9/64
[PASS] 2q1, the heterozygote share as Hardy prints it computed = 15/32 claimed = 15/32
[PASS] the children sum to 1: p1 + 2q1 + r1 computed = 1 claimed = 1
[PASS] Hardy's condition now holds exactly: q1^2 - p1*r1 computed = 0 claimed = 0
[PASS] generation 2 equals generation 1, AA share computed = 25/64 claimed = 25/64
[PASS] generation 2 equals generation 1, Aa share computed = 15/32 claimed = 15/32
[PASS] generation 2 equals generation 1, aa share computed = 9/64 claimed = 9/64
[PASS] allele frequency of A in the parents, p + q computed = 5/8 claimed = 5/8
[PASS] allele frequency of A in the children computed = 5/8 claimed = 5/8
[PASS] allele frequency of A in the grandchildren computed = 5/8 claimed = 5/8
[PASS] the dominant allele does not spread: all three are equal computed = True claimed = True
[PASS] class of 640: AA children computed = 250 claimed = 250
[PASS] class of 640: Aa children computed = 300 claimed = 300
[PASS] class of 640: aa children computed = 90 claimed = 90
[PASS] class of 640: the three counts sum to 640 computed = 640 claimed = 640
--- 5. Turing's decibans: three clues, added rather than multiplied
[PASS] a 4 to 1 clue, in decibans computed = 6.0206 claimed = 6.0206 (tol 0.0001)
[PASS] a 5 to 1 clue, in decibans computed = 6.9897 claimed = 6.9897 (tol 0.0001)
[PASS] a 2 to 1 clue, in decibans computed = 3.0103 claimed = 3.0103 (tol 0.0001)
[PASS] the three added together, in decibans computed = 16.0206 claimed = 16.0206 (tol 0.0001)
[PASS] converting the total back to odds: 10^(16.0206/10) computed = 40.0 claimed = 40.0 (tol 0.0001)
[PASS] and multiplying the odds directly gives the same 40 computed = 40 claimed = 40
[PASS] 20 decibans as an odds ratio computed = 100.0 claimed = 100.0 (tol 1e-09)
--- 6. Counts and intervals stated in the chapter prose
[PASS] Blackwell: papers per year over the ten Howard years, 20 / 10 computed = 2 claimed = 2
[PASS] Blackwell: years of the ten Howard years he was NOT chair, 10 - 7 computed = 3 claimed = 3
[PASS] Blackwell: years from the 1941 PhD to the 1965 NAS election computed = 24 claimed = 24
[PASS] Blackwell: letters written in 1942 computed = 105 claimed = 105
[PASS] Fasenmyer: age at the June 1946 doctorate, born 4 October 1906 computed = 39 claimed = 39
[PASS] Fasenmyer: years between A=B's 1945 dissertation and the June 1946 degree computed = 1 claimed = 1
[PASS] Fasenmyer: years from the 1946 doctorate to Zeilberger's 1978 reading computed = 32 claimed = 32
[PASS] Fasenmyer: papers published in her life computed = 2 claimed = 2
[PASS] Snow: days from the Board of Guardians to the handle coming off computed = 1 claimed = 1
[PASS] Snow: days from the handle coming off to the first map exhibited computed = 87 claimed = 87
[PASS] Snow: the outbreak's ten days, 31 August to 9 September inclusive computed = 10 claimed = 10
[PASS] Du Bois: LOT 11931 items minus boards computed = 16 claimed = 16
[PASS] Du Bois: the LOC count minus the Data by Design count computed = 9 claimed = 9
[PASS] Du Bois: board area in square centimetres, 71 by 56 computed = 3976 claimed = 3976
[PASS] Du Bois: board area in square metres computed = 0.3976 claimed = 0.3976 (tol 1e-09)
[PASS] Du Bois: cleared images in images/IMAGE-MANIFEST.md, IMG-044 to IMG-055 computed = 12 claimed = 12
[PASS] Du Bois: charts as a share of the 500 photographs shipped, 72 vs 500 computed = 18/125 claimed = 18/125
[PASS] Noether: years from the 1915 Gottingen invitation to habilitation computed = 4 claimed = 4
[PASS] Noether: years from habilitation to dismissal, 1919 to April 1933 computed = 14 claimed = 14
[PASS] Noether: months from the April 1933 dismissal to the October sailing computed = 6 claimed = 6
[PASS] Boole Stott: her age when Hinton showed her the cubes, born 8 June 1860 computed = 18 claimed = 18
[PASS] Boole Stott: age in 1914 before her 8 June birthday computed = 53 claimed = 53
[PASS] Boole Stott: age in 1914 on or after 8 June, as verify_domainG.py has it computed = 54 claimed = 54
[PASS] Boole Stott: regular four-dimensional polytopes she worked out computed = 6 claimed = 6
[PASS] Bourbaki: years from the Cafe Capoulade meeting to the AMS application computed = 15 claimed = 15
[PASS] Bourbaki: founders in the room on 10 December 1934 computed = 6 claimed = 6
[PASS] Bourbaki: years from Husson's c. 1923 prank to the 1934 adoption computed = 11 claimed = 11
[PASS] Erdos: distance-1 plus distance-2 people, August 2025 computed = 14296 claimed = 14296
[PASS] Erdos: the site's 14,297 minus that total is Erdos himself computed = 1 claimed = 1
[PASS] Erdos: the 1998 memorial's own two co-author figures differ by computed = 15 claimed = 15
[PASS] Lovelace: months from Menabrea's October 1842 original to 1843 computed = 1 claimed = 1
[PASS] Taussky: years at the National Physical Laboratory, 1943 to 1946 computed = 3 claimed = 3
[PASS] Kublanovskaya and Francis: days between the two QR submissions computed = 250 claimed = 250
[PASS] Rosenbluth: years from the 1953 paper to Marshall's 2003 account computed = 50 claimed = 50
[PASS] Leontief: multiplications for a 450 by 450 elimination, n^3/3 computed = 30375000 claimed = 30375000
[PASS] Illiac Suite: movements not played at the 9 August 1956 premiere computed = 1 claimed = 1
--- 7. Floros 2018, the prevalence figures, as stated
[PASS] width of the reported North American range, 2.1% to 2.6% computed = 0.5 claimed = 0.5 (tol 1e-09)
[PASS] width of the reported Oceania range, 0.2% to 4.4% computed = 4.2 claimed = 4.2 (tol 1e-09)
[PASS] width of the reported European range, 0.2% to 12.3% computed = 12.1 claimed = 12.1 (tol 1e-09)
[PASS] the European range is the widest of the three computed = 12.1 claimed = 12.1
[PASS] the top of the European range as a multiple of the top of the North American range computed = 4.730769 claimed = 4.7308 (tol 0.0001)
==============================================================================
TOTAL: 87 passed, 0 failed
==============================================================================E.9 The mathematics of Chapter 9: A womb, an archer, and one man's exclamation point
This chapter has no theorem to prove, so the shown work is a different kind. Below are three first-use claims, each one of the sort a textbook prints in a single confident sentence, and each one taken apart into the chain of documents that would have to hold for the sentence to be true. Every link is scored: opened, or not opened. What this section is really measuring is the price of a fact.
The five links are the same each time.
- The tertiary claim. A textbook, an encyclopaedia, or a dictionary says it. Cheap, everywhere, and traceable to nothing.
- Miller's citation. Jeff Miller's Earliest Known Uses gives a person, a work, a volume, and usually a page. Free, curated, widely cited by historians, and not peer reviewed.
- Cajori's volume, section, and page. Florian Cajori's A History of Mathematical Notations, 1928 and 1929, is where most of Miller's notation entries ultimately come. Cajori gives his own footnote to the original.
- The original work. The book or paper the footnote names, opened and read.
- The mark on the page. A page image, looked at by a human being, showing the glyph or the sentence in type.
9.7.1 Chain A. "Sylvester coined the word matrix in 1850"
| Link | Status | What was done |
|---|---|---|
| 1. Tertiary claim | opened | Etymonline glosses the mathematical sense of matrix as "a set of components into which quantities can be set," and gives the Latin matrix, "womb," from mater, "mother" (S306). |
| 2. Miller's citation | opened | Miller names the paper: "Additions to the Articles 'On a new class of theorems' and 'On Pascal's theorem'," Philosophical Magazine, 1850, pp. 363 to 370, reprinted in the Collected Mathematical Papers, vol. 1 (Cambridge, 1904), pp. 145 to 151, at page 150, and quotes the sentence (S302, m.html). |
| 3. Cajori | not applicable | Cajori treats Sylvester's umbral notation at sect. 462 and matrix notations at sect. 468 (S301, vol. 2, both sections read), but this book did not go through Cajori for the coinage. The chain skips this link legitimately. |
| 4. The original work | opened, at one remove | The 1904 collected edition was opened on the Internet Archive and the sentence read in the scan's own text index at leaf 168, which that item's _page_numbers.json maps to printed page 150, exactly Miller's locator (S307). The 1850 Philosophical Magazine printing itself was not opened, so the wording could in principle differ between the two (S307, section 10). |
| 5. The mark on the page | not opened | The page image was not fetched. The capital M on "Matrix," the thing that shows Sylvester knew he was coining a word, has not been seen in type by anyone in this book (S307, section 10). It is cleared as IMG-015 and can be fetched. |
Links opened: 3 of 5. This is the strongest chain in the whole domain, and it still has two holes in it. 9.7.2 Chain B. "Kramp introduced n! in 1808"
| Link | Status | What was done |
|---|---|---|
| 1. Tertiary claim | opened | Etymonline dates the English noun "factorial" to 1816 and gives the Latin factor, "a doer, a maker" (S306). It does not date the symbol. |
| 2. Miller's citation | not reached | The factorial symbol is not on Miller's symbols-of-operation page. That page was queried for it, for the binomial bracket, and for nCr and nPr, and has none of them (S305, section 3, recorded as a negative result). |
| 3. Cajori | opened | Cajori, vol. 2, sect. 448, p. 72, quotes Kramp in Kramp's own French: "Je me sers de la notation tres simple n! pour designer le produit," and footnotes it to Elemens d'arithmetique universelle (Cologne, 1808), section "Notations" (S301). |
| 4. The original work | not opened | Kramp's 1808 book was never located or opened in this book. The sentence is quoted here through Cajori, who quotes it (this book's register of words, "What could not be reached"). |
| 5. The mark on the page | not opened | Worse than not opened. The optical character recognition of the Cajori scan renders the exclamation point in Kramp's sentence as a backslash, so even Cajori's printing of the glyph has not been seen (S301, section 10). The page is cleared as IMG-031 and IMG-032. |
Links opened: 2 of 5. The most famous notation date in Unit 2 rests, in this book, entirely on one 1929 monograph read through a machine transcription that cannot render the symbol under discussion. 9.7.3 Chain C. "The binomial coefficient bracket is Ettingshausen, 1827"
| Link | Status | What was done |
|---|---|---|
| 1. Tertiary claim | opened, and wrong | The dispatching brief and a great deal of popular writing give 1826. No source for the 1826 date was ever identified in this book (S301, section 10). |
| 2. Miller's citation | not reached | Not on Miller's operation page (S305). |
| 3. Cajori | opened | Cajori, vol. 2, sect. 439, p. 63: "The notation which has become the more common was introduced in 1827 by von Ettingshausen," footnoted to Vorlesungen uber hohere Mathematik, vol. 1 (Vienna, 1827), p. 38 (S301). |
| 4. The original work | opened | The Internet Archive scan of the book itself was opened. The title page reads Wien, 1827. The preface is dated "Wien, im Sommer 1827." Volume 2 is also 1827. Page 38 is about the binomial theorem, matching Cajori's page (S387). |
| 5. The mark on the page | not opened | The OCR does not render a two-line bracket, so the glyph on page 38 has not been seen (S387, section 10). This item is not in this book's image manifest, so the page image has not even been rights-cleared yet. |
Links opened: 3 of 5, and the one that matters most in a dispute, the original work, is one of them. This is why records the conflict as resolved: 1827, from the title page and the preface, and 1826 should never be printed again. 9.7.4 The cost, added up
From verify/ch09_output.txt, section 9:
matrix, Sylvester 1850 links opened = 3 of 5
n!, Kramp 1808 links opened = 2 of 5
binomial bracket, Ettingshausen 1827 links opened = 3 of 5
[PASS] total links opened across the three chains computed = 8 claimed = 8
[PASS] links still missing across the three chains computed = 7 claimed = 7Three of the best-documented notation claims in the entire course, and the project has opened eight of the fifteen links. Not one of the three has been followed all the way to a page image of the mark itself.
The same arithmetic explains the tag distribution in section 9.6. Seventeen rows carry [PRIMARY], which sounds like seventeen confirmed coinages. It is not. Twelve of those seventeen cite S301, which is Cajori's own scan: the tag records that somebody opened Cajori, not that somebody opened Kramp or Jarrett or Whitworth. Five rows cite a mathematician's own text, and they are worth naming because they are the only ones: Matrix, Menge, Scalar, Vector and Versor.
[PASS] rows carrying [PRIMARY] in the merged table computed = 17 claimed = 17
[PASS] of the [PRIMARY] rows, how many cite Cajori's own scan computed = 12 claimed = 12
[PASS] of the [PRIMARY] rows, how many cite the mathematician's own text computed = 5 claimed = 5One more piece of arithmetic, because it underpins every Cajori page number in this chapter. Cajori's volume 2 was read through the Internet Archive's search-inside index, which returns a scan leaf and not a printed page, so every page number here was converted with that item's own leaf-to-page map. The offset is constant, printed page equals leaf minus 24, and the researcher verified it at eleven separate leaves (S301). Every citation in this chapter is re-derived from its leaf in verify/ch09.py section 1:
[PASS] Cajori sect. 448, Kramp's own sentence: leaf 96 computed = 72 claimed = 72
[PASS] Cajori sect. 449, Jarrett: leaf 98 computed = 74 claimed = 74
[PASS] Cajori sect. 439, binomial bracket: leaf 87 computed = 63 claimed = 63
[PASS] Sylvester: leaf 168 computed = 150 claimed = 150Those page numbers have not been read off page images either. They are derived, and the derivation is stated so that anyone can check it. That is the honest version of a citation.
The whole script reports:
TOTAL: 94 passed, 0 failedWhat verify/ch09.py printed
========================================================================
1. Leaf-to-printed-page offsets used for the citations
========================================================================
Cajori vol. 2 (Internet Archive item b29980343_0002): the researcher
states printed page = leaf minus 24, verified at eleven leaves (S-301).
Every Cajori page cited in this chapter is re-derived from its leaf here.
[PASS] Cajori sect. 439, binomial bracket: leaf 87 computed = 63 claimed = 63
[PASS] Cajori sect. 445, Kramp on combinatorial notations: leaf 90 computed = 66 claimed = 66
[PASS] Cajori sect. 448, Factorial 'n' opens: leaf 95 computed = 71 claimed = 71
[PASS] Cajori sect. 448, Kramp's own sentence: leaf 96 computed = 72 claimed = 72
[PASS] Cajori sect. 448, Legendre's Gamma: leaf 97 computed = 73 claimed = 73
[PASS] Cajori sect. 449, Jarrett: leaf 98 computed = 74 claimed = 74
[PASS] Cajori sect. 449, Durrande and n-admiration: leaf 99 computed = 75 claimed = 75
[PASS] Cajori sect. 450, subfactorial: leaf 101 computed = 77 claimed = 77
[PASS] Cajori sect. 452, Goodwin's nPr: leaf 103 computed = 79 claimed = 79
[PASS] Cajori sect. 452, nCr, nVr, nPr: leaf 104 computed = 80 claimed = 80
[PASS] Cajori sect. 462, Sylvester's umbral notation: leaf 121 computed = 97 claimed = 97
[PASS] Cajori sect. 468, matrix notations: leaf 126 computed = 102 claimed = 102
[PASS] Cajori sect. 468, Cullis: leaf 127 computed = 103 claimed = 103
Sylvester, Collected Papers vol. 1 (item collectedmathem01sylvrich):
the item's own _page_numbers.json gives leaf 168 = printed page 150 (S-307).
[PASS] Sylvester: leaf 166 computed = 148 claimed = 148
[PASS] Sylvester: leaf 167 computed = 149 claimed = 149
[PASS] Sylvester: leaf 168 computed = 150 claimed = 150
[PASS] Sylvester: leaf 169 computed = 151 claimed = 151
========================================================================
2. The factorial notation fight, every interval
========================================================================
[PASS] Euler's capital M (1751) to Kramp's n! computed = 57 claimed = 57
[PASS] Ruffini's small pi (1799) to Kramp's n! computed = 9 claimed = 9
[PASS] Legendre's Gamma (1808) against Kramp (1808) computed = 0 claimed = 0
[PASS] Kramp (1808) to Durrande's complaint (1816) computed = 8 claimed = 8
[PASS] Kramp (1808) to Sarrus (1819) computed = 11 claimed = 11
[PASS] Kramp (1808) to Jarrett's rival sign (1827) computed = 19 claimed = 19
[PASS] Kramp (1808) to Warburton's !n! (1847) computed = 39 claimed = 39
[PASS] Kramp (1808) to Weber's Pi(m) (1893) computed = 85 claimed = 85
[PASS] Jarrett (1827) to Goodwin's use (1846) computed = 19 claimed = 19
[PASS] Jarrett (1827) to Goodwin's article in print (1849) computed = 22 claimed = 22
[PASS] Jarrett (1827) to Todhunter, about 1860 computed = 33 claimed = 33
Cajori writes that after Jarrett 'for a quarter of a century the notation
was neglected' and then dates the revival to Goodwin, 1846. A quarter of a
century is 25 years; 1846 minus 1827 is 19. The two do not agree, and this
chapter prints the dates rather than the phrase. (S-301, sect. 449, p. 74.)
[PASS] a quarter of a century, in years computed = 25 claimed = 25
[PASS] Cajori's own dated gap, 1827 to 1846 computed = 19 claimed = 19
[PASS] size of the discrepancy in Cajori's own paragraph computed = 6 claimed = 6
========================================================================
3. Ettingshausen: 1826 against 1827
========================================================================
[PASS] the error, in years, in the widely printed 1826 computed = 1 claimed = 1
[PASS] Kramp's n! (1808) to Ettingshausen's bracket (1827) computed = 19 claimed = 19
[PASS] Ettingshausen (1827) to Raabe's use (1851) computed = 24 claimed = 24
[PASS] Ettingshausen's bracket, age in 2026 computed = 199 claimed = 199
========================================================================
4. Unit 1's vocabulary, and how old it is in 2026
========================================================================
[PASS] intersection, Webster's New International Dictionary computed = 117 claimed = 117
[PASS] disjoint, Keyser in Science computed = 117 claimed = 117
[PASS] universal class, Whitehead and Russell computed = 116 claimed = 116
[PASS] union, Pierpont computed = 114 claimed = 114
[PASS] complement (set complementary to), Chittenden computed = 112 claimed = 112
[PASS] empty set, McAtee computed = 107 claimed = 107
[PASS] 'set theory' in English, Frink computed = 100 claimed = 100
[PASS] earliest year in the Unit 1 group computed = 1909 claimed = 1909
[PASS] latest year in the Unit 1 group computed = 1926 claimed = 1926
[PASS] span of the whole Unit 1 group, in years computed = 17 claimed = 17
[PASS] every Unit 1 item above is at most 120 years old computed = True claimed = True
For contrast, the older layer of the same unit:
[PASS] Cantor fixes Menge and Mengenlehre (1883), age in 2026 computed = 143 claimed = 143
[PASS] Cantor's Menge (1883) to 'set theory' in English (1926) computed = 43 claimed = 43
[PASS] Peano's membership epsilon (1889), age in 2026 computed = 137 claimed = 137
========================================================================
5. The eigen-words
========================================================================
[PASS] Cauchy's equation caracteristique (1840) to Hilbert (1904) computed = 64 claimed = 64
[PASS] Sylvester's latent roots (1883) to Hilbert's Eigenwert (1904) computed = 21 claimed = 21
[PASS] Sylvester's latent roots (1883) to Aitken's latent vectors (1937) computed = 54 claimed = 54
[PASS] Hilbert's Eigenwert (1904) to Courant and Hilbert's Eigenvektor (1924) computed = 20 claimed = 20
[PASS] Hilbert (1904) to Halmos's surrender (1967) computed = 63 claimed = 63
[PASS] Halmos's complaint (1958) to Halmos's surrender (1967) computed = 9 claimed = 9
[PASS] 63 years is over sixty computed = True claimed = True
========================================================================
6. Two symbols younger than they look
========================================================================
[PASS] Bourbaki's printed empty set glyph (1939), age in 2026 computed = 87 claimed = 87
[PASS] Bourbaki (1939) to Weil's memoir claiming it (1992) computed = 53 claimed = 53
[PASS] Jeffreys's conditional bar (1931) to Kolmogorov's P(A) (1933) computed = 2 claimed = 2
[PASS] Jeffreys (1931) to Feller making it standard (1950) computed = 19 claimed = 19
[PASS] Jeffreys's bar (1931), age in 2026 computed = 95 claimed = 95
[PASS] Zermelo's braces (1908), age in 2026 computed = 118 claimed = 118
[PASS] Wilson's bold type for vectors (1901), age in 2026 computed = 125 claimed = 125
========================================================================
7. Stochastic, Markov, and the two long sleeps
========================================================================
[PASS] Bernoulli's Stochastice (1713) to Bortkiewicz's Stochastik (1917) computed = 204 claimed = 204
[PASS] Bernoulli (1713) to Chuprov's English 'stochastical' (1923) computed = 210 claimed = 210
[PASS] English 'stochastic' (1662) to the statistical sense (1923) computed = 261 claimed = 261
[PASS] Kolmogorov's processo stocastico (1932) to Doob's English (1934) computed = 2 claimed = 2
[PASS] Markov's chains (1906) to the English phrase (1938) computed = 32 claimed = 32
[PASS] Markov's death (1922) to the English phrase (1938) computed = 16 claimed = 16
[PASS] Romanovsky's French (1929) to the English phrase (1938) computed = 9 claimed = 9
Sylvester's own two dates, and the word that outlived its reason:
[PASS] Sylvester coins Matrix (1850) to the womb sentence (1851) computed = 1 claimed = 1
[PASS] Sylvester's Matrix (1850) to Cayley's memoir (1858) computed = 8 claimed = 8
[PASS] Sylvester's Matrix (1850), age in 2026 computed = 176 claimed = 176
[PASS] Gauss's determinans (1801) to Cauchy's borrowing, read 1812 computed = 11 claimed = 11
Two words that are one man, al-Khwarizmi:
[PASS] al-jabr (c. 825) to Leibniz's 'algorithm' (1684) computed = 859 claimed = 859
[PASS] al-jabr (c. 825), approximate age in 2026 computed = 1201 claimed = 1201
NOTE: 825 is Miller's 'c. 825'. The 1201 above is arithmetic on an
approximate year and the chapter prints it as 'about 1,200 years'.
========================================================================
8. The section 9.6 table, counted from the source files
========================================================================
[PASS] word rows in ETYMOLOGY.md computed = 107 claimed = 107
[PASS] glyph rows in SYMBOLS.md computed = 35 claimed = 35
[PASS] rows in the merged 9.6 table computed = 142 claimed = 142
[PASS] rows carrying [UNVERIFIED] in ETYMOLOGY.md computed = 20 claimed = 20
[PASS] rows carrying [UNVERIFIED] in SYMBOLS.md computed = 7 claimed = 7
[PASS] rows carrying [UNVERIFIED] in the merged table computed = 27 claimed = 27
[PASS] rows carrying [PRIMARY] in the merged table computed = 17 claimed = 17
[PASS] rows carrying [CONTESTED] in the merged table computed = 7 claimed = 7
[PASS] rows carrying [1-SOURCE] in the merged table computed = 6 claimed = 6
[PASS] of the [PRIMARY] rows, how many cite Cajori's own scan computed = 12 claimed = 12
[PASS] of the [PRIMARY] rows, how many cite the mathematician's own text computed = 5 claimed = 5
share of the merged table tagged UNVERIFIED: 19.0%
[PASS] UNVERIFIED share, one decimal place computed = 19.0 claimed = 19.0
========================================================================
9. What a verified first-use claim costs: link counts for section 9.7
========================================================================
Each chain is scored on five links: (1) a tertiary claim, (2) Miller's
citation, (3) Cajori's volume, section and page, (4) the original work,
(5) the glyph or sentence seen on a page image. A link is 1 if somebody in
this project opened it, 0 if not.
matrix, Sylvester 1850 links opened = 3 of 5
n!, Kramp 1808 links opened = 2 of 5
binomial bracket, Ettingshausen 1827 links opened = 3 of 5
[PASS] links opened in the matrix chain computed = 3 claimed = 3
[PASS] links opened in the factorial chain computed = 2 claimed = 2
[PASS] links opened in the bracket chain computed = 3 claimed = 3
[PASS] total links opened across the three chains computed = 8 claimed = 8
[PASS] total links possible across the three chains computed = 15 claimed = 15
[PASS] links still missing across the three chains computed = 7 claimed = 7
Three of the best-documented notation claims in the course, and this
project has opened 8 of the 15 links. Not one of the three has been
followed all the way to a page image of the mark itself.
========================================================================
TOTAL: 94 passed, 0 failed
========================================================================Appendix F
Disagreements between sources
48 places where the sources contradict each other, and 21 of them are still open. That is the number worth sitting with: on 21 of these questions, the honest state of the field is that nobody knows. 8 of them carry more than two positions, because more than two respectable places give different answers.
An open disagreement is not a failure of research. It usually means the evidence needed to settle it does not survive, or survives in a manuscript nobody has edited into a modern edition. The column headed What would settle it names the document that would, which is the useful part: it turns "we do not know" into a specific request a librarian could act on.
| State | The question | The positions | What would settle it | What this book does |
|---|---|---|---|---|
| Open | When the Nine Chapters was compiled | Claim A: 200 BC to 100 BC. Source: S122, MacTutor matrices topic. Claim B: 200 BC to 50 AD, and shortly after 200 BC from the units argument. Source: S123, S124. Claim C: 100 BC to 100 AD. Source: S126, Lam Lay Yong. Claim D: over 2000 years ago. Source: S120, Grcar. | The excavated Suan shu shu bamboo strips of about 186 BC read against the internal unit evidence, in a specialist critical edition such as Chemla and Guo's. Nobody in this book could open one. | Unresolved and italicized. records that the only reasoned account available is MacTutor's units argument, and instructs that the range be printed and not a number. The row carries the widest range any source defends, 200 BCE to 50 CE. |
| Resolved | The date of Pingala's Chandahsastra | Claim A: c. 200 BCE. Source: S011, Miller's Earliest Uses. Claim B: 2nd century BCE. Source: S002, Shah. | A critical edition of Chandahsastra book 8 with manuscript variants. | Resolved to 2nd c. BCE. The two claims are compatible: c. 200 BCE falls inside the 2nd century BCE, so this is one dating written two ways, not two datings. |
| Open | Mahavira's century | Claim A: 9th century CE for the Ganita-sara-sangraha. Source: S009. Claim B: a Mahavira algorithm placed in a 7th century position in Shah's chronology. Source: S002. | Checking whether Shah means the same author, and the Rashtrakuta court dating of Mahavira. | Unresolved and italicized. Neither source argues its dating; they may not even be talking about the same man. |
| Open | When the binomial theorem appears in Persia and China | Claim A: around 1100. Source: S011, Miller. Claim B: al-Karaji was dead by about 1029 and the construction is credited to him (S003), and Jia Xian flourished about 1050 (S014). Source: S003, S014. | Distinguishing a table of coefficients from the binomial theorem stated as a theorem. Miller may mean the second. | Unresolved and italicized. The row is kept because Miller's date is in print and is widely repeated, and the reader needs to see it against the earlier dates. |
| Open | Whether De Morgan's laws are Ockham's, c. 1323 | Claim A: the laws occur explicitly in the Summa Totius Logicae. Source: S055, Miller citing Kneale and Kneale 1962. Claim B: De Morgan states them in 1850 and the phrase De Morgan's Laws is not used until a 1945 index. Source: S055. | Summa Logicae Part II, chapters 32 and 33, in Latin. The Logic Museum carries only a summary. | Unresolved and italicized. The Latin was not read in this book, so the c. 1323 row stands as a reported claim, not a fact. |
| Resolved | Cardano's life dates | Claim A: 1501 to 1575. Source: S072, Gorroochurn. Claim B: 24 September 1501 to 21 September 1576. Source: S094, MacTutor, and Todhunter gives 1576 at S070. | A Roman burial record. | Resolved in favor of the full dates. Two sources give 1576 and one of them gives exact days, so the timeline prints birth 24 September 1501 and death 21 September 1576. |
| Open | Tartaglia's death year | Claim A: 13 December 1557, Venice. Source: S019, Britannica, which also gives an age. Claim B: 1547. Source: S011, Miller, in the parenthesis 1499 to 1547. | A Venetian record, Tartaglia's will, or Masotti's standard biography. | Unresolved and italicized. Britannica's full date and age is the stronger form, but neither source cites a document, so the year is not settled. |
| Open | When Cardano wrote the Liber de ludo aleae | Claim A: probably completed by 1563. Source: S094, MacTutor, no document cited. Claim B: the manuscript date is unknown. Source: S070, Todhunter, art. 3. Claim C: compiled over an approximate forty year span. Source: S073, Bowman, unsourced. | Ore's 1953 study, which used the manuscript tradition, or the statement of the editor of the 1663 Opera Omnia. | Unresolved and italicized. conflict 1 says to print no composition date without one of those two, so the row is kept only as MacTutor's hedged claim. |
| Open | When Galileo wrote Sopra le scoperte dei dadi | Claim A: his 1620 probability paper. Source: S072, Gorroochurn, no source given. Claim B: no date at all. Source: S071, the York transcription of Thorne's translation. | The editorial apparatus of the 1898 Barbera Opere, volume 8. | Unresolved and italicized. Do not print 1620 on Gorroochurn's authority alone. |
| Resolved | Pascal's Traite, 1654 or 1665 | Claim A: 1654, the composition date given by later reference works. Source: S011. Claim B: 1665, the printing date on the title page. Source: S001. | Nothing; they are two events. | Resolved by splitting into a composition row and a printing row. |
| Open | The year of a Wallis title carrying the word Combinations | Claim A: 1673. Source: S011, Miller, c.html. Claim B: 1685, the usual date of A Treatise of Algebra, Both Historical and Practical. Source: no source opened in this book. | The 1673 and 1685 title pages. | Unresolved and italicized. conflict 9 flags this so that the book does not print 1673 without a check. |
| Resolved | Thomas Bayes's birth year | Claim A: 1702, flat. Source: S094, MacTutor, no reasoning. Claim B: probably between July 1701 and April 1702. Source: S084, Bellhouse p. 4, with archival reasoning. | Nothing currently known; Bellhouse's is the only argument on offer. | Resolved in favor of the window,. The table prints 1701 to 1702 and says it is a window, not a birth year. |
| Open | When Bayes entered the University of Edinburgh | Claim A: 1720. Source: S084, Bellhouse p. 7, who says he used the matriculation rolls. Claim B: 1719. Source: S094, MacTutor. | The Edinburgh matriculation rolls. | Unresolved and italicized. Bellhouse is the archival historian and is the better bet, but he does not quote the roll entry, so the year is flagged. |
| Resolved | The date of Euler's Konigsberg paper | Claim A: 1736, in almost all popular retellings. Source: widespread, no source registered. Claim B: presented 26 August 1735, printed 1741, in a Commentarii volume dated 1736. Source: S421, S422. | Already settled by the two catalog records. | Resolved into three separate rows: read in 1735, printed in 1741, in the volume for 1736. Stop calling it Euler's 1736 paper without qualification. |
| Open | What the 1748 first edition of Maclaurin's Treatise of Algebra contained | Claim A: the 1748 Treatise of Algebra contained the first published results on determinants, proving Cramer's rule for 2 by 2 and 3 by 3 systems and indicating the 4 by 4 case. Source: S122, MacTutor matrices and determinants topic. Claim B: nothing read in this book certifies the 1748 text: the copy opened is a later edition with Lawson's appendix, and Muir's volume 1 does not mention Maclaurin at all. Source: S132, section 10, and S129. | A page image of the 1748 first edition, at the chapter of general theorems for exterminating unknown quantities. | Unresolved and italicized. The S132 note records the claim as contested and partly unverified, because the exterminating chapter was read in a later edition. CH-04 section 4.10 item 3 carries the same open question as the Maclaurin against Cramer priority. |
| Resolved | Thomas Bayes's death date | Claim A: 7 April 1761. Source: S084, Bellhouse p. 27, from The Public Advertiser and the Whitehall Evening Post plus the Bunhill Fields registers RG 4/3982 and RG 4/3983. Claim B: 17 April 1761. Source: S094, MacTutor, with no source attached. | The Bunhill Fields burial register itself, at The National Archives. | Resolved to 7 April 1761. One side has two named contemporary newspapers and archival register references; the other has nothing. Do not print 17 April. |
| Resolved | Bayes's Essay, 1763 or 1764 | Claim A: the volume is dated 1763 and the article carries Read Dec. 23, 1763. Source: S083. Claim B: read on 23 December 1763 and published in 1764. Source: S085, S094. | Nothing; they are two events. | Resolved by splitting: a reading row at 1763 and a printing row at 1764. |
| Open | When Euler's Lettres a une princesse d'Allemagne was published | Claim A: 1768, in three volumes. Source: S051, Bennett; S058, SEP. Claim B: between 1768 and 1772. Source: S052, Klyve. | The imprint dates on the title pages of the three original French volumes. | Unresolved and italicized. The start year is agreed; the end year is not, and the difference matters for who saw the logic letters when. |
| Open | The year of Gauss's Pallas memoir, 1810 or 1811 | Claim A: 1810 or 1811 for the Disquisitio de elementis ellipticis Palladis. Source: S120, S134, recorded as contested. Claim B: 1811 for Gauss's elimination in the Pallas memoir. Source: S302, Miller, g.html. | The printed Commentationes volume carrying the memoir. | Unresolved and italicized. The row prints both years rather than choosing. |
| Resolved | Cauchy's determinant memoir, 1812 or 1815 | Claim A: 1812. Source: S122, MacTutor. Claim B: addressed 30 November 1812, first printed 1815 in Journal de l'Ecole Polytechnique, cahier 17, tome X. Source: S134, S129. | Nothing, because both are true of different events. | Resolved by splitting: the timeline carries a row for the reading in 1812 and a row for the printing in 1815, and never prints one date alone. |
| Open | Cauchy's eigenvalue work, 1826 or 1829 | Claim A: the eigenvalue results, the word tableau, and the diagonalization of every real symmetric matrix belong to 1826, and 1829 is not mentioned. Source: S180, MacTutor. Claim B: secular attaches to Cauchy's 1829 work on symmetric determinants. Source: S176, Miller. | The 1826 and 1829 items in Cauchy's Oeuvres completes, Serie 2, or Hawkins, Cauchy and the spectral theory of matrices, Historia Mathematica 2 (1975). Gallica is unreachable from the research environment and Hawkins is only on ScienceDirect, which is blocked. | Unresolved and both rows italicized. These are probably two papers in one body of work, but that is an inference and the timeline does not print inferences as facts. |
| Resolved | Ettingshausen's binomial coefficient bracket, 1826 or 1827 | Claim A: 1827, Vorlesungen uber die hohere Mathematik, volume 1, Vienna, p. 38. Source: S301, Cajori vol. 2, sect. 439, p. 63. Claim B: 1826, as given in much popular writing. Source: no source identified. | The title page of volume 1, which was read in the gap-closing pass. | Resolved to 1827. The title page reads Wien, 1827 and the preface is dated Wien, im Sommer 1827; volume 2 is 1827 as well, so the 1826 to 1827 set reconciliation does not apply (S387). |
| Resolved | Who first drew a polar area diagram | Claim A: Florence Nightingale, in the near-universal popular account. Source: no source. Claim B: Guerry's courbes circulaires of 1829 are the first known use; Nightingale was the first to use such charts for political persuasion. Source: S267, Friendly, Statistical Science, p. 8, footnote. | Nothing; Friendly is a peer reviewed journal article with a citation. | Resolved to Guerry, 1829. Nightingale did not invent the polar area diagram. |
| Resolved | When Poisson coined la loi des grands nombres | Claim A: 1835, in a Comptes Rendus note. Source: S088, Miller. Claim B: 1837, on page 7 of the Recherches sur la probabilite des jugements. Source: S080, Seneta, who quotes the sentence. | Reading both items. | Resolved by carrying both rows. These are almost certainly two uses by the same man, and conflict 6 instructs that the timeline say so. |
| Open | Whether Cantor's birth date is Old Style or New Style | Claim A: born in St Petersburg on 3 March 1845, calendar not stated. Source: S062, MacTutor. Claim B: the same date, calendar not stated. Source: S041, Jourdain. | A Russian civil or church register entry, or a Cantor biography that states its convention. | Unresolved and italicized. Russia used the Julian calendar until 1918, and this book's sourcing rule requires the label, so the year is flagged until somebody supplies it. |
| Open | The year of De Morgan's laws paper, 1850 or 1856 | Claim A: 1850, Trans. Camb. Phil. Soc. 9, pp. 79 to 127. Source: S055, Miller. Claim B: the scanned volume 9 carries an 1856 imprint. Source: S069. | The read on date printed at the head of the paper. | Unresolved and italicized. Cambridge Philosophical Society volumes were assembled from parts read on different dates, so both dates are probably right about different things, but nobody here has seen the head of the paper. |
| Open | The date and setting of Je le vois, mais je ne le crois pas | Claim A: an 1877 letter to Dedekind about a correspondence between a line and p-dimensional space. Source: S062, MacTutor quotations page. Claim B: the line does not appear in Dauben's paper at all. Source: S053. | The Cantor to Dedekind Briefwechsel edited by Noether and Cavailles (1937), or Cantor's Gesammelte Abhandlungen. | Unresolved and italicized. MacTutor gives no archive, no printed edition and no exact day. The popular claim that the line is about the diagonal argument is separately wrong. |
| Open | Frobenius on bilinear forms, 1877 or 1878 | Claim A: 1877, a 63-page paper in Crelle. Source: S137, Hawkins. Claim B: 1878, Uber lineare Substitutionen und bilineare Formen. Source: S122, MacTutor. | The title page and receipt date of Crelle volume 84. | Unresolved and italicized. The row is placed at 1878 and names Hawkins's 1877. |
| Resolved | Venn's system, 1880 or 1881 | Claim A: 1881, Symbolic Logic. Source: S058, SEP. Claim B: July 1880, Philosophical Magazine X, pp. 1 to 18. Source: S051, S055. | Nothing; they are two publications. | Resolved by splitting: the paper is 1880 and the book is 1881, and each has its own row. |
| Open | What Peano's cup and cap meant in 1888 | Claim A: Peano introduced the union and intersection symbols in Calcolo geometrico, 1888. Source: S054, Cajori via Miller; S064, MacTutor. Claim B: in the 1889 Arithmetices principia, read directly, the same glyphs are glossed in Latin as et and vel. Source: S050. | An openly readable scan of the 1888 Italian original. | Unresolved and italicized. records that no open copy exists in reach and that Kennedy's translation is lending-restricted. Do not print Peano invented the union and intersection symbols in 1888 as a bare fact. |
| Resolved | How long after 1879 Kempe's proof was broken | Claim A: eleven years, in 1890. Source: S423, S424. Claim B: the Royal Society only realized ten years later. Source: S427, Gonthier, p. 1382. | Arithmetic: 1890 minus 1879 is 11. | Resolved to eleven years. Gonthier is writing loosely. |
| Resolved | Arthur Cayley's death year | Claim A: 1895. Source: S143, Dictionary.com American entry. Claim B: 1893. Source: S143, Dictionary.com British entry, in the same source. | Any standard reference. | Resolved to 26 January 1895. The British entry contradicts the American entry inside one source and is simply wrong. Do not use 1893. |
| Open | Whether Zermelo found the paradox before Russell | Claim A: Zermelo discovered a similar contradiction between 1897 and 1902, possibly first. Source: S056, SEP on Russell's paradox. Claim B: MacTutor's Zermelo biography does not mention it at all. Source: S065. | A dated document from Zermelo or the Gottingen circle. | Unresolved and italicized. The span itself is only as good as the one source that gives it. |
| Resolved | Eugenio Beltrami's death year | Claim A: 1899. Source: S170, Stewart's abstract. Claim B: 18 February 1900, in Rome. Source: S201, MacTutor. Claim C: 1835 to 1900. Source: S204, Dictionary.com. | A primary record from the Accademia dei Lincei or an Italian biographical dictionary. | Resolved to 18 February 1900. Two sources to one, and only one of the three gives a full date and a place. |
| Resolved | How many Du Bois charts there are | Claim A: 63 poster-sized charts. Source: S281, Data by Design. Claim B: 72 items on 56 poster boards. Source: S282, Library of Congress LOT 11931. | The Library of Congress item-level plate list, which the gap-closing pass reached. | Resolved as a counting-units question, not a disagreement: the boards are two-sided, so 72 items on 56 boards is consistent, and the Library of Congress states No known restrictions on publication (S380, S381, S382, S383). |
| Resolved | The Ehrenfests' statistical mechanics article | Claim A: 1912. Source: S222, MacTutor. Claim B: commissioned by Klein in 1906, and commonly cited as 1911 in the physics literature. Source: S223, van der Heijden. | The fascicle itself, Encyklopadie der mathematischen Wissenschaften, Band IV, Teil 32. | Resolved by carrying only what is sourced: the 1906 commission and the separate 1907 urn paper. No row claims a publication year for the encyclopedia article. |
| Resolved | Whether Markov's excommunication request was granted | Claim A: it was granted. Source: S210, Hayes, p. 93. Claim B: it was refused; the Synod resolved that Markov had seceded from God's Church. Source: S217, Sheynin, following Emeliakh's 1954 archival study. | Emeliakh, The case of the excommunication of academician A. A. Markov from the Church, Voprosy Istorii Religii i Ateizma 2 (1954), 397 to 411, which quotes the archival file. | Resolved in favor of Sheynin: the request was refused. Markov was not excommunicated, and the 1912 row says so. |
| Resolved | Banach's axiomatic approach, 1920 or 1922 | Claim A: the 1920 Lwow doctoral dissertation. Source: S179, MacTutor. Claim B: the 1922 Fundamenta Mathematicae 3 paper. Source: S173, Dorier p. 254. | Reading both documents; the 1922 paper could not be opened here. | Resolved as two different documents. The timeline carries both and does not merge them. |
| Resolved | Camille Jordan's death year | Claim A: 1838 to 1921. Source: S170, Stewart's abstract. Claim B: 1838 to 1922. Source: S144, Althoen and McLaughlin, read in full. Claim C: 1838 to 1922. Source: S134, Miller's Earliest Uses, entry GAUSS-JORDAN METHOD. | A French academy obituary or a standard biographical dictionary entry for Jordan. None was opened in this book. | Resolved to 1922. Two sources to one, and the single source for 1921 is an abstract, which this book's sourcing rule forbids reporting as a full-text read. The same abstract gives Beltrami's death as 1899, which the Beltrami entry in this log settles at 1900, so the life dates in that abstract are unreliable as a class. this book's people register conflict entries 13 and 14 and this book's own working files defect D-003 had already settled the chapters and the people register at 1922; this entry brings the timeline into line and records the change instead of making it quietly. No source read here gives a day or a month, so the row prints the year alone. |
| Open | Who first printed the phrase Markov chain | Claim A: S. N. Bernstein, Mathematische Annalen 97 (1926), 1 to 59. Source: S212, Basharin, Langville and Naumov, ref. 3. Claim B: Romanovsky, Sur les chaines de Markoff, 1929, for the French, and American Mathematical Monthly 45 (1938), p. 410, for the English. Source: S216, S244. Claim C: first recorded in 1940 to 1945. Source: S245, Dictionary.com, discarded. | A human eye on the page images of Mathematische Annalen 97. The Gottingen digitization serves images with no text layer and the alternative host closed on 31 December 2025. | Unresolved and italicized,. Dictionary.com's 1940 to 1945 is contradicted by Miller's own page citation and is discarded. Do not print a bare the phrase was first used in 1926. |
| Resolved | Kolmogorov's analytic methods paper, 1931 or 1938 | Claim A: Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Annalen 104 (1931), 415 to 458, dated by Kolmogorov 26 July 1930. Source: S236, S216, S246. Claim B: Analytic methods in probability theory, 1938. Source: S237, MacTutor. | Kolmogorov's bibliography in the Selected Works. | Resolved to 1931 for the Mathematische Annalen paper. These are almost certainly two items, the German paper and a later Russian version. Do not print 1938 for the Mathematische Annalen paper. |
| Resolved | Onsager's reciprocal relations, 1931 or 1932 | Claim A: 1931. Source: S231, Yablonsky, Gorban and co-authors, a peer reviewed letter. Claim B: 1932. Source: S230, Gorban's workshop slides, by one of the same authors. | The papers themselves, Physical Review 37, 405 to 426 and 38, 2265 to 2279. | Resolved to 1931, which is what the co-authored peer reviewed letter says. |
| Open | The year of Fasenmyer's doctorate | Claim A: a 1945 dissertation. Source: S255, Petkovsek, Wilf and Zeilberger, pp. 18 and 55. Claim B: PhD conferred June 1946, University of Michigan. Source: S254, Agnes Scott. Claim C: PhD 1946, university ambiguously implied as Pittsburgh. Source: S253, MacTutor. | The University of Michigan dissertation record or the thesis title page. | Unresolved and italicized. The likely reconciliation is a 1945 dissertation with a June 1946 degree, but that is inference, so the year stays flagged. |
| Resolved | Kublanovskaya's doctorate, 1948 or 1955 | Claim A: she obtained her doctorate in 1948. Source: S193, Golub and Uhlig. Claim B: she completed her degree at Leningrad State University in 1948 and received the candidate's degree in 1955. Source: S194, MacTutor. | LOMI or St Petersburg University records. | Resolved in favor of MacTutor, which distinguishes the two things and is internally coherent. Both rows are carried, and neither is italicized. |
| Resolved | Who attached Gauss's name to elimination | Claim A: Bessel, in an 1838 report on the East Prussian survey. Source: S134, Miller, g.html. Claim B: George Forsythe in 1953, for the English phrase. Source: S120, Grcar. | Bessel's 1838 report, and Forsythe 1953 for the English phrase. | Resolved for the English phrase to Forsythe, 1953. The Bessel line is recorded in this book's register of words as a garbled extraction that must not be printed as fact, so no 1838 row appears in this table. |
| Resolved | The dates of the ILLIAC Suite | Claim A: first movement August 1956, complete November 1956. Source: S272, MIT OpenCourseWare. Claim B: premiere of the first three movements 9 August 1956. Source: S273, S241, S242. Claim C: the score is conventionally dated 1957, the publication year. Source: no source read. | Nothing; these are three different events. | Resolved by splitting. The timeline carries the 9 August 1956 premiere and the complete four-movement work of November 1956, and says which is which. |
| Resolved | When Euler's Latin square conjecture fell | Claim A: 1959 to 1960. Source: S442, received 10 April 1959; S443, New York Times 26 April 1959. Claim B: later disproven in 1970. Source: S421, the Euler Archive E530 catalog page. | Nothing needs settling; the catalog page contradicts two dated documents. | Resolved to 1959. Do not cite S421 for that date. |
| Resolved | When Doeblin's sealed envelope was opened | Claim A: the spring of 2000. Source: S232, Bru, who ran the operation. Claim B: May 2000. Source: S233, MacTutor. Claim C: 1991. Source: S234, Zentralblatt feature. | The December 2000 Comptes rendus special volume, or Bru and Yor's 2002 survey in Finance and Stochastics, both out of reach. | Resolved to May 2000. May is in spring, so Bru and MacTutor agree, and Bru was in the room. 1991 is not printed anywhere in this timeline. |
Appendix G
Stories that check out, and stories that do not
122 claims you will meet in a textbook, a lecture or a video, checked against the documents. 82 are wrong, 22 are contested, 12 are legends whose earliest telling comes long after the event, and 6 could not be settled here either way.
