The History of Calculus
Calculus was not invented in a single year by a single genius.
It was assembled, in pieces, by people who mostly never met,
and two of them got their names on it.
Start hereWelcome, and how to read this book
This is the story of where calculus came from, written to be read alongside your Derivatives and Integrals courses.
Most textbooks give the history two sentences. Newton and Leibniz invented calculus, they had a fight about it, now here is the power rule. Both sentences are true and both are badly misleading. The tangent problem and the area problem had been worked on for roughly two thousand years before either man was born, in Syracuse, in Kerala, in Baghdad, in Oxford, in Paris. People solved them, published them, and in several cases got no credit at all.
What Newton and Leibniz added was not the derivative and not the area under a curve. Both of those had been done. What was new was the algorithm: seeing that the two problems are inverse operations, and then building a symbolic machine that made the connection routine, so a competent person who was not a genius could turn the crank and get the answer. That is the thing worth a name on a subject, and it is a much better story than the one about the apple.
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 page. Both ways work, and nothing here assumes you have already read the rest.
Three parts, and they are used differently:
Part I, the story. Twelve chapters, in order, about two hours and twenty minutes of reading end to end. Chapter 1 is two minutes and Chapter 5 is seventeen; that unevenness is deliberate, not an oversight.
Part II, the master timeline. 327 dated events across ten eras. It is built for looking things up, not reading straight through. You arrive at it to place somebody.
Part III, reference. Eleven appendices and a glossary: 212 people, 191 terms and symbols, 129 logged disagreements between sources, 255 sources, and the worked mathematics re-derived 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 both, rather than picking a favorite and sounding confident. 58 of the 129 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."
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; the full list lives in the Glossary, Appendix M.
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 Derivatives and Integrals are the one exception, since they go to two other books.
Quick questions, quick answers
Do I have to read it in order?
No. Part I rewards reading in order, and 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 every story in this book to the exact section of Derivatives or Integrals it belongs with, 89 placements in all, each one 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 the word calculus means "pebble" 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 129 of them with both readings side by side.
Can I check the math for myself?
Yes, and you should. Appendix E re-derives every calculation in this book, step by step, and each one was verified in code before it shipped.
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. The style key just below is here for whenever you want to know what a mark means.
Ways to read this book
The same book, in two places, both free. Each one does something the other cannot.
-
In your browser You are here
The Listen buttons and the hover definitions on underlined terms run here, and nowhere else.
-
Download it free from Apple Books
Yours to keep, with no copy protection. On an iPhone, iPad or Mac you can highlight, take notes and bookmark as you read, which this page cannot do. Reading aloud there is your device's own voice, and every formula carries a spoken label so it says the mathematics correctly.
Front matterThe style key: what every mark means
This book uses color and shape on purpose, not for decoration. Here is the full key, so nothing is a mystery.
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 friendly, jargon-free opening that sets a scene before any names or dates arrive. Filled soft sky-blue panel.
The precise meaning of a term, at the moment it first matters. The word being defined is shown in bold navy. Iris bar.
A claim you can rely on, with the evidence behind it named. Steel-blue bar on a pale panel.
A question placed before the passage that answers it, using a wrong answer a competent person once published as the distractor. Guess first, then open the reveal. Twelve 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. Twelve of these too.
A story you have probably heard that is wrong, or a claim that gets repeated everywhere and should not be. Amber bar, the book's one caution color.
The sources genuinely disagree and the disagreement is not settled. Both readings are given. 58 of this book's 129 logged disagreements are still open, and Appendix F lists every one of them. Steel bar with the scales mark.
The link back to your own textbook: the exact section of Derivatives or Integrals where this story does the most good. Appendix I holds all 89 placements.
The actual mathematics behind a historical claim, worked through. Everything in Appendix E is this: re-derived from scratch and checked in code before it shipped.
Dig Deeper Optional higher reach▸
The deeper reach: a longer argument, a primary text in the original language, or a scholarly dispute in full. Collapsed until you open it, and nothing else depends on it.
The two or three things to carry forward. Appears at the end of a long chapter.
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 255 sources are listed. It is a box, not a superscript, so it is big enough to tap on a phone. |
| Disputed: c. 490 BCEDisputed | Sources disagree and it is not settled. The word "Disputed" is written out rather than left to italics, because a screen reader does not announce italics and the large-print setting removes them. |
| 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. |
| calculus | A word in its original language, tagged so a screen reader pronounces it in that language rather than in English. |
| Math set in a serif | All symbols and equations are typeset, for example . |
Words you may not know
A word you may not know is explained in the prose, at the point the book first uses it, and as this book uses it rather than in full generality. The Glossary collects those words in one place and, where the term register records one, adds the origin: the language the word came from and what it literally meant.