Why a whole appendix for this. The history of this subject is told, in most classrooms, through a handful of tidy anecdotes, and a striking number of them are wrong. Not exaggerated: wrong. Venn did not invent the Venn diagram and said so himself. Pascal's triangle was in print in five civilizations before Pascal. Euler's founding paper on graphs contains no graph. Markov did not build his chains to model language. Each of those is repeated in print by people who had every reason to know better, which is the actual lesson: a claim can be respectable, widely printed, and false at the same time, and the way you find out is by going to the source.
Read the middle column, not the label. A Busted row is not a gotcha. It is a better story than the one it replaces, nearly every time.
| Verdict | The version you have heard | What the record shows | Chapter | Sources |
|---|---|---|---|---|
| Busted | Kronecker's bullying drove Cantor into the asylum. | Dauben, the standard biographer: "such events had little to do with its underlying cause." A treating psychiatrist diagnosed cyclic manic depression. Kronecker's attacks were real and vicious and were also serious mathematics. | 1 | S053 |
| Busted | Venn invented the Venn diagram. | Circles for syllogisms are in print in Sturm, 1661; in Leibniz's manuscript by about 1686; and in Euler's bestseller of 1768. Venn himself calls what he inherited "the familiar Eulerian scheme." His actual contribution is the fixed primary frame plus shading. | 1 | S051, S048 |
| Busted | Even Venn thought his diagrams were new. | He counted. Of 60 logic treatises he consulted, 34 already used diagrams, nearly all Eulerian. | 1 | S051 |
| Busted | Cantor's famous 1874 paper contains the diagonal argument. | It does not. The word "diagonal" never appears. The 1874 proof is a nested-intervals argument. The diagonal came in 1891. | 1 | S040 |
| Busted | Cantor invented set theory single-handed. | Zermelo said it was "created by Cantor and Dedekind." Dedekind's own 1888 book credits Cantor 1878 and Bolzano 1851 for the definition of infinity. The chain is three deep. | 1 | S059, S043 |
| Busted | The empty set symbol is a zero, or a Greek phi. | It is a letter of the Norwegian and Danish alphabet, first printed by Bourbaki in 1939 and claimed personally by Andre Weil in 1992. | 1 | S054 |
| Busted | Set theory vocabulary is ancient. | "Union" in English, 1912. "Complement," 1914. "Empty set," 1919. "Set theory," 1926. "Disjoint," 1909. | 1 | S055 |
| Busted | Pascal invented the triangle. | It is in India in the tenth century, in Baghdad before 1029, in China around 1050, in Marrakesh before 1228, and in Nuremberg in 1544. Pascal's treatise was composed in 1654 and printed in 1665. | 2 | S002, S003, S014, S015, S018, S001 |
| Busted | Pascal called it Pascal's triangle. | He called it the triangle arithmetique. His name appears nowhere on it. The eponym starts with Montmort in 1708, forty six years after Pascal died. | 2 | S001, S011 |
| Busted | "Pascal's triangle" is an old English phrase. | Plain English "Pascal's triangle" first appears in 1886, in Chrystal's Algebra. | 2 | S011 |
| Busted | Everyone calls it Pascal's triangle. | Italy says Tartaglia's triangle, China says Yang Hui's triangle, and the Indian name is meru-prastara, the Staircase of Mount Meru. | 2 | S011, S013 |
| Busted | Old mathematicians did not make mistakes. | Varahamihira's perfume total of 174720 is wrong by a factor of 4, and his 1884 translator says so in a footnote. | 2 | S010 |
| Busted | Stifel printed Pascal's triangle for counting. | He printed it in 1544 for extracting roots. Calling the array in a 1544 book "Pascal's triangle" is an anachronism the modern write-up adopts for convenience. | 2 | S018 |
| Busted | The word "combinatorics" is ancient. | The noun is twentieth century: F. W. Levi, 1940. Leibniz's adjective "combinatorial" is 1666, and "combinatory analysis" as a subject title is MacMahon's, 1915. | 2 | S011, S008, S007 |
| Busted | Bayes wrote Bayes' rule. | The 1763 Essay contains no such formula and no vertical bar. The formula students learn is Laplace's Sixth Principle of 1814, with the law of total probability in its denominator. Miller: the rule "is not in the Essay but comes from Laplace." | 3 | S083, S086, S088 |
| Busted | The Chevalier de Mere is named in Pascal's letter. | Todhunter, who is the source of "a reputed gamester," also prints a page noting that in the passage quoted "the name de Mere is not given... a blank occurs," and that de Mere "was not the person alluded to by Pascal." | 3 | S070 |
| Busted | Chebyshev proved Chebyshev's inequality first. | Bienayme derived it in 1853, fourteen years earlier. Chebyshev's 1867 paper never mentions him. Liouville's journal reprinted Bienayme's 1853 paper immediately before Chebyshev's. | 3 | S092, S091, S094 |
| Busted | Kolmogorov cleared away centuries of confusion in 1933. | Shafer and Vovk: "its real originality was rhetorical and philosophical." Kolmogorov's own preface says the approach "has been common for some time." Frechet in 1937 said everything needed had come together by 1909. | 3 | S089, S090 |
| Busted | Cardano's book started probability theory. | It was printed in 1663, long after his death in 1576, and had already, in Bowman's phrase, "tended to collect dust on the back of library shelves." Todhunter treats it as "a gambler's manual." | 3 | S070, S073 |
| Busted | Bayes died on 17 April 1761. | 7 April 1761. Bellhouse works from two named contemporary newspapers, The Public Advertiser and the Whitehall Evening Post, and from the Bunhill Fields burial registers, RG 4/3982 and RG 4/3983. MacTutor's 17 April cites nothing. | 3 | S084, S094 |
| Busted | Bayes was born in 1702. | Nobody knows. Bellhouse: "all that can be said about Bayes's birth date is that it is probably between July of 1701 and April of 1702." | 3 | S084 |
| Busted | The notation P(A given B) is old. | The vertical stroke is Jeffreys, 1931, made popular by Feller in 1950. P(A) itself is Kolmogorov's, 1933. The word "Bayesian" dates from about 1950. | 3 | S088 |
| Busted | Pascal and Fermat invented the problem of points. | It is in Italian manuscripts by 1380 and in print in Pacioli's Summa of 1494, where Pacioli divides the stake 5 to 3, which is wrong. Cardano's rule b(b+1): a(a+1) gives 6 to 1, also wrong. The right answer is 7 to 1. | 3 | S072, S075, S093 |
| Busted | Nicolaus Bernoulli edited Ars Conjectandi. | Calling him the editor is "a massive overstatement," says Mattmuller: he supplied an errata list and a preface, three pages in total, after Varignon appealed to him in May 1713. | 3 | S079 |
| Busted | Gauss invented Gaussian elimination. | Gauss called the method eliminatio vulgaris, common elimination, in 1809. The method is in the Nine Chapters. The English phrase was printed by Forsythe in 1953, who, Grcar says, "misattributed 'high school' elimination to Gauss". | 4 | S120, S121, S134 |
| Busted | The method is European. | The Chinese treatments were the "only treatments of general linear problems until early modern Europe", and the two traditions arose independently. | 4 | S120, S121 |
| Busted | Cayley and Hamilton proved the Cayley-Hamilton theorem together. | A full-text search of the 1858 memoir finds no "Hamilton" and no "quaternion". Bocher printed "Hamilton-Cayley equation" in 1907; Turnbull printed "the Cayley-Hamilton theorem" in 1929. | 4 | S136, S134 |
| Busted | The Jordan of Gauss-Jordan is Camille Jordan. | It is Wilhelm Jordan, 1842 to 1899, a geodesist, who published it in a surveying handbook. Camille Jordan gets the Jordan normal form. | 4 | S134, S142, S144 |
| Busted | Vandermonde discovered the Vandermonde determinant. | It appears nowhere in his four mathematical papers; MacTutor attributes the naming to a misreading of his notation. | 4 | S430 |
| Busted | Matrices came first, determinants second. | Backwards. Determinants were being computed for more than 150 years before anybody named the rectangle. Sylvester's rectangle exists in order to make determinants. | 4 | S122, S127, S129, S135 |
| Busted | Negative numbers are a European idea. | Chapter 8 of the Nine Chapters states the zhengfu rule, and the board used red rods for positive and black for negative. | 4 | S123, S124, S125 |
| Busted | Lester Hill built the first matrix cipher machine. | The patent lists Louis Weisner as first-named inventor and Hill second. | 4 | S141 |
| Busted | Turing invented the LU factorization. | Von Neumann and Goldstine introduced it in 1947. Turing stated the existence and uniqueness condition and introduced the name "LU" in 1948. He did name the condition number. | 5 | S188, S189 |
| Busted | The Eckart-Young theorem is Eckart and Young's. | Erhard Schmidt proved it in 1907, in a paper on integral equations. Stewart: they "rediscovered Schmidt's approximation theorem, which is often (and incorrectly) called the Eckart-Young theorem." Miller: "The result, however, is much older." | 5 | S170, S176 |
| Busted | "Eigenvalue" is an old, settled mathematical word. | Eigenwert is Hilbert's, 1904. The earliest English "eigenvalue" is Eddington's letter to Nature of 23 July 1927. Before that: latent, characteristic, proper, secular. Halmos conceded in 1967. | 5 | S176, S202 |
| Busted | Grassmann was ignored because rivals were jealous. | Mobius said he could not follow the philosophy and declined to review it. Dorier's reason is the book's strict Euclidean order, "which did not permit a partial reading." Kummer's report killed the chair. | 5 | S172, S173, S181 |
| Busted | Grassmann was famous for linear algebra. | In his lifetime the fame came from Sanskrit. The honorary doctorate from Tubingen in 1876 was for the linguistics, and Grassmann's Law was still being refined in Language in 1966. | 5 | S181, S203 |
| Busted | The QR algorithm was invented once, in the West. | John Francis submitted on 29 October 1959 and Vera Kublanovskaya on 5 July 1960, 250 days apart, both starting from Rutishauser 1958, neither knowing of the other. | 5 | S193, S194 |
| Busted | Peano's 1888 axioms were immediately influential. | Kennedy: the achievement "aroused no attention at the time." Dorier: "his approach was not taken up immediately." Weyl added the dimensional axiom thirty years later, in 1918. | 5 | S173, S182 |
| Busted | Google uses 0.85 and the textbooks use 0.15, so they disagree. | The same number in complementary conventions: . Brin and Page write the damping factor ; Bryan and Leise write . | 5 | S197, S198 |
| Busted | Vector notation won because it was better mathematics. | The quaternion tradition had 594 publications to the Grassmannian 217. Crowe's reasons for the outcome are electrical engineering, textbook editions, personal authority, Maxwell's ambivalence, and splitting the product in two. | 5 | S177 |
| Busted | "Tensor" has always meant a multi-index array. | Hamilton coined it in October 1846 for the absolute value of a quaternion, the stretching factor, and nothing else. | 5 | S174 |
| Busted | Perron set out to prove a famous theorem. | It is in section 14 of a Habilitationsschrift on Jacobi's continued fractions. MacCluer: he proved "what he apparently thought to be a mere technical lemma." | 5 | S185 |
| Busted | The SVD was born in high theory. | Beltrami published it in 1873 in a journal for university students, as an exercise to make undergraduates comfortable with bilinear forms. | 5 | S170, S201 |
| Busted | Nobody serious computes with matrices until the computer age. | Cauchy diagonalized every real symmetric matrix in the 1820s, and Gauss ran an elimination on six equations in six unknowns from the Pallas observations between 1803 and 1809. | 5 | S180 |
| Busted | Markov was excommunicated. | He asked, in February 1912, and the Synod refused. It resolved that he had seceded from God's Church, and an internal comment records that excommunication "would be too honourable for Markov". | 6 | S217, S210 |
| Busted | Markov invented the Markov chain and called it that. | His own phrase is "the connection of samples in chains". The French "chaines de Markoff" is Romanovsky's, 1929; the English is attested in 1938, sixteen years after his death. | 6 | S214, S216, S244 |
| Busted | Markov studied Markov processes. | "The term comes from the analogy with Markov chain; Markov did not study Markov processes." | 6 | S216 |
| Busted | Markov built chains to model language. | He built them to break Nekrasov's free-will argument, in 1906. The Onegin count came seven years later, as a demonstration on real data. | 6 | S211, S212, S220 |
| Busted | The urn model is Ehrenfest's. | The journal's own contents page for 6 May 1907 reads "P. u. T. Ehrenfest". Kac credits both. Klein's 1906 commission named both. | 6 | S224, S221, S223 |
| Busted | Metropolis invented the Metropolis algorithm. | Marshall Rosenbluth, a co-author, said Metropolis "played no role in its development other than providing computer time", and that Arianna Rosenbluth wrote the MANIAC code from scratch. | 6 | S256, S257 |
| Busted | The 1953 paper is built on detailed balance. | The paper never uses the phrase, or the word reversible. It argues ergodicity. Reversibility is explicit in Hastings, 1970. | 6 | S226, S228 |
| Busted | Shannon read Markov. | He cited Frechet's Methode des fonctions arbitraires of 1938, and none of Markov's papers. | 6 | S215 |
| Busted | A stationary chain is a reversible chain. | The three-state cycle with entries and is doubly stochastic, so the uniform distribution is stationary, and its two cycle products are and , so no reversible distribution exists. | 6 | |
| Busted | "Markov chain" first appears in English in 1940 to 1945. | Miller cites American Mathematical Monthly 45 (1938), p. 410, and Merriam-Webster gives 1938. Dictionary.com's range is later and has no page behind it. | 6 | S216, S244, S245 |
| Busted | Euler drew the first graph in 1736. | He drew a map with letters on it, four capitals for land masses and seven lowercase for bridges. No dots and lines. The modern picture is a later invention. | 7 | S420, S422 |
| Busted | It is "Euler's 1736 paper". | Read to the St Petersburg Academy on 26 August 1735, printed in 1741, in the Commentarii volume dated 1736. All three dates are real and none of them is the whole answer. | 7 | S420, S421, S422 |
| Busted | Euler took the bridge problem seriously from the start. | He told Ehler it "bears little relationship to mathematics" and asked why a mathematician should be expected to solve it, and told Marinoni the question "is so banal". | 7 | S422 |
| Busted | "Graph" is a mathematical word. | Sylvester took it from chemistry in 1878, in the same sentence as his own coinage "chemicograph", and named a Kekule diagram as the model. | 7 | S302 |
| Busted | Kempe proved the four color theorem in 1879. | The proof stood eleven years and then Heawood found the hole in the reducibility argument in 1890. Kempe acknowledged it to the London Mathematical Society and was knighted anyway, in 1912. | 7 | S423, S424, S425 |
| Busted | The 1976 proof took 1200 hours of computer time, as the authors reported. | That figure is MacTutor's. The announcement was read in full for this row and gives no hours and no machine name, so the tag records what a primary source does not say. | 7 | S423, S426 |
| Busted | Karinthy coined "six degrees of separation" in 1929. | He proposes at most five intermediaries, which is six links. The story was read in full for this row and the phrase is not in it, so the tag records what a primary source does not say. | 7 | S439 |
| Busted | Milgram proved that everyone is six steps apart. | Kleinfeld, from Milgram's papers at Yale: "only three of 60 letters - 5 percent - made it" in the unpublished pilot, and "less than 30 percent of the folders got through" in the published studies. The famous number is a median over the chains that finished. | 7 | S440 |
| Busted | Graph theory begins in Europe in 1736. | Two full 8 by 8 knight's tours are in an Arabic manuscript of about 840, one of them reentrant, and a half-board tour is encoded in a Kashmiri poem of about 900. A knight's tour is a Hamiltonian path. | 7 | S431 |
| Busted | Euler was wrong about the 36 officers. | He was right about order 6 and wrong about every larger order of the form 4k+2. All 9408 reduced Latin squares of order 6 were generated here and none has an orthogonal partner. | 7 | S442 |
| Busted | Florence Nightingale invented the polar area diagram. | "Nightingale was not the inventor of polar-area diagrams, even though this is almost always credited to her. Rather, she was arguably the first to use this and other statistical graphs for political persuasion and popular impact." Guerry's courbes circulaires of 1829 came first, showing seasonal wind direction. | 8 | S267 |
| Busted | John Snow drew a map, saw the cluster, and worked out that cholera came from water. | "The map did not give rise to the insight, but rather it tended to confirm theories already held." Snow addressed the Board of Guardians on 7 September 1854, the handle came off on 8 September, and the first spot map was exhibited on 4 December, 87 days later. | 8 | S269 |
| Busted | Nicholas Metropolis invented the Metropolis algorithm. | Marshall Rosenbluth, a co-author, said Metropolis "played no role in its development other than providing computer time", and that Arianna Rosenbluth wrote the MANIAC code from scratch. Chapter 6 carries the algorithm itself. | 8 | S256, S257 |
| Busted | Du Bois drew 63 charts. | The Library of Congress counts "72 items... on 56 poster boards" and its collection description says "72 drawings". Some boards were drawn on both sides. The two counts are counting different units, not disagreeing about a fact. | 8 | S282, S380, S381, S382 |
| Busted | Ada Lovelace's Jacquard loom sentence is in Note G, beside the first computer program. | "The Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves" is in Note A. Note G carries the table, which the historians call "a table often described as 'the first computer program'", reporting the description rather than endorsing it. | 8 | S260, S259 |
| Busted | The exclamation point in n! means excitement, or emphasis. | It is a printer's mark. In the trade the exclamation point was the "note of admiration," and Cajori records that English texts of his day still suggested reading n! as "n-admiration." | 9 | S301 |
| Busted | Mathematical notation is ancient and was handed down. | Union in the set sense is 1912, intersection is a 1909 dictionary entry, disjoint is 1909, universal class is 1910, empty set is 1919, and "set theory" reaches English in 1926. | 9 | S302 |
| Busted | The binomial coefficient bracket is 1826. | 1827. The title page of volume 1 says Wien 1827 and the preface is dated "Wien, im Sommer 1827." Volume 2 of the same edition is 1827 too. | 9 | S387, S301 |
| Busted | Kramp's n! was obviously the right answer and swept the field. | Eight years later Durrande was complaining in print that "a notation so simple and consequently so useful has not yet been universally adopted." England used a rival bar-and-corner sign for decades. | 9 | S301 |
| Busted | The bar in P(A|B) is as old as probability. | Jeffreys wrote it in 1931 and Feller made it standard in 1950. The letter P settles with Kolmogorov in 1933. | 9 | S303 |
| Busted | Cayley invented matrices, so the word is his. | Sylvester coined it in 1850, eight years before Cayley's memoir, and meant a womb. | 9 | S307, S135 |
| Busted | "Determinant" has always meant what it means now. | Gauss's 1801 determinans is the bb minus ac of a quadratic form, which is our discriminant. Cauchy took the word in 1812 and pointed it at a different object, crediting Gauss while changing the referent. | 9 | S302 |
| Busted | "Stochastic" is a modern technical word. | It is in English by the 1660s meaning "pertaining to conjecture," and Bernoulli names a subject with it in 1713. It then leaves mathematics for 204 years and returns through German statistics in 1917. | 9 | S302, S306 |
| Busted | Euler chose sigma for a deep reason. | S is the first letter of summa, and he had already used Delta for difference. He says exactly that: "Just as we used the symbol Delta to signify a difference, so we use the symbol Sigma to indicate a sum." | 9 | S305 |
| Busted | Bold type for vectors is a mathematical convention. | It is a printing decision. Wilson set vectors in Clarendon, a heavy slab-serif type, in his 1901 write-up of Gibbs's lectures. You cannot write Clarendon with a pencil, which is why the arrow and the underline exist. | 9 | S304 |
| Busted | A vector has always meant an arrow with a direction. | In Harris's Lexicon Technicum of 1704 the vector is the astronomical radius vector, a line from a planet to a center, with a length and no direction of its own. Hamilton took the word and gave it direction. | 9 | S302 |
| Busted | "Eigenvalue" was always the English word. | It beat "proper value," "characteristic value," and Sylvester's "latent root" over sixty-three years. Halmos, who fought for "proper value," conceded in print in 1967: "eigenvalues have won it." | 9 | S302 |
| Contested | Euler was the first to use closed curves for syllogisms. | The Stanford Encyclopedia says so; Bennett, the specialist, shows Sturm 1661 in print and Leibniz c. 1686 in manuscript. The encyclopedia claim is probably too strong. | 1 | S058, S051 |
| Contested | "I see it, but I do not believe it" is Cantor on the diagonal argument, or on sizes of infinity. | MacTutor places it in an 1877 letter to Dedekind about a one to one correspondence between a line segment and p-dimensional space. Dimension does not change how many points you have. That is what he could not believe. | 1 | S062 |
| Contested | De Morgan's laws are De Morgan's. | He states them in 1850. Kneale and Kneale, through Miller, report that they "occur explicitly" in Ockham's Summa Totius Logicae. The phrase "De Morgan's Laws" is not printed until a 1945 index. | 1 | S055 |
| Contested | Peano invented the union and intersection symbols in 1888 as set operations. | Cajori, through Miller, says 1888. But in the 1889 Arithmetices principia, read directly here, the same glyphs are glossed in Latin as et and vel, logical "and" and "or." No open copy of the 1888 book was reachable. | 1 | S054, S050 |
| Contested | Pingala invented the triangle around 200 BCE. | Halayudha says so, a thousand years later. Shah reads the actual sutras and finds a doubling recursion, not a triangle: "the sūtra by itself has no information for carrying out the construction described by Halāyudha." Miller states the traditional version, crediting Cooke and Edwards. | 2 | S002, S011 |
| Contested | Pascal invented mathematical induction. | Simonson credits Levi ben Gershon in 1321 with "the earliest rigorous use"; Bajri, Hannah, and Montelle argue al-Samaw'al's diagrams carry inductive force around 1150. The dispute is really about how you define the term. | 2 | S004, S003, S001 |
| Contested | The triangle traveled from China to India to the Arab world to Europe. | MAA Convergence says this. Indian prosodic combinatorics is older than Jia Xian, and al-Karaji died about twenty years before Jia Xian flourished. The honest reading is largely independent discovery with unclear contact. | 2 | S017, S002, S003 |
| Contested | Cayley invented matrices in 1858. | Sylvester coined the word in 1850. Hawkins: the memoir's significance "has been grossly exaggerated" and it "went generally unnoticed, especially outside of England, until the 1880's". | 4 | S135, S137 |
| Contested | Cramer invented Cramer's rule. | Leibniz had the idea in 1693 in an unpublished letter, Maclaurin's posthumous 1748 book proves the small cases, and Cramer states the general rule in an appendix without proof in 1750. | 4 | S122, S129, S130, S132 |
| Contested | The Nine Chapters dates from a particular year. | It is a compilation. Specialists defend 200 BCE to 100 BCE, 200 BCE to 50 CE, and 100 BCE to 100 CE. Only Liu Hui's commentary of 263 CE is firm. | 4 | S120, S122, S123, S126 |
| Contested | Doeblin's envelope was opened in 1991. | Bru, who ran the opening, says spring 2000; MacTutor says May 2000. Print May 2000. | 6 | S232, S233, S234 |
| Contested | The ILLIAC Suite used Markov chain Monte Carlo. | The MIT course notes, following Hiller's own description, say "Markov chains (zero and first order) for interval and harmony selection". MCMC in the Metropolis sense is a 1953 idea and calling the 1956 work MCMC is an anachronism. | 6 | S241, S242 |
| Contested | Appel and Haken checked 1,936 cases. | Their own announcement says an unavoidable set of "fewer than 2000 configurations, each of ring size fourteen or smaller". MacTutor's topic page says about 1500 configurations; MacTutor's Kempe biography says 1,936 cases. Three sources, three quantities. | 7 | S423, S424, S426 |
| Contested | Euler's Latin square conjecture stood 175 years. | 1782 to 1959 is 177 years. | 7 | S442, S443 |
| Contested | Euler's Latin square conjecture fell in 1970. | The Euler Archive catalog page says so and is wrong. The paper was received on 10 April 1959 and the newspaper story ran on 26 April 1959. | 7 | S421, S442, S443 |
| Contested | The third of Euler's spoilers is E. C. Parker. | MacTutor prints "E. C."; the Crossref record for the 1960 Canadian Journal of Mathematics paper gives E. T. Parker. | 7 | S443 |
| Contested | Xenakis pioneered Markov chains in music. | Carvalho writes that he "has pioneered the use of Markov chains in music composition", but the same article dates Analogique A and B to 1958 to 1959 and the Illiac Suite to 1956 to 1957. The defensible claim is narrower: Xenakis fixed his transition probabilities in advance instead of estimating them from a corpus. | 8 | S272, S273, S274 |
| Contested | Andre Weil was nearly shot as a spy in Finland. | MacTutor reports it through Nevanlinna's own later memoir. No corroborating archival record was reached. The calling cards, the Russian letters, the arrest, and the release on 12 December 1939 are all sourced; the firing squad is not. | 8 | S277 |
| Contested | Princeton barred Blackwell from lectures. | Three tellings exist and they escalate: Doob "had to intervene to ensure him privileges" (peer reviewed); the president objected to honorary faculty membership and wrote of "abusing the hospitality of the University"; the president wrote of "abusing the University's hospitality by admitting a black" and "organized a great protestation". The archive letter has not been located. | 8 | S250, S251, S252 |
| Contested | The empty set symbol is a zero with a line through it, or a Greek phi. | It is a letter of the Norwegian and Danish alphabet, according to Andre Weil, who says he chose it because he was the only Bourbaki member who read Norwegian. His account is a 1992 memoir with no contemporary corroboration. | 9 | S054 |
| Contested | Peano invented the union and intersection symbols in 1888. | Miller credits the 1888 Calcolo geometrico, but the 1889 Arithmetices principia, read directly in this book, glosses the same glyphs in Latin as et and vel, logical "and" and "or." No open copy of the 1888 original could be found. | 9 | S054, S050 |
| Contested | Markov called them Markov chains. | He wrote about variables connected in a chain. The French "les chaines de Markoff" is Romanovsky, 1929, and the English phrase appears in 1938. Who printed it first is not settled: three modern historians say Bernstein used it in 1926, and this book could not open Bernstein's paper. | 9 | S302 |
| Legend | A gambler noticed the 24 versus 25 throw discrepancy at the table. | The probabilities are 0.4914 and 0.5055. Ore: detection would take "at least 100 sequences of trials, which in turn would involve several thousand individual throws," and the story "seems very unlikely." De Mere was a courtier and moral philosopher, 1607 to 1684. | 3 | S075, S093 |
| Legend | De Moivre predicted the day of his own death by summing an arithmetic progression. | The story is absent from Bellhouse and Genest's annotated translation of Maty's near-contemporary biography. MacTutor prints it with an exclamation mark and no citation. The earliest attestation is unknown. | 3 | S082, S094 |
| Legend | Hamilton carved his formula on Broome Bridge and it is still there. | The carving does not survive. The only evidence is Hamilton's own letter to his son of 5 August 1865, written twenty-two years after the event. Hamilton calls it Brougham Bridge. | 4 | S139 |
| Legend | Hamilton's carving is still on Broom Bridge. | The only evidence is Hamilton's own letter of 5 August 1865, written 7,964 days after the walk. He spells it "Brougham Bridge." No source read here confirms that any cut survives. | 5 | S175 |
| Legend | He asked in solidarity with Tolstoy. | Tolstoy was excommunicated in 1901 and died in 1910, so the 1912 timing needs another account. Sheynin proposes the Beilis prosecution in Kiev, as his own inference and not as a documented motive. | 6 | S217 |
| Legend | Erdos said a mathematician is a machine for turning coffee into theorems. | MacTutor gives the line to Renyi, and records Turan's better follow-up that weak coffee is fit only for lemmas. No dated source in either man's hand was found. | 7 | S438 |
| Legend | The Du Bois exhibit won a gold medal at the Exposition. | Not in either Du Bois source read for this book. No award is mentioned in the Library of Congress record or in the Data by Design chapter. | 8 | S281, S282 |
| Legend | Bourbaki's colleagues sent out a wedding announcement for his daughter Betti. | The announcement appears in every popular account of Bourbaki and in none of the three sources read here: the peer reviewed article in full, and both MacTutor topic pages. | 8 | S275, S276, S277 |
| Legend | Hilbert told the Gottingen senate it was not a bathing establishment. | The most repeated sentence in the Noether story is not on the MacTutor page, and no documentary attestation was found. The printed course listing under Hilbert's name is documented and is a better fact. | 8 | S278 |
| Legend | Erdos arrived at your door saying "my brain is open." | Not in the memorial article read for this book. No documentary source was located and no earliest attestation established. | 8 | S263 |
| Legend | F. N. David was named after Florence Nightingale because her parents knew the family. | Her own long interview does not say so. Widely repeated, unsourced here. | 8 | S265 |
| Legend | Monte Carlo was named for the casino because gambling is random. | Metropolis suggested the name, reportedly after an uncle who would borrow money because he "just had to go to Monte Carlo." The earliest attestation is Metropolis's own later recollection. | 9 | S302 |
| Unverified | A large language model is a Markov chain. | An n-gram model is one, exactly, and Shannon built one by hand in 1948. A transformer is not, in any useful sense. Markov chains are used as controlled training data for studying transformers, which is a claim taken from the Makkuva et al. abstract: the abstract was read for this book and the body of the paper was not. | 6 | S225, S240 |
| Unverified | Hamiltonian cycles come from Hamilton's Icosian game of 1857, which he sold to a games dealer. | Nothing about the Icosian game or the Icosian calculus could be reached in this book. Everything in that story is unverified here. | 7 | |
| Unverified | De Bruijn sequences are de Bruijn's. | De Bruijn himself published a note in 1975 acknowledging the priority of C. Flye Sainte-Marie, 1894. That note 404s at both paths tried, so the correct attribution is unverified here, and it matters. | 7 | S444 |
| Unverified | The Sanskrit mnemonic yamatarajabhanasalagam is an ancient de Bruijn sequence. | The combinatorics of the modern transcription checks out exactly. The history does not: no manuscript, no dated attestation, no scholarly edition was opened. Often repeated, not verified here. | 7 | S444 |
| Unverified | Huffman coding was a class assignment its author solved instead of sitting the exam. | This book reached only an image scan with no text layer, yielding the citation Proceedings of the IRE 40, September 1952, pp. 1098 onwards. The student-assignment story is unverified here. | 7 | |
| Unverified | Hardy and Weinberg discovered the principle together. | Hardy's letter is read in full and does not mention Weinberg. Weinberg's 1908 German paper and Edwards's 2008 specialist article on the simultaneity were both unreachable. Nothing about Weinberg's priority may be printed as read. | 8 | S270, S388 |
Appendix H
Pictures, and who owns them
Two very different lists, kept apart on purpose, because confusing them is how a book ends up printing something it has no right to print.