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 Storytwelve chapters, about 2 hours 20 minutes end to end▸
- Opener and Objectives
- Chapter 1. Two problems, and why they turned out to be one
- Chapter 2. Greece: a method that could check, but could not find
- Chapter 3. Elsewhere: the parts most textbooks skip
- Chapter 4. Medieval Europe: somebody invents the graph
- Chapter 5. The seventeenth century: everybody gets close
- Chapter 6. Newton
- Chapter 7. Leibniz
- Chapter 8. The war
- Chapter 9. The eighteenth century: attacked, defended, and finally taught
- Chapter 10. The nineteenth century: making it true
- Chapter 11. The people history skipped
- Chapter 12. Calculus now
Part II: The Master Timeline327 dated events, ten eras▸
- Opener and Objectives
- How to read the timeline
- Antiquity to 500 CE
- 500 to 1400: the wider world and the medieval schools
- 1400 to 1637: Kerala, print, and the algebraic turn
- 1637 to 1665: the race begins
- 1665 to 1687: Newton and Leibniz
- 1688 to 1734: publication, textbooks, and the war
- 1734 to 1799: attack, repair, and the Enlightenment
- 1800 to 1872: the age of rigor
- 1873 to 1913: measure, set theory, and the modern integral
- 1914 to the present: vindication and the machine age
Part III: Reference212 people, 191 terms, 255 sources▸
- Opener and Objectives
- Appendix A. People register
- Appendix B. Symbols, words, and eponyms
- Appendix C. Pronunciation quick reference
- Appendix D. Whose name is on it, and who did the work
- Appendix E. Worked mathematics, re-derived and checked in code
- Appendix F. Disagreements between sources
- Appendix G. Stories that check out, and stories that do not
- Appendix H. Image catalog and rights
- Appendix I. Where each story goes in your two textbooks
- Appendix J. Bibliography
- Appendix L. How this book was checked
- Appendix M. Glossary
Back Matterhow this book was built▸
The story: twenty-three centuries, and about thirty people
Calculus was not invented in 1666. It was assembled, in pieces, by people who mostly never met. They worked across Syracuse and Kerala and Baghdad and Oxford and Paris, over roughly twenty-three centuries. Two of them got their names on it. This part is the whole cast, in order, with the disagreements left visible instead of smoothed over.
- Say what Newton and Leibniz added, and what was already done before they arrived.
- Name at least four places outside Europe where calculus ideas were worked out first, and what each contributed.
- Explain why the priority war happened, who ran the committee, and what the modern verdict is.
- Describe how the subject was made rigorous in the nineteenth century, and what Berkeley had complained about.
- Name people the standard story leaves out, and say what each of them did.
- Connect a historical episode to the section of your own course where it belongs.
Chapter 1
Two problems, and why they turned out to be one
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
Calculus exists because two questions refused to go away for two thousand years.
The first question is about steepness at an instant. A car's speedometer says 50 miles an hour. It has not traveled 50 miles, and it will not travel for an hour. So what is that number describing? Judith Grabiner opens her article on the origins of rigor with exactly this exchange between a student and a teacher. Then comes the student's follow-up: "How in the world did anybody ever think of such an answer?" (S360, p. 185). That question is the whole of differential calculus.
The second question is about totals of things that keep changing. How much wine is in a barrel whose sides curve? How far has something traveled if its speed changed the whole time? What is the area under a curve? That is integral calculus.
For most of recorded history these looked like separate crafts. Finding tangents was one trade. Finding areas was another. People were good at both, in Syracuse and in Kerala and in Baghdad and in Oxford, centuries before anyone had a symbol for either.
Tangents were being found and areas were being found, in several countries, for centuries before 1670. So the field already worked when Newton and Leibniz walked into it. Name the one thing they added that nobody had.
I have a guess▸
The algorithm. A set of rules you can hand to a competent person who does not have to be a genius, and they get the answer. Seeing that tangents and areas are inverse operations, and then building a symbolic machine that made the connection routine, is the part that was new.
If you guessed "the derivative" or "the area under a curve", you gave the answer most textbooks give. Both had been done, and the next four chapters are the people who did them.
That distinction matters for teaching, because it changes who the story is about. It stops being about two men, Newton and Leibniz. It becomes about thirty people across a dozen centuries. Most of them never met, several of them got no credit, and a few got someone else's.
The word itself
Calculus is Latin for pebble. Specifically a small limestone counter, from calx, limestone. Roman reckoning boards used them. A teacher of arithmetic in Rome had two different job titles depending on his legal status. The title was calculo if he was a slave, calculator if he was freeborn (S453). Quintilian records that Demosthenes practiced speaking with pebbles in his mouth, using the same word (S455, at Quintilian 11.3.54).