H.1 The 34 figures in this book, which this book drew
Every picture you have seen in these chapters was drawn from scratch for this book by a script, in this book's own colors. Nothing in them was downloaded, traced or copied, so no third party licence applies to any of them and you may reuse them under the same terms as the rest of the book.
They are not facsimiles, and this book never captions one as though it were. The Konigsberg map in Chapter 7 is our map, not Euler's plate. The tally in Chapter 6 is a reconstruction of how Markov counted, not a photograph of his page. That distinction is load-bearing: a drawing of a historical thing is evidence about our understanding, not evidence about the past.
| Figure | What it shows | Where it belongs | Caption |
|---|---|---|---|
| FIG-001 | The seven bridges of Konigsberg, 1736 | Chapter 7, course section 4.2 States | Konigsberg in 1736, drawn as a map because Euler drew a map. Four land masses, lettered A for the Kneiphof island, B for the north bank, C for the south bank, and D for the ground between the two branches of the Pregel. Seven bridges, lettered a to g, exactly as Euler letters them. Count the bridges at each letter: A has 5, and B, C, and D have 3 each. All four counts are odd, and that is the whole answer. |
| FIG-002 | One picture, three ways: the Konigsberg bridges as a map, a graph, and a 4 by 4 matrix | Chapter 7, course section 3.3 Matrix Multiplication, 4.1 Transition Matrices | The same seven bridges, three times. On the left the map Euler worked. In the middle the same information with the land shrunk to four dots and the bridges stretched to seven lines, which is a graph. On the right the same information again as a 4 by 4 table: the entry in row A, column B is 2 because two bridges join A to B. The tinted arrow follows those two bridges from the map, through the two lines in the graph, into the entry 2. Row sums are the bridge counts, 5, 3, 3, 3, and every entry of the table sums to 14, which is twice the seven bridges. |
| FIG-003 | Why the entry 4 is there: the four two bridge walks from A to D | Chapter 7, course section 3.3 Matrix Multiplication, 3.4 Matrix Powers | Square the bridge table and the entry in row A, column D reads 4. Here are the four walks it is counting, one to a panel: a then f, b then f, c then g, and d then g. Two of them go by way of B and two by way of C, because two bridges lead from A to B and two lead from A to C. Nothing goes A to A or A to D to D, which is why the sum is 0 times 1 plus 2 times 1 plus 2 times 1 plus 1 times 0. |
| FIG-004 | The eighth bridge of 1875, and the walk it makes possible | Chapter 7, course section 4.2 States | In 1875 the townspeople built an eighth bridge joining B to C. That is exactly the bridge the puzzle was waiting for. With it, only A and D still have an odd number of bridges, and a walk that crosses every bridge once becomes possible. The numbers 1 to 8 follow one such walk: A, B, A, C, A, D, B, C, D. |
| FIG-005 | A three state chain and its transition matrix, side by side | Chapter 7, course section 4.1 Markov Chain Properties and Transition Matrices | The same trick as the bridge table, with probabilities instead of counts. Nine arrows, nine entries, and they line up: the arrow from Sunny to Rainy carries , and so does the entry in row 1, column 3. Every row of the matrix sums to exactly 1, because tomorrow has to be something. In the long run the weather settles at sunny, cloudy, and rainy, and that is the row vector that the matrix leaves unchanged. |
| FIG-006 | All eight spanning trees of the simple Konigsberg graph | Chapter 7, course section 3.5 Inverses, 3.6 Solving Matrix Systems | Throw away the repeated bridges and Konigsberg leaves a graph on four points with five edges. A spanning tree keeps every point and just enough edges to hold them together, which here means three. There are eight of them, drawn here in full, with the discarded edge of each drawn faint and dotted. Kirchhoff's theorem gets the same 8 out of a determinant, without drawing anything. |
| FIG-007 | Five people are not enough: the pentagon and pentagram coloring of K5 | Chapter 7, course section 2.2 Properties of Probability, 2.5 Independence | Take five people and join every pair, acquainted or not. Color the five outer edges one way and the five crossing edges the other, and neither color contains a triangle. So five people can avoid three mutual acquaintances and three mutual strangers at the same time. Add a sixth person and it becomes impossible, which is what R(3,3) = 6 means. All 1,024 colorings of five points and all 32,768 colorings of six were checked by machine. |
| FIG-008 | Rewiring a ring: how a few long edges make a small world | Chapter 7, course section 4.2 States | Twenty dots on a ring, each joined to its four nearest neighbors. Every path between distant dots has to walk the whole way round. Now move a handful of edges to random partners, drawn here dashed and heavier. The clustering barely changes, because most edges are still local, but the typical distance across the network collapses. Watts and Strogatz printed the effect in three real networks in 1998, and the ratios are in the panel below. |
| FIG-009 | The ring 00010111 and its eight windows | Chapter 7, course section 2.1 Counting | Eight digits on a ring. Read three at a time, moving one step each time, and you get all eight of the three digit binary words exactly once: 000, 001, 010, 101, 011, 111, 110, 100. Nothing repeats and nothing is missing. The standard way to build one of these is to take an Euler circuit through a small graph, which is Euler's 1736 argument doing paid work two hundred and ten years later. |
| FIG-010 | Euler's 36 officers, and why the parade cannot be formed | Chapter 7, course section 2.1 Counting | Six ranks from six regiments, one officer of each pairing, to be drawn up six by six so that no rank and no regiment repeats in any row or any column. Rank is shown here by shape and regiment by color, so every cell has to be unique twice over. The highlighted row and column show the rule that has to hold everywhere at once. It cannot be done. All 9,408 reduced Latin squares of order 6 were generated for this book and not one of them has a partner. |
| FIG-011 | How Markov counted: one square of a hundred, out of two hundred | Chapter 6, course section 4.1 Markov Chain Properties and Transition Matrices | Markov took the first 20,000 letters of Eugene Onegin, wrote them out in 200 squares of a hundred, and counted by hand. This is the shape of one square: a hundred cells, a running tally down the right edge, and a total at the foot. The shaded cells show the overall share he found, 8,638 vowels in 20,000 letters, which is 43.19 in every hundred. It is a reconstruction of the method, not a facsimile of any one of his squares, and the letters are deliberately not printed, because no copy of the Russian text was opened for this book. |
| FIG-012 | Markov's Onegin chain: two states, four arrows, one matrix | Chapter 6, course section 4.1 Markov Chain Properties and Transition Matrices, 4.3 Stationary Distribution | The whole of Markov's 1913 result, drawn. Two states, vowel and consonant, and four numbers taken straight from his hand counts. After a vowel, the chance of another vowel is 1,104 in 8,638, which is 0.128. After a consonant, the chance of a vowel is 7,534 in 11,362, which is 0.663. Those four numbers are the matrix on the right. Run the chain forever and it settles at 0.4319 vowels, which is exactly the share he counted in the first place. |
| FIG-013 | The Ehrenfest urn, with both names on it | Chapter 6, course section 4.1 Transition Matrices, 4.4 Reversibility | Two boxes and four numbered balls. At each tick, pick one ball out of the four at random and move it to the other box. The state is just how many balls are in the left box, so the whole model is the five circles underneath. From state 1 the chance of moving right is , because three of the four balls are on the right. From state 3 it is . The long run distribution is 1, 4, 6, 4, 1 over 16, which is a row of the arithmetic triangle from Chapter 2, and the all in one box state comes back on average once every 16 ticks. |
| FIG-014 | The sealed envelope, 26 February 1940 | Chapter 6, course section 4.0 Unit 4 Opener | Wolfgang Doeblin was a soldier with an exercise book. On 26 February 1940 he posted his work to the Academy of Sciences in Paris as a sealed envelope, docket number 11668, to be opened only by him or after his death. He died 116 days later, at 25. The envelope was opened in May 2000, sixty years on, and the exercise book inside was titled Sur l'equation de Kolmogoroff. Two dates do all the work in this figure. |
| FIG-015 | Reversible or not, in two loops | Chapter 6, course section 4.4 Reversibility | Kolmogorov's criterion, with no algebra. On the left a three state chain in which going round one way costs three times over, which is , and going round the other way costs three times over, which is . Those two numbers differ by a factor of eight, so no reversible distribution can exist, even though the uniform distribution is stationary. On the right the Ehrenfest chain, where every edge balances: , , , , in both directions. One picture fails the test and the other passes it. |
| FIG-016 | The Mount Meru spread, built the way Halayudha describes it | Chapter 2, course section 2.1 Counting | Halayudha, writing in tenth century India, gives the construction as an instruction: one cell at the top, two below it extending half way on each side, and every cell after that the sum of the two above it. The name is meru-prastara, the Mount Meru spread, and the shape is the reason for the name. The faint arrows show one cell being made: 10 is 4 plus 6, and nothing else has to be known to get it. |
| FIG-017 | Mahavira's two row device for choosing r from n | Chapter 2, course section 2.1 Counting | Rule 218 of the Ganita-sara-sangraha is a recipe, not a formula. Write 1 up to n along the top. Underneath, write the same numbers the other way round. Take the last r columns, multiply the top three, multiply the bottom three, and divide. For 3 chosen from 8 that is 8 times 7 times 6 over 1 times 2 times 3, which is 56. This is the binomial coefficient, written as an instruction for a hand and a board. |
| FIG-018 | You place on a board one, and one below it | Chapter 2, course section 2.1 Counting, A.2 Exponent Rules | Al-Karaji's instruction is the whole construction: one, and one below it. Al-Samaw'al printed the table to its twelfth row around 1150 in al-Bahir fi al-Jabr. The drawing is deliberately bare, with a row counter down the left edge and no other labels, because nothing about the shape of the diagrams in the manuscript has been rights cleared or even described in the sources read for this book. |
| FIG-019 | Choose 3 colors from 28: 3,276 ways | Chapter 2, course section 2.1 Counting | Ibn Mun'im opened his combinatorics in Marrakesh with silk threads. How many ways can a weaver choose three colors out of twenty eight? The answer is 3,276, and the counting argument behind it is the one your course calls 28C3. The sources read for this book do not describe the shape of his table, so the figure draws the threads instead of guessing at the page. |
| FIG-020 | The same table, five names, five places | Chapter 2, course section 2.1 Counting | One array of numbers, drawn five times. India called it meru-prastara, the Mount Meru spread. Baghdad wrote it on a board and al-Samaw'al printed twelve rows of it about 1150. China had it by about 1050 and by 1303 the printed book already called it the Old Method. Italy calls it Tartaglia's triangle. Pascal called it the triangle arithmetique and put his rows on the diagonal, which is the fifth drawing here. His name was attached to it by Montmort in 1708, forty six years after Pascal died, and plain English Pascal's triangle waited until Chrystal in 1886. |
| FIG-021 | The priority strip: circles for syllogisms, 1555 to 1918 | Chapter 1, course section 1.2 Unions, Intersections, and Complements | Venn's contribution is the method, not the picture. Circles for testing a syllogism are in print in Sturm in 1661, they are in a Leibniz manuscript by about 1686, and they are in a Euler bestseller in 1768. Venn's own paper is 1880 and the phrase Dr Venn's diagrams is 1884. The dotted line is the point of this figure: Leibniz wrote his down about 1686 and nobody printed it until 1903, which is 217 years of nothing. |
| FIG-022 | Euler versus Venn: what is true, and what you know | Chapter 1, course section 1.2 Unions, Intersections, and Complements | Two circles, twice. On the left, the Euler way: the circle for A sits inside the circle for B, and the picture can only be drawn once you already know that all A are B. On the right, Venn's way: the two circles always overlap in the same fixed frame, and the part of A outside B is shaded out to record what you have been told. The left picture shows what is true. The right picture shows what you know. |
| FIG-023 | Dedekind's ladder: a set matched with a part of itself | Chapter 1, course section 1.4 Cardinality | Definition 64, 1888: a system S is said to be infinite when it is similar to a proper part of itself, and in the contrary case S is said to be a finite system. Here is what that means. Every counting number on the left is matched with its double on the right, nothing on the left is missed, nothing on the right is used twice, and the odd numbers on the left have no arrow coming back to them. The right hand column is a part of the left hand column, and it is the same size. |
| FIG-024 | Galileo counts to 216: nine against ten | Chapter 3, course section 2.2 Properties of Probability | Three dice, 216 ordered outcomes, drawn one cell each. The first die picks the block down the page, the second picks the block across, and the third picks the cell inside the block. Shade the cells that total nine and the cells that total ten and you get 25 against 27. Nine and ten each have six unordered combinations, which is why the gamblers looked wrong, and they were not wrong: the gap is one throw in 108, and they felt it by playing. |
| FIG-025 | Ten thousand people, and the denominator that catches everybody | Chapter 3, course section 2.4 Bayes' Rule and the Law of Total Probability | A disease one person in a hundred has, a test that catches 99 in 100 of the people who have it, and a test that clears 95 in 100 of the people who do not. Take ten thousand people and split them. A hundred have it and 9,900 do not. Of the hundred, 99 test positive. Of the 9,900, 495 test positive anyway. So 594 people test positive and only 99 of them are ill, which is one in six. Laplace's Sixth Principle says exactly this: the numerator is one cause, the denominator is every cause added up. |
| FIG-026 | The Sixth Principle, and the formula underneath it | Chapter 3, course section 2.4 Bayes' Rule and the Law of Total Probability | Laplace printed this in 1814 with no symbols in it at all. Read the underlined clause and then look straight down: the denominator of the modern formula sits in the same column as the words that describe it. Bayes wrote no such formula and no vertical bar. The formula you learn is Laplace's, and Cournot attached Bayes's name to it in 1843. |
| FIG-027 | The board in action: the paddy problem, three states, and the same thing in modern notation | Chapter 4, course section A.4 Solving Systems of Linear Equations, 3.6 Solving Matrix Systems | A Han clerk laid each equation out as a column of rods, filling the board from the right, then multiplied a whole column through and subtracted the right hand column as many times as it would go. Watch the top row empty out: 3, 2, 1 becomes 3, 0, 0. Every entry stays a whole number until the very last division, which is the whole reason for doing it this way with sticks. The same three steps in your course notation are on the right, and the only difference is that fractions arrive early instead of late. |
| FIG-028 | The Message Protector: a matrix product turned by hand | Chapter 4, course section 3.3 Matrix Multiplication | Louis Weisner and Lester Hill filed the patent on 14 February 1929 and it was granted on 16 February 1932 as United States patent 1,845,947. Inside the box, gear trains of different sizes take three letters in and put three letters out, and what the gearing computes is a matrix product modulo 26. It is the first machine built to do linear algebra on letters, and the mathematics in it is the mathematics in section 3.3. |
| FIG-029 | The carving nobody has seen | Chapter 5, course section 3.1 Vectors | On 16 October 1843 William Rowan Hamilton walked along the Royal Canal and, by his own account, cut the quaternion relations into a stone of the bridge. This drawing shows what he says he cut. Handle the story with all three words attached: it is documented, it is retrospective, and it is self reported. The only evidence is a letter he wrote to his son on 5 August 1865, which is 7,964 days later, twenty one years and about ten months. He spells the place Brougham Bridge. Modern Dublin calls it Broom Bridge. No source read for this book confirms that any cut survives. |
| FIG-030 | One product, or two: the vector war in a single line of algebra | Chapter 5, course section 3.1 Vectors | Hamilton's quaternion product of two pure vectors carries both pieces of information at once: a number and a direction, welded together with a minus sign in front of the number. Gibbs and Heaviside split the same product in two and gave each half its own symbol. Nothing was lost and nothing was added. The split won, and the tally on the right is what the argument looked like at its loudest: 8 journals, 12 scientists, 38 publications, in the five years from 1890 to 1894. |
| FIG-031 | The MANIAC console, and the person who wrote the code | Chapter 8, course section 4.3 Stationary Distribution, 4.4 Reversibility | There is no usable portrait of Arianna Rosenbluth, so this is a drawing of the machine instead. She took a physics doctorate at Harvard in 1949 at 22, and at Los Alamos she wrote from scratch the code that ran the 1953 equation of state calculation on the MANIAC. Her obituary calls that the first implementation of Markov chain Monte Carlo. Her co-author Marshall Rosenbluth said in 2003 that Nicholas Metropolis played no role in the development other than providing computer time. |
| FIG-032 | Unit 1 is younger than the aeroplane | Chapter 9, course section 1.0 Unit 1 Opener, 1.2 Unions, Intersections, and Complements | The strip runs from 1880 to 2026. Seven words that Unit 1 treats as ancient furniture land in a single seventeen year window: intersection and disjoint in 1909, the universal class in 1910, union in 1912, set complementary to in 1914, empty set in 1919, and set theory in 1926. Cantor's German Menge is fixed by 1883 and sits just outside the cluster, because the mathematics is older than the English words for it. The whole argument of the scene is the shape of that cluster, and the strip makes it in a second. |
| FIG-033 | The factorial fight: Kramp's mark against Jarrett's | Chapter 9, course section B.1 Notation and Symbols, 2.1 Counting | Two marks for the same thing. Christian Kramp printed n! at Cologne in 1808 and his whole justification was that it is simple. Thomas Jarrett proposed a printer's rule bent at a right angle in 1827 at Cambridge. Kramp did not win quickly: in 1816 Durrande was still complaining in print that a notation so simple and consequently so useful had not yet been universally adopted. Jarrett's sign sat unused for about a quarter of a century, was revived by Harvey Goodwin in 1846, and spread only when Todhunter put it in his textbooks about 1860. One of the two is on your calculator. |
| FIG-034 | Where the mathematics in this book happened, with dates | Chapter 8, course section 1.0, 2.0, 3.0, and 4.0 Unit Openers | Twenty one places, from the second century BCE to 1953. The schematic is not a survey map and the positions are approximate, but the spread is the point: the arithmetic triangle is written in India, Baghdad, Marrakesh, and China long before it reaches Nuremberg or Paris, the determinant is reached in Edo before it is printed in Geneva, and the transition matrix arrives in St Petersburg. Europe gets its own enlarged panel because eleven of the twenty one sit inside it. |
H.2 The 57 historical images, which this book does not reproduce
These are real primary source scans: manuscript pages, printed title pages, plates. Their rights were read at the repository's own page and are quoted below verbatim, including spelling and capitalization, because an archive's own wording is the only thing that governs what may be reproduced.
None of them is printed in this book, and that is a deliberate choice. The bytes were never fetched here, so nobody who built this book has seen a single one of them, and a book that reproduces an image its makers have not looked at is making a claim it cannot support. What you get instead is a description and a link to the page that holds it. A described image you can go and look at is worth more than a reproduced image nobody checked.
Two of them are non commercial only and would have to be replaced if this book were ever sold or bundled with anything sold. They are listed like the rest, with their licence on the row.
| What it shows | Repository | Rights status | Rights wording, verbatim |
|---|---|---|---|
| The page of Pascal's 1665 treatise that opens with his definition, "Appelle Triangle Arithmetique, une figure dont la construction est telle...", the passage that names and builds the triangle. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| Pascal's own contents list for the treatise, naming the four uses of the Triangle Arithmetique, including using it "pour determiner les partis qu'on doit faire entre deux joueurs". | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| The page carrying Pascal's first consequence, "En tout Triangle Arithmetique, toutes les cellules du...", the start of his numbered list of properties of the table. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| The page carrying Pascal's consequence that "chaque cellule egale la somme de toutes celles du rang perpendiculaire", one cell equals the sum of the cells in the column above it. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| The page carrying Pascal's consequence that "la somme des cellules de chaque base est double de celles de la base" preceding it, the doubling rule down the triangle. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| The Latin title page of Ars Conjectandi, Basel 1713, the book that founded mathematical probability. | Internet Archive (Smithsonian Libraries) | Public Domain (repository statement) | Public domain. The Library considers that this work is no longer under copyright protection |
| The page where Bernoulli defines his subject: "Ars Conjectandi sive Stochastice nobis definitur ars metiendi quam fieri potest exactissime probabilitates rerum...". | Internet Archive (Smithsonian Libraries) | Public Domain (repository statement) | Public domain. The Library considers that this work is no longer under copyright protection |
| Mikami's page on Zhu Shijie's Siyuan yujian (Precious Mirror, 1303), where Zhu "gives what may be called an arithmetical triangle, which he calls a 'diagram for the 8th and lower powers'". | Internet Archive | Public Domain Mark 1.0 (PDM) | http://creativecommons.org/publicdomain/mark/1.0/ (the item's licenseurl field) |
| Mikami's page stating that "Chu Shih-chieh's recording of an arithmetical triangle... consists of the binomial coefficients", tying the Chinese diagram to nCr. | Internet Archive | Public Domain Mark 1.0 (PDM) | http://creativecommons.org/publicdomain/mark/1.0/ (the item's licenseurl field) |
| The page carrying "Fig. 5. The sangi or computing rods. Nineteenth century specimens.", a photograph of the Japanese counting rods used to solve simultaneous equations. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page carrying "Fig. 6. The general form of the sangi board, from a work of 1698.", the ruled grid the rods were laid on. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page explaining that "the number 1267, represented by the sangi without the ruled board, is shown in Fig. 7". | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page stating that "Seki also expands this array of coefficients, practically the determinant that is the eliminant of the equations", and describing his removal of a common factor from a row or column. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page stating that "the Chinese and Japanese method of writing a set of simultaneous equations was such that it is rather remarkable that no predecessor of Seki's discovered the idea of the determinant". | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| Printed page 150, where Sylvester writes that an array "is, as it were, a Matrix out of which we may form various systems of determinants", the sentence that coins the word. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| Printed page 247, where Sylvester writes "I have in previous papers defined a Matrix as a rectangular array of terms, out of which different systems of determinants may be engendered". | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page instructing the reader to "form the rectangular matrix consisting of n rows and (n + 1) columns" and noting that all the n+1 determinants formed by deleting one column are zero. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The first page image of the Internet Archive item for Cayley's "A Memoir on the Theory of Matrices", Philosophical Transactions 148 (1858), pp. 17 to 37. | Internet Archive (Royal Society of London, Philosophical Transactions) | Public Domain Mark 1.0 (PDM) | Public Domain Mark 1.0 |
| The opening page of Cramer's appendix, headed "APPENDICE.", the section that carries what everyone now calls Cramer's rule. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public domain |
| Printed page 657 of the appendix, inside the passage where Cramer sets out the determinant formula for solving simultaneous linear equations. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public domain |
| Printed page 658 of the appendix, continuing Cramer's rule for systems of three and more unknowns. | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public domain |
| The page where Cramer points the reader forward, saying the rule "se trouve parmi les Traites imprimes ordinairement a la suite de la Geometrie de Descartes" and refers to "l'Appendice N. g". | Internet Archive (European Libraries; University of Lausanne copy, Google scan) | Public Domain Mark 1.0 (PDM) | Public domain |
| The page of Disquisitiones Arithmeticae where Gauss writes "numerum bb - ac... determinantem huius formae vocabimus", the first printed use of the word determinant in mathematics. | Internet Archive (Smithsonian Libraries) | Public Domain (repository statement) | Public domain. The Library considers that this work is no longer under copyright protection |
| The first page image of the Internet Archive item for Bayes's "An Essay towards Solving a Problem in the Doctrine of Chances", Philosophical Transactions 53 (1763). | Internet Archive (Royal Society of London, Philosophical Transactions) | Public Domain Mark 1.0 (PDM) | http://creativecommons.org/publicdomain/mark/1.0/ (the item's licenseurl field) |
| The title page of the second edition of John Snow's On the Mode of Communication of Cholera, London 1855. | Internet Archive (Wellcome Library) | Public Domain Mark 1.0 (PDM) | This work is available under the Creative Commons, Public Domain Mark |
| The opening page of the text, headed "MODE OF COMMUNICATION OF CHOLERA.", where Snow starts building his case. | Internet Archive (Wellcome Library) | Public Domain Mark 1.0 (PDM) | This work is available under the Creative Commons, Public Domain Mark |
| The page of Peano's table of signs carrying "Signum e significat est. Ita a e b legitur a est quoddam b", where the epsilon that became the set membership symbol is introduced. | Internet Archive (Harvard University copy, Google scan) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page carrying "Signum a significat falsum, sive absurdum" and Peano's quantifier notation for "quaecumque sunt x, y,..., a propositione". | Internet Archive (Harvard University copy, Google scan) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page headed "ARITHMETICES PRINCIPIA.", the opening of the section that states Peano's axioms for the natural numbers. | Internet Archive (Harvard University copy, Google scan) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The title page of Augustus De Morgan's Formal Logic, or, The Calculus of Inference, Necessary and Probable, London 1847. | Internet Archive (Robarts, University of Toronto) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| Printed page 72, opening Cajori's section "448. Factorial 'n'", on the frequency of n factorial in algebra and the notations invented for it. | Internet Archive (Wellcome Library) | Public Domain Mark 1.0 (PDM) | This work is available under the Creative Commons, Public Domain Mark |
| The page stating that "a sign for n-factorial arises as a special case of a more general notation also in Christian Kramp", crediting Kramp with the notation. | Internet Archive (Wellcome Library) | Public Domain Mark 1.0 (PDM) | This work is available under the Creative Commons, Public Domain Mark |
| The page beginning "Relating to the origin of the notation for 'n-factorial', nothing has been given in histories", and tracking the neglected corner-bracket notation to Harvey Goodwin in 1846. | Internet Archive (Wellcome Library) | Public Domain Mark 1.0 (PDM) | This work is available under the Creative Commons, Public Domain Mark |
| The title page leaf carrying "Die Ausdehnungslehre von 1844", the second printing that Grassmann prepared shortly before he died. | Internet Archive (Harvard University copy, Google scan) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page where Grassmann notes that other mathematicians later reached, by other routes, results "die schon in meiner Ausdehnungslehre von 1844 behandelt waren", already treated in his 1844 book. | Internet Archive (Harvard University copy, Google scan) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The title page of the third edition of Laplace's Theorie analytique des probabilites, Paris 1820. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The title leaf of MacMahon's Combinatory Analysis, volume one, Cambridge 1915, the book that named the subject. | Internet Archive (Gerstein, University of Toronto) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The first page of the preface, stating that "the object of this work is, in the main, to present to mathematicians an account of theorems in combinatory analysis which are of a perfectly general character" and that the subject "occupies the ground between algebra, properly so called, and the higher arithmetic". | Internet Archive (Gerstein, University of Toronto) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The title page of the third edition of Abraham de Moivre's The Doctrine of Chances, carrying the imprint "Printed for A. Millar, in the Strand.", London 1756. | Internet Archive (University of California Libraries) | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page introducing "the Chevalier de Mere (a reputed gamester)" who "proposed to the recluse of Port Royal" the problem that started the Pascal to Fermat correspondence. | Internet Archive | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page where Todhunter corrects the story, noting that in Pascal's letter "the name de Mere is not given in the passage we have quoted... a blank occurs", and that de Mere "was not the person alluded to by Pascal". | Internet Archive | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The page carrying the word "Eigenwert" in Hilbert's statement that the equations have a solution of convergent square sum exactly when lambda is an Eigenwert. | Internet Archive | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| The contents entry "16. Eigenwert- und Eigenfunktionentheorie der Differentialgleichung", showing the theory named after the word. | Internet Archive | Not in copyright (repository statement); public domain | NOT_IN_COPYRIGHT |
| Du Bois's hand drawn data portrait titled "The Georgia Negro A social study / / By W. E. Burghardt Du Bois." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] The states of the United States according to their Negro population." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Age distribution of Georgia Negroes compared with France." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Slaves and free Negroes." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Value of land owned by Georgia Negroes." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Assessed valuation of all taxable property owned by Georgia Negroes." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Occupations of Georgia Negroes. Males over 10." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[The Georgia Negro] Darien, McIntosh Co., Ga. Distribution of 1000 Negro inhabitants." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[A series of statistical charts illustrating the condition of the descendants of former African slaves now in residence in the United States of America] The rise of the Negroes from slavery to freedom in one generation." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[A series of statistical charts...] Proportion of freemen and slaves among American Negroes." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[A series of statistical charts...] City and rural population among American Negroes in the former slave states." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Du Bois's hand drawn data portrait titled "[A series of statistical charts...] Conjugal condition of American Negroes according to age periods." | Library of Congress, Prints and Photographs Division (LOT 11931) | No Known Restrictions on Publication (Library of Congress) | No known restrictions on publication. |
| Florence Nightingale's polar area diagram, "Diagram of the Causes of Mortality in the Army in the East", as printed in 1859. | Internet Archive (David Rumsey Map Collection, Stanford Libraries) | CC BY-NC-SA 3.0. NON COMMERCIAL ONLY. Not licensed for anything sold, or bundled with anything sold. | Images may be downloaded and used following Creative Commons CC BY-NC-SA 3.0 license. Image credit should be given to 'David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries.' |
| Map 1 from John Snow's On the Mode of Communication of Cholera, lithographed by C. F. Chefins, showing streets, pumps, and a black mark for each cholera death. | Internet Archive (David Rumsey Map Collection, Stanford Libraries) | CC BY-NC-SA 3.0. NON COMMERCIAL ONLY. Not licensed for anything sold, or bundled with anything sold. | Images may be downloaded and used following Creative Commons CC BY-NC-SA 3.0 license. Image credit should be given to 'David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries.' |
Appendix I
Where each story goes in your course
All 44 sections of the course, in order, with the chapter of this book that belongs beside each one. 33 of the 44 have history material. Every section number below is a live link that opens the course book at that exact section.
11 sections have nothing here yet, and they are listed rather than quietly skipped. Most are mixed practice, recaps and unit openers, which have no content of their own to attach a story to. Two are real gaps: A.1 Fraction Rules and A.3 Order of Operations. Both have genuine histories, and neither is written yet.
How to use this. You do not have to read this book in order, or at all, to get something from it. Arrive from a link in your course book, read one section, and go back. Each pairing answers a question the course has just made you ask, rather than a question the history happens to be able to answer.
About the Minutes column. It is an estimate of how long the story takes to tell out loud, made by the researchers who wrote it, not something measured with a stopwatch. Treat it as the difference between a two minute aside and a twelve minute one, and nothing finer than that.
Where exactly links straight to the passage. Every entry in that column is the heading you will land on, so you can also just search for its words.