The chain runs: limestone, then little stone, then counter on a reckoning board. From there it goes on to the act of reckoning, then a method of reckoning, then this method of reckoning. The subject you are taking is named after a slave's pebble.
And the kidney stone is the same word, arriving by a separate route from the same source. There is no metaphor connecting them beyond the shape (S455).
A Roman could have held a calculus in his hand. What was it?
Show the answer▸
A pebble. A small limestone counter, from calx, limestone, pushed around a reckoning board (S453). The chain runs limestone, little stone, counter, the act of reckoning, a method of reckoning, this method of reckoning.
That is the name of the subject your whole Derivatives course belongs to. Remember it when you open Derivatives 1.0: the subject with the frightening name is named after the cheapest object in the room.
Chapter 2
Greece: a method that could check, but could not find
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The crisis that started it
Greek mathematics had a theory of ratio built on counting. Two lengths have a ratio because some common unit fits a whole number of times into each. Then the Pythagoreans found that the diagonal of a square and its side have no common unit. The entire theory of proportion broke.
Eudoxus of Cnidus (Disputed: c. 408 to c. 355 BCEDisputed) repaired it. The repair is one of the great pieces of mathematical engineering. Instead of asking what unit fits both, he compared all the multiples. Two ratios are equal when, for every pair of whole numbers, the multiples fall on the same side of each other. That definition survives as Euclid, Elements V, Definition 5, and it is why Greek geometry could continue at all (S008).
The idea he made rigorous was not his. Antiphon of Athens (Disputed: c. 480 to c. 411 BCEDisputed) got there first. He was not a mathematician at all: he was a sophist, a professional arguer. He proposed inscribing a polygon in a circle and going on taking more and more sides. MacTutor's entry on Eudoxus calls that the germ of the method of exhaustion and credits Eudoxus with building on it (S008).
So the pattern that runs through this entire book shows up in its first chapter. One person has the idea and another person makes it safe to use. Antiphon supplied "take more sides"; Eudoxus supplied the reason you are allowed to believe the answer.
Out of the same repair came the method Eudoxus is remembered for. Squeeze the unknown quantity between figures you can measure, from inside and outside. Then show that any proposed answer other than the right one leads to a contradiction.
The name "method of exhaustion" is a seventeenth-century coinage. No Greek called it that, and it does not exhaust anything: it brackets (S008). The name has been misleading students for three hundred years.
Archimedes, and the difference between finding and proving
Archimedes of Syracuse (c. 287 to 212 BCE) is where this chapter earns its place, and not for the bathtub.
He proves that a parabolic segment is four thirds of the triangle drawn inside it. The proof runs on the sum 1 + + + and so on forever. He gets the right answer, and he gets it without one move you would make. Name the move he refuses.
I have a guess▸
Taking the limit. He writes the finite sum and the leftover term, and stops there. He never writes that the infinite series equals (S001). And he does not need to: the leftover can be made as small as he likes. That is enough to rule out every wrong answer by contradiction.
If you expected the limit, you were reading a proof from 250 BCE with a habit that did not exist until the 1820s. He would not put in print a step he could not defend.
In Quadrature of the Parabola he proves that a parabolic segment has exactly four thirds the area of the triangle inscribed in it. The proof rests on summing a geometric series, and he does not take a limit. He proves a finite identity and then argues by contradiction. In modern notation, the identity is:
Both the finite identity and the limit check out symbolically (Appendix E.1). Archimedes has the first line and the second. He does not write the third, because he will not take a limit in print. He does not have to: the second line lets him make the leftover as small as he likes. That is enough for a proof by contradiction.
Then in 1906 we found out how he was really working.
The palimpsest
Johan Ludvig Heiberg found it because of a footnote. A catalog notice from 1899 mentioned a mathematical palimpsest (a manuscript whose original text was scraped off so the parchment could be reused) in Constantinople. Heiberg read a few printed lines of the Greek, concluded it "must contain something by Archimedes", traveled there, and was right (S002). The lesson for a classroom: read the boring catalog.
What he found was The Method of Mechanical Theorems, a letter from Archimedes to Eratosthenes. Eratosthenes was the man who measured the Earth with a stick and a shadow. In it Archimedes explains how he finds his results in practice. He slices figures into infinitely thin sections and balances them on an imaginary lever, using mechanics to discover the answer. Only afterwards does he construct the rigorous geometric proof. He tells Eratosthenes plainly that publishing only the polished proof wastes everyone's time (S002).
This is the oldest surviving argument for showing your working, and it is from around 250 BCE.
Two more things in it are worth the whole chapter:
He credits Democritus. The preface turns to the cone being one third of its cylinder. It says the credit should go "to Democritus who was the first to make the assertion... though he did not prove it" (S002). The preface to On the Sphere and Cylinder credits only Eudoxus. So the recovered text changed the story. Democritus, the atom