There is no images column here, and that is deliberate. The research behind this table lists a picture for most of these sections, and this book reproduces none of them, because nobody who built it has seen the files and their rights are the archives' to grant. Appendix H describes every one of those images and links to the archive that holds it.
| Course section | Title | Chapter here | Where exactly | Minutes | What it gives you |
|---|---|---|---|---|---|
| 1.0 | Unit 1 Opener & Objectives | Ch 1, Ch 8 | Ch 1 section 1.0; Seventy-two drawings on fifty-six boards | 5 | The words are younger than you are. "Union" reaches English in 1912, "complement" in 1914, "empty set" in 1919, and "set theory" in 1926, so your great-grandparents were alive before this subject had its English name (S055). |
| 1.1 | Sets, the Empty Set, and Subsets | Ch 1, Ch 8, Ch 9 | A country house in Bohemia, and a book nobody read, The letter that broke Frege, and the seven rules that fixed it, A Latin booklet in Turin, and a Norwegian letter in Paris; A mathematician who never existed, and a man with no address; Two symbols younger than they look, and only one of them documented | 8 | The empty set symbol is a letter of the Norwegian and Danish alphabet, printed by Bourbaki in 1939 and claimed personally by Andre Weil in 1992 because he was the only member of the group who read Norwegian (S054). |
| 1.2 | Unions, Intersections, and Complements | Ch 1, Ch 8, Ch 9 | Two books, one year, and a law that may not be his, Everybody's circles; Seventy-two drawings on fifty-six boards; Unit 1 is younger than the aeroplane | 10 | Everybody's circles. Circles for syllogisms are in print in Sturm 1661, in a Leibniz manuscript by about 1686, and in a Euler bestseller of 1768. Venn calls what he inherited "the familiar Eulerian scheme" and contributes the fixed frame plus shading (S051, S048). |
| 1.3 | Partitions | Ch 1, Ch 7, Ch 8 | The letter that broke Frege, and the seven rules that fixed it; Eleven years of a wrong proof, and then a proof nobody could read; Seventy-two drawings on fifty-six boards | 8 | A map coloring is a partition, and the four color question ran from a student's query to Augustus De Morgan on 23 October 1852 to a computer proof in 1976, which is 124 years (S423, S426). |
| 1.4 | Cardinality | Ch 1, Ch 2, Ch 8 | Five days in December 1873, The proof you have seen is not the proof that happened, Halle, and what the record says; A mathematician who never existed, and a man with no address | 12 | Five days in December 1873. On 2 December Cantor writes to Dedekind that he does not know whether the reals can be matched with the counting numbers, and on 7 December he writes again with the proof that they cannot (S053, S040). |
| 1.5 | Sample Spaces and Events | Ch 1, Ch 3, Ch 8 | The letter that broke Frege, and the seven rules that fixed it; The book in the drawer, Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom | 8 | The book in the drawer. Cardano wrote the first definition of classical probability, "consider the whole circuit", in about the 1560s, and it was printed in 1663, 87 years after he died, and nobody read it (S072, S070). |
| 1.P | Mixed Practice | none | none | none yet | None yet |
| 1.R | Recap: Check Yourself | none | none | none yet | None yet |
| 2.0 | Unit 2 Opener & Objectives | Ch 3 | Ch 3 section 3.0; Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom | 5 | You will learn five axioms, three rules, and one famous formula in about a week. It took the human race roughly three hundred years, from a doctor scribbling about dice in Milan to a 62 page book in German in 1933 (S090, S089). |
| 2.1 | Counting | Ch 2, Ch 7, Ch 8, Ch 9 | A poet counts syllables, and a fact is born by repetition to 2.8; Euler is wrong for the first time in this chapter, and it makes the front page; A doctorate with two dates; The factorial fight, 1808 to about 1860, A bracket from Vienna, and a year that is wrong everywhere | 12 | The table was already old. Zhu Shijie printed the arithmetic triangle in 1303 and the book calls it "the Old Method", 320 years before Pascal was born (S013, S017). |
| 2.2 | Properties of Probability | Ch 3, Ch 7, Ch 8 | Galileo counts to 216, Two books, five years apart: a price on a chance, and a pile of the dead, Twenty-four years of not publishing, and an inequality named after the wrong man, Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom; Esther Klein asks about five dots, and Erdos proves things exist without building them, Networks snap, and the most famous experiment in social science mostly failed; hooks 2, 9, 10 | 10 | Galileo counts to 216. Nine and ten each have six unordered combinations, so the gamblers looked wrong, and counting orderings gives 25 against 27, a gap of one throw in 108 that players had felt at the table (S071, S070). |
| 2.3 | Conditional Probability | Ch 3, Ch 8 | Four months of letters, and one very famous story that will not survive, The paper Bayes never published, and what is not in it; One hundred and five letters, Seventy-two drawings on fifty-six boards; hook 3 | 10 | Snow's map came 87 days late. He put his case to the Board of Guardians on 7 September 1854, the pump handle came off on 8 September, and he first exhibited a spot map on 4 December: "The map did not give rise to the insight" (S269). |
| 2.4 | Bayes' Rule & the Law of Total Probability | Ch 3, Ch 8 | The paper Bayes never published, and what is not in it, Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom; hook 4 | 12 | Bayes did not write Bayes' rule. The 1763 Essay contains no such formula and no vertical bar. The formula you learn is Laplace's Sixth Principle of 1814, and Cournot attached Bayes's name to it in 1843 (S083, S086, S088). |
| 2.5 | Independence and Conditional Independence | Ch 3, Ch 7, Ch 8 | Four months of letters, and one very famous story that will not survive, The refugee at the coffee house table; Esther Klein asks about five dots, and Erdos proves things exist without building them; hooks 4, 7, 9 | 10 | The Chevalier de Mere probably never noticed anything. The probabilities are 0.4914 at 24 throws and 0.5055 at 25, and Ore calculates that detecting that gap by play would take "several thousand individual throws" (S075, S093). |
| 2.P | Mixed Practice | none | none | none yet | None yet |
| 2.R | Recap: Check Yourself | none | none | none yet | None yet |
| 3.0 | Unit 3 Opener & Objectives | Ch 4, Ch 5 | Ch 4 section 4.0; Ch 5 section 5.0 | 5 | The algorithm came first and the word came last. Han clerks ran this elimination on a board of bamboo sticks two thousand years before Gauss, and Sylvester did not coin the word "matrix" until 1850 (S120, S135). |
| 3.1 | Vectors | Ch 5, Ch 8 | The schoolteacher, and 600 copies of waste paper, Four words, two magazine articles, three months apart, The axioms, in Italian, in a book about somebody else's book, "Behold how these vectorists love one another"; No school, no post, no toilet | 10 | Six hundred copies as waste paper. Grassmann published the founding book of linear algebra in 1844, and in 1864 about 600 copies of it "were in 1864 used for waste paper" (S177, S171). |
| 3.2 | Matrix Terminology & Operations | Ch 4, Ch 8, Ch 9 | The man they would not let graduate, "I have not thought it necessary"; Note G, and the argument that will not end, No school, no post, no toilet; The word that means womb | 8 | Matrix means womb. Sylvester coined it mid-sentence in 1850 and explained a year later that determinants are engendered from the array "as from the womb of a common parent" (S307, S135). |
| 3.3 | Matrix Multiplication | Ch 4, Ch 7 | Two men, one afternoon in 1812, "I have not thought it necessary", A cipher, and a machine to run it; The founding paper of graph theory has no graph in it, Networks snap, and the most famous experiment in social science mostly failed.2 | 10 | Matrix multiplication is route counting and always was. Square the Konigsberg bridge table and the entry for A to D reads 4, and you can name all four two-bridge walks (S420, S422). |
| 3.4 | Matrix Powers | Ch 7 | The founding paper of graph theory has no graph in it; section 7.7.2 | 6 | Cube the same table and you count three-step walks. The trace of the cube counts closed three-step walks, which is six times the number of triangles (`verify/domainJ.py` J-2.12, J-2.13). |
| 3.5 | Inverses (Left, Right, and General) | Ch 4, Ch 5, Ch 7 | Seki in a closed country, Leibniz in a letter, The most famous rule in the course, printed in an appendix, Two men, one afternoon in 1812, "I have not thought it necessary", A cipher, and a machine to run it; Aircraft that shook themselves apart, and the year the machines arrived; The second bridge from a picture to a matrix, from the same small town; hooks 5, 6 | 10 | Seki in a closed country, Leibniz in a letter. Seki reached the determinant in Edo in 1683, Leibniz wrote it to the Marquis de l'Hopital on 28 April 1693, and nobody read the letter for 157 years (S127, S129). |
| 3.6 | Solving Matrix Systems | Ch 4, Ch 5, Ch 9 | Rice, rods, and a book with no date, The most famous rule in the course, printed in an appendix, Gauss calls it common; Aircraft that shook themselves apart, and the year the machines arrived; Two of the biggest words in mathematics are the same man; hook 5 | 12 | Gauss called it common. In Theoria motus of 1809 he names the method eliminatio vulgaris, and George Forsythe first printed the phrase "Gaussian elimination" in 1953, 144 years later (S134, S121, S120). |
| 3.7 | Eigenvalues & Eigenvectors | Ch 5, Ch 8, Ch 9 | Will the solar system fall apart?, A footnote about continued fractions, and the twenty five billion dollar eigenvector, Two strangers, one algorithm, and a decomposition named after the wrong people; Aircraft flutter, a review journal, and an algorithm invented twice; Eigenvalue: a word half-translated, and the word that lost | 10 | Half a German word. Hilbert wrote Eigenwert in 1904, English translated the second half and kept the first, and the earliest English "eigenvalue" is a letter to Nature of 23 July 1927 (S176, S202). |
| 3.P | Mixed Practice | none | none | none yet | None yet |
| 3.R | Recap: Check Yourself | none | none | none yet | None yet |
| 4.0 | Unit 4 Opener & Objectives | Ch 6 | Ch 6 section 6.0; A proof of free will, and the man who broke it | 6 | All four of this unit's skills exist because a fifty year old number theorist in St Petersburg wanted to destroy another man's argument about free will and needed a counterexample to do it (S212, S211). |
| 4.1 | Markov Chain Properties & Transition Matrices | Ch 6, Ch 7, Ch 8, Ch 9 | A proof of free will, and the man who broke it, Twenty thousand letters, counted by hand, The church that would not throw him out, Shannon opens a book at random; Networks snap, and the most famous experiment in social science mostly failed.3; One hundred and five letters; Stochastic: a word for aiming at something you cannot hit; hooks 7, 8 | 12 | Twenty thousand letters, counted by hand. Markov sorted the first 20,000 letters of Eugene Onegin into 8,638 vowels and 11,362 consonants, in 200 tables of ten by ten, and then did it again on 100,000 letters of Aksakov (S214, S213). |
| 4.2 | States | Ch 6, Ch 7 | Two authors, one urn, and why heat only flows one way, Detailed balance, and a rule one line long; The founding paper of graph theory has no graph in it, Paths older than the picture: knights, an Icosian game, and a ring of noughts and ones; section 6.7.4 | 8 | Euler thought it was beneath him. He told Marinoni on 13 March 1736 that "This question is so banal", and then answered it by counting letters instead of trying routes (S422, S421). |
| 4.3 | Stationary Distribution | Ch 5, Ch 6, Ch 8 | A footnote about continued fractions, and the twenty five billion dollar eigenvector, Twenty thousand letters, counted by hand, The envelope, An equation with two names, one of them borrowed; The woman who wrote the code | 10 | Perron's technical lemma. He proved it in section 14 of a 1907 paper about continued fractions, thinking it "a mere technical lemma", and it is the theorem that makes PageRank well posed (S185, S198). |
| 4.4 | Reversibility | Ch 6, Ch 8 | Two authors, one urn, and why heat only flows one way, Detailed balance, and a rule one line long; The woman who wrote the code | 10 | The 1953 paper never says "detailed balance" or "reversible". It argues ergodicity, and Marshall Rosenbluth said Metropolis "played no role in its development other than providing computer time" while Arianna Rosenbluth wrote the MANIAC code from scratch (S226, S256, S257). |
| 4.P | Mixed Practice | none | none | none yet | None yet |
| 4.R | Recap: Check Yourself | none | none | none yet | None yet |
| A.0 | Algebra Opener | Ch 9 | Two of the biggest words in mathematics are the same man | 5 | Algebra is a word about setting bones. Al-jabr is "the restoring, the reunion of broken parts", from al-Khwarizmi's title of about 825, and in fifteenth and sixteenth century English "algebra" also meant bone-setting (S306, S302). |
| A.1 | Fraction Rules | none | none | none yet | None yet |
| A.2 | Exponent Rules | Ch 2 | Baghdad: "you place on a board one and one below it" | 3 | Al-Samaw'al proved the index law you use without noticing. Around 1150 in Baghdad he demonstrates what we write as for and , in the same book that prints twelve rows of the arithmetic triangle (S003). |
| A.3 | Order of Operations (PEMDAS) | none | none | none yet | None yet |
| A.4 | Solving Systems of Linear Equations | Ch 4 | Rice, rods, and a book with no date, Gauss calls it common; sections 4.7.1, 4.7.2 | 12 | The legal moves, in wood. A Han clerk laid each equation out as a column of bamboo rods, multiplied a whole column through, and subtracted the right column as many times as possible, with red rods for positive and black for negative (S124, S125, S122). |
| B.0 | Study Tools Opener | none | none | none yet | None yet |
| B.1 | Notation & Symbols | Ch 9, Ch 1, Ch 6 | The word that means womb, The factorial fight, 1808 to about 1860, A bracket from Vienna, and a year that is wrong everywhere, Eigenvalue: a word half-translated, and the word that lost, Two symbols younger than they look, and only one of them documented; A Latin booklet in Turin, and a Norwegian letter in Paris | 10 | Kramp's whole justification for in 1808 was that it is simple, and eight years later Durrande was still complaining in print that "a notation so simple and consequently so useful has not yet been universally adopted" (S301). |
| B.2 | Glossary | Ch 9, Ch 7 | Unit 1 is younger than the aeroplane, Stochastic: a word for aiming at something you cannot hit; The word "graph" comes out of a chemistry lab | 8 | "Graph" came out of a chemistry lab. Sylvester borrowed it in Nature on 7 February 1878 in the same sentence as his own coinage "chemicograph", naming a Kekule diagram as the model (S302). |
| B.3 | Methods | Ch 4, Ch 1, Ch 6 | Gauss calls it common; Everybody's circles; Twenty thousand letters, counted by hand | 8 | Two of the fourteen methods carry names that are not honest. Gaussian elimination is Forsythe's phrase from 1953, and "reading a Venn diagram word problem" is a 1880 procedure sitting under a 1661 picture (S120, S051). |
| B.4 | Formulas | Ch 3, Ch 1, Ch 6 | Laplace writes the formula, Cournot names it after the wrong man, Kolmogorov makes it an axiom; section 1.7(b); section 6.7.2 | 8 | Formula 9 and formula 8 are one sentence of Laplace's, printed in 1814 with no symbols in it at all: the numerator is one cause, the denominator is "the sum of the similar probabilities relative to all the causes" (S086). |
| B.5 | Properties & Rules | Ch 1, Ch 4 | Two books, one year, and a law that may not be his, A Latin booklet in Turin, and a Norwegian letter in Paris; "I have not thought it necessary" | 8 | A named law is a bookmark, not a birth certificate. De Morgan states his laws in 1850, Kneale and Kneale report them in Ockham five centuries earlier, and the phrase "De Morgan's Laws" is not printed until a 1945 index (S055, S047). |
Appendix J
Bibliography
266 sources in APA 7, grouped by how much weight each can carry. The access column is the honesty control: a source marked not opened supports no claim anywhere in this book, and is listed so you know it exists and know it is not load-bearing.
Every source pointer in the text, the small boxed S numbers, links here. Click one and you land on the row.
Tier 1: primary sources and critical editions of them
74 sources.
| ID | Reference (APA 7) | Repository | Access |
|---|---|---|---|
| S393 | Mathematische Annalen (Vol. 97). (1927). Berlin. Gottinger Digitalisierungszentrum, PPN235181684_0097. http://resolver.sub.uni-goettingen.de/purl?PPN235181684_0097Cited in 1910 to 1935. | Niedersachsische Staats- und Universitatsbibliothek Gottingen (GDZ) | not opened |
| S426 | Appel, K., & Haken, W. (1976). Every planar map is four colorable. Bulletin of the American Mathematical Society, 82(5), 711-712. https://www.ams.org/journals/bull/1976-82-05/S0002-9904-1976-14122-5/S0002-9904-1976-14122-5.pdfCited in Chapter 7 7 times and 1960 to now. | American Mathematical Society | read in full |
| S083 | Bayes, T., & Price, R. (1763). An essay towards solving a problem in the doctrine of chances. By the late Rev. Mr. Bayes, F.R.S. Communicated by Mr. Price, in a letter to John Canton, A.M. F.R.S. Philosophical Transactions of the Royal Society of London, 53, 370-418. Internet Archive, philtrans09948070. https://archive.org/details/philtrans09948070Cited in Chapter 3 3 times and 1650 to 1800 twice. | Internet Archive | partial (The article heading, Price's covering letter and its date, the statement of the Problem, Definition 5 and Proposition 3 were read from the OCR) |
| S006 | Bernoulli, J. (1713). Jacobi Bernoulli, profess. Basil. & utriusque societ.... Ars conjectandi, opus posthumum: Accedit Tractatus de seriebus infinitis, et epistola Gallice scripta de ludo pilae reticularis. Impensis Thurnisiorum, Fratrum. [Digitised copy, Smithsonian Libraries and Archives; Internet Archive identifier jacobibernoulli00bern, ark:/13960/t89h2nc8c]. https://archive.org/details/jacobibernoulli00bernCited in Chapter 9 and 1650 to 1800. | Internet Archive, scanned by Smithsonian Libraries and Archives | partial (Item metadata and title page read in full; the OCR text layer read only from the start of the volume through the front matter, Nicolaus Bernoulli's...) |
| S044 | Bolzano, B. (1851). Paradoxien des Unendlichen (F. Prihonsky, Ed.). C. H. Reclam sen. Internet Archive, paradoxiendesun00bolzgoog. https://archive.org/details/paradoxiendesun00bolzgoogCited in Chapter 1 twice, 1800 to 1850 twice and 1850 to 1880. | Internet Archive (Google Books scan) | partial (Title page, Prihonsky's editorial foreword, and the early sections including section 4 were read from the OCR text; the archive text stream truncated...) |
| S045 | Boole, G. (1847/2011). The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning. Macmillan, Barclay, & Macmillan; George Bell. Project Gutenberg ebook 36884. https://www.gutenberg.org/ebooks/36884Cited in Chapter 1 and 1800 to 1850. | Project Gutenberg | read in full |
| S046 | Boole, G. (1854/1951). An investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. Dover Publications. Internet Archive, investigationofl00bool. https://archive.org/details/investigationofl00boolCited in 1850 to 1880. | Internet Archive | partial (Title page, front matter, and Chapter I with the early parts of Chapter II were read from the OCR text; the archive text stream truncated before the...) |
| S442 | Bose, R. C., & Shrikhande, S. S. (1960). On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler. Transactions of the American Mathematical Society, 95(2), 191-209. https://doi.org/10.1090/S0002-9947-1960-0111695-3Cited in Chapter 7 8 times, 1650 to 1800, 1935 to 1960 and 1960 to now. | American Mathematical Society | read in full |
| S198 | Brin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual web search engine. Computer Science Department, Stanford University. http://infolab.stanford.edu/~backrub/google.htmlCited in Chapter 5 3 times and 1960 to now. | Stanford University InfoLab | partial (title, authorship, the PageRank definition in section 2.1.1, the random surfer statement, and the index size figures; the crawler, indexer and...) |
| S069 | Cambridge Philosophical Society. (1856). Transactions of the Cambridge Philosophical Society (Vol. 9). Internet Archive, transactionsofca09camb. https://archive.org/details/transactionsofca09cambCited in Chapter 1 and 1850 to 1880 twice. | Internet Archive | not opened |
| S040 | Cantor, G. (1874). Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal fur die Reine und Angewandte Mathematik, 77, 258-262. Facsimile PDF. https://jamesrmeyer.com/pdfs/cantor-1874-ueber-eine-eigenschaft-des-inbegriffes.pdfCited in Chapter 1 twice and 1850 to 1880. | independently hosted facsimile at jamesrmeyer.com (the journal itself, Crelle vol. 77, is the source of the... | read in full |
| S041 | Cantor, G. (1895/1897/1915). Contributions to the founding of the theory of transfinite numbers (P. E. B. Jourdain, Trans.). Open Court. Dover reprint. Internet Archive, contributionstof00cant. https://archive.org/details/contributionstof00cantCited in 1800 to 1850 and 1880 to 1910. | Internet Archive | partial (The OCR text stream returned by the archive server stopped near page 40, so I read Jourdain's Introduction to about p. 39 and the table of contents) |
| S042 | Cantor, G. (1895/2024). Contributions to the founding of the theory of transfinite numbers, sections 1-6 (J. R. Meyer, Trans.). Retrieved August 20, 2026, from https://jamesrmeyer.com/infinite/cantor-contributions-transfiniteCited in Chapter 1 and 1880 to 1910. | jamesrmeyer.com | read in full |
| S136 | Cayley, A. (1858). A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17-37. https://doi.org/10.1098/rstl.1858.0002Cited in Chapter 4 13 times and 1850 to 1880 twice. | the Royal Society's own copy at royalsocietypublishing.org returned HTTP 403 from this environment; the... | read in full |
| S021 | Colebrooke, H. T. (Trans.). (1817). Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhascara. John Murray. [Digitised copy; Internet Archive identifier 1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-br]. https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-brNot cited in the story. It backs a register entry. | Internet Archive | not opened |
| S130 | Cramer, G. (1750). Introduction a l'analyse des lignes courbes algebriques. Chez les Freres Cramer & Cl. Philibert. Internet Archive, bub_gb_HzcVAAAAQAAJ (digitised from Google Books). https://archive.org/details/bub_gb_HzcVAAAAQAAJCited in Chapter 4 6 times and 1650 to 1800. | Internet Archive | partial (I located and read the Appendix passage in which the rule is stated, at scan leaves 749 to 752, together with the section heading and the sign rule) |
| S047 | De Morgan, A. (1847). Formal logic, or, The calculus of inference, necessary and probable. Taylor and Walton. Internet Archive, formallogicorthe00demouoft. https://archive.org/details/formallogicorthe00demouoftCited in Chapter 1 and 1800 to 1850. | Internet Archive | partial (Title page, preface, and the early chapters through roughly p. 40 were read from the OCR text, plus the contents) |
| S043 | Dedekind, R. (1872/1888/1901). Essays on the theory of numbers: I. Continuity and irrational numbers, II. The nature and meaning of numbers (W. W. Beman, Trans.). Open Court. Project Gutenberg ebook 21016. https://www.gutenberg.org/ebooks/21016Cited in Chapter 1 twice, 1850 to 1880 and 1880 to 1910 twice. | Project Gutenberg | read in full |
| S224 | Ehrenfest, P., & Ehrenfest, T. (1907). Uber zwei bekannte Einwande gegen das Boltzmannsche H-Theorem. Physikalische Zeitschrift, 8, 311-314. [Contents page of the volume consulted in the Internet Archive scan of No. 9, 6 May 1907.]. https://archive.org/details/per_physikalische-zeitschrift_physikalische-zeitschrift_1907-05-01_9Cited in Chapter 6 4 times and 1880 to 1910. | Internet Archive | partial (The issue's own contents listing was read in full) |
| S433 | Erdos, P., & Szekeres, G. (1935). A combinatorial problem in geometry. Compositio Mathematica, 2, 463-470. http://www.numdam.org/item/CM_1935__2__463_0/Cited in Chapter 7 5 times, Chapter 8 and 1935 to 1960. | Numdam (Numerisation de documents anciens mathematiques) | read in full |
| S435 | Erdos, P. (1947). Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4), 292-294. https://www.ams.org/journals/bull/1947-53-04/S0002-9904-1947-08785-1/S0002-9904-1947-08785-1.pdfCited in Chapter 7 twice and 1935 to 1960. | American Mathematical Society | read in full |
| S436 | Erdos, P., & Renyi, A. (1959). On random graphs I. Publicationes Mathematicae Debrecen, 6, 290-297. Offprint hosted at the Alfred Renyi Institute of Mathematics. https://www.renyi.hu/~p_erdos/1959-11.pdfCited in Chapter 7 3 times and 1935 to 1960 twice. | Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences | read in full |
| S437 | Erdos, P., & Renyi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences (Matematikai Kutato Intezet Kozlemenyei), 5(A/1-2), 17-61. Offprint hosted at the Alfred Renyi Institute of Mathematics. https://www.renyi.hu/~p_erdos/1960-10.pdfCited in Chapter 7 3 times, 1935 to 1960 and 1960 to now. | Alfred Renyi Institute of Mathematics, Hungarian Academy of Sciences | read in full |
| S387 | Ettingshausen, A. von. (1827). Vorlesungen uber die hohere Mathematik: Erster Band [Vorlesungen über die höhere Mathematik]. Wien: C. Gerold. Internet Archive, vorlesungenberd00ettigoog. https://archive.org/details/vorlesungenberd00ettigoogCited in Chapter 9 5 times and 1800 to 1850. | Internet Archive | partial (Title page and preface read from the OCR text; the notation on p. 38 could NOT be read, because the OCR does not render the bracket symbol) |
| S420 | Euler, L. (1741). Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, 8, 128-140. (Presented to the St Petersburg Academy 26 August 1735.) Latin text transcribed at Wikisource. https://la.wikisource.org/wiki/Solutio_problematis_ad_geometriam_situs_pertinentisCited in Chapter 7 9 times and 1650 to 1800. | Wikisource (Latin) | partial (The transcription carries only the opening of the paper: the first paragraph and part of the second) |
| S049 | Euler, L. (1768/1802). Letters of Euler on different subjects in physics and philosophy: Addressed to a German princess (H. Hunter, Trans., 2nd ed., Vol. 1). Murray and Highley. Internet Archive, lettersofeuleron01eule. https://archive.org/details/lettersofeuleron01euleCited in Chapter 1 and 1800 to 1850. | Internet Archive | partial (Title page, front matter and the full table of contents were read; the OCR stream truncated around Letter LXXXIII, so the logic letters themselves...) |
| S074 | Fermat, P. de, & Pascal, B. (1654/1929). Fermat and Pascal on probability (V. Sanford & M. Merrington, Trans.). In Oeuvres de Fermat (P. Tannery & C. Henry, Eds., Vol. 2, pp. 288-314). Gauthier-Villars, 1894. Translation as printed in D. E. Smith, A source book in mathematics, and transcribed at the University of York History of Statistics pages. https://www.york.ac.uk/depts/maths/histstat/pascal.htmCited in Chapter 3 3 times and 1650 to 1800 4 times. | University of York, Department of Mathematics | read in full |
| S071 | Galilei, G. (n.d./1898). Sopra le scoperte dei dadi [Concerning an investigation on dice] (E. H. Thorne, Trans.). In Le opere di Galileo Galilei (Vol. 8, pp. 591-594). Barbera. Reprinted in F. N. David, Games, gods and gambling. Transcribed at the University of York History of Statistics pages. https://www.york.ac.uk/depts/maths/histstat/galileo.htmCited in Chapter 3 and 1400 to 1650. | University of York, Department of Mathematics, History of Statistics pages | read in full |
| S133 | Gauss, C. F. (1801). Disquisitiones arithmeticae. In commiss. apud Gerh. Fleischer, jun. Internet Archive, disquisitionesa00gaus (Smithsonian Libraries copy). https://archive.org/details/disquisitionesa00gausCited in Chapter 4 twice and 1800 to 1850. | Internet Archive (Smithsonian Libraries) | partial (I searched for and read the passages containing "determinans" and "determinantem", including the defining sentence) |
| S171 | Grassmann, H. (1844/1878). Die Ausdehnungslehre von 1844, oder Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, dargestellt und durch Anwendungen auf die ubrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erlautert (2nd, textually unchanged ed.). Verlag von Otto Wigand. Internet Archive, dieausdehnungsl04grasgoog. https://archive.org/details/dieausdehnungsl04grasgoogCited in Chapter 5 5 times and 1800 to 1850. | Internet Archive | partial (title page, Vorrede zur ersten Auflage opening, Vorrede zur zweiten Auflage opening, read through the OCR text file; the mathematical body was not...) |
| S077 | Graunt, J. (1662). Natural and political observations, mentioned in a following index, and made upon the bills of mortality. Printed by Tho. Roycroft, for John Martin, James Allestry, and Tho. Dicas. Internet Archive, 2356015R.nlm.nih.gov. https://archive.org/details/2356015R.nlm.nih.gov; Graunt, J. (1676). Natural and political observations made upon the bills of mortality (5th ed.), Chapter 1. Wikisource. https://en.wikisource.org/wiki/Natural_and_Political_Observations_Made_upon_the_Bills_of_Mortality_(Graunt_1676)/Chapter_1Cited in Chapter 3 twice, 1400 to 1650 3 times and 1650 to 1800. | Internet Archive, from the US National Library of Medicine, digitized 2019; and English Wikisource | partial (Chapter 1 of the 1676 text read in full; the 1662 scan opened at the item page only, not read as text) |
| S174 | Hamilton, W. R. (1844-1850/2000). On quaternions; or on a new system of imaginaries in algebra (D. R. Wilkins, Ed.). Originally published in 18 installments in The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd series), vols. xxv-xxxvi. School of Mathematics, Trinity College Dublin. https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/OnQuat.pdfCited in Chapter 5 8 times and 1800 to 1850 twice. | School of Mathematics, Trinity College Dublin, History of Mathematics archive | partial (the transcription's editorial note on the installments, and articles 18 and 19, which carry the coinages of "scalar", "vector", "tensor" and "versor") |
| S139 | Hamilton, W. R. (1865/1885). Letter to Archibald H. Hamilton, 5 August 1865. In R. P. Graves, Life of Sir William Rowan Hamilton (Vol. 2, ch. 28). Transcribed by D. R. Wilkins, School of Mathematics, Trinity College Dublin. https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.htmlCited in 1800 to 1850. | School of Mathematics, Trinity College Dublin | read in full |
| S175 | Hamilton, W. R. (1865/2000). Letter to Rev. Archibald H. Hamilton, 5 August 1865 [and an extract of a letter of 15 October 1858] (D. R. Wilkins, Ed.). Transcribed from R. P. Graves, Life of Sir William Rowan Hamilton, Vol. II, ch. XXVIII. School of Mathematics, Trinity College Dublin. https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.htmlCited in Chapter 5 6 times and 1800 to 1850 twice. | School of Mathematics, Trinity College Dublin, History of Mathematics archive | read in full |
| S270 | Hardy, G. H. (1908). Mendelian proportions in a mixed population. Science, New Series, 28(706), 49-50. http://www.esp.org/foundations/genetics/classical/hardy.pdfCited in 1880 to 1910 twice. | Electronic Scholarly Publishing Project (esp.org), Foundations of Classical Genetics series | read in full |
| S388 | Hardy, G. H. (1908). Mendelian proportions in a mixed population. Science, 28(706), 49-50. Reproduced by Electronic Scholarly Publishing. http://www.esp.org/foundations/genetics/classical/hardy.pdfCited in 1880 to 1910 twice. | esp.org (Electronic Scholarly Publishing) | read in full |
| S190 | Hestenes, M. R., & Stiefel, E. (1952). Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards, 49(6), 409-436 (Research Paper 2379). https://nvlpubs.nist.gov/nistpubs/jres/049/6/V49.N06.A08.pdfCited in Chapter 5 and 1935 to 1960. | NIST Virtual Library (nvlpubs.nist.gov) | partial (title, affiliations, sponsorship footnote, abstract, opening of the introduction and the termination statement; the body of the analysis was not read...) |
| S184 | Hilbert, D. (1904-1910/1912). Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen (Fortschritte der Mathematischen Wissenschaften in Monographien, Heft 3). Druck und Verlag von B. G. Teubner. Internet Archive, grundzugeallg00hilbrich. https://archive.org/details/grundzugeallg00hilbrichCited in Chapter 5 3 times, Chapter 9 3 times and 1880 to 1910. | Internet Archive | partial (title page, Vorwort opening, Inhaltsverzeichnis, and the statement of where and when the six Mitteilungen originally appeared; the mathematical body...) |
| S140 | Hill, L. S. (1929). Cryptography in an algebraic alphabet. The American Mathematical Monthly, 36(6), 306-312. https://doi.org/10.1080/00029890.1929.11986963Cited in Chapter 4 8 times and 1910 to 1935. | Universidad Autonoma Metropolitana course server (mirror) | read in full |
| S076 | Huygens, C. (1657/1714). Christiani Hugenii libellus de ratiociniis in ludo aleae. Or, the value of all chances in games of fortune... mathematically demonstrated (W. Browne, Trans.). Internet Archive, bim_eighteenth-century_christiani-hugenii-libel_huygens-christiaan-van_1714. https://archive.org/details/bim_eighteenth-century_christiani-hugenii-libel_huygens-christiaan-van_1714Cited in Chapter 3 twice and 1650 to 1800 twice. | Internet Archive (British Library "bim" eighteenth century collection); Universiteit Leiden | partial (Title page, dedication, postulate, Propositions I to III and the shape of the closing five problems read from the OCR; the Leiden page read for the...) |
| S138 | Jacobi, C. G. J. (1841/1896). Uber die Bildung und die Eigenschaften der Determinanten; De formatione et proprietatibus determinantium (P. Stackel, Ed.) [Ostwalds Klassiker der exakten Wissenschaften]. W. Engelmann. Internet Archive, ueberdiebildungu00jacouoft. https://archive.org/details/ueberdiebildungu00jacouoftCited in Chapter 4 3 times and 1800 to 1850. | Internet Archive (University of Toronto, Gerstein collection) | partial (I read the returned passages containing "1841" and "alternantibus", including Stackel's editorial note at scan p. 70) |
| S439 | Karinthy, F. (1929/2006). Chain-links (A. Makkai, Trans.). In Everything is different (original collection, 1929). PDF transcription of the Makkai translation. http://vadeker.net/articles/Karinthy-Chain-Links_1929.pdfCited in Chapter 7 4 times and 1910 to 1935. | vadeker.net (transcription host); the translation is the one reprinted in the network science literature | read in full |
| S066 | Kolmogorov, A. N. (1933/1950). Foundations of the theory of probability (N. Morrison, Trans.). Chelsea Publishing Company. Internet Archive, foundationsofthe00kolm. https://archive.org/details/foundationsofthe00kolmCited in 1910 to 1935 and 1935 to 1960. | Internet Archive | partial (Title page, preface and Chapter I with the axioms were read from the OCR text; the archive stream truncated before the later chapters) |
| S090 | Kolmogorov, A. N. (1933/1950). Foundations of the theory of probability (N. Morrison, Trans. and Ed.). Chelsea Publishing Company. (Original work published 1933 as Grundbegriffe der Wahrscheinlichkeitsrechnung.) Internet Archive, foundationsofthe00kolm. https://archive.org/details/foundationsofthe00kolmCited in Chapter 3 3 times, 1910 to 1935 and 1935 to 1960. | Internet Archive | partial (Title page, Editor's Note, Preface, Chapter I section 1 (the axioms) and Chapter I section 4 (conditional probability, total probability, Bayes) were...) |
| S087 | Laplace, P. S. de. (1812/1820). Theorie analytique des probabilites (3rd ed.). Mme Ve Courcier. Internet Archive, theorieanaldepro00laplrich. https://archive.org/details/theorieanaldepro00laplrichCited in 1800 to 1850. | Internet Archive, California Digital Library, from a University of California Libraries copy | not opened |
| S086 | Laplace, P. S. de. (1814/1951). A philosophical essay on probabilities (F. W. Truscott & F. L. Emory, Trans.). Dover Publications. (Translated from the sixth French edition; original Essai philosophique sur les probabilites first published 1814.) Internet Archive, philosophicaless00lapl. https://archive.org/details/philosophicaless00laplCited in Chapter 3 3 times and 1800 to 1850. | Internet Archive, from the University of Florida, George A. Smathers Libraries | partial (Title page, the demon passage, the First and Second Principles, the Sixth Principle, and the rule of succession passage were read from the OCR) |
| S008 | Leibniz, G. W. (1666). Dissertatio de arte combinatoria, in qua ex arithmeticae fundamentis complicationum ac transpositionum doctrina nouis praeceptis exstruitur... Praefixa est synopsis totius tractatus, & additamenti loco demonstratio existentiae Dei, ad mathematicam certitudinem exacta. Apud Joh. Simon. Fickium et Joh. Polycarp. Seuboldum. [Digitised copy, Biblioteca Nazionale Centrale di Firenze; Internet Archive identifier ita-bnc-mag-00000844-001, ark:/13960/t5m933z56]. https://archive.org/details/ita-bnc-mag-00000844-001Cited in 1650 to 1800. | Internet Archive, from the Biblioteca Nazionale Centrale di Firenze (Early European Books) | partial (Item metadata read in full; the OCR text layer read from the beginning through the Synopsis and the early Problemata) |
| S132 | Maclaurin, C. (1748/1796). A treatise of algebra, in three parts (with an appendix by J. Lawson). Internet Archive, bub_gb_8t82AAAAMAAJ (digitised from Google Books). https://archive.org/details/bub_gb_8t82AAAAMAAJCited in Chapter 4 and 1650 to 1800 twice. | Internet Archive | partial (I located and read the section headings and returned passages on pages 79, 94, 97 to 102, and 236 of the scan) |
| S007 | MacMahon, P. A. (1915). Combinatory analysis (Vol. 1). Cambridge University Press. [Digitised copy; Internet Archive identifier combinatoryanal01macmuoft, ark:/13960/t27945q0m]. https://archive.org/details/combinatoryanal01macmuoftCited in 1910 to 1935. | Internet Archive | partial (Item metadata, the Introduction, and the full table of contents read from the OCR text layer. The body chapters were not read) |
| S009 | Mahavira. (c. 850/1912). The Ganita-sara-sangraha of Mahaviracarya, with English translation and notes (M. Rangacharya, Ed. and Trans.). [Page-level scan and OCR hosted by Wisdom Library]. https://www.wisdomlib.org/hinduism/book/ganita-sara-sangraha-by-mahavira-acharyaCited in Chapter 2 twice and 500 to 1400. | Wisdom Library (wisdomlib.org) page-level OCR of the 1912 edition | partial (The combinatorics passage in Chapter VI (Mixed Problems), rules and examples 217 to 221, read in full on scan pages 338 and 339) |
| S214 | Markov, A. A. (1913/2006). An example of statistical investigation of the text Eugene Onegin concerning the connection of samples in chains (G. Custance & D. Link, Trans.). Science in Context, 19(4), 591-600. (Original work published 1913). https://alpha60.de/research/markov/DavidLink_AnExampleOfStatistical_MarkovTrans_2007.pdfCited in Chapter 6 5 times and 1910 to 1935. | David Link's research site, alpha60.de | read in full |
| S332 | Martineau, H., & Nightingale, F. (1859). Diagram of the causes of mortality in the army in the East [Chart]. Smith Elder & Co. David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries, via the Internet Archive. https://archive.org/details/dr_diagram-of-the-causes-of-mortality-in-the-army-in-the-east-10563002Not cited in the story. It backs a register entry. | David Rumsey Map Collection, Stanford Libraries, hosted on the Internet Archive | partial (item metadata read in full, including the rights statement, description, and complete file list; the image itself was not viewed) |
| S260 | Menabrea, L. F. (1842/1843). Sketch of the Analytical Engine invented by Charles Babbage (A. A. L. [Ada Augusta, Countess of Lovelace], Trans., with notes by the translator). Scientific Memoirs, 3, 666-731. (Original work published in Bibliotheque Universelle de Geneve, October 1842, No. 82.) Transcription retrieved August 20, 2026, from https://www.fourmilab.ch/babbage/sketch.htmlCited in Chapter 8 7 times and 1800 to 1850 twice. | Fourmilab (fourmilab.ch) | partial (The transcription was returned only as far as Note C and the surrounding translator's notes across two separate fetches) |
| S226 | Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087-1092. https://doi.org/10.1063/1.1699114Cited in Chapter 6 6 times and 1935 to 1960. | Washington University in St Louis, bayes.wustl.edu | read in full |
| S081 | Moivre, A. de. (1718). The doctrine of chances: Or, a method of calculating the probability of events in play. W. Pearson. Internet Archive, bim_eighteenth-century_the-doctrine-of-chances_moivre-abraham-de_1718. https://archive.org/details/bim_eighteenth-century_the-doctrine-of-chances_moivre-abraham-de_1718; Moivre, A. de. (1756). The doctrine of chances: Or, a method of calculating the probabilities of events in play (3rd ed.). A. Millar. Internet Archive, doctrineofchance00moiv. https://archive.org/details/doctrineofchance00moivCited in Chapter 3 3 times and 1650 to 1800 twice. | Internet Archive; the 1756 copy from University of California Libraries (SRLF_UCLA) | partial (both scans: Title pages, dedications, prefaces and the opening of the Introduction were read) |
| S218 | Nekrasov, P. A. (2004). Theory of probability: Central limit theorem; method of least squares; reactionary views; teaching of probability theory; further developments (O. Sheynin, Trans. & Ed.). Berlin: Oscar Sheynin. https://www.probabilityandfinance.com/sheynin/004_Nekrasov.pdfCited in Chapter 6 8 times, 1880 to 1910 twice and 1910 to 1935 3 times. | Probability and Finance (Shafer and Vovk's site), Sheynin collection | read in full |
| S385 | Nekrasov, P. A. (2004). Theory of probability: Central limit theorem; method of least squares; reactionary views; teaching of probability theory; further developments (O. Sheynin, Trans.). Berlin. https://www.probabilityandfinance.com/sheynin/004_Nekrasov.pdfNot cited in the story. It backs a register entry. | probabilityandfinance.com | partial (The foreword and the opening technical material were returned; the extraction did not reach the later sections, and the fetch tool truncates long...) |
| S001 | Pascal, B. (1665). Traite du triangle arithmetique avec quelques autres petits traitez sur la mesme matiere. G. Desprez. [Digitised copy, University of Lausanne, via Google Books; Internet Archive identifier bub_gb_UqgUAAAAQAAJ, ark:/13960/t3rv3m35d]. https://archive.org/details/bub_gb_UqgUAAAAQAAJCited in Chapter 2 twice and 1650 to 1800. | Internet Archive (European Libraries collection; scanning copy from the University of Lausanne) | read in full |
| S050 | Peano, G. (1889). Arithmetices principia, nova methodo exposita. Fratres Bocca. Internet Archive, arithmeticespri00peangoog. https://archive.org/details/arithmeticespri00peangoogCited in Chapter 1 twice and 1880 to 1910 twice. | Internet Archive (Google Books scan) | read in full |
| S239 | Rabiner, L. R. (2015, January 12). First-hand: The hidden Markov model. Engineering and Technology History Wiki, IEEE History Center. https://ethw.org/First-Hand:The_Hidden_Markov_ModelCited in 1960 to now twice. | Engineering and Technology History Wiki (IEEE) | read in full |
| S225 | Shannon, C. E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27, 379-423, 623-656. https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdfCited in Chapter 6 7 times and 1935 to 1960. | Harvard University, people.math.harvard.edu | read in full |
| S330 | Snow, J. (1855). On the mode of communication of cholera (2nd ed.). John Churchill. Internet Archive, b28985266, digitised by the Wellcome Library. https://archive.org/details/b28985266Not cited in the story. It backs a register entry. | Internet Archive, contributed and sponsored by the Wellcome Library | partial (rights metadata read in full; text located through the item's search-inside OCR index; scan leaves 5 and 13 identified, not viewed as images) |
| S333 | Snow, J. (1855/1936). Map 1. C.F. Chefins, Lith. London. (to accompany) On the mode of communication of cholera [Map]. Commonwealth Fund. David Rumsey Map Collection, David Rumsey Map Center, Stanford Libraries, via the Internet Archive. https://archive.org/details/dr_map-1-cf-chefins-lith-london-to-accompany-on-the-mode-of-communicat-13187008Not cited in the story. It backs a register entry. | David Rumsey Map Collection, Stanford Libraries, hosted on the Internet Archive | partial (item metadata read in full, including the rights statement, description, and file list; the image itself was not viewed) |
| S307 | Sylvester, J. J. (1850/1904). Additions to the articles "On a new class of theorems" and "On Pascal's theorem." In The collected mathematical papers of James Joseph Sylvester (Vol. 1, pp. 145-151). Cambridge University Press. Internet Archive, collectedmathem01sylvrich. https://archive.org/details/collectedmathem01sylvrichCited in Chapter 9 3 times and 1850 to 1880. | Internet Archive | partial (The passage was located and read in this scan's own OCR through the Internet Archive search-inside endpoint, which returned the matched sentence and...) |
| S135 | Sylvester, J. J. (1904). The collected mathematical papers of James Joseph Sylvester (Vol. 1, 1837-1853; H. F. Baker, Ed.). Cambridge University Press. Internet Archive, collectedmathem01sylvrich. https://archive.org/details/collectedmathem01sylvrichCited in Chapter 4 8 times, Chapter 9 3 times and 1850 to 1880 3 times. | Internet Archive | partial (I read the table of contents in full, and the passages returned by full-text searches on "Matrix", "womb", and "homaloidal", which include the two...) |
| S191 | Taussky, O. (1988). How I became a torchbearer for matrix theory. The American Mathematical Monthly, 95(9), 801-812. https://doi.org/10.1080/00029890.1988.11972101Cited in Chapter 5 7 times, Chapter 8 5 times, 1910 to 1935 and 1935 to 1960 4 times. | copy hosted on Andres Caicedo's mathematics blog ( | read in full |
| S091 | Tchebychef, P. L. (1867/n.d.). Des valeurs moyennes [On mean values] (R. J. Pulskamp, Trans.). Journal de mathematiques pures et appliquees (Liouville), 2nd series, 12, 177-184. https://probabilityandfinance.com/pulskamp/Chebyshev/1866.pdfCited in Chapter 3 twice and 1850 to 1880. | probabilityandfinance.com, Pulskamp translations | read in full |
| S261 | Turing, A. M. (c. 1941-1942/2015). The applications of probability to cryptography (I. Taylor, Ed.). The National Archives, HW 25/37. arXiv:1505.04714. https://arxiv.org/abs/1505.04714Cited in 1935 to 1960. | The National Archives, Kew (original, HW 25/37); arXiv (typeset edition) | read in full |
| S010 | Varahamihira. (6th c./1884). Brihat Samhita (N. Chidambaram Iyer, Trans.), Chapter 77, "Preparation of Perfumes (gandhayukti)". [Text hosted by Wisdom Library]. https://www.wisdomlib.org/hinduism/book/brihat-samhita/d/doc229339.htmlCited in Chapter 2 twice, 500 to 1400 and 1880 to 1910. | Wisdom Library | read in full |
| S048 | Venn, J. (1881). Symbolic logic. Macmillan and Co. Internet Archive, symboliclogic00venngoog. https://archive.org/details/symboliclogic00venngoogCited in Chapter 1 and 1880 to 1910. | Internet Archive (Google Books scan) | partial (Title page, preface and introduction to about p. xxx, plus the table of contents including the description of Chapter V, were read from the OCR text...) |
| S067 | von Mises, R. (1928/1957). Probability, statistics and truth (2nd rev. English ed., H. Geiringer, Ed.). George Allen and Unwin; The Macmillan Company. Internet Archive,.ernet.dli.2015.189506. https://archive.org/details/in.ernet.dli.2015.189506Not cited in the story. It backs a register entry. | Internet Archive (Digital Library of India collection) | partial (Title page, front matter and the first two lectures through about p. 20 were read from the OCR text; the archive stream truncated after that) |
| S188 | von Neumann, J., & Goldstine, H. H. (1947). Numerical inverting of matrices of high order. Bulletin of the American Mathematical Society, 53(11), 1021-1099. https://www.ams.org/journals/bull/1947-53-11/S0002-9904-1947-08909-6/S0002-9904-1947-08909-6.pdfCited in Chapter 5 6 times and 1935 to 1960. | American Mathematical Society, open Bulletin archive | partial (title, front matter, preface, statement of purpose, and the headline error bounds; the 79 pages of analysis were not read line by line) |
| S441 | Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of small-world networks. Nature, 393(6684), 440-442. Copy hosted for teaching at Stanford. http://snap.stanford.edu/class/cs224w-readings/watts98smallworld.pdfCited in Chapter 7 5 times and 1960 to now. | Stanford SNAP course reading list (publisher of record: Nature) | read in full |
| S141 | Weisner, L., & Hill, L. S. (1932). Message protector (U.S. Patent No. 1,845,947). United States Patent and Trademark Office. https://patents.google.com/patent/US1845947Cited in Chapter 4 4 times and 1910 to 1935 twice. | Google Patents (mirroring the USPTO record) | read in full |
Tier 2: peer reviewed articles and academic press books
78 sources.
| ID | Reference (APA 7) | Repository | Access |
|---|---|---|---|
| S015 | African Mathematical Union Commission on the History of Mathematics in Africa. (n.d.). AMUCHMA Newsletter, 15. Hosted by the Mathematicians of the African Diaspora site, University at Buffalo. https://www.math.buffalo.edu/mad/AMU/amuchmapdf/amuchma15.pdfCited in Chapter 2 twice and 500 to 1400. | University at Buffalo, Mathematicians of the African Diaspora archive | read in full |
| S144 | Althoen, S. C., & McLaughlin, R. (1987). Gauss-Jordan reduction: A brief history. The American Mathematical Monthly, 94(2), 130-142. https://doi.org/10.1080/00029890.1987.12000605Cited in Chapter 4 4 times, 1880 to 1910 twice and 1910 to 1935. | Academia.edu (uploaded copy) | read in full |
| S003 | Bajri, S., Hannah, J., & Montelle, C. (2015). Revisiting Al-Samaw'al's table of binomial coefficients: Greek inspiration, diagrammatic reasoning and mathematical induction. Archive for History of Exact Sciences, 69(6), 537-576. https://doi.org/10.1007/s00407-015-0156-xCited in Chapter 2 3 times and 500 to 1400 3 times. | University of Canterbury Research Repository (ir.canterbury.ac.nz) | read in full |
| S275 | Barany, M. J. (2020). Impersonation and personification in mid-twentieth century mathematics. History of Science, 58(4), 417-436. https://doi.org/10.1177/0073275320924571Cited in Chapter 8 6 times, 1910 to 1935 twice and 1935 to 1960. | SAGE Journals | read in full |
| S212 | Basharin, G. P., Langville, A. N., & Naumov, V. A. (2004). The life and work of A. A. Markov. Linear Algebra and Its Applications, 386, 3-26. https://doi.org/10.1016/j.laa.2003.12.041Cited in Chapter 6 8 times, 1880 to 1910 twice and 1910 to 1935 3 times. | University of Florida, College of Engineering | read in full |
| S084 | Bellhouse, D. R. (2004). The Reverend Thomas Bayes FRS: A biography to celebrate the tercentenary of his birth. Statistical Science, 19(1), 3-43. Preprint version read: Bellhouse, D. R. (2001). The Reverend Thomas Bayes FRS: a biography to celebrate the tercentenary of his birth [Preprint]. Department of Statistical and Actuarial Sciences, University of Western Ontario. Copy hosted by Georgia Institute of Technology. https://www2.isye.gatech.edu/isyebayes/bank/bayesbiog.pdfCited in Chapter 3 twice and 1650 to 1800 5 times. | Georgia Institute of Technology, ISyE Bayesian statistics pages | read in full |
| S082 | Bellhouse, D. R., & Genest, C. (2007). Maty's biography of Abraham De Moivre, translated, annotated and augmented. Statistical Science, 22(1), 109-136. https://doi.org/10.1214/088342306000000268 Author's arXiv version: https://arxiv.org/abs/0708.3965Cited in Chapter 3 3 times and 1650 to 1800 8 times. | arXiv | read in full |
| S051 | Bennett, D. (2015). Origins of the Venn diagram. In M. Zack & E. Landry (Eds.), Research in history and philosophy of mathematics (pp. 105-119). Springer International Publishing. https://logic-teaching.github.io/pred/texts/Bennett%202015%20-%20Origins%20of%20the%20Venn%20Diagram.pdfCited in Chapter 1, 1400 to 1650, 1650 to 1800 6 times and 1880 to 1910. | logic-teaching.github.io course materials mirror | read in full |
| S251 | Bickel, P. J. (2020). David Blackwell, 1919-2010: An explorer in mathematics and statistics. Proceedings of the National Academy of Sciences, 117(47), 29391-29393. https://doi.org/10.1073/pnas.2020164117Cited in Chapter 8 8 times and 1960 to now. | UC Berkeley Department of Statistics | read in full |
| S073 | Bowman, N. (n.d.). Reading in context: The reception of Gerolamo Cardano's Liber de Ludo Aleae [Student research paper, Department of Mathematics, Western Carolina University, supervised by S. Despeaux]. Hosted by the Mathematical Association of America. https://old.maa.org/sites/default/files/pdf/upload_library/46/Bowman_Cardano_ed.pdfCited in Chapter 3 twice and 1400 to 1650. | Mathematical Association of America | read in full |
| S269 | Brody, H., Rip, M. R., Vinten-Johansen, P., Paneth, N., & Rachman, S. (2000). Map-making and myth-making in Broad Street: The London cholera epidemic, 1854. The Lancet, 356(9223), 64-68. https://doi.org/10.1016/S0140-6736(00)02442-9Cited in 1850 to 1880 6 times and 1960 to now. | John Snow Archive and Research Companion, Matrix, Michigan State University | read in full |
| S232 | Bru, B. (2009). Review of L'equation de Kolmogoroff: Vie et mort de Wolfgang Doeblin, un genie dans la tourmente nazie by Marc Petit (O. Pekonen, Ed.). The Mathematical Intelligencer, 31(2), 61-65. https://www.scaillet.ch/cours_processus/references/Doeblin_Petit.pdfCited in Chapter 6 10 times, 1910 to 1935 3 times, 1935 to 1960 twice and 1960 to now. | scaillet.ch | read in full |
| S197 | Bryan, K., & Leise, T. (2006). The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3), 569-581. https://doi.org/10.1137/050623280Cited in Chapter 5 twice. | University of Arizona (mirror); Rose-Hulman Institute of Technology (author repository) | read in full |
| S300 | Cajori, F. (1928). A history of mathematical notations: Vol. 1. Notations in elementary mathematics. The Open Court Publishing Company. Internet Archive, historyofmathema031756mbp. https://archive.org/details/historyofmathema031756mbpNot cited in the story. It backs a register entry. | Internet Archive | partial (The OCR text stream truncates at about paragraph 33 in this environment, so continuous reading was impossible) |
| S301 | Cajori, F. (1929). A history of mathematical notations: Vol. 2. Notations mainly in higher mathematics. The Open Court Publishing Company. Internet Archive, b29980343_0002 (Wellcome Collection scan). https://archive.org/details/b29980343_0002Cited in Chapter 9 17 times, 1650 to 1800 twice, 1800 to 1850 8 times and 1850 to 1880. | Internet Archive (Wellcome Collection copy) | partial (continuous reading of the OCR stream truncates in this environment) |
| S274 | Carvalho, H. T. de. (2019). An introduction to Markov chains in music composition and analysis. Journal MusMat, 3(2), 18-43. https://musmat.org/wp-content/uploads/2019/12/06-Carvalho.pdfCited in 1935 to 1960. | MusMat, Federal University of Rio de Janeiro | read in full |
| S177 | Crowe, M. J. (2002). A history of vector analysis [Lecture]. University of Louisville, Autumn Term 2002. https://worrydream.com/refs/Crowe_2002_-_History_Of_Vector_Analysis.pdf (based on Crowe, M. J. (1967). A history of vector analysis: The evolution of the idea of a vectorial system. University of Notre Dame Press.)Cited in Chapter 5 21 times, 1850 to 1880 6 times, 1880 to 1910 5 times and 1910 to 1935. | Bret Victor's worrydream.com reference archive (the only reachable copy from this environment) | read in full |
| S053 | Dauben, J. W. (1993). Georg Cantor and the battle for transfinite set theory. In Proceedings of the 9th ACMS Conference. Association of Christians in the Mathematical Sciences. https://pillars.taylor.edu/acms-1993/8/Cited in Chapter 1 3 times, 1850 to 1880 3 times and 1880 to 1910 3 times. | Taylor University Pillars institutional repository; the copy read was a mirrored PDF | read in full |
| S229 | Diaconis, P. (2009). The Markov chain Monte Carlo revolution. Bulletin of the American Mathematical Society, 46(2), 179-205. https://doi.org/10.1090/S0273-0979-08-01238-XNot cited in the story. It backs a register entry. | American Mathematical Society | read in full |
| S189 | Dopico, F. M. (2013). Alan Turing and the origins of modern Gaussian elimination. Arbor, 189(764), a084. https://doi.org/10.3989/arbor.2013.764n6007Cited in Chapter 5 5 times and 1935 to 1960 4 times. | Arbor, Consejo Superior de Investigaciones Cientificas (CSIC), fully open access | read in full |
| S173 | Dorier, J.-L. (1995). A general outline of the genesis of vector space theory. Historia Mathematica, 22(3), 227-261. https://doi.org/10.1006/hmat.1995.1024 (open copy at the University of Geneva repository, https://archive-ouverte.unige.ch/unige:16642)Cited in Chapter 5 9 times, 1800 to 1850 and 1910 to 1935 twice. | Archive ouverte, University of Geneva | read in full |
| S172 | Dorier, J.-L. (2000). Originalite et posterite: L'Ausdehnungslehre de Hermann Gunther Grassmann (1844). Philosophia Scientiae, 4(1), 3-45. https://www.numdam.org/item/PHSC_2000__4_1_3_0/Cited in Chapter 5 7 times, 1800 to 1850 and 1850 to 1880. | Numdam (Numerisation de documents anciens mathematiques), fully open | read in full |
| S263 | Erdelyi, T., & Vertesi, P. (1998). In memoriam: Paul Erdos (1913-1996). Journal of Approximation Theory, 94(1), 1-41. https://history-of-approximation-theory.com/fpapers/erdos.pdfCited in Chapter 8 6 times and 1960 to now. | history-of-approximation-theory.com | read in full |
| S279 | Feferman, S. (1994). Julia Bowman Robinson, 1919-1985. In Biographical memoirs (Vol. 63). National Academy Press. https://www.nasonline.org/wp-content/uploads/2024/06/robinson-julia.pdfCited in 1910 to 1935 and 1960 to now 3 times. | National Academy of Sciences (nasonline.org) | read in full |
| S389 | Floros, G. D. (2018). Gambling disorder in adolescents: Prevalence, new developments, and treatment challenges. Adolescent Health, Medicine and Therapeutics, 9, 43-51. https://doi.org/10.2147/AHMT.S135423Cited in 1960 to now. | PubMed Central | read in full |
| S267 | Friendly, M. (2008). The Golden Age of statistical graphics. Statistical Science, 23(4), 502-535. https://doi.org/10.1214/08-STS268 (arXiv:0906.3979)Cited in 1800 to 1850, 1850 to 1880 and 1960 to now. | arXiv | read in full |
| S023 | Garcia, P. (2006). The life and work of Major Percy Alexander MacMahon [Doctoral thesis, The Open University]. Open Research Online. https://doi.org/10.21954/ou.ro.0000e97eCited in 1850 to 1880. | Open Research Online (The Open University); copy also on arXiv as 1607.01321 per search results, and mirrored... | read in full |
| S193 | Golub, G. H., & Uhlig, F. (2009). The QR algorithm: 50 years later, its genesis by John Francis and Vera Kublanovskaya, and subsequent developments. IMA Journal of Numerical Analysis, 29(3), 467-485. https://doi.org/10.1093/imanum/drp012Cited in Chapter 5 11 times, Chapter 8 5 times, 1910 to 1935, 1935 to 1960 twice and 1960 to now 3 times. | UNICAMP course mirror; canonical version at Oxford Academic | read in full |
| S427 | Gonthier, G. (2008). Formal proof: The four-color theorem. Notices of the American Mathematical Society, 55(11), 1382-1393. https://www.ams.org/notices/200811/tx081101382p.pdfCited in Chapter 7 4 times, 1850 to 1880 and 1960 to now 3 times. | American Mathematical Society | read in full |
| S230 | Gorban, A. N. (2014). Detailed balance and complex balance: Modern history of 130 year old laws [Conference presentation paper]. CRNT Workshop, University of Portsmouth. https://reaction-networks.net/crnt-portsmouth-2014/downloads/gorban-detailed_balance_and_complex_balance-modern_history_of_130_year_old_laws.pdfCited in Chapter 6 4 times, 1850 to 1880, 1880 to 1910 twice and 1910 to 1935 twice. | reaction-networks.net (Chemical Reaction Network Theory workshop archive) | read in full |
| S093 | Gorroochurn, P. (2011). Errors of probability in historical context. The American Statistician, 65(4), 246-253. Author copy hosted by Columbia University. http://www.columbia.edu/~pg2113/index_files/Gorroochurn-Errors%20of%20Probability.pdfCited in Chapter 3 and 1400 to 1650. | Columbia University (author page) | read in full |
| S072 | Gorroochurn, P. (2012). Some laws and problems of classical probability and how Cardano anticipated them. Chance, 25(4), 13-20. Author copy hosted by Columbia University. https://www.columbia.edu/~pg2113/index_files/Gorroochurn-Some%20Laws.pdfCited in Chapter 3 twice, 500 to 1400, 1400 to 1650 4 times and 1650 to 1800 twice. | Columbia University (author page) | read in full |
| S120 | Grcar, J. F. (2011). Mathematicians of Gaussian elimination. Notices of the American Mathematical Society, 58(6), 782-792. https://www.ams.org/notices/201106/rtx110600782p.pdfCited in Chapter 4 12 times, Before 500, 1650 to 1800, 1800 to 1850 twice, 1880 to 1910 and 1935 to 1960 3 times. | American Mathematical Society | read in full |
| S121 | Grcar, J. F. (2011). How ordinary elimination became Gaussian elimination. Historia Mathematica, 38(2), 163-218. arXiv preprint arXiv:0907.2397. https://arxiv.org/abs/0907.2397Cited in Chapter 4 4 times, 1650 to 1800 and 1800 to 1850. | arXiv (Cornell) | partial (abstract read verbatim; the PDF was read through structured extraction and I could reach the abstract, section 1 and 2 material, the footnote on...) |
| S256 | Gubernatis, J. E. (2005). Marshall Rosenbluth and the Metropolis algorithm. Physics of Plasmas, 12(5), 057303. https://doi.org/10.1063/1.1887186Cited in Chapter 6 3 times, Chapter 8 5 times, 1935 to 1960 and 1960 to now. | Universita degli Studi di Cagliari (dsf.unica.it) | read in full |
| S125 | Guo, S. (2013). The Nine Chapters on the Mathematical Procedures and Liu Hui's mathematical theory. In E. Knobloch, H. Komatsu, & D. Liu (Eds.), Seki, founder of modern mathematics in Japan: A commemoration on his tercentenary (Springer Proceedings in Mathematics & Statistics, Vol. 39, pp. 63-90). Springer. https://legacy-www.math.harvard.edu/~knill/teaching/math22a2018/exhibits/ninechapters/Guo_Shuchun_2013.pdfCited in Chapter 4 5 times and Before 500. | Harvard Mathematics Department course page (mirror) | partial (I reached the fangcheng and zhengfu passages (pp. 67, 69) and the assessment of Liu Hui (p. 63)) |
| S137 | Hawkins, T. (1975). The theory of matrices in the 19th century. In Proceedings of the International Congress of Mathematicians, Vancouver, 1974 (Vol. 2, pp. 561-570). Canadian Mathematical Congress. https://silo.tips/download/the-theory-of-matrices-in-the-19th-century-thomas-hawkinsCited in Chapter 4 3 times, 1800 to 1850, 1850 to 1880 twice and 1880 to 1910. | third-party document host (silo.tips); the official proceedings are published by the Canadian Mathematical... | partial (I read the sections on Cayley's memoir and its reception, on Eisenstein and Gauss, on Weierstrass and Kronecker, and on Frobenius) |
| S210 | Hayes, B. (2013). First links in the Markov chain. American Scientist, 101(2), 92-97. https://www.americanscientist.org/sites/americanscientist.org/files/201321152149545-2013-03Hayes.pdfCited in Chapter 6 3 times, 1880 to 1910 and 1910 to 1935. | americanscientist.org (publisher's site) | read in full |
| S271 | Hayes, B. (2013). First links in the Markov chain. American Scientist, 101(2), 92-97. https://www.americanscientist.org/sites/americanscientist.org/files/201321152149545-2013-03Hayes.pdfCited in 1850 to 1880, 1880 to 1910 and 1910 to 1935 twice. | American Scientist / Sigma Xi | read in full |
| S178 | Higham, N. J. (2007). Cayley, Sylvester, and early matrix theory (MIMS EPrint 2007.119). Manchester Institute for Mathematical Sciences, University of Manchester. https://eprints.maths.manchester.ac.uk/863/1/cay_syl_07.pdf (Published in Linear Algebra and its Applications, 2008.)Cited in Chapter 5 4 times, 1850 to 1880 twice and 1880 to 1910. | Manchester eScholar / MIMS EPrints | read in full |
| S227 | Hitchcock, D. B. (2003). A history of the Metropolis-Hastings algorithm. The American Statistician, 57(4), 254-257. https://doi.org/10.1198/0003130032413Cited in Chapter 6 4 times, 1935 to 1960 and 1960 to now 3 times. | tommasorigon.github.io | read in full |
| S258 | Hollings, C., Martin, U., & Rice, A. (2017). The Lovelace-De Morgan mathematical correspondence: A critical re-appraisal. Historia Mathematica, 44(3), 202-231. https://doi.org/10.1016/j.hm.2017.04.001Cited in Chapter 8 6 times and 1800 to 1850. | University of Edinburgh Research Explorer (Pure) | read in full |
| S259 | Hollings, C., Martin, U., & Rice, A. (2017). The early mathematical education of Ada Lovelace. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 32(3), 221-234. https://doi.org/10.1080/17498430.2017.1325297Cited in Chapter 8 3 times and 1800 to 1850. | University of St Andrews / British Society for the History of Mathematics | read in full |
| S131 | Joffredo, T. (2019). Une analyse genetique de l'Introduction a l'analyse des lignes courbes algebriques de Gabriel Cramer (1750). Revue d'histoire des mathematiques, 25(2), 235-289. https://www.numdam.org/item/10.24033/rhm.226.pdfCited in Chapter 4 5 times and 1650 to 1800. | Numdam (Societe Mathematique de France) | partial (I read the structural description of Cramer's book, the passage locating the rule, and the Cramer-paradox section) |
| S221 | Kac, M. (1947). Random walk and the theory of Brownian motion. The American Mathematical Monthly, 54(7, Part 1), 369-391. https://doi.org/10.1080/00029890.1947.11990189Cited in Chapter 6 7 times, 1880 to 1910 and 1935 to 1960 twice. | Universidad de Buenos Aires (dm.uba.ar) | read in full |
| S182 | Kennedy, H. C. (2002). Peano: Life and works of Giuseppe Peano (Definitive ed.). Peremptory Publications. https://w2.cs.uni-saarland.de/op/f/logics/Kennedy_Peano.pdfCited in Chapter 5 10 times and 1880 to 1910. | Saarland University (w2.cs.uni-saarland.de) | partial (the sections on the 1888 Calcolo geometrico and the definition of a linear system; the rest of the biography was not read line by line) |
| S281 | Klein, L. F., et al. (n.d.). Between data and truth: W. E. B. Du Bois's "data portraits". In Data by design: An interactive history of data visualization. Digital Humanities Lab, Emory University. Retrieved August 20, 2026, from https://dataxdesign.io/chapters/duboisCited in Chapter 8 7 times and 1880 to 1910. | Digital Humanities Lab, Emory University (dataxdesign.io) | read in full |
| S440 | Kleinfeld, J. S. (2002, March). Six degrees: Urban myth? Psychology Today. https://www.psychologytoday.com/us/articles/200203/six-degrees-urban-mythCited in Chapter 7 5 times and 1960 to now. | Psychology Today | read in full |
| S283 | Kohli, M. C. (2001). Leontief and the U.S. Bureau of Labor Statistics, 1941-54: Developing a framework for measurement. History of Political Economy, 33(Suppl. 1), 190-212. (Read as the BLS Office of Survey Methods Research working paper, 2002, https://www.bls.gov/osmr/research-papers/2002/pdf/st020190.pdf)Cited in 1910 to 1935 and 1935 to 1960 twice. | U.S. Bureau of Labor Statistics | read in full |
| S265 | Laird, N. M. (1989). A conversation with F. N. David. Statistical Science, 4(3), 235-246. https://www.dcscience.net/conversation-with-Florence-Nightingale-David.pdfCited in Chapter 8 8 times, 1880 to 1910, 1910 to 1935 and 1960 to now. | dcscience.net (David Colquhoun's site); the journal of record is Statistical Science, Institute of... | read in full |
| S126 | Lam, L. Y. (1987). The Chinese rod numeral legacy and its impact on mathematics [Presidential address, Singapore Mathematical Society, 20 March 1987]. Mathematical Medley, 15(2). https://sms.math.nus.edu.sg/smsmedley/Vol-15-2/The%20Chinese%20rod%20numeral%20legacy%20and%20its%20impact%20on%20mathematics(Lam%20Lay%20Yong).pdfCited in Chapter 4 4 times and Before 500 twice. | Singapore Mathematical Society / NUS | partial (The extraction returned the opening sections on the rod system and the dating of the Nine Chapters) |
| S203 | Langendoen, D. T. (1966). A restriction on Grassmann's law in Greek. Language, 42(1), 7-9. https://dingo.sbs.arizona.edu/~langendoen/GrassmannsLaw.pdfCited in Chapter 5 and 1850 to 1880. | University of Arizona (dingo.sbs.arizona.edu) | read in full |
| S235 | Levin, D. A., & Peres, Y. (2017). Markov chains and mixing times (2nd ed.; with contributions by E. L. Wilmer). American Mathematical Society. https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdfNot cited in the story. It backs a register entry. | University of Oregon, pages.uoregon.edu/dlevin | read in full |
| S213 | Link, D. (2006). Traces of the mouth: Andrei Andreyevich Markov's mathematization of writing. History of Science, 44(145), 321-348. https://doi.org/10.1177/007327530604400302Cited in Chapter 6 7 times and 1910 to 1935. | California Institute of Technology (its.caltech.edu) | read in full |
| S215 | Link, D. (2006). Chains to the West: Markov's theory of connected events and its transmission to Western Europe. Science in Context, 19(4), 561-589. https://doi.org/10.1017/S0269889706001062Cited in Chapter 6 5 times and 1910 to 1935 twice. | David Link's research site, alpha60.de | read in full |
| S192 | Luchins, E. H., & McLoughlin, M. A. (1996). In memoriam: Olga Taussky-Todd. Notices of the American Mathematical Society, 43(8), 838-847. https://www.ams.org/notices/199608/taussky.pdfCited in Chapter 5 6 times, Chapter 8 5 times, 1880 to 1910, 1910 to 1935, 1935 to 1960 twice and 1960 to now twice. | American Mathematical Society, open Notices archive | read in full |
| S185 | MacCluer, C. R. (2000). The many proofs and applications of Perron's theorem. SIAM Review, 42(3), 487-498. https://doi.org/10.1137/S0036144599359449 (copy consulted: https://sites.oxy.edu/lengyel/m372/papers/SIAMREVIEW_APPL_OF_FROBENIUS_PERRON.pdf)Cited in Chapter 5 8 times, 1880 to 1910 twice and 1910 to 1935. | Occidental College course page (mirror). The authoritative version is at SIAM | read in full |
| S262 | MacKay, D. J. C. (2003). Information theory, inference, and learning algorithms (pp. 265-280). Cambridge University Press. https://www.inference.org.uk/mackay/itprnn/ps/265.280.pdfNot cited in the story. It backs a register entry. | Inference Group, University of Cambridge (inference.org.uk) | read in full |
| S240 | Makkuva, A. V., Bondaschi, M., Girish, A., Nagle, A., Jaggi, M., Kim, H., & Gastpar, M. (2024). Attention with Markov: A framework for principled analysis of transformers via Markov chains (arXiv:2402.04161). https://arxiv.org/abs/2402.04161Cited in Chapter 6. | arXiv | not opened |
| S079 | Mattmuller, M. (2014). The difficult birth of stochastics: Jacob Bernoulli's Ars Conjectandi (1713). Historia Mathematica, 41(2), 277-290. https://doi.org/10.1016/j.hm.2014.04.001 Full text hosted by the Bernoulli-Euler Zentrum, Universitat Basel. https://bez.unibas.ch/fileadmin/user_upload/bez/The-difficult-birth-of-stochastics-Jacob-Bernoulli-s-Ars-Conjectandi-1713-_2014_Historia-Mathematica.pdfCited in Chapter 3 3 times and 1650 to 1800 4 times. | Bernoulli-Euler Zentrum, University of Basel | read in full |
| S075 | Ore, O. (1960). Pascal and the invention of probability theory. The American Mathematical Monthly, 67(5), 409-419. Copy hosted by the University of Kentucky Department of Mathematics. https://www.ms.uky.edu/~dmu228/ma320/pascal_invention_probability.pdfCited in Chapter 3 3 times, 500 to 1400, 1400 to 1650 3 times and 1650 to 1800 3 times. | University of Kentucky, Department of Mathematics course pages | read in full |
| S255 | Petkovsek, M., Wilf, H. S., & Zeilberger, D. (1996). A=B (Foreword by D. E. Knuth). A K Peters. Retrieved August 20, 2026, from https://sites.math.rutgers.edu/~zeilberg/AeqB.pdfCited in Chapter 8 5 times and 1935 to 1960. | Rutgers University, Department of Mathematics (mirrors at Penn and the University of Ljubljana) | partial (front matter, pp. 18 and 23, and the opening of Chapter 4 (p. 55) were returned and read) |
| S228 | Robert, C., & Casella, G. (2011). A short history of Markov chain Monte Carlo: Subjective recollections from incomplete data. Statistical Science, 26(1), 102-115. arXiv:0808.2902. https://arxiv.org/abs/0808.2902Cited in Chapter 6 3 times, 1935 to 1960 twice and 1960 to now. | arXiv | read in full |
| S211 | Seneta, E. (2006). Markov and the creation of Markov chains [Paper prepared for the Markov Anniversary Meeting, Charleston, SC, June 12-14, 2006]. School of Mathematics and Statistics, University of Sydney. https://www.maths.usyd.edu.au/u/eseneta/senetamcfinal.pdfCited in Chapter 6 6 times, 1850 to 1880, 1880 to 1910 3 times and 1910 to 1935 twice. | University of Sydney, School of Mathematics and Statistics | read in full |
| S080 | Seneta, E. (2013). A tricentenary history of the law of large numbers. Bernoulli, 19(4), 1088-1121. https://doi.org/10.3150/12-BEJSP12 Author's arXiv version: https://arxiv.org/abs/1309.6488Cited in Chapter 3 4 times, 1650 to 1800 4 times, 1800 to 1850 twice, 1850 to 1880 twice and 1880 to 1910. | arXiv | read in full |
| S089 | Shafer, G., & Vovk, V. (2006). The sources of Kolmogorov's Grundbegriffe. Statistical Science, 21(1), 70-98. https://doi.org/10.1214/088342305000000467 Authors' arXiv version: https://arxiv.org/abs/math/0606533Cited in Chapter 3 twice, 1880 to 1910, 1910 to 1935 7 times and 1935 to 1960. | arXiv | read in full |
| S236 | Shafer, G., & Vovk, V. (2018). The origins and legacy of Kolmogorov's Grundbegriffe (The Game-Theoretic Probability and Finance Project, Working Paper No. 4; first posted 8 February 2003, revised 29 December 2018). http://www.probabilityandfinance.com/articles/04.pdfCited in Chapter 6 5 times and 1910 to 1935. | probabilityandfinance.com | read in full |
| S002 | Shah, J. (n.d.). A history of Pingala's combinatorics [PDF]. Northeastern University, Boston. Hosted by the Department of Sanskrit Studies, University of Hyderabad, "Algorithms in Ancient India" course materials. https://sanskrit.uohyd.ac.in/Algorithms_in_Ancient_India/Material/Pingala.pdfCited in Chapter 2 4 times, Before 500 twice and 500 to 1400 5 times. | University of Hyderabad, Department of Sanskrit Studies teaching site | read in full |
| S386 | Sheynin, O. (2017). Theory of probability: A historical essay (Rev. and enl. ed.). Berlin. https://www.probabilityandfinance.com/sheynin/010_double.pdfNot cited in the story. It backs a register entry. | probabilityandfinance.com | partial (front matter and table of contents read; the fetch truncated before Chapter 14) |
| S217 | Sheynin, O. (n.d.). Why Markov had requested to be excommunicated from the Russian Orthodox Church? [English and Russian text, item 99]. Retrieved August 20, 2026, from https://www.probabilityandfinance.com/sheynin/099_excomm_Russ_Engl.pdfCited in Chapter 6 6 times, 1910 to 1935 3 times and 1935 to 1960. | Probability and Finance (Shafer and Vovk's site), Sheynin collection | read in full |
| S384 | Sheynin, O. (n.d.). Antistigler [Unpublished paper]. Retrieved August 20, 2026, from https://www.probabilityandfinance.com/sheynin/031_antistigler.pdfNot cited in the story. It backs a register entry. | probabilityandfinance.com (the Shafer and Vovk site hosts Sheynin's archive) | read in full |
| S004 | Simonson, S. (2000). The mathematics of Levi ben Gershon. Mathematics Teacher, 93(8), 659-663. https://old.maa.org/sites/default/files/images/upload_library/46/NCTM/mt2000-11-659a.pdfCited in Chapter 2 twice and 500 to 1400 3 times. | Mathematical Association of America (legacy site), NCTM article archive | read in full |
| S170 | Stewart, G. W. (1992). On the early history of the singular value decomposition (Technical Report UMIACS-TR-92-31 / CS-TR-2855). Institute for Advanced Computer Studies and Department of Computer Science, University of Maryland. https://drum.lib.umd.edu/items/30400812-51b4-4c66-9e21-57713c98d76b (Published version: Stewart, G. W. (1993). On the early history of the singular value decomposition. SIAM Review, 35(4), 551-566.)Cited in Chapter 5 7 times, 1800 to 1850 twice, 1850 to 1880 twice, 1880 to 1910 5 times, 1910 to 1935 twice and 1935 to 1960. | DRUM, the Digital Repository at the University of Maryland | read in full |
| S085 | Stigler, S. M. (2013). The true title of Bayes's essay. Statistical Science, 28(3), 283-288. https://doi.org/10.1214/13-STS438 Author's arXiv version: https://arxiv.org/abs/1310.0173Cited in Chapter 3 and 1650 to 1800 twice. | arXiv | read in full |
| S005 | Sylla, E. D. (2013). Tercentenary of Ars Conjectandi (1713): Jacob Bernoulli and the founding of mathematical probability. In Proceedings of the 59th ISI World Statistics Congress, 25-30 August 2013, Hong Kong (Session IPS008, pp. 91-96). International Statistical Institute. [A version of this paper carries the DOI 10.1111/insr.12050, International Statistical Review, 2014, John Wiley & Sons.]. https://scispace.com/pdf/tercentenary-of-ars-conjectandi-1713-jacob-bernoulli-and-the-11m41deb9q.pdfCited in 1650 to 1800 4 times. | SciSpace mirror of the PDF | read in full |
| S223 | van der Heijden, M. (2024). More is known about him than about her: Tatiana Ehrenfest-Afanassjewa. Physics Today, 77(1), 40-46. https://doi.org/10.1063/pt.tfso.pxvmCited in Chapter 6 5 times, 1880 to 1910 and 1910 to 1935. | American Institute of Physics, Physics Today | read in full |
| S243 | von Hilgers, P., & Langville, A. N. (2006). The five greatest applications of Markov chains [Paper prepared for the Markov Anniversary Meeting, Charleston, SC]. https://langvillea.people.charleston.edu/MCapps7.pdfCited in Chapter 6 twice and 1960 to now twice. | College of Charleston, Amy Langville's faculty page | read in full |
| S231 | Yablonsky, G. S., Gorban, A. N., Constales, D., Galvita, V. V., & Marin, G. B. (2011). Reciprocal relations between kinetic curves. EPL (Europhysics Letters), 93(2), 20004. arXiv:1008.1056. https://arxiv.org/pdf/1008.1056Cited in Chapter 6 3 times, 1880 to 1910 and 1910 to 1935 twice. | arXiv | read in full |
Tier 3: reference works, catalogues, and data services
96 sources.
| ID | Reference (APA 7) | Repository | Access |
|---|---|---|---|
| S422 | Leonard Euler's solution to the Konigsberg bridge problem. (2006). Convergence, 3. Mathematical Association of America. Retrieved August 20, 2026, from https://old.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problemCited in Chapter 7 11 times, 1650 to 1800 3 times and 1850 to 1880. | Mathematical Association of America, Convergence (legacy old.maa.org) | reference entry read in full |
| S272 | Ariza, C. (2010). Meeting 9, History: Lejaren Hiller [Lecture notes]. 21M.380 Music and Technology: Algorithmic and Generative Music, Spring 2010. MIT OpenCourseWare. https://ocw.mit.edu/courses/21m-380-music-and-technology-algorithmic-and-generative-music-spring-2010/Cited in 1935 to 1960 twice. | MIT OpenCourseWare | reference entry read in full |
| S428 | Baker, A. (2025). Non-deductive methods in mathematics. In E. N. Zalta & U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy. Stanford University. (First published August 17, 2009; substantive revision August 29, 2025.) https://plato.stanford.edu/entries/mathematics-nondeductive/Cited in Chapter 7 3 times and 1960 to now twice. | Stanford Encyclopedia of Philosophy | reference entry read in full |
| S257 | Betancourt, M. (2021, February 23). Obituary: Arianna Rosenbluth, 1927-2020. IMS Bulletin, Institute of Mathematical Statistics. https://imstat.org/2021/02/23/obituary-arianna-rosenbluth-1927-2020/Cited in Chapter 6, Chapter 8 7 times, 1910 to 1935, 1935 to 1960 twice and 1960 to now. | Institute of Mathematical Statistics | reference entry read in full |
| S014 | Breard, A. (n.d.). Jia Xian. In Encyclopaedia Britannica. Retrieved August 20, 2026, from https://www.britannica.com/biography/Jia-XianCited in Chapter 2 twice, 500 to 1400 twice, 1400 to 1650 and 1800 to 1850. | Encyclopaedia Britannica online | reference entry read in full |
| S447 | Campbell, M. (n.d.). Frigyes; Szekeres; Markov. In Behind the Name and Behind the Name: Surnames. Retrieved August 21, 2026, from https://www.behindthename.com/name/frigyes; https://surnames.behindthename.com/name/szekeres; https://surnames.behindthename.com/name/markovNot cited in the story. It backs a register entry. | Behind the Name (Mike Campbell) | reference entry read in full |
| S068 | Dictionary.com. (n.d.). Entries for Dedekind, Bolzano, Peano, Frege, Weierstrass, Kronecker, Weil, Boole, Venn diagram, Cantor. Retrieved August 20, 2026, from https://www.dictionary.com/browse/dedekind and the companion entries listed below.Cited in 1880 to 1910. | Dictionary.com | reference entry read in full |
| S143 | Dictionary.com. (n.d.). Cauchy; Cayley; Jacobi; Leibniz; Sylvester. Retrieved August 20, 2026, from https://www.dictionary.com/browse/cauchy, /cayley, /jacobi, /leibniz, /sylvesterCited in 1880 to 1910. | Dictionary.com | reference entry read in full |
| S204 | Dictionary.com. (n.d.). Grassman; Peano; Gibbs; Heaviside; Hilbert; Beltrami; Leontief; Cauchy. Retrieved August 20, 2026, from https://www.dictionary.com/browse/grassmann (and the companion entries)Cited in 1880 to 1910. | Dictionary.com | reference entry read in full |
| S016 | Djebbar, A. (2008). Mathematics in the medieval Maghrib: General survey on mathematical activities in North Africa (C. Nizamoglu, Ed.). Muslim Heritage. http://muslimheritage.com/mathematics-in-the-medieval-maghrib-general-survey-on-mathematical-activities-in-north-africa/Cited in Chapter 2 3 times and 500 to 1400. | Muslim Heritage (Foundation for Science, Technology and Civilization) | reference entry read in full |
| S019 | Encyclopaedia Britannica. (n.d.). Niccolo Fontana Tartaglia. Retrieved August 20, 2026, from https://www.britannica.com/biography/Niccolo-Fontana-TartagliaCited in Chapter 2 twice and 1400 to 1650 3 times. | Encyclopaedia Britannica online | reference entry read in full |
| S200 | Gander, W., & Joss, J. (2024). Computing the SVD [Manuscript/talk notes]. Department of Computer Science, ETH Zurich. https://people.inf.ethz.ch/gander/talks/SVDnew.pdfCited in Chapter 5 twice and 1960 to now 3 times. | ETH Zurich, Department of Computer Science | partial (the historical and algorithmic-lineage sections; the numerical derivations were not read line by line) |
| S057 | Hallett, M. (n.d.). Zermelo's axiomatization of set theory. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved August 20, 2026, from https://plato.stanford.edu/entries/zermelo-set-theory/Cited in Chapter 1 and 1880 to 1910 twice. | Stanford Encyclopedia of Philosophy | read in full |
| S196 | Hammarling, S., & Higham, N. J. (2016, October 4). Remembering James Hardy Wilkinson. SIAM News. https://www.siam.org/publications/siam-news/articles/remembering-james-hardy-wilkinson/Cited in Chapter 5 twice. | SIAM News | reference entry read in full |
| S306 | Harper, D. (n.d.). Online Etymology Dictionary. Retrieved August 20, 2026, from https://www.etymonline.com/Cited in Chapter 4 twice, Chapter 9 16 times, 1650 to 1800 and 1800 to 1850. | etymonline.com | reference entry read in full |
| S186 | Higham, N. J. (2021, July 13). What is the Perron-Frobenius theorem? Nick Higham's blog. https://nhigham.com/2021/07/13/what-is-the-perron-frobenius-theorem/Cited in Chapter 5 5 times, Chapter 6 3 times and 1910 to 1935. | the author's own site | reference entry read in full |
| S013 | Hosch, W. L. (2026, July 15). Pascal's triangle. In Encyclopaedia Britannica. https://www.britannica.com/science/Pascals-triangleCited in Chapter 2 5 times and 500 to 1400 twice. | Encyclopaedia Britannica online | reference entry read in full |
| S242 | Illinois Distributed Museum. (n.d.). ILLIAC Suite. University of Illinois Urbana-Champaign. Retrieved August 20, 2026, from https://distributedmuseum.illinois.edu/exhibit/illiac-suite/Cited in 1935 to 1960 twice. | University of Illinois Urbana-Champaign | reference entry read in full |
| S273 | Illinois Distributed Museum. (n.d.). ILLIAC Suite. University of Illinois Urbana-Champaign. Retrieved August 20, 2026, from https://distributedmuseum.illinois.edu/exhibit/illiac-suite/Cited in 1935 to 1960 twice. | University of Illinois Urbana-Champaign | reference entry read in full |
| S056 | Irvine, A. D., & Deutsch, H. (n.d.). Russell's paradox. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved August 20, 2026, from https://plato.stanford.edu/entries/russell-paradox/Cited in Chapter 1 and 1880 to 1910 3 times. | Stanford Encyclopedia of Philosophy | read in full |
| S020 | Jawalgekar, P., Sooryanarayan, D. G., & Ramasubramanian, K. (2023, January). Ganitakaumudi of Narayana Pandita. Bhavana, 7(1). https://bhavana.org.in/ganitakaumudi-of-narayana-pandita/Cited in Chapter 2 and 500 to 1400. | Bhavana (bhavana.org.in) | reference entry read in full |
| S431 | Jelliss, G. P. (n.d.). Knight's tour notes: Early history. Mayhematics. Retrieved August 20, 2026, from https://www.mayhematics.com/t/1a.htmCited in Chapter 7 6 times, 500 to 1400 3 times and 1650 to 1800. | mayhematics.com (G. P. Jelliss) | reference entry read in full |
| S052 | Klyve, D. (2020, December). Euler's Letters to a German Princess: Translation and betrayal. MAA Convergence. https://old.maa.org/press/periodicals/convergence/euler-s-letters-to-a-german-princess-translation-and-betrayalCited in 1650 to 1800 twice. | MAA Convergence | reference entry read in full |
| S142 | Knill, O. (2018). About Gauss-Jordan elimination [Course exhibit, Mathematics 22a]. Harvard University Department of Mathematics. https://legacy-www.math.harvard.edu/~knill/teaching/math22a2018/exhibits/gaussjordan/index.htmlCited in Chapter 4 3 times and 1880 to 1910 twice. | Harvard Mathematics Department | reference entry read in full |
| S241 | Massachusetts Institute of Technology. (2010). Meeting 9, History: Lejaren Hiller [Lecture notes, 21M.380 Music and Technology: Algorithmic and Generative Music, Spring 2010]. MIT OpenCourseWare. https://ocw.mit.edu/courses/21m-380-music-and-technology-algorithmic-and-generative-music-spring-2010/Cited in 1935 to 1960 4 times. | MIT OpenCourseWare | reference entry read in full |
| S017 | Mathematical Association of America. (n.d.). Mathematical treasure: The precious mirror of Zhu Shijie. Convergence. Retrieved August 20, 2026, from https://old.maa.org/press/periodicals/convergence/mathematical-treasure-the-precious-mirror-of-zhu-shijieCited in Chapter 2 twice and 500 to 1400. | Mathematical Association of America, Convergence (legacy site) | reference entry read in full |
| S018 | Mathematical Association of America. (n.d.). Mathematical treasures: Michael Stifel's Arithmetica Integra. Convergence. Retrieved August 20, 2026, from https://old.maa.org/press/periodicals/convergence/mathematical-treasures-michael-stifels-arithmetica-integraCited in Chapter 2 and 1400 to 1650. | MAA Convergence (legacy site) | reference entry read in full |
| S202 | Merriam-Webster. (n.d.). Eigenvalue; Vector; Scalar; Hilbert space; Cauchy sequence; Heaviside layer. In Merriam-Webster.com dictionary. Retrieved August 20, 2026, from https://www.merriam-webster.com/dictionary/eigenvalue (and the companion entries)Cited in Chapter 5 7 times, 1800 to 1850, 1850 to 1880 and 1910 to 1935. | Merriam-Webster.com | reference entry read in full |
| S011 | Miller, J. (n.d.). Pascal's triangle. In Earliest known uses of some of the words of mathematics (P). Retrieved August 20, 2026, from https://jeff560.tripod.com/p.htmlCited in Chapter 2 8 times, Before 500, 500 to 1400 twice, 1400 to 1650, 1650 to 1800 5 times, 1800 to 1850, 1850 to 1880, 1880 to 1910 and 1935 to 1960. | Jeff Miller's personal site, hosted on Tripod | reference entry read in full |
| S054 | Miller, J. (n.d.). Earliest uses of symbols of set theory and logic. Retrieved August 20, 2026, from https://jeff560.tripod.com/set.htmlCited in Chapter 1 4 times, Chapter 9 6 times, 1880 to 1910 6 times, 1935 to 1960 and 1960 to now. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S055 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/s.html, https://jeff560.tripod.com/e.html, https://jeff560.tripod.com/d.html, https://jeff560.tripod.com/v.html, https://jeff560.tripod.com/c.html, https://jeff560.tripod.com/u.htmlCited in Chapter 1 6 times, Chapter 9, 500 to 1400, 1650 to 1800 twice, 1800 to 1850, 1850 to 1880 twice, 1880 to 1910 4 times, 1910 to 1935 8 times and 1935 to 1960 twice. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S088 | Miller, J. (n.d.). Earliest uses of symbols in probability and statistics. Retrieved August 20, 2026, from https://jeff560.tripod.com/stat.html; Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/b.html, https://jeff560.tripod.com/i.html, and https://jeff560.tripod.com/l.htmlCited in Chapter 3 6 times, 1650 to 1800, 1800 to 1850 4 times, 1880 to 1910, 1910 to 1935 3 times and 1935 to 1960 3 times. | Jeff Miller's site | reference entry read in full |
| S134 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/c.html, https://jeff560.tripod.com/d.html, https://jeff560.tripod.com/g.html, and the MacTutor mirror https://mathshistory.st-andrews.ac.uk/Miller/mathword/m/Cited in Chapter 4 17 times, 1800 to 1850 5 times, 1850 to 1880 4 times, 1880 to 1910 3 times and 1910 to 1935 twice. | jeff560.tripod.com and the MacTutor mirror at St Andrews | reference entry read in full |
| S176 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/e.html, https://jeff560.tripod.com/l.html and https://jeff560.tripod.com/s.htmlCited in Chapter 5 14 times, 1400 to 1650, 1650 to 1800, 1800 to 1850 3 times, 1850 to 1880 twice, 1880 to 1910 twice, 1910 to 1935 twice, 1935 to 1960 twice and 1960 to now. | Jeff Miller's personal site, hosted on Tripod | reference entry read in full |
| S216 | Miller, J. (n.d.). MARKOV CHAIN; MARKOV PROCESS; MONTE CARLO. In Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/m.htmlCited in Chapter 6, 1910 to 1935 4 times and 1935 to 1960 4 times. | Jeff Miller's Earliest Uses pages | reference entry read in full |
| S302 | Miller, J. (n.d.). Earliest known uses of some of the words of mathematics. Retrieved August 20, 2026, from https://jeff560.tripod.com/mathword.htmlCited in Chapter 7 12 times, Chapter 9 35 times, 500 to 1400, 1400 to 1650 twice, 1650 to 1800 11 times, 1800 to 1850 15 times, 1850 to 1880 6 times, 1880 to 1910 12 times, 1910 to 1935 11 times, 1935 to 1960 5 times and 1960 to now. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S303 | Miller, J. (n.d.). Earliest uses of symbols in probability and statistics. Retrieved August 20, 2026, from https://jeff560.tripod.com/stat.htmlCited in Chapter 9 6 times, 1910 to 1935 twice and 1935 to 1960 twice. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S304 | Miller, J. (n.d.). Earliest uses of symbols for matrices and vectors. Retrieved August 20, 2026, from https://jeff560.tripod.com/matrices.htmlCited in 1800 to 1850, 1880 to 1910 and 1910 to 1935 twice. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S305 | Miller, J. (n.d.). Earliest uses of symbols of operation. Retrieved August 20, 2026, from https://jeff560.tripod.com/operation.htmlCited in 1650 to 1800. | Jeff Miller's personal site on Tripod | reference entry read in full |
| S012 | O'Connor, J. J., & Robertson, E. F. (2003, April). Percy Alexander MacMahon. MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/MacMahon/Cited in 1850 to 1880 and 1910 to 1935 twice. | MacTutor History of Mathematics Archive, University of St Andrews | reference entry read in full |
| S060 | O'Connor, J. J., & Robertson, E. F. (n.d.). George Boole. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Boole/Cited in 1800 to 1850 twice and 1850 to 1880 3 times. | MacTutor, University of St Andrews | reference entry read in full |
| S061 | O'Connor, J. J., & Robertson, E. F. (n.d.). Mary Everest Boole. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Boole_Mary/Cited in 1800 to 1850, 1850 to 1880 twice and 1910 to 1935. | MacTutor, University of St Andrews | reference entry read in full |
| S062 | O'Connor, J. J., & Robertson, E. F. (n.d.). Georg Ferdinand Ludwig Philipp Cantor. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/ and https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/quotations/Cited in Chapter 1, 1800 to 1850, 1850 to 1880 twice, 1880 to 1910 and 1910 to 1935. | MacTutor, University of St Andrews | reference entry read in full |
| S063 | O'Connor, J. J., & Robertson, E. F. (n.d.). Augustus De Morgan. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/De_Morgan/Cited in 1800 to 1850 twice and 1850 to 1880 twice. | MacTutor, University of St Andrews | reference entry read in full |
| S064 | O'Connor, J. J., & Robertson, E. F. (n.d.). Giuseppe Peano. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Peano/Cited in Chapter 1, 1850 to 1880, 1880 to 1910 3 times and 1910 to 1935. | MacTutor, University of St Andrews | reference entry read in full |
| S065 | O'Connor, J. J., & Robertson, E. F. (n.d.). Ernst Friedrich Ferdinand Zermelo. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Zermelo/Cited in Chapter 1, 1850 to 1880 twice, 1880 to 1910 3 times and 1935 to 1960. | MacTutor, University of St Andrews | reference entry read in full |
| S094 | O'Connor, J. J., & Robertson, E. F. (n.d.). MacTutor history of mathematics archive. University of St Andrews. Entries read on 2026-08-20: Cardan (https://mathshistory.st-andrews.ac.uk/Biographies/Cardan/), De Moivre (https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre/), Bayes (https://mathshistory.st-andrews.ac.uk/Biographies/Bayes/), Saunderson (https://mathshistory.st-andrews.ac.uk/Biographies/Saunderson/), Bernstein_Sergi (https://mathshistory.st-andrews.ac.uk/Biographies/Bernstein_Sergi/), Bienayme (https://mathshistory.st-andrews.ac.uk/Biographies/Bienayme/), Chebyshev (https://mathshistory.st-andrews.ac.uk/Biographies/Chebyshev/); Merriam-Webster.com dictionary. Entries read 2026-08-20: Huygens, Laplace, Kolmogorov, Cardan, Fermat.; Dictionary.com. Entries read 2026-08-20: Bernoulli, de Moivre, Borel, Huygens, Laplace, Frechet, Tchebycheff equation.Cited in Chapter 3 4 times, 1400 to 1650 4 times, 1650 to 1800 9 times, 1800 to 1850, 1850 to 1880 twice, 1880 to 1910 twice, 1910 to 1935 and 1960 to now. | University of St Andrews (MacTutor); Merriam-Webster.com; Dictionary.com | reference entry read in full |
| S122 | O'Connor, J. J., & Robertson, E. F. (n.d.). Matrices and determinants. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/Cited in Chapter 4 13 times, Before 500 twice, 1400 to 1650, 1650 to 1800 5 times, 1800 to 1850 6 times, 1850 to 1880 3 times and 1880 to 1910 twice. | University of St Andrews | reference entry read in full |
| S123 | O'Connor, J. J., & Robertson, E. F. (n.d.). Nine chapters on the Mathematical Art. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/Cited in Chapter 4 10 times, Before 500 3 times and 500 to 1400. | University of St Andrews | reference entry read in full |
| S128 | O'Connor, J. J., & Robertson, E. F. (n.d.). Takakazu Seki; James Joseph Sylvester; William Rowan Hamilton; Georg Ferdinand Frobenius. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Seki/, /Sylvester/, /Hamilton/, /Frobenius/Cited in Chapter 4 11 times, 1650 to 1800 twice and 1800 to 1850. | University of St Andrews | reference entry read in full |
| S179 | O'Connor, J. J., & Robertson, E. F. (n.d.). Abstract linear spaces. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces/Cited in Chapter 5 4 times, 1850 to 1880 twice, 1880 to 1910 and 1910 to 1935. | MacTutor, University of St Andrews | reference entry read in full |
| S180 | O'Connor, J. J., & Robertson, E. F. (n.d.). Matrices and determinants. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/Cited in Chapter 5 6 times, 1800 to 1850 3 times, 1850 to 1880 4 times and 1880 to 1910. | MacTutor, University of St Andrews | reference entry read in full |
| S181 | O'Connor, J. J., & Robertson, E. F. (n.d.). Hermann Grassmann. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Grassmann/Cited in Chapter 5 11 times, 1800 to 1850 4 times and 1850 to 1880. | MacTutor, University of St Andrews | reference entry read in full |
| S183 | O'Connor, J. J., & Robertson, E. F. (n.d.). Giuseppe Peano. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Peano/Cited in Chapter 5 4 times, 1850 to 1880 and 1880 to 1910 twice. | MacTutor, University of St Andrews | reference entry read in full |
| S187 | O'Connor, J. J., & Robertson, E. F. (n.d.). Ferdinand Georg Frobenius. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Frobenius/Cited in Chapter 5 3 times, 1800 to 1850 and 1910 to 1935. | MacTutor, University of St Andrews | reference entry read in full |
| S194 | O'Connor, J. J., & Robertson, E. F. (n.d.). Vera Nikolaevna Kublanovskaya. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Kublanovskaya/Cited in Chapter 5 4 times, Chapter 8 4 times, 1910 to 1935, 1935 to 1960 twice and 1960 to now twice. | MacTutor, University of St Andrews | reference entry read in full |
| S195 | O'Connor, J. J., & Robertson, E. F. (n.d.). James Hardy Wilkinson. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Wilkinson/Cited in Chapter 5 twice, 1910 to 1935, 1935 to 1960 3 times and 1960 to now 3 times. | MacTutor, University of St Andrews | reference entry read in full |
| S201 | O'Connor, J. J., & Robertson, E. F. (n.d.). Eugenio Beltrami. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Beltrami/Cited in Chapter 5, 1800 to 1850 and 1880 to 1910. | MacTutor, University of St Andrews | reference entry read in full |
| S219 | O'Connor, J. J., & Robertson, E. F. (n.d.). Andrei Andreyevich Markov. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Markov/Cited in Chapter 6 6 times, 1850 to 1880, 1880 to 1910 3 times and 1910 to 1935 twice. | University of St Andrews | reference entry read in full |
| S222 | O'Connor, J. J., & Robertson, E. F. (n.d.). Tatiana Alexeyevna Afanassjewa. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Ehrenfest-Afanassjewa/Cited in Chapter 6 3 times, 1850 to 1880, 1880 to 1910, 1910 to 1935 twice and 1960 to now. | University of St Andrews | reference entry read in full |
| S233 | O'Connor, J. J., & Robertson, E. F. (n.d.). Wolfgang Doeblin. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Doeblin/Cited in Chapter 6 7 times, 1910 to 1935, 1935 to 1960 twice and 1960 to now. | University of St Andrews | reference entry read in full |
| S237 | O'Connor, J. J., & Robertson, E. F. (n.d.). Andrey Nikolaevich Kolmogorov. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Kolmogorov/Cited in Chapter 6 3 times, 1880 to 1910 and 1960 to now. | University of St Andrews | reference entry read in full |
| S238 | O'Connor, J. J., & Robertson, E. F. (n.d.). Sydney Chapman. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Chapman/Cited in Chapter 6 4 times, 1880 to 1910 and 1960 to now. | University of St Andrews | reference entry read in full |
| S250 | O'Connor, J. J., & Robertson, E. F. (n.d.). David Blackwell. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Blackwell/Cited in Chapter 8 9 times, 1910 to 1935, 1935 to 1960 3 times and 1960 to now twice. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S253 | O'Connor, J. J., & Robertson, E. F. (n.d.). Mary Celine Fasenmyer. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Fasenmyer/Cited in Chapter 8 9 times, 1880 to 1910, 1935 to 1960 and 1960 to now twice. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S266 | O'Connor, J. J., & Robertson, E. F. (n.d.). Alicia Boole Stott. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Stott/Cited in Chapter 8 8 times, 1850 to 1880 twice, 1880 to 1910 twice, 1910 to 1935 and 1935 to 1960 twice. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S276 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bourbaki: The pre-war years. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_1/Cited in Chapter 8 5 times, 1850 to 1880 and 1910 to 1935. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S277 | O'Connor, J. J., & Robertson, E. F. (n.d.). Andre Weil. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Weil/Cited in Chapter 8 7 times, 1880 to 1910, 1935 to 1960 and 1960 to now. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S278 | O'Connor, J. J., & Robertson, E. F. (n.d.). Emmy Amalie Noether. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/Cited in Chapter 8 7 times, 1880 to 1910 3 times, 1910 to 1935 5 times and 1935 to 1960. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S280 | O'Connor, J. J., & Robertson, E. F. (n.d.). Dame Mary Lucy Cartwright. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Cartwright/Cited in 1880 to 1910, 1935 to 1960 and 1960 to now twice. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S284 | O'Connor, J. J., & Robertson, E. F. (n.d.). Madhava of Sangamagramma. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/Cited in 1400 to 1650 twice. | School of Mathematics and Statistics, University of St Andrews | reference entry read in full |
| S334 | O'Connor, J. J., & Robertson, E. F. (n.d.). Copyright. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Miscellaneous/copyright/Not cited in the story. It backs a register entry. | University of St Andrews, School of Mathematics and Statistics | reference entry read in full |
| S423 | O'Connor, J. J., & Robertson, E. F. (n.d.). The four colour theorem. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/HistTopics/The_four_colour_theorem/Cited in Chapter 7 13 times, 1850 to 1880 5 times, 1880 to 1910 3 times, 1910 to 1935 twice and 1960 to now 3 times. | MacTutor History of Mathematics Archive | reference entry read in full |
| S424 | O'Connor, J. J., & Robertson, E. F. (n.d.). Alfred Bray Kempe. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Kempe/Cited in Chapter 7 7 times, 1800 to 1850, 1850 to 1880, 1880 to 1910 and 1910 to 1935 twice. | MacTutor History of Mathematics Archive | reference entry read in full |
| S425 | O'Connor, J. J., & Robertson, E. F. (n.d.). Percy John Heawood. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Heawood/Cited in Chapter 7 6 times, 1850 to 1880, 1880 to 1910 and 1935 to 1960. | MacTutor History of Mathematics Archive | reference entry read in full |
| S429 | O'Connor, J. J., & Robertson, E. F. (n.d.). Gustav Robert Kirchhoff. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Kirchhoff/Cited in Chapter 7 8 times, 1800 to 1850 3 times and 1880 to 1910. | MacTutor History of Mathematics Archive | reference entry read in full |
| S430 | O'Connor, J. J., & Robertson, E. F. (n.d.). Alexandre-Theophile Vandermonde. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Vandermonde/Cited in Chapter 4 5 times, Chapter 7 3 times and 1650 to 1800 3 times. | MacTutor History of Mathematics Archive | reference entry read in full |
| S432 | O'Connor, J. J., & Robertson, E. F. (n.d.). Frank Plumpton Ramsey. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Ramsey/Cited in Chapter 7 6 times, 1880 to 1910 and 1910 to 1935 3 times. | MacTutor History of Mathematics Archive | reference entry read in full |
| S434 | O'Connor, J. J., & Robertson, E. F. (n.d.). George Szekeres. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Szekeres/Cited in Chapter 7 7 times, 1910 to 1935, 1935 to 1960 twice and 1960 to now. | MacTutor History of Mathematics Archive | reference entry read in full |
| S438 | O'Connor, J. J., & Robertson, E. F. (n.d.). Alfred Renyi. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Renyi/Cited in 1910 to 1935, 1935 to 1960 and 1960 to now. | MacTutor History of Mathematics Archive | reference entry read in full |
| S443 | O'Connor, J. J., & Robertson, E. F. (n.d.). Raj Chandra Bose. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/Bose_Raj/Cited in Chapter 7 6 times, 1880 to 1910, 1935 to 1960 and 1960 to now twice. | MacTutor History of Mathematics Archive | reference entry read in full |
| S444 | O'Connor, J. J., & Robertson, E. F. (n.d.). Nicolaas Govert de Bruijn. MacTutor History of Mathematics Archive, University of St Andrews. Retrieved August 20, 2026, from https://mathshistory.st-andrews.ac.uk/Biographies/De_Bruijn/Cited in Chapter 7 5 times, 1910 to 1935, 1935 to 1960 and 1960 to now. | MacTutor History of Mathematics Archive | reference entry read in full |
| S092 | Pulskamp, R. J. (n.d.). Irenee-Jules Bienayme [Bibliography and English translations]. Probability and Finance. Retrieved August 20, 2026, from https://probabilityandfinance.com/pulskamp/Bienayme/bienayme.htmlCited in Chapter 3, 1800 to 1850 twice and 1850 to 1880. | probabilityandfinance.com (Glenn Shafer and Vladimir Vovk's site), Pulskamp translations | reference entry read in full |
| S059 | Reck, E. (n.d.). Dedekind's contributions to the foundations of mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved August 20, 2026, from https://plato.stanford.edu/entries/dedekind-foundations/Cited in Chapter 1 twice, 1850 to 1880 and 1880 to 1910. | Stanford Encyclopedia of Philosophy | read in full |
| S254 | Riddle, L. (n.d.). Sister Mary Celine Fasenmyer. Biographies of Women Mathematicians, Agnes Scott College. Retrieved August 20, 2026, from https://mathwomen.agnesscott.org/women/celine.htmCited in Chapter 8 8 times, 1880 to 1910, 1935 to 1960 twice and 1960 to now. | Agnes Scott College, Decatur, Georgia | reference entry read in full |
| S078 | Rosenfeld, B. (n.d.). History of statistics 1: The Bills of Mortality, and the beginning of statistical thinking [Classroom activity]. Vermont Mathematics Initiative, distributed by the American Statistical Association. https://higherlogicdownload.s3.amazonaws.com/AMSTAT/1484431b-3202-461e-b7e6-ebce10ca8bcd/UploadedImages/Classroom_Activities/HS_1__John_Graunt_and_the_Bills_of_Mortality.pdfCited in Chapter 3 twice, 1400 to 1650 and 1650 to 1800. | American Statistical Association classroom activities library | reference entry read in full |
| S124 | Schwartz, R. K. (2018, December). A classic from China: The Nine Chapters. Convergence (Mathematical Association of America). https://old.maa.org/press/periodicals/convergence/a-classic-from-china-the-nine-chapters (PDF copy consulted at https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/gaussjordan/SchwarzNineChapters.pdf)Cited in Chapter 4 6 times and Before 500 twice. | Mathematical Association of America; PDF mirror at Harvard | partial (The Harvard PDF gave me the fangcheng and negative-number passages, the problem statement, the dating sentence, and the reference list) |
| S058 | Shin, S.-J., Lemon, O., & Mumma, J. (n.d.). Diagrams. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved August 20, 2026, from https://plato.stanford.edu/entries/diagrams/Cited in 1650 to 1800. | Stanford Encyclopedia of Philosophy | read in full |
| S199 | The Nobel Foundation. (n.d.). Wassily Leontief: Facts. The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1973. NobelPrize.org. Retrieved August 20, 2026, from https://www.nobelprize.org/prizes/economic-sciences/1973/leontief/facts/Cited in Chapter 5 twice, 1880 to 1910 and 1960 to now twice. | NobelPrize.org, the Nobel Foundation | reference entry read in full |
| S252 | Williams, S. W. (n.d.). David Blackwell. Mathematicians of the African Diaspora, Department of Mathematics, State University of New York at Buffalo. Retrieved August 20, 2026, from https://math.buffalo.edu/mad/PEEPS/blackwell_david.htmlCited in Chapter 8 6 times, 1935 to 1960 twice and 1960 to now. | University at Buffalo, Department of Mathematics | reference entry read in full |
| S234 | Zentralblatt MATH. (n.d.). The mathematician Wolfgang Doeblin (1915-1940) [Year of Mathematics feature, item 11]. FIZ Karlsruhe. https://www.zentralblatt-math.org/year-of-mathematics/yom-11.pdfCited in Chapter 6 8 times, 1910 to 1935 twice, 1935 to 1960 3 times and 1960 to now. | zentralblatt-math.org | reference entry read in full |
| S220 | Markov, Andrei Andreevich. (n.d.). In Encyclopedia of Mathematics. European Mathematical Society and Springer. Retrieved August 20, 2026, from https://encyclopediaofmath.org/wiki/Markov,_Andrei_AndreevichCited in Chapter 6 3 times, 1850 to 1880 and 1910 to 1935. | encyclopediaofmath.org | reference entry read in full |
| S244 | Markov chain. (n.d.). In Merriam-Webster.com dictionary. Retrieved August 20, 2026, from https://www.merriam-webster.com/dictionary/Markov%20chainCited in Chapter 6 twice, 1910 to 1935 and 1935 to 1960. | Merriam-Webster | reference entry read in full |
| S245 | Markov chain; Onsager; Kolmogorov. (n.d.). In Dictionary.com. Retrieved August 20, 2026, from https://www.dictionary.com/browse/markov-chain, https://www.dictionary.com/browse/onsager, https://www.dictionary.com/browse/kolmogorovCited in 1880 to 1910 and 1960 to now twice. | Dictionary.com | reference entry read in full |
| S445 | Euler; Konigsberg; Kirchhoff. (n.d.). In Dictionary.com. Retrieved August 20, 2026, from https://www.dictionary.com/browse/euler; https://www.dictionary.com/browse/konigsberg; https://www.dictionary.com/browse/kirchhoffCited in Chapter 7 4 times, 1800 to 1850 and 1880 to 1910. | Dictionary.com | reference entry read in full |
| S446 | Guthrie, Woodrow Wilson; Guthrie. (n.d.). In The American Heritage dictionary of the English language (5th ed.). HarperCollins. Retrieved August 21, 2026, from https://www.ahdictionary.com/word/search.html?q=GuthrieNot cited in the story. It backs a register entry. | American Heritage Dictionary online (HarperCollins) | reference entry read in full |
Tier 4: everything else, typed as it was registered
18 sources.
| ID | Reference (APA 7) | Repository | Access |
|---|---|---|---|
| S391 | Allen Institute for AI. (n.d.). Semantic Scholar Academic Graph API [Data service]. Retrieved August 20, 2026, from https://api.semanticscholar.org/graph/v1/paper/DOI:10.1080/00031305.1983.10483122Not cited in the story. It backs a register entry. | Semantic Scholar | reference entry read in full |
| S381 | Atlanta University students. (ca. 1900). A series of statistical charts illustrating the condition of the descendants of former African slaves now in residence in the United States of America [Drawing]. LOT 11931, no. 37 (M) [P&P]. Library of Congress Prints and Photographs Division, Washington, DC. https://www.loc.gov/item/2005676836/Cited in Chapter 8 5 times and 1880 to 1910. | Library of Congress, Prints and Photographs Division | reference entry read in full |
| S392 | Crossref. (n.d.). Crossref REST API [Data service]. Retrieved August 20, 2026, from https://api.crossref.org/works?query.bibliographic=Statistical+regularity+and+free+will+Quetelet+Nekrasov+Seneta&rows=3Not cited in the story. It backs a register entry. | Crossref | reference entry read in full |
| S282 | Du Bois, W. E. B. (ca. 1900). Charts and graphs showing the condition of African Americans at the turn of the century exhibited at the Paris Exposition Universelle in 1900 [Graphic]. Prints and Photographs Division, Library of Congress, LOT 11931. https://www.loc.gov/pictures/item/2005679642/Cited in Chapter 8 4 times, 1850 to 1880 and 1880 to 1910. | Library of Congress, Prints and Photographs Division, Washington, D.C | partial (catalog record read in full; the item-level plate list and the Rights and Restrictions page returned HTTP 403) |
| S331 | Du Bois, W. E. B. (ca. 1900). [The Georgia Negro] and [A series of statistical charts illustrating the condition of the descendants of former African slaves now in residence in the United States of America] [Drawings]. Library of Congress, Prints and Photographs Division, LOT 11931. https://www.loc.gov/item/2013650420/Not cited in the story. It backs a register entry. | Library of Congress, Prints and Photographs Division | reference entry read in full |
| S380 | Du Bois, W. E. B. (ca. 1900). [The Georgia Negro] Occupations [Drawing]. LOT 11931, no. 28 [P&P]. Library of Congress Prints and Photographs Division, Washington, DC. https://www.loc.gov/item/2013650448/Cited in Chapter 8 4 times and 1880 to 1910. | Library of Congress, Prints and Photographs Division | reference entry read in full |
| S246 | European Digital Mathematics Library. (n.d.). Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung [Record urn:eudml:doc:159476]; Zur Theorie der Markoffschen Ketten [Record urn:eudml:doc:159823]. Retrieved August 20, 2026, from https://eudml.org/doc/159476 and https://eudml.org/doc/159823Cited in Chapter 6 3 times, 1910 to 1935 and 1935 to 1960. | European Digital Mathematics Library | not opened |
| S268 | Friendly, M., Dray, S., Li, P., & Bellhouse, D. (n.d.). Nightingale: Florence Nightingale's data on deaths in the Crimean War [Documentation for the HistData R package, version 1.0.1]. Retrieved August 20, 2026, from https://friendly.github.io/HistData/reference/Nightingale.htmlCited in 1850 to 1880 twice. | HistData project pages (friendly.github.io); the package itself is on CRAN | reference entry read in full |
| S264 | Grossman, J. W. (2025). The Erdos Number Project: Information about the Erdos Number Project. Oakland University. Retrieved August 20, 2026, from https://sites.google.com/oakland.edu/grossman/home/the-erdoes-number-project/information-about-the-erdoes-number-projectCited in Chapter 8 3 times and 1960 to now twice. | Oakland University, Rochester, Michigan | reference entry read in full |
| S382 | Library of Congress. (n.d.). African American photographs assembled for 1900 Paris Exposition [Collection record]. Prints and Photographs Online Catalog. Retrieved August 20, 2026, from https://www.loc.gov/pictures/collection/anedub/?fo=jsonCited in Chapter 8 twice and 1880 to 1910. | Library of Congress, Prints and Photographs Division | reference entry read in full |
| S383 | Library of Congress. (n.d.). African American photographs assembled for 1900 Paris Exposition [Search results, query "charts"]. Retrieved August 20, 2026, from https://www.loc.gov/collections/african-american-photographs-1900-paris-exposition/?q=charts&fo=json&c=5&at=resultsCited in Chapter 8 3 times and 1880 to 1910. | Library of Congress, Prints and Photographs Division | reference entry read in full |
| S022 | Mikami, Y. (1913). The development of mathematics in China and Japan, with 67 figures in the text. Stechert & Co. [Digitised copy, University of Michigan, digitised by Google; Internet Archive identifier DevelopmentOfMat01MikaUMic, ark:/13960/s2g91gbjrg1]. https://archive.org/details/DevelopmentOfMat01MikaUMicNot cited in the story. It backs a register entry. | Internet Archive | not opened |
| S129 | Muir, T. (1906). The theory of determinants in the historical order of development (Vol. 1, 2nd ed.). Macmillan. Internet Archive, theoryofdetermin01muiruoft. https://archive.org/details/theoryofdetermin01muiruoftCited in Chapter 4 20 times, 1650 to 1800 5 times and 1800 to 1850 twice. | Internet Archive (University of Toronto scan) | partial (I read the introduction in full through MacTutor's transcription, and I read the returned passages from scan pages 13 to 40, 71 to 155, 276, 318...) |
| S390 | OurResearch. (n.d.). Unpaywall REST API (v2) [Data service]. Retrieved August 20, 2026, from https://api.unpaywall.org/v2/Cited in Chapter 5 and 1935 to 1960. | Unpaywall / OurResearch | reference entry read in full |
| S394 | Peano, G. (1888/2000). Geometric calculus: According to the Ausdehnungslehre of H. Grassmann (L. C. Kennedy, Trans.). Birkhauser. Internet Archive, geometriccalculu0000pean. https://archive.org/details/geometriccalculu0000peanNot cited in the story. It backs a register entry. | Internet Archive | not opened |
| S127 | Smith, D. E., & Mikami, Y. (1914). A history of Japanese mathematics. The Open Court Publishing Company. Internet Archive, historyofjapanes00smitiala. https://archive.org/details/historyofjapanes00smitialaCited in Chapter 4 5 times and 1650 to 1800. | Internet Archive (scanned from a Smithsonian Libraries copy) | partial (I searched the full text and read the returned passages from pages 135 to 141 and 295) |
| S421 | The Euler Archive. (n.d.). E53: Solutio problematis ad geometriam situs pertinentis; E530: Recherches sur une nouvelle espece de quarres magiques; Correspondence: Giovanni Marinoni. Retrieved August 20, 2026, from http://eulerarchive.maa.org/pages/E053.html; http://eulerarchive.maa.org/pages/E530.html; http://eulerarchive.maa.org/correspondence/correspondents/Marinoni.htmlCited in Chapter 7 10 times and 1650 to 1800 4 times. | The Euler Archive (Mathematical Association of America), legacy site | reference entry read in full |
| S070 | Todhunter, I. (1865). A history of the mathematical theory of probability: From the time of Pascal to that of Laplace. Macmillan. Internet Archive, ofmathemahistory00todhrich. https://archive.org/details/ofmathemahistory00todhrichCited in Chapter 3 6 times, 1400 to 1650, 1650 to 1800 3 times and 1850 to 1880. | Internet Archive, from the University of California Libraries copy | partial (The OCR stream is served from the beginning, so Chapter I (Cardan and Galileo) and Chapter II (Pascal and Fermat) were read) |
Appendix K
What is still open, and what to request
128 questions this book could not close, one list, by chapter. Every one names what was tried and what document would settle it.
This is the most useful page in the book for anybody who wants to do some history. Several of these are one library request away. A university library card, an interlibrary loan form and an afternoon would close more of them than another year of searching online, because the documents that would settle them are printed objects sitting in buildings.
8 of the 266 sources in the bibliography were never opened. They are registered because they exist and because a later reader may be able to get them, and they are marked not opened so that nothing in this book can rest on one.
| What is open | Chapter | What was tried, and what would settle it | Sources |
|---|---|---|---|
| Is Cantor's birth date Old Style or New Style? | 1 | He was born in St Petersburg on 3 March 1845, and Russia used the Julian calendar until 1918. Neither MacTutor (S062) nor Jourdain's introduction (S041, p. 23) says which calendar the date is. CONVENTIONS section 4 requires this to be settled before the timeline is final. What would settle it: a Russian civil or church register entry, or a Cantor biography that states the convention. Currently as unverified. | S062, S041 |
| What did Peano's cup and cap mean in 1888? | 1 | Claim A: Peano introduced the union and intersection symbols in Calcolo geometrico (Cajori via Miller, S054; MacTutor, S064). Claim B: in the 1889 Arithmetices principia, read directly, the same glyphs are glossed et and vel (S050). The gap-closing pass could not locate any open copy of the 1888 original; Kennedy's 2000 English translation is in copyright and lending-restricted. What would settle it: an openly readable scan of the 1888 Italian. Do not print "Peano invented the union and intersection symbols in 1888" as a bare fact. | S054, S064, S050 |
| Where is the "Je le vois" letter? | 1 | MacTutor gives the French and English and dates it to an 1877 letter to Dedekind (S062, quotations page), but supplies no archive, no printed edition, and no exact date. Dauben's paper does not contain the line at all (S053). What would settle it: the Cantor-Dedekind Briefwechsel edited by Noether and Cavailles (1937), or Cantor's Gesammelte Abhandlungen. Both were out of reach. as contested as to date and wording; the popular claim about its subject is separately wrong. | S062, S053 |
| 1850 or 1856 for De Morgan's paper? | 1 | Miller dates it 1850 (S055); the scanned volume 9 of the Transactions carries an 1856 imprint (S069). Cambridge Philosophical Society volumes were assembled from parts read on different dates, so both are probably right about different things. What would settle it: the "read on" date printed at the head of the paper. | S055, S069 |
| Are De Morgan's laws Ockham's? | 1 | Kneale and Kneale (1962), through Miller, say they "occur explicitly" in the Summa Totius Logicae (S055). The Latin of Part II was not read in this book; the Logic Museum carries only a summary. What would settle it: Summa Logicae Part II, chapters 32 and 33, in Latin. | S055 |
| When were Euler's letters published? | 1 | 1768 as a three volume book (Bennett, S051; SEP, S058) against "published between 1768 and 1772 as a three-volume book" (Klyve, S052). Bennett also gives 1795 for Hunter's first English translation, while the copy read here is Hunter's second edition of 1802 (S049). What would settle it: the imprint dates on the three original French volumes and a copy of the 1795 English imprint. | S051, S058, S052, S049 |
| Did Weil really choose the empty set symbol? | 1 | The only account is Weil's own, written in 1992 about a decision of the 1930s (S054). No contemporary document has been produced. What would settle it: the Bourbaki archives. | S054 |
| Did Zermelo find the paradox before Russell? | 1 | The Stanford Encyclopedia entry on Russell's paradox says Zermelo discovered a similar contradiction between 1897 and 1902, possibly first (S056); MacTutor's Zermelo biography does not mention it (S065). What would settle it: a dated document from Zermelo or the Gottingen circle. | S056, S065 |
| The Boole water-cure story. | 1 | MacTutor states that Mary Everest Boole's remedy of throwing water over her husband worsened his pneumonia, and gives no source (S060). Tagged as a legend; earliest attestation not established. This book does not repeat it. What would settle it: Desmond MacHale's biography of Boole. | S060 |
| Cantor's 1891 diagonal paper was never opened here. | 1 | Uber eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der DMV 1, pp. 75-78. No open scan was reachable this session. The chapter therefore names it and quotes nothing from it. All ten belong in the master conflict log. | |
| Did Pingala have the triangle? | 2 | Claim A: yes, the tradition begins with Pingala around 200 BC and the meru-prastara was put into triangular form by the tenth century (S011, following Cooke and Edwards). Claim B: no, Halayudha's construction cannot be derived from sutras 8.34 to 8.35, which state a doubling recursion, and the sutra that genuinely gives the binomial coefficients is one Halayudha does not quote (S002). What would settle it: a critical edition of Chandahsastra book 8 with manuscript variants, read against Halayudha's commentary. | S011, S002 |
| Who first used something we would call mathematical induction? | 2 | Pascal, 1654 to 1665, two lemmas and "a l'infiny" (S001); Levi ben Gershon, 1321, "the earliest rigorous use" (S004); al-Samaw'al, c. 1150, inductive force carried diagrammatically (S003). What would settle it: agreeing a definition of "rigorous induction" first. This is a definitional dispute dressed as a factual one, which is exactly why it belongs in a classroom. | S001, S004, S003 |
| When did Tartaglia die? | 2 | 13 December 1557 in Venice (S019, Britannica, which also gives an age) versus 1547 (S011, Miller, in the parenthesis "(1499-1547)"). Neither cites a document. What would settle it: a Venetian record, Tartaglia's will, or Masotti's standard biography. | S019, S011 |
| Which century is Mahavira's? | 2 | Ninth century for the Ganita-sara-sangraha (S009); Shah discusses a "Mahavira" algorithm in a seventh century position in his chronology (S002). What would settle it: checking whether Shah means the same author, and the Rashtrakuta court dating of Mahavira. | S009, S002 |
| Which Yang Hui book carries the triangle? | 2 | Xiangjie jiuzhang suanfa, 1261, which absorbed two thirds of Jia Xian's problems (S014), versus Jiuzhang suanfa zuanlei, said to contain "the oldest representation" of the array (Britannica, "Yang Hui"). What would settle it: the Chinese bibliographic literature, or Lam Lay Yong's studies of Yang Hui. | S014 |
| How did the triangle travel? | 2 | "Passed from China to India, then via Arab sources to Europe by the 16th century" (S017) versus no such chain, since Indian prosodic combinatorics predates Jia Xian and al-Karaji does too (S002, S003). Nothing available settles it. The honest answer for a textbook is largely independent discovery with unclear contact, and S017's sentence should not be repeated. | S017, S002, S003 |
| When does the binomial theorem appear in Persia and China? | 2 | "Around 1100" (S011) versus al-Karaji dead by about 1029 and Jia Xian flourishing about 1050 (S003, S014). What would settle it: distinguishing "a table of coefficients" from "the binomial theorem stated as a theorem." Miller may mean the latter. | S011, S003, S014 |
| Is Ibn Mun'im's the first book with a whole chapter on combinatorics? | 2 | Yes (S016) versus untested against Sanskrit prosody manuals, several of which treat the six pratyayas systematically and are far older (S002). What would settle it: deciding whether a prosody manual counts as a book of mathematics. | S016, S002 |
| Is Varahamihira's 174720 an error? | 2 | The text says 174720 (S010, ch. 77 v. 17); the translator says the arithmetic is wrong and the answer is 43680, or 28392 with restrictions (S010, translator's note). What would settle it: a second translation, plus a decision about whether repeated ingredients are allowed. The arithmetic itself is settled: the discrepancy is exactly a factor of 4 in the arrangement count. | S010 |
| Which chapter of the Brhat Samhita is the perfume chapter? | 2 | Iyer's translation gives 77; some listings give 76 (S010). What would settle it: a second printed edition. | S010 |
| 1654 or 1665 for the Traite? | 2 | Not a real conflict: 1654 is the composition date given by later reference works (S011), 1665 is the printing date on the title page (S001). Every timeline entry must say which event it means. | S011, S001 |
| "Combinations" in an English title, 1673 or 1685? | 2 | Miller gives 1673 for Wallis's Treatise of Algebra (S011); the book is usually dated 1685. Nobody in this book opened either title page, so this chapter does not print the year as a fact. | S011 |
| When Cardano wrote the Liber de ludo aleae. | 3 | MacTutor says "probably completed by 1563" with no document cited (S094); Todhunter says the manuscript date is unknown (S070, art. 3); Bowman says it was compiled over an approximate forty year span, also unsourced (S073). What would settle it: Ore's 1953 study, which used the manuscript tradition, or the statement of the editor of the 1663 Opera Omnia. Neither was reachable. Print no composition date without one. | S094, S070, S073 |
| When Galileo wrote the dice piece. | 3 | Gorroochurn calls it "his 1620 probability paper" with no source (S072); the York transcription of Thorne's translation gives no date at all (S071). What would settle it: the editorial apparatus of the 1898 Barbera Opere, volume 8. | S072, S071 |
| Thomas Bayes's birth date. | 3 | Bellhouse can only give the window July 1701 to April 1702 (S084); MacTutor prints 1702 flat with no reasoning (S094). What would settle it: nothing currently known. Print the window. The death date, by contrast, is settled at 7 April 1761 and is recorded as such. | S084, S094 |
| When Poisson coined "law of large numbers." | 3 | Miller says 1835, in a Comptes Rendus note (S088); Seneta quotes the sentence from page 7 of the 1837 Recherches (S080). Probably two uses by the same man. What would settle it: reading both. | S088, S080 |
| How many axioms Kolmogorov states. | 3 | Morrison's translation has five in Chapter I plus a continuity axiom in Chapter II (S090, pp. 2 to 3); Shafer and Vovk speak of six axioms (S089, p. 24). What would settle it: the 1933 German original, unread in this book. Until then, write "five, plus a sixth axiom of continuity in Chapter II." | S090, S089 |
| Who named the law of total probability. | 3 | Untraced. Miller has no entry on either the words page or the symbols page, both read in full for the relevant sections (S088). Kolmogorov's 1933 section heading, in translation, is the earliest attestation available (S090). What would settle it: a systematic search of nineteenth and early twentieth century textbooks. | S088, S090 |
| Who first gave the pairwise-but-not-mutual independence example. | 3 | It is standard to credit Sergei Bernstein. MacTutor confirms a fourth edition of his Probability Theory in 1946 (S094), but neither the book nor the specialist literature on the example could be opened. The mathematics is verified from first principles in `verify/domainC.py` Section 12; the attribution is. | S094 |
| Whether Bayes was the first to state his own theorem. | 3 | There is a minority view that the case for Bayes is weaker than people assume, argued by Stephen Stigler in 1983. We have not been able to read that paper: two independent aggregators record it as closed access, and every open route failed (S390, S391). So we are telling you the argument exists and not what it says. What would settle it: a library copy of The American Statistician 37(4), 290 to 296. | S390, S391 |
| The earliest telling of the de Moivre sleeping story. | 3 | Absent from Bellhouse and Genest (S082); present in MacTutor with no citation (S094). What was tried: a targeted search for an earlier printed source, which returned only modern retellings. What would settle it: a nineteenth century history that prints it with a reference. | S082, S094 |
| Todhunter on de Mere, two pages, two positions. | 3 | The note file records the page where Todhunter calls de Mere "a reputed gamester" who proposed the problem (S070, art. 11); the image manifest records a later page where he writes that "the name de Mere is not given in the passage we have quoted... a blank occurs" and that de Mere "was not the person alluded to by Pascal" (IMG-041). Both pages are Todhunter's. What would settle it: reading the whole of his Chapter II, which the OCR stream in this book delivered only in part. | S070 |
| 1763 or 1764 for the Essay. | 3 | The volume is dated 1763 and the article says "Read Dec. 23, 1763" (S083); Stigler and MacTutor say it was published in 1764 (S085, S094). Not a real conflict, but never print one date alone: separate the reading from the printing. | S083, S085, S094 |
| Russian dates. | 3 | MacTutor gives Chebyshev's and Bernstein's birth and death dates without Old Style or New Style labels, which this book's sourcing rule requires for Russian dates before 1918 (S094). Unresolved. | S094 |
| Cardano in his own words. | 3 | Ore's translation of the Liber de ludo aleae is a lending-only item on the Internet Archive, so every Cardano quotation in this chapter reaches us through Gorroochurn or Todhunter. Nothing here is quoted from the Latin. | |
| The date of the Nine Chapters. | 4 | Claim A: 200 BCE to 100 BCE (S122). Claim B: 200 BCE to 50 CE, with the units argument putting it shortly after 200 BCE (S123, and S124 agrees). Claim C: 100 BCE to 100 CE (S126). Claim D: "over 2000 years ago" (S120). What would settle it: the excavated Suan shu shu bamboo strips of about 186 BCE read against the internal unit evidence in a specialist critical edition, such as Chemla and Guo's, which nobody in this book could open. The instruction that stands is to print the range and say why it is a range. | S122, S123, S124, S126, S120 |
| Seki or Leibniz. | 4 | Claim A: Seki "was the first person to study determinants in 1683" and "Seki's version was the more general", stated by MacTutor with no source attached (S128). Claim B: Smith and Mikami praise Seki in detail and never mention Leibniz, so they make no priority claim at all (S127). Claim C: Muir starts the European story with Leibniz in 1693 and does not discuss Japan (S129). What would settle it: Hayashi's "The 'Fukudai' and Determinants in Japanese Mathematics", cited at S127 p. 136, read against the 1693 letter. | S128, S127, S129 |
| Maclaurin or Cramer. | 4 | Claim A: Maclaurin's 1748 Treatise of Algebra "contained the first published results on determinants" (S122). Claim B: Muir's volume 1 goes Leibniz, Fontaine, Cramer, and never mentions Maclaurin (S129). Claim C: Cramer states the general n by n rule, without proof, in 1750 (S130). What would settle it: a page image of the 1748 first edition of the "exterminating unknown quantities" chapter. The copy read for this book is a later edition, dated 1796 in the Internet Archive record (S132). | S122, S129, S130, S132 |
| Who first attached Gauss's name. | 4 | Claim A: Bessel, in an 1838 report on the East Prussian survey (S134). Claim B: George Forsythe in 1953 for the English phrase (S120). These may both be right about different languages and different objects. Note also that the project's own conflict log in `this book's register of words` marks the Bessel line as a garbled extraction and instructs writers not to print it as fact. This chapter therefore prints Forsythe and leaves Bessel here. What would settle it: reading Bessel's 1838 report, and checking Forsythe 1953 for the phrase. | S134, S120 |
| Did Cayley found matrix theory in 1858? | 4 | Claim A: the memoir gives the first abstract definition of a matrix and is the founding document (S122, and most textbooks). Claim B: its significance "has been grossly exaggerated" and it went unnoticed outside England until the 1880s, the ideas being already in Gauss 1801 and Eisenstein 1844 (S137). What would settle it: the citation study of the memoir before 1880 that Hawkins says he carried out. Present both. | S122, S137 |
| Frobenius, 1877 or 1878? | 4 | Claim A: a 63-page Crelle paper in 1877 (S137). Claim B: "Uber lineare Substitutionen und bilineare Formen", 1878 (S122). What would settle it: the title page and receipt date of Crelle volume 84. | S137, S122 |
| Clasen's initials. | 4 | Claim A: "J. B." (S134, citing Householder). Claim B: "B.-I." (S142). Claim C: no initials given (S144). What would settle it: the 1888 volume of the Annales de la Societe Scientifique de Bruxelles, pages 251 to 281. | S134, S142, S144 |
| What fangcheng means. | 4 | Claim A: "divided rectangle" (S124). Claim B: "rectangular array procedure", or simply "the system of linear equations" (S125). Claim C: "Calculation by square tables" (S123). All three are glosses, not translations of a settled term. What would settle it: a philological treatment of 方程. | S124, S125, S123 |
| Which chapter 8 problem is the classic one. | 4 | Grcar calls the grain-yield problem the first problem of chapter 8 (S120); Schwartz quotes the same three yields, 39, 34, and 26 dou, as problem 7 (S124). What would settle it: the Shen, Crossley, and Lun translation, which nobody here could open. This chapter therefore names the numbers and not the problem number. | S120, S124 |
| Gauss's own worked example, and the sign in it. | 4 | The note taken from the 1801 scan reads Gauss's example form as (3, 7, -8) and flags the sign as unread, because the OCR renders it "(3, 7, -. 8)" (S133). The verification script computes with (3, 7, 8) and gets bb minus ac equal to 25 (`verify/domainD_output.txt`, section 10). Those two cannot both be describing the same form. Nothing from that Latin page is quoted in this chapter without this flag attached, and no number from Gauss's example is printed in 4.7. What would settle it: a human eye on the page image at leaf 195 of the Internet Archive scan, which is the image cleared as IMG-023. | S133 |
| How long before his own paper did Hill file the patent. | 4 | Domain D's digest says "Five months before the issue appeared"; the S141 note says "Five days before that issue was even in print". Those cannot both be right, and neither is a documented fact. The documented facts are the filing date of 14 February 1929 (S141) and the June-July 1929 issue (S140), which are 107 days apart if the issue is dated 1 June (`verify/ch04.py`, section 5). This chapter prints the dates and the word "months". | S141, S140 |
| Hill's own alphabet table and his own worked example. | 4 | The OCR of the 1929 scan is internally inconsistent, since it reads n = 13 and 13 shares a factor with 26, so it cannot be one of Hill's twelve primary letters (S140). His worked example could not be recomputed. What would settle it: a clean scan or a library copy of The American Mathematical Monthly 36(6). Until then no letter-to-number table from Hill is printed anywhere in this book. | S140 |
| Cayley's death year. | 4 | Dictionary.com's American entry says 1895 and its British entry says 1893, in the same source (S143). The standard date is 26 January 1895, so the British entry is simply wrong. This chapter prints 1895 and the 1893 figure should never be used. | S143 |
| Cauchy: 1826 or 1829? | 5 | Claim A: the eigenvalue results, the word "tableau", and the diagonalization of every real symmetric matrix belong to 1826, and MacTutor does not mention 1829 at all (S180). Claim B: "secular" attaches to "Cauchy's 1829 work on symmetric determinants" (S176), and the standard citation is his 1829 memoir "Sur l'equation a l'aide de laquelle on determine les inegalites seculaires des mouvements des planetes". These are probably two papers in one body of work, but that is an inference. What would settle it: the 1826 and 1829 items in Cauchy's Oeuvres completes, Serie 2, or Thomas Hawkins's "Cauchy and the spectral theory of matrices" (Historia Mathematica 2, 1975). Gallica is unreachable from the research environment and Hawkins is only on ScienceDirect, which is blocked. | S180, S176 |
| Beltrami's death year. | 5 | Claim A: 1899 (Stewart's abstract, S170). Claim B: 18 February 1900 in Rome (MacTutor, S201). Claim C: 1835 to 1900 (Dictionary.com, S204). Two sources to one, and only one of them gives a full date and a place. This chapter prints 18 February 1900. What would settle it: a primary record from the Accademia dei Lincei or an Italian biographical dictionary. | S170, S201, S204 |
| Banach: 1920 or 1922? | 5 | Claim A: the fully axiomatic approach is in the 1920 doctoral dissertation (S179). Claim B: the 1922 Fundamenta Mathematicae paper established "sets of elements of which I will postulate certain properties" (S173, p. 254). These are two different documents and both may be right. The 1922 paper itself could not be opened: the ICM virtual library serves a JavaScript wrapper and the EuDML direct URL returns 404. What would settle it: reading both. | S179, S173 |
| Kublanovskaya's doctorate: 1948 or 1955? | 5 | Claim A: "She obtained her doctorate in 1948" (S193). Claim B: she completed her degree at Leningrad State University in 1948 and received the candidate's degree, the Soviet doctorate, in 1955 (S194). MacTutor's version distinguishes the two things and is internally more coherent, and this chapter follows it. What would settle it: LOMI or St Petersburg University records. | S193, S194 |
| Krokino or Krokhono? | 5 | Two transliterations of one village name in Vologda Oblast (S193 versus S194). Not a conflict of substance; both are recorded in 5.5. | S193, S194 |
| Golub and Kahan's page range. | 5 | Claim A: SIAM J. Numer. Anal. 2, 202 to 224 (S200). Claim B: 205 to 224, which most of the citing literature gives. One of the two is a typo, so this chapter prints no page range for that item. What would settle it: the printed volume; epubs.siam.org is disallowed by robots.txt here. | S200 |
| The reception of Grassmann's 1862 edition. | 5 | Claim A: it "met with no more success than the first one" (S173). Claim B: it "fared no better" (S181). Claim C: it "met with even less attention than the first" (S177). All three agree it failed and Crowe's is the strongest, so this chapter attributes it to Crowe rather than asserting it flat. | S173, S181, S177 |
| What is in chapter 9 of Peano 1888. | 5 | Nobody working on this book has opened the Italian original: the Google Books full-view scan, digitized from Sapienza University of Rome, returned metadata only; mathematica.sns.it presents a self-signed certificate; and Kennedy's English translation is in copyright and lending-restricted (S182). Kennedy's own page reference came back as "141-142" on one fetch of the same PDF and "38-39" on another, so no page number for the passage is printed here. Also unresolved: MacTutor and Dorier both say Peano defined dimension and linear independence in the 1888 book, but Kennedy's four translated definitions do not include them, so they are probably elsewhere in the same chapter (S173, S179, S182). What would settle it: a library copy of the 1888 book or of Kannenberg's Birkhauser translation. | S182, S173, S179 |
| Kummer's verdict: one report or two? | 5 | Claim A: Kummer reported that Grassmann's 1846 prize essay contained "commendably good material expressed in a deficient form", and that this ended his chances of a chair (S181, and S172 agrees on the wording). Claim B: "Ernst Kummer's 1847 report judged it 'a failure; for... it lacks everywhere a suitable organization of its content'" (S173, p. 243), where "it" reads as the Ausdehnungslehre. Those may be two renderings of one 1847 report on the prize essay, or two separate judgements on two different objects. This chapter quotes the first, attributes the second, and does not merge them. What would settle it: the text of Kummer's report. | S181, S172, S173 |
| How long did Hamilton search? | 5 | Claim A: the problem had haunted him for roughly fifteen years (S175, his own letter). Claim B: he "searched for thirteen years" (S177). This chapter avoids the number and gives the dates instead. | S175, S177 |
| Grassmann's Law: where was it published? | 5 | No source opened in this pass gives the citation of Grassmann's 1863 linguistics paper. Langendoen confirms the law is real, applies to Sanskrit and Greek, and was still an active research object in 1966, but cites only secondary literature (S203); MacTutor names the law without a date or a paper (S181). The usual citation, Zeitschrift fur vergleichende Sprachforschung 12 (1863), 81 to 138, is unverified here and is deliberately not printed. | S203, S181 |
| Where does Eigenwert first appear? | 5 | The 1912 Teubner collected volume was read only in its front matter, through OCR, so the sentence in the 1904 first Mitteilung in which Eigenwert first occurs was NOT read (S184). The claim that the coinage is there rests on Miller (S176); the pagination, 1904, pages 49 to 91, is independently confirmed from the book's own front matter (S184). What would settle it: the GDZ page images of the 1904 Nachrichten article. IMG-042 is a page of the 1912 collected volume, not of the 1904 original, and the caption must say so. | S184, S176 |
| Turing 1948, unread. | 5 | Everything this chapter says about the contents of "Rounding-off errors in matrix processes" comes through Dopico (S189). The paper is nominally open access but the publisher's link serves a landing page and an abstract only, confirmed twice (S390). Two bibliographic facts are safe and are used: the receipt date of 4 November 1947 and the National Physical Laboratory affiliation. Nothing from the body of the paper is quoted. | S189, S390 |
| Sylvester 1883, journal volume. | 5 | Higham gives the paper's title, Miller gives the year, and the two agree (S176, S178), but neither source captured here gives the full journal reference, so none is printed. Stewart, separately, does not connect Sylvester's 1883 "latent roots" to his 1889 SVD papers, and that relationship is unsettled (S170). | S176, S178, S170 |
| Beltrami's 1873 pages. | 5 | Stewart gives 98 to 106; other secondary literature sometimes gives 98 to 108 (S170). The Italian journal itself was not reachable. | S170 |
| The 1854 Grassmann and Cauchy priority committee. | 5 | MacTutor asserts the 1853 "clefs algebriques" episode and the 1854 committee with no source inside the entry, and comments only "We still await the committee's report!" (S179). It is repeated here as MacTutor's claim, not as an established fact, and it needs a scholarly citation before it goes to print. | S179 |
| "Linear algebra" in English: 1870 or 1881? | 5 | Claim A: Miller's l.html, as read for S176, dates the defining sentence in the American Journal of Mathematics to 1870 and credits the OED. Claim B: the same page, as read for S302, places that sentence in volume 4 of the same journal, dated 1881 (S302, l.html; this book's register of words, Conflicts). The two cannot both describe one printing. What would settle it: the OED entry, or a second reading of Miller's l.html. The 5.6 row is tagged as contested and prints both. | S176, S302 |
| Who first printed the phrase "Markov chain". | 6 | Claim A: S. N. Bernstein, "Sur l'extension du theoreme limite du calcul des probabilites", Mathematische Annalen 97 (1926), 1 to 59, per Basharin, Langville, and Naumov, who write that "as early as 1926, just twenty years after his initial discoveries, a paper by Russian mathematician S. N. Bernstein used the phrase 'Markov chain'" (S212, ref. 3). Claim B: Romanovsky, 1929, for the French, and American Mathematical Monthly 45 (1938), p. 410, for the English (S216, S244). Claim C: Dictionary.com's "First recorded in 1940-45" (S245), which is contradicted by Miller's page citation and should be discarded. What would settle it: a human eye on the page images of Mathematische Annalen 97. The Gottingen digitization exists but serves images with no text layer, and the alternative host closed on 31 December 2025. Do not print a bare "the phrase was first used in 1926". | S212, S216, S244, S245 |
| When Doeblin's envelope was opened. | 6 | Claim A: "the spring of 2000" (Bru, S232, who ran the operation). Claim B: May 2000 (MacTutor, S233). Claim C: 1991, following Bru's discovery of Doeblin's reference to it (Zentralblatt, S234). May is in spring, so A and B agree and this chapter prints May 2000. What would settle it: the December 2000 Comptes rendus special volume, or Bru and Yor's 2002 survey in Finance and Stochastics, both paywalled. | S232, S233, S234 |
| Bru's "two months". | 6 | Bru writes "Two months later, 21 June 1940, the soldier Doblin killed himself" after giving the deposit date as 26 February 1940 (S232). Those two dates are 116 days apart (`verify/ch06.py`, section 7). Either the phrase is loose or it counts from a different event. This chapter prints the two dates and the day count. | S232 |
| Was Markov's excommunication request granted. | 6 | Claim A: granted (Hayes, S210, p. 93). Claim B: refused, with the Synod resolving that he had seceded and an internal comment saying it "would be too honourable for Markov" (Sheynin, S217). Sheynin has the archives, through Emeliakh's 1954 study, and this chapter follows him. What would settle it: reading Emeliakh, Voprosy Istorii Religii i Ateizma 2 (1954), 397 to 411. | S210, S217 |
| Why February 1912. | 6 | Claim A: solidarity with Tolstoy (the popular framing, and Hayes puts the two in one sentence, S210). Claim B: the Beilis prosecution in Kiev (Sheynin's proposal, S217), which is his inference, not a documented statement of Markov's motive. What would settle it: the text of Markov's own letter, quoted in Emeliakh. | S210, S217 |
| Hayes's pair counts. | 6 | Claim A: 1,104 vowel-vowel, 3,827 consonant-consonant, and 15,069 mixed (S210, p. 95). Claim B: a 20,000-letter run has 19,999 overlapping pairs, so the mixed count is 15,068 (F-2.1 to F-2.3). What would settle it: Markov's own table in the 1913 paper, which this book has not seen in the original Russian. The chapter uses only Markov's , , , and . | S210 |
| Kolmogorov 1931 or 1938. | 6 | Claim A: "Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung", Math. Annalen 104 (1931), 415 to 458, dated 26 July 1930 (S236, S216, S246). Claim B: MacTutor's "Analytic methods in probability theory... 1938" (S237). Almost certainly two items, the German paper and a later Russian version. Do not print 1938 for the Mathematische Annalen paper. | S236, S216, S246, S237 |
| Which paper carries Kolmogorov's cycle criterion. | 6 | It is usually attributed to "Zur Theorie der Markoffschen Ketten", Math. Ann. 112 (1936), 155 to 160, and sometimes to "Zur Umkehrbarkeit der statistischen Naturgesetze", Math. Ann. 113 (1937), 766 to 772. Neither scan was opened in this book (S246). The criterion is stated in this chapter as mathematics, verified in code, and not attributed to a specific paper. | S246 |
| Does Chapman deserve half the name. | 6 | Claim A: the name is standard and Shafer and Vovk use it (S236) while citing no Chapman paper. Claim B: MacTutor's biography of Chapman does not mention the equation and lists his fields as gas dynamics, geomagnetism, and the ionosphere (S238). What would settle it: Chapman, S. (1928), Proc. Roy. Soc. A 119(781), 34 to 54, which returns 403 to this book's fetcher. | S236, S238 |
| Onsager's reciprocal relations, 1931 or 1932. | 6 | Claim A: 1931 (Yablonsky, Gorban, and co-authors, S231, a peer reviewed letter). Claim B: 1932 (Gorban's workshop slides, S230, by one of the same authors). 1931 is better supported and is what this chapter prints. What would settle it: Physical Review 37, 405 to 426, and 38, 2265 to 2279. | S231, S230 |
| The Ehrenfests' joint encyclopedia article. | 6 | Claim A: 1912 (MacTutor, S222). Claim B: commissioned by Klein in 1906 (van der Heijden, S223), and commonly cited as 1911 in the physics literature. What would settle it: the fascicle itself, Encyklopadie der mathematischen Wissenschaften, Band IV, Teil 32. This chapter gives only the 1906 commission and the 1907 urn paper, both of which are sourced. | S222, S223 |
| The ILLIAC Suite's fourth movement. | 6 | Claim A: it used "the Markov Chain Monte Carlo method" (Illinois Distributed Museum, S242). Claim B: it used "Markov chains (zero and first order) for interval and harmony selection" (MIT OpenCourseWare, following Hiller's own description, S241). B is almost certainly right and A is an anachronism. What would settle it: Hiller and Isaacson, Experimental Music (McGraw-Hill, 1959). | S242, S241 |
| Nekrasov's own words. | 6 | Nothing quoted in this chapter is Nekrasov's own statement of the free-will argument. The reconstruction is Basharin, Langville, and Naumov's (S212, preprint pp. 6 and 11), and Seneta is careful to say Markov "interpreted" Nekrasov that way (S211). What would settle it: Seneta, E. (2003), "Statistical regularity and free will: Quetelet and Nekrasov", International Statistical Review 71(2), 319 to 334, which two independent aggregators confirm is closed access, or Nekrasov's 1902 text itself. | S212, S211 |
| A slip in this book's own note file, now corrected downstream. | 6 | The S221 note records Kac's balanced-state return time as "about 100/sqrt(pi) seconds, about 175 seconds", and `sources/-markov-chains.md` repeated it until it was corrected on 2026-08-20. Those two are not the same number: is 56.4, and is 177.2. The exact value, divided by , is 177.25 seconds, so the multiplication is the right reading and the division is a transcription slip somewhere between Kac's page and the digest. What would settle it: Kac's printed formula on p. 379 region of S221. This chapter prints the exact 177.25 and Kac's "about 175". | S221 |
| The bold pi. | 6 | No source reached in this book dates the reuse of pi for a stationary distribution (`this book's register of glyphs`). The symbol is in Appendix B.1 and its history is blank. | |
| The image clearance. | 6 | See 6.9. Domain F's single cleared image is not cleared, and until somebody clears the 1907 contents page from a license-stating repository this chapter is illustrated entirely by commissioned drawings. | |
| How many configurations are in the Appel and Haken unavoidable set. | 7 | Claim A: "fewer than 2000 configurations, each of ring size fourteen or smaller", the authors' own words (S426, p. 711). Claim B: "approximately 1500 configurations", with 1200 hours of computer time (S423). Claim C: the proof "analyzed 1,936 cases" (S424). What would settle it: the two 1977 papers in the Illinois Journal of Mathematics 21, which were not read because Project Euclid served an Incapsula challenge. Until then this book quotes only the authors' phrase and names who says what else. | S426, S423, S424 |
| Ten years or eleven. | 7 | Claim A: Heawood found the error in 1890, eleven years after 1879 (S423, S424). Claim B: the Royal Society "only realized ten years later" (S427, p. 1382). The arithmetic settles it at eleven (J-13.4); Gonthier is writing loosely. Recorded because the discrepancy is in print. | S423, S424, S427 |
| The title of Heawood's 1890 paper. | 7 | Claim A: "Map colour theorems" (S425). Claim B: "Map colouring theorem" (S423). The standard citation is "Map colour theorem", singular, Quarterly Journal of Pure and Applied Mathematics 24 (1890), 332 to 338. What would settle it: the journal. It was not read. | S425, S423 |
| Sylvester's 1878 note itself. | 7 | The "graph" claim rests entirely on Jeff Miller's Earliest Uses, which quotes Sylvester's sentence with a locator in the Collected Mathematical Papers III, pp. 103 to 104 (S302). The Nature page was reached and returned only the publisher's summary paragraph, which is an abstract and not the note, so under this book's rules it is not registered as read. What would settle it: Nature 17, p. 284, or the Collected Papers volume. | S302 |
| Kirchhoff's 1847 paper. | 7 | The matrix tree theorem is verified as mathematics against brute force on six graphs (J-4). Its attribution to Kirchhoff's paper in Annalen der Physik und Chemie 72, pp. 497 to 508, is unverified: an OpenAlex lookup returned HTTP 429 twice, Wiley was unreachable, and the Internet Archive timed out. MacTutor's biography of Kirchhoff does not mention trees or graph theory at all (S429). What would settle it: the journal volume. | S429 |
| Cayley or Sylvester for the word "tree". | 7 | Claim A: Cayley 1857, which is the OED2 attestation through Miller (S302). Claim B: Sylvester coined it, a claim Miller records but sources only to an unnamed internet site (S302). This book prints Cayley 1857. What would settle it: Sylvester's Collected Mathematical Papers, vol. III, and the 1878 American Journal of Mathematics memoir, neither read here. | S302 |
| The Icosian game. | 7 | Everything about Hamilton's Icosian game of 1857, the Icosian calculus, and the story that he sold the game to a dealer is unverified in this book. MacTutor's Hamilton biography carries nothing beyond an image link and the Trinity College Dublin pages strip their links in the fetcher. What would settle it: the Trinity College Dublin Hamilton archive, or Biggs, Lloyd, and Wilson, Graph Theory 1736-1936. | |
| Who found de Bruijn sequences first. | 7 | Claim A: de Bruijn, 1946 (S444). Claim B: C. Flye Sainte-Marie, 1894, which is the attribution de Bruijn himself acknowledged in a note of 1975. That note 404s at both paths tried. This matters, because Claim B appears to be correct and the name on the object is Claim A. What would settle it: de Bruijn's 1975 note, or Flye Sainte-Marie's original. | S444 |
| The Sanskrit mnemonic. | 7 | The arithmetic of yamatarajabhanasalagam is verified exactly (J-12.4, J-12.5) and the history is not verified at all: no manuscript, no dated attestation, no scholarly edition. Subhash Kak's note in the Indian Journal of History of Science 35(2) (2000), 123 to 127, 404s at the path tried, and Rachel Hall's "Math for Poets and Drummers", which was opened, does not mention the mnemonic (S444). What would settle it: Kak's note, and a critical edition of the prosodic tradition. Print the arithmetic; flag the history. | S444 |
| Milgram's own numbers. | 7 | Kleinfeld's archival figures were read and are printed here (S440). Travers and Milgram (1969) and Milgram (1967) were reached as image scans with no text layer, so their reported medians and starter counts are unverified here. Do not print "5.2 intermediaries" or "296 starters" from this book. What would settle it: a text copy of Sociometry 32, 425 to 443. | S440 |
| Parker's initials. | 7 | Claim A: E. C. Parker (S443). Claim B: E. T. Parker, from the Crossref record for the 1960 Canadian Journal of Mathematics paper. Use E. T. | S443 |
| How long Euler's Latin square conjecture stood, and when it fell. | 7 | MacTutor says 175 years; 1782 to 1959 is 177 (J-13.8). The Euler Archive's E530 page says the conjecture "was later disproven in 1970", which contradicts a paper received on 10 April 1959 and a newspaper story of 26 April 1959 (S421, S442, S443). Nothing needs settling; the catalog page is wrong and must not be cited for that date. | S421, S442, S443 |
| The New York Times front page of 26 April 1959. | 7 | Not seen. The claim rests on MacTutor alone, and the headline and byline are unverified (S443). | S443 |
| The other random graph model. | 7 | Gilbert's 1959 paper "Random graphs", Annals of Mathematical Statistics 30, 1141 to 1144, which defines the model where each edge appears independently with probability p, was not read: Project Euclid served an Incapsula challenge. Anatol Rapoport's earlier work on random nets was not reached either. Both are usually skipped in retellings, and both were skipped here for lack of access rather than lack of interest. | |
| Two page ranges and one phrase, in the Erdos and Renyi papers. | 7 | The page ranges 290 to 297 and 17 to 61 come from the standard citations, not from the offprints' own pagination, which was not legible (S436, S437). And whether the authors themselves write "giant component" in the 1960 paper was not confirmed from the scan, so this book uses the phrase as later standard usage and does not attribute it to them. | S436, S437 |
| Hamming 1950 and Huffman 1952. | 7 | Neither is registered as a source in this book. Two hosts for Hamming's Bell System Technical Journal paper failed robots.txt; the Huffman file reached is an image scan with no text layer, yielding only the citation Proceedings of the IRE 40, September 1952, pp. 1098 onwards. The story that Huffman produced his code as a student assignment, in place of a final examination, is unverified here. What would settle it: library copies of both. | |
| Esther Klein Szekeres's life dates. | 7 | Her death date, 28 August 2005, is sourced (S434). Her birth date is not: the two MacTutor URLs for her both return 404. What would settle it: an obituary or the Australian Academy of Science memoir. | S434 |
| The coffee line. | 7 | Claim A: it is Renyi's (S438). Claim B: popular usage overwhelmingly gives it to Erdos. No dated source in either man's hand was located. Tagged as a legend in student prose. All eighteen belong in the master conflict log. | S438 |
| When was Fasenmyer's doctorate? | 8 | Claim A: a "doctoral dissertation of 1945", bibliography key [Fase45] (S255, pp. 18 and 55). Claim B: PhD conferred June 1946 at the University of Michigan, doctoral study autumn 1942 to June 1946 (S254). Claim C: PhD 1946, with the university ambiguously implied as Pittsburgh (S253). The likely reconciliation is a 1945 dissertation and a June 1946 degree, but that is inference. What would settle it: the University of Michigan dissertation record or the thesis title page. | S255, S254, S253 |
| What did Princeton do to Blackwell in 1941? | 8 | Three tellings, escalating: S251 p. 2 (Doob intervened to secure privileges), S250 (the president objected to honorary faculty membership, "abusing the hospitality of the University"), S252 (the president wrote of "abusing the University's hospitality by admitting a black" and "organized a great protestation"). This chapter prints the mildest and reports the others. What would settle it: the correspondence in the Institute for Advanced Study archive. | S251, S250, S252 |
| How old was Arianna Rosenbluth at her Harvard PhD? | 8 | Claim A: 22 (S257). Claim B: 21, from the dispatch brief with no source. Born 15 September 1927 with a 1949 degree, both are arithmetically possible, and `verify/domainG.py` confirms it: 21 if conferred before 15 September 1949, 22 on or after. What would settle it: the Harvard conferral date. | S257 |
| How many co-authors did Erdos have? | 8 | Claim A: "almost 500" (S263, p. 1). Claim B: 485 (S263, p. 32). Claim C: 514 as of August 2025 (S264). Resolved by dating, not by choosing: joint papers appeared after his death. Any figure printed must carry its date. | S263, S264 |
| Weinberg, 1908. | 8 | Hardy's letter is read in full (S388). Weinberg's German paper was not reached, and neither was Edwards's 2008 Genetics article on the simultaneity: esp.org returned robots.txt failures on three attempts, Europe PMC's REST endpoint was rejected by the proxy, and NCBI's efetch reports that the publisher blocks XML full text. What would settle it: a library copy of either. Until then this chapter does not say Weinberg discovered it independently, and gives him no date, journal, or month. | S388 |
| Note G itself. | 8 | Nobody in this book has read Note G or seen its table. The Fourmilab transcription was fetched twice and truncated at Note C both times, and archive.org was refused by the egress proxy in the research for this book (S260). What would settle it: a scan of Scientific Memoirs, volume 3. The page range 666 to 731 given in 8.11 comes from standard bibliographies, not from anything read here. | S260 |
| Who wrote the 1843 Notes. | 8 | Claim A: Bromley and Stein, that Lovelace was mathematically incompetent and Babbage wrote them. Claim B: Misa (2016) and Hollings, Martin, and Rice (2017), that it was a real collaboration (S258, p. 204). What would settle it: the Babbage-Lovelace working correspondence of summer 1843 in the British Library, which those authors use and this book did not open. | S258 |
| Was Weil nearly executed in Finland? | 8 | MacTutor reports it through Nevanlinna's own later memoir (S277). No corroborating archival record was reached, and some historians treat the detail as embellished. What would settle it: a Finnish archival record of the arrest. | S277 |
| Which event is "the Illiac Suite, 1956"? | 8 | Claim A: movement 1 composed August 1956, complete November 1956 (S272). Claim B: premiere of the first three movements 9 August 1956, complete by the end of 1956 (S273). Claim C: the score is conventionally dated 1957, which is its publication year, and no source read states that explicitly. These are three different events. This chapter names the premiere and its date and says so. | S272, S273 |
| The Kerala school is deliberately not in this book. | 8 | Madhava of Sangamagrama, 1350 to 1425, worked out series equivalent to the Maclaurin expansions for sine, cosine, and arctangent around 1400, roughly 250 years before Newton and Leibniz, and every one of his mathematical manuscripts is lost; we know what he did because Nilakantha and Jyesthadeva wrote it down (S284). The research pass investigated whether that connects to this course and concluded that it does not: his achievement is infinite series, which belongs to the prehistory of calculus, not to probability, matrices, or counting. Including it here would have been decorative, which is exactly the kind of tokenism a serious book should refuse. The genuinely relevant Indian material is in Chapter 2, where Pingala, Varahamihira, Mahavira, and the Ganita Kaumudi carry real weight (S002, S009, S010, and S020, listed in Chapter 2's sources). | S284, S002, S009, S010, S020 |
| F. N. David's death date, and her name. | 8 | Her birth is 23 August 1909 (S265); the 1989 interview does not give a death date and no source read here supplies one. The widely repeated claim that she was named after Florence Nightingale because her parents were friends of the Nightingale family is also absent from the interview. What would settle both: an obituary in a statistics journal. | S265 |
| What movement IV of the Illiac Suite used. | 8 | Claim A: "Markov chains (zero and first order) for interval and harmony selection" (S272). Claim B: "the Markov Chain Monte Carlo method" (S273). These are not the same thing: generating notes from a transition matrix is not MCMC. This chapter prints Claim A. What would settle it: Hiller and Isaacson's Experimental Music (1959), not opened here. | S272, S273 |
| Who coined "polytope". | 8 | MacTutor credits Alicia Boole Stott (S266). Jeff Miller's Earliest Uses, which this book uses for every other etymology, was not checked for this word. What would settle it: the relevant Miller entry, or her 1900 paper. | S266 |
| Cartwright's wartime equation. | 8 | The standard account says Mary Cartwright and Littlewood were working on the van der Pol equation. MacTutor as read says only "modelling radio and radar work" and does not name it (S280). This chapter therefore does not name it either, and does not otherwise use her story, which belongs with the dynamical-systems material rather than here. | S280 |
| The bracket on Ettingshausen's page 38. | 9 | Claim A: the year is 1827, from the title page and the preface dateline "Wien, im Sommer 1827," and Cajori's page 38 is about the binomial theorem (S387; S301, vol. 2, sect. 439, p. 63). Claim B: nothing. The date is settled. What is not settled is the glyph, because the OCR of an 1827 German scan cannot render a two-line bracket, so the symbol on that page has never been seen (S387, section 10). What would settle it: the page image, once the item is rights-cleared. | S387, S301 |
| Kramp's 1808 sentence, in Kramp. | 9 | Claim A: Kramp wrote "Je me sers de la notation tres simple n! pour designer le produit" in the "Notations" section of Elemens d'arithmetique universelle (Cologne, 1808), per Cajori (S301, vol. 2, sect. 448, p. 72). Claim B: none, but no copy of Kramp's book was located in this book, and the Cajori OCR renders the exclamation point as a backslash, so nobody has seen the mark in either book (S301, section 10; this book's register of words, "What could not be reached"). What would settle it: a scan of the 1808 Cologne edition, or the page image of Cajori vol. 2, p. 72, cleared as IMG-031. | S301 |
| Cajori's own arithmetic on the Jarrett sign. | 9 | Claim A: after 1827 "for a quarter of a century the notation was neglected" (S301, vol. 2, sect. 449, p. 74). Claim B: the same paragraph dates the revival to Goodwin in 1846, nineteen years later, and the spread to Todhunter about 1860, thirty-three years later (S301, same page). Twenty-five, nineteen, and thirty-three are three different numbers (`verify/ch09_output.txt`, section 2). What would settle it: nothing. The phrase is loose and the dates are precise, so this book prints the dates. | S301 |
| Peano's cup and cap in 1888. | 9 | Claim A: Peano introduced the union and intersection symbols in Calcolo geometrico (1888), per Miller citing Cajori vol. 2, p. 298 (S054). Claim B: the 1889 Arithmetices principia, read directly in this book, glosses the same glyphs in Latin as et and vel, logical "and" and "or," which is not union and intersection of sets (S050). What would settle it: the 1888 Italian original, which records as unreachable; no open copy exists within reach and Kennedy's English translation is in copyright and lending-restricted (S394). Do not print "Peano invented the union and intersection symbols in 1888" as a bare fact. | S054, S050, S394 |
| The empty set glyph. | 9 | Claim A: Andre Weil chose it from the Norwegian alphabet, being the only Bourbaki member who read Norwegian, stated in his 1992 autobiography at p. 114 (S054). Claim B: the only printed evidence is Bourbaki, Elements de mathematique (Paris, 1939), p. 4, fifty-three years earlier, and no contemporary document supports the attribution; Miller's own note says the memory is of a slightly earlier decision than the 1939 fascicule (S054, section 10). What would settle it: Bourbaki's internal minutes in the Archives Bourbaki. | S054 |
| The first printing of "Markov chain." | 9 | Claim A: Miller gives Romanovsky's French "les chaines de Markoff" in 1929 and the English at American Mathematical Monthly 45 (1938), p. 410 (S302, m.html). Claim B: three modern historians of the subject say Bernstein used the phrase in Mathematische Annalen 97 (1926). What would settle it: a human eye on the page images of that volume, or a library copy. records the Gottingen digitization as image-only with no readable text layer, and instructs this book not to print a bare "first used in 1926." | S302 |
| Cajori's volume 2 and its own year. | 9 | Claim A: the Internet Archive record for the Wellcome copy gives 1928 (S301, item record). Claim B: volume 2 is conventionally 1929, with volume 1 in 1928. What would settle it: the title page image of that copy. This chapter dates volume 1 to 1928 and volume 2 to 1929, and flags it here. | S301 |
| The year of the Port-Royal Logic. | 9 | Claim A: probability in its modern sense appears in the last chapter of Arnauld and Nicole, La Logique, ou l'Art de Penser, 1662 (S302, p.html). Claim B: the same Miller entry, as retrieved, gives 1682 in another place (S302, p.html). What would settle it: the title page of the edition Miller cites. 1662 is the standard first-edition date, and the row in 9.6 is tagged as contested on the year. | S302 |
| Euler's sigma, page 23 or page 27. | 9 | Claim A: Institutiones calculi differentialis (St Petersburg, 1755), cap. I, sect. 26, at page 23 (S305). Claim B: other sources give page 27 for the same paragraph (S305, section 10). What would settle it: the 1755 St Petersburg edition, or the Opera Omnia. | S305 |
| Double vertical bars around a matrix. | 9 | Claim A: Cayley introduced them in 1843 and 1845 (S301, vol. 2, sect. 468, p. 102). Claim B: they arrive in English with MacDuffee's The Theory of Matrices (1933) and Wedderburn's Lectures on Matrices (1934) (S304). What would settle it: nothing, because these answer different questions, first invention against adoption in English textbooks. Both rows stay, and any sentence built on them must say which question it is answering. | S301, S304 |
| "Scalar," and "set." | 9 | For scalar, claim A: the mathematical noun is Hamilton's, 1846 (S306, scalar); claim B: Viete used magnitudines scalares in 1646, two hundred years earlier, in a different sense, "of a ladder, in geometric proportion" (S302, s.html). For set, claim A: Frend's "set of numbers," 1796; claim B: Hamilton's "set" and "theory of sets" of 1853, meaning an n-tuple; claim C: E. H. Moore fixing the modern equivalence in 1901 (all S302, s.html). What would settle either: nothing. They answer different questions, and the book has to say which one it is asking. | S306, S302 |
| Bernoulli's page 213. | 9 | Claim A: Ars Conjectandi (Basel, 1713), p. 213, carries "Ars Conjectandi sive Stochastice...", cited by Chuprov in 1923 and quoted by Miller from Chuprov (S302, s.html). Claim B: none, but the 1713 scan on the Internet Archive was searched twice for "Stochastice" and "Stochastic" and returned nothing, which is an OCR failure on early modern Latin type and not evidence of absence (S302, section 10). What would settle it: the page image, which is cleared as IMG-007. | S302 |
| Seven glyphs with no date at all. | 9 | Lambda for an eigenvalue, bold pi for a stationary distribution, the complement superscript, the vertical bars for cardinality, the double-subscript entry notation, the arrow over a handwritten vector, and the double-bar norm. Each was queried against Miller's four symbol pages and against Cajori volume 2, and none of those sources dates it (this book's register of glyphs, "What could not be dated"). Four of the seven are in Appendix B.1, which means the course uses symbols this book cannot date. What would settle it: Cajori volume 2, paragraphs 490 to 520, which were never searched. | |
| Three words on Miller's letter C page. | 9 | CONDITIONAL PROBABILITY, CHANCE, and COROLLARY were never reached, because the page truncates before them in both the Tripod original and the St Andrews mirror (S302, section 10). Conditional probability is a glossary headword and a Unit 2 section title. What would settle it: a working copy of that page, or the Cajori and OED entries behind it. | S302 |
| The Bessel line. | 9 | An extraction from Miller's g.html appeared to attribute the first naming of "Gaussian elimination" to Bessel in 1838, but the line did not read cleanly and the page truncated before it could be re-checked (S302, section 10). It is tagged as unverified in this book's register of words with the instruction not to print it, and this chapter does not. What would settle it: a second reading of Miller's g.html. | S302 |
Part III
Glossary
142 terms a 14 to 18 year old reader, or a teacher from another subject, may not know. Mathematical terms are defined as this book uses them, not in full generality, and the parenthesis after each word is where it came from.
(Latin)
Not established for this book; Miller has no entry for the Markov-chain sense
IdeaChapter 9
(Latin)
1690s in English in the general sense; no mathematical first use established here
IdeaChapter 9
(the first letter of the Hebrew alphabet)
Beitrage zur Begrundung der transfiniten Mengenlehre, Mathematische Annalen XLVI, p. 492, written "Alef-null"
IdeaChapter 1
(Arabic proper name)
Used for a rule of calculation in "Nova Methodus pro maximis et minimis," Acta Eruditorum 3, 467-473
IdeaChapter 2
(an arrow)
Not established for this book
IdeaChapter 9
(Greek)
In English from 1485 in the sense "a proposition that commends itself to general acceptance"
IdeaChapter 1
(English proper name plus Latin regula)
"La regle de Bayes" in Exposition de la Theorie des Chances et des Probabilites, pp. 158-159
IdeaChapter 3
(Latin)
The English phrase in The Philosophical Transactions (From the Year 1720, to the Year 1732), Vol. VI, part I
IdeaChapter 2
(a large parenthesis holding two stacked numerals)
Cajori: "The notation which has become the more common was introduced in 1827 by von Ettingshausen," footnoted to Vorlesungen uber hohere Mathematik, vol. 1 (Vienna, 1827), p. 38; used by Raabe in 1851
IdeaChapter 9
(Clarendon)
Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics, founded on the lectures of J. Willard Gibbs, using Clarendon type for vectors and ordinary type for scalars
IdeaChapter 9
(the sixteenth letter of the Greek alphabet)
Not established for this book
IdeaChapter 6
(printers' braces)
Untersuchungen uber die Grundlagen der Mengenlehre, p. 263, submitted 1907, published 1908
IdeaChapter 1
(Latin)
Latin use by Glareanus, 1538; first English citation Richard Percival, Bibliotheca Hispanica, 1591
IdeaChapter 9
(Greek kharakter)
L'equation caracteristique in "Memoire sur l'integration des equations lineaires," Exercices d'analyse et de physique mathematique, vol. 1, p. 53
IdeaChapter 5
(Latin)
Pascal's letter to Fermat of 29 July 1654, "all the possible combinations of 4 letters"; in English, John Wallis, Treatise of Algebra, 1673
PeopleChapter 2
(Latin)
Dissertatio de Arte Combinatoria; the English "combinatorial analysis" in P. Nicholson, Essays on the Combinatorial Analysis, 1818
IdeaChapter 2
(Latin)
"Set complementary to" in E. W. Chittenden, "Relatively Uniform Convergence of Sequences of Functions," Transactions of the AMS 15, 197-201
IdeaChapter 1
(letters and diacritics)
Not established for this book; the word "complement" is dated to Chittenden 1914 but the glyph is not dated in any source reached
IdeaChapter 9
(Latin)
Gauss uses determinantem in Disquisitiones arithmeticae, but for what we now call the discriminant of a quadratic form
IdeaChapter 4
(as above)
Cauchy's memoir of 1812, published 1815, Journal de l'Ecole Polytechnique, XVIIe Cahier, Tome X, where he says Gauss "designated these same functions by the name determinants"
PeopleChapter 4
(the vertical rule)
Cambridge Mathematical Journal vol. II, 267-271, reprinted Papers vol. 1, p. 1
IdeaChapter 9
(Greek)
Attributed to Heron of Alexandria for the first definition, "the straight line drawn from angle to angle"
IdeaChapter 1
(Latin)
Not established as a mathematical first use for this book; the English word is late 14c. and the sense "component of a situation" is 1929
IdeaChapter 5
(Latin)
Sir Henry Billingsley's English Euclid, "Two contrary kynds of quantity; quantity discrete or number"
IdeaChapter 6
(Latin)
C. J. Keyser, "The Thesis of Modern Logistic," Science 30, no. 783: "two classes are disjoint if neither includes a term of the other"
IdeaChapter 7
(subscript numerals)
Not established for this book
IdeaChapter 9
(two vertical rules)
Cajori: Cayley introduced them for a matrix in 1843 and 1845; the English textbook tradition takes them up with MacDuffee (1933) and Wedderburn (1934)
IdeaChapter 9
(German plus English)
Eigenwert, in "Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen," Nachrichten der Gesellschaft der Wissenschaften zu Gottingen, 49-91
PeopleChapter 5
(German plus English)
Eigenvektor in the finite-dimensional exposition of Courant and Hilbert, Methoden der Mathematischen Physik
IdeaChapter 5
(Latin)
Cantor's German Element in "Uber unendliche, lineare Punktmannichfaltigkeiten," Mathematische Annalen XX, p. 114
PeopleChapter 1
(Old English plus Latin)
Used without explanation in J. E. McAtee, "Modular Invariants of a Quadratic Form for a Prime Power Modulus," American Journal of Mathematics 41, p. 237
IdeaChapter 1
(a letter of the Norwegian and Danish alphabet)
N. Bourbaki, Elements de mathematique, Paris, p. 4, in the phrase "la partie vide de E, et designee par la notation"
IdeaChapter 1
(Latin)
Cantor's first Acta Mathematica translation, "Une contribution a la theorie des ensembles," Acta Mathematica 2, 311-328
PeopleChapter 9
(Latin)
The opening sentence of De Moivre's The Doctrine of Chances: "The Probability of an Event is greater or less..."
IdeaChapter 1
(the printers' "note of admiration")
Elemens d'arithmetique universelle (Cologne), in the section "Notations": "Je me sers de la notation tres simple n! pour designer le produit..."
IdeaChapter 9
(Latin)
Expectatio in van Schooten's Latin translation of Huygens's De Ratiociniis in Ludo Aleae; neither Pascal nor Huygens used the word
PeopleChapter 3
(Latin)
The English noun is dated 1816; Kramp's French factorielle belongs to his 1808 "Notations"
IdeaChapter 2
(two exclamation points)
Henry Warburton of Cambridge, England
IdeaChapter 2
(a printer's rule bent at a right angle)
Suggested by Thomas Jarrett, newly graduated B.A. of St Catherine's College, Cambridge; neglected for a quarter of a century; adopted by Todhunter about 1860
IdeaChapter 2
(the third letter of the Greek alphabet)
Legendre's 1808 article, continued in his integral calculus of 1811, so that Gamma(n+1) stands for n-factorial
IdeaChapter 2
(the sixteenth letter of the Greek alphabet)
Gauss wrote Pi(n) for n-factorial; Jacobi and H. Weber (1893) followed
IdeaChapter 2
(German proper name plus Latin)
George Forsythe "appears to have been the first to call it 'Gaussian elimination'", and in doing so misattributed schoolroom elimination to Gauss, per Grcar
IdeaChapter 4
(German proper name plus Latin)
Gauss described the method in Disquisitio de Elementis Ellipticis Palladis (Werke 6, sect. 13); in 1809 he had called it eliminatio vulgaris, ordinary elimination
IdeaChapter 4
(Greek)
Greek antiquity; the English word appears in a 14th century manuscript treatise
ManuscriptsChapter 3
(Greek)
"Chemistry and Algebra," a note in Nature: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph"
IdeaChapter 7
(a capital letter)
Wedderburn, Lectures on Matrices, p. 8, and MacDuffee, The Theory of Matrices, use I for the identity and O for the zero matrix
IdeaChapter 9
(Latin)
L. E. Dickson, "Representations of the General Symmetric Group as Linear Groups in Finite and Infinite Fields," Transactions of the AMS 9, no. 2
IdeaChapter 9
(modified less-than and greater-than signs)
Ernst Schroder, Vorlesungen uber die Algebra der Logik, vol. 1, replacing the earlier use of < and >; Gergonne had used a C for containment in 1817
IdeaChapter 9
(Latin)
De Moivre, The Doctrine of Chances: "Two Events are independent, when they have no connexion one with the other"
IdeaChapter 3
(Latin)
Not traced for this book
IdeaChapter 9
(Latin)
The set-theoretic sense recorded in Webster's New International Dictionary of 1909
IdeaChapter 1
(Latin)
"The inverse or reciprocal matrix," in "A Memoir on the Theory of Matrices," Collected Mathematical Papers I, p. 480
IdeaChapter 3
(Latin)
"Irreducible invariant" in "A Second Memoir upon Quantics," Philosophical Transactions
IdeaChapter 5
(Latin)
"On the Equation to the Secular Inequalities in the Planetary Theory," Philosophical Magazine 16, p. 267
IdeaChapter 5
(Old English plus suffix)
R. A. Fisher's technical sense, "On the 'Probable Error' of a Coefficient of Correlation Deduced from a Small Sample," Metron 1, 3-32
IdeaChapter 9
(Latin plus Arabic)
The phrase in the title "On the uses and transformations of linear algebra," American Acad. Proc. 2
IdeaChapter 4
(Latin plus Arabic)
"An algebra in which every expression is reducible to the form of an algebraic sum of terms, each of which consists of a single letter with a quantitative coefficient," in the American Journal of Mathematics; Miller's l.html as read for S176 dates that sentence 1870 and credits the OED, and the same page as read for S302 gives American Journal of Mathematics 4 (1881), p. 107
IdeaChapter 4
(Latin)
J. J. Sylvester, "On a Theory of the conjugate relations of two rational integral functions," Abstracts of Papers communicated to the Royal Society of London, 1850-1854 volume
IdeaChapter 5
(Russian proper name plus Latin catena)
The French "les chaines de Markoff" in V. Romanovsky, "Sur les chaines de Markoff," Comptes Rendus de l'Academie des Sciences de l'U.R.S.S., no. 9, 203-208; Markov himself introduced chains in 1906
IdeaChapter 5
(Latin)
"This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants," Collected Mathematical Papers vol. 1, p. 150
IdeaChapter 4
(the Greek letter epsilon)
Arithmetices principia, nova methodo exposita (Turin), pp. vi and x; Peano said the symbol abbreviated the Latin est, "is"
IdeaChapter 1
(German)
Defined as a totality whose arrangement is a matter of indifference, Paradoxien des Unendlichen, section 4, p. 3
PeopleChapter 1
(German)
Technical use in von Staudt's Geometrie der Lage, 2nd ed., 1856; adopted as Cantor's settled term in the Grundlagen note of 1883 and fixed in the Beitrage of 1895
PeopleChapter 1
(Monegasque place name)
N. Metropolis and S. Ulam, "The Monte Carlo Method," Journal of the American Statistical Association 44, 335-341
IdeaChapter 6
(letter-and-subscript compounds)
The triple appears in the practice of Potts, Whitworth, and Chrystal; Cajori's footnote is to W. A. Whitworth, Choice and Chance (Cambridge), p. 121
Idea
(as above)
Goodwin used nPr for "the number of permutations of n things taken r at a time"
IdeaChapter 9
(English)
The sense "chance or balance of probability in favour of something happening" by the 1580s; the OED's earliest quotation for "lay odds" is 1560
IdeaChapter 7
(Greek)
Thomas Digges, A geometrical practice named Pantometria: "the Orthogonall, the Obtuse and the Acute Angle"
IdeaChapter 7
(a capital letter)
Grundbegriffe der Wahrscheinlichkeitsrechnung; H. Cramer carried P(A) into English in 1937
Idea
(printers' parentheses)
Used for matrices by many writers, including Maxime Bocher, Introduction to Higher Algebra, and G. Kowalewski, Einfuhrung in die Determinantentheorie
IdeaChapter 2
(Latin)
Thomas Strode, A Short Treatise of the Combinations, Elections, Permutations & Composition of Quantities: "By Variations, permutation or changes of the Places of Quantities"
IdeaChapter 2
(French)
"This unit element will henceforth be called the pivotal element," E. T. Whittaker and G. Robinson, The Calculus of Observations, p. 71
PeopleChapter 9
(Latin)
The a posteriori contrast in Jacob Bernoulli, Ars Conjectandi, Part IV, chapter 4
PeopleChapter 3
(Latin)
The modern sense in the last chapter of Arnauld and Nicole, La Logique, ou l'Art de Penser
IdeaChapter 1
(a two-letter abbreviation)
An Introduction to Probability Theory and its Applications, vol. 1; later editions use P{A}
Idea
(Old French)
George Boole, An Investigation of the Laws of Thought: "random distribution of stars over the celestial vault"
IdeaChapter 5
(English compound)
Karl Pearson, "The Problem of the Random Walk," Nature LXXII, p. 294, 27 July 1905
IdeaChapter 9
(Frankish or Germanic)
German Rang, in "Uber homogene totale Differentialgleichungen," Journal fur die reine und angewandte Mathematik 86, p. 1
IdeaChapter 5
(Latin)
Not established for this book; Miller's r.html has no entry for the Markov-chain sense
IdeaChapter 9
(Latin)
Miller's entry names von Mises 1919 (Merkmalraum context), Neyman and Pearson 1933, Feller 1938 and 1950, as the chain by which the English phrase settled
IdeaChapter 1
(Latin)
"we shall call it therefore the scalar part, or simply the scalar of the quaternion," "On quaternions," Philosophical Magazine xxix, art. 18, pp. 26-31
IdeaChapter 5
(Latin)
Viete used magnitudines scalares in 1646 for quantities in geometrical proportion; the modern mathematical noun, a real number, is Hamilton's, dated 1846
IdeaChapter 5
(a capital letter in a script face)
Choice and Chance, fifth edition, uses a large script E, but the symbol and its calculus did not settle in English until much later; H. L. Rietz uses E in Mathematical Statistics (1927)
IdeaChapter 9
(Latin)
From celestial mechanics through Cauchy's 1829 paper "Sur l'equation a l'aide de laquelle on determine les inegalites seculaires des mouvements des planetes"
PeopleChapter 5
(Old English and Old French)
William Frend, The Principles of Algebra, uses "set of numbers" in 1796; Hamilton uses "set" and even "theory of sets" in Lectures on Quaternions (1853) but meaning an n-tuple; E. H. Moore fixes the modern equivalence in 1901
Idea
(the eighteenth letter of the Greek alphabet)
Institutiones calculi differentialis (St Petersburg), cap. I, sect. 26: "Just as we used the symbol Delta to signify a difference, so we use the symbol Sigma to indicate a sum"
IdeaChapter 4
(Old English)
Not established for this book; Miller has no entry
IdeaChapter 5
(German)
Carried into English as "trace" by H. L. Brose's 1922 translation of Weyl, which prints "the trace (spur) of a matrix"
IdeaChapter 9
(Latin)
Not established as a probability term of art here; etymonline dates the quantum-physics usage to 1913
IdeaChapter 1
(Latin)
"Stationary stochastic process" in A. Khintchine, "Korrelationstheorie der stationaren stochastischen Prozesse," Mathematische Annalen 109, p. 604
IdeaChapter 6
(Greek)
Jacob Bernoulli, Ars Conjectandi, p. 213: "Ars Conjectandi sive Stochastice nobis definitur ars metiendi quam fieri potest exactissime probabilitates rerum"; the word re-enters probability through Bortkiewicz's Stochastik in 1917 and Chuprov's English "stochastical" in 1923
PeopleChapter 3
(an inverted version of the Jarrett factorial sign)
Messenger of Mathematics vol. VII, p. 145; see also Choice and Chance (Cambridge, 1886), preface p. xxxiii
IdeaChapter 9
(Latin prefix plus English)
Not established for this book; Miller's s.html has no subset entry
IdeaChapter 1
(a capital letter set as a superscript)
C. C. MacDuffee, The Theory of Matrices, p. 5, with MacDuffee's own hope that "the present notation is in keeping with a systematic notation which, it is hoped, may find favour"
IdeaChapter 9
(Latin)
Hamilton's quaternion sense, "the tensor of the quaternion Q," Philosophical Magazine XXIX, p. 27
IdeaChapter 5
(as above)
Woldemar Voigt, Die fundamentalen physikalischen Eigenschaften der Krystalle, taken up by Einstein and Grossmann in 1913
IdeaChapter 5
(Greek)
Robert Recorde, The Pathwaie to Knowledge: "The Theoremes, (whiche maye be called approued truthes) seruing for proof"
IdeaChapter 1
(Latin)
H. L. Brose's translation of Weyl's Raum, Zeit, Materie as Space-Time-Matter: "the trace (spur) of a matrix"
IdeaChapter 9
(Latin)
Not established for this book; Miller's t.html has no entry for the Markov-chain sense
IdeaChapter 9
(Latin)
"A matrix compounded with the transposed matrix gives rise to symmetrical" results, "A Memoir on the Theory of Matrices," Philosophical Transactions CXLVIII, p. 32
IdeaChapter 4
(three vertical rules)
Whitworth "uses in one place triple vertical lines to indicate that three determinant equations may be independently formed from the matrix"
IdeaChapter 9
(Latin)
The set-theoretic sense in James Pierpont, Lectures on the Theory of Functions of Real Variables, vol. 2, p. 22; the previous term was "sum"
IdeaChapter 1
(rounded brackets rotated)
Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann; larger versions for general unions and intersections in Formulario mathematico, tomo V (Turin, 1908), p. 82
IdeaChapter 1
(Latin)
"The class determined by a function which is always true is called the universal class," Whitehead and Russell, Principia Mathematica vol. I, p. 30
IdeaChapter 1
(Latin)
"The universe of a proposition, or of a name," Transactions of the Cambridge Philosophical Society VIII, p. 380, presented as a new technical term
IdeaChapter 1
(Latin)
"may be called the vector part, or simply the vector of the quaternion," "On quaternions," Philosophical Magazine xxix, art. 18, pp. 26-31; the sense was presented to the Royal Irish Academy on 11 November 1844 and printed in the Proceedings vol. 3, pp. 1-16
IdeaChapter 5
(Latin)
"A Line supposed to be drawn from any Planet moving round a Center... called the Vector," J. Harris, Lexicon Technicum I
IdeaChapter 5
(English proper name plus Greek)
John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings," Philosophical Magazine and Journal of Science X, 1-18, formalized in Symbolic Logic (1881) ch. V
IdeaChapter 1
(Latin)
"we propose to call this quotient the versor," "On quaternions," Philosophical Magazine xxix, art. 19, pp. 326-328
ManuscriptsChapter 5
(Latin)
"Every Radial Quotient is a Versor. A Versor has thus, in general, a plane, an axis, and an angle," Elements of Quaternions ii. i., vol. ii, p. 133
ManuscriptsChapter 5
(the vertical rule)
H. Jeffreys, Scientific Inference, writing P(p given q); made popular by Feller's Pr{A given B} in 1950
IdeaChapter 3
(the vertical rule)
Not established for this book
IdeaChapter 9
(Arabic)
The English word is c. 1600; Cajori records that early Babylonians had no symbol for zero at all
IdeaChapter 1
Back matter
Provenance and reproducibility
Where every number on these pages comes from, and how to check it.
This book is generated, not typed. 11 registers hold the structured data, 9 narrative files hold the prose, and a build script turns them into the page you are reading. No table in this book is hand written. If a count appears in a sentence, it was computed from the register at build time, which is why the sentence cannot go stale when the register changes.
| Register | Rows |
|---|---|
| Dated events | 583 |
| People | 306 |
| Sources | 266 |
| Words and glyphs | 142 |
| Questions still open | 128 |
| Claims checked against the documents | 122 |
| Figures and images | 91 |
| Hooks into another subject | 54 |
| Disagreements between sources | 48 |
| Course sections mapped | 44 |
| Worked mathematics, by chapter | 9 |
The arithmetic. 1,157 checks across 22 scripts, and 0 failures. Every one runs on this machine before the book is written, and Appendix E prints what each one printed. One of those scripts does not run at all on a machine without numpy, and when that happens the build stops rather than quietly reporting a smaller total. A gate that cannot run is a failure, never a skip, and a suite that reports 1,106 where the real number is 1,157 is a suite that has lost 51 checks without telling anybody.
The check that matters most is the one on the checks. A gate that passes because it is looking in the wrong place is worse than no gate, because it produces confidence instead of doubt. So the gates are mutation tested: the build breaks the book on purpose, one way per gate, and confirms the gate goes red. 43 mutations across 31 gates, and every one of them was caught. The register gate carries 12 checks of its own and twelve of them went red under their own mutation. The readability gate carries nine probes on the readability gate, each one a sentence it must still refuse. There are ten checks in the narrative lint. A gate that stays green while its own subject is broken is reported as blind and does not count as passing.
The links out. Every link into the course book is checked against that course book's own live page, not against a scheme somebody wrote down. That matters more than it sounds: a plausible anchor like #s-10 for a unit opener does not exist on the site, and a link built from a rule rather than from evidence lands nowhere and looks fine.
Known limits, stated plainly.
- 21 disagreements between sources are unresolved. Appendix F lists every one with what would settle it.
- 8 sources in the bibliography were registered but never opened. Nothing in this book rests on any of them, and Appendix J marks each one.
- 206 of the 306 people have a pronunciation that is an approximation rather than a sourced one. Every one of them says so in its own row.
- No portrait of anybody in this book is cleared for reuse, so the book prints none.
- 57 historical images have had their rights read and quoted, and none of them is reproduced here, because nobody who built this book has seen the file. Appendix H gives the description and the link instead.
- Two sections of the course, A.1 Fraction Rules and A.3 Order of Operations, have no history material yet. Appendix I says so on their own rows rather than leaving them out.
Back matter
About this book
Written as the free companion reader for Megan Warren's Discrete Math and Linear Algebra course, 2026-2027. It exists because the one-sentence version of this history is not merely thin, it is misleading in a way that costs students understanding. A reader who believes probability began with Pascal, or that linear algebra began with matrices, will never see why the two halves of this course belong in one room.
The course itself is Megan Warren's design. This book was written to that design and keyed to it section by section: Appendix I maps all {s['placements']} of its sections to the history that belongs beside them.
Every claim traces to a source. {s['sources']} sources went in, and where a source could not be obtained the book says so in the reader's own language rather than in a note to the author. {s['conflicts']} disagreements between sources are logged and {s['open_conflicts']} are still open. Every calculation in Appendix E was re-run in code before it shipped, and that appendix is the scripts' own output rather than a transcription of it.
Design and voice: hers as well, navy with a tangerine accent. Math is typeset with KaTeX and fully embedded, so the book renders with no internet connection. All {s['figures']} figures were drawn for this book and carry no third party licence.
Like what you've seen here? Megan takes on a small number of commissions at a time. She builds custom interactive textbooks like this one, and custom software for classrooms and small teams, on commission from Boston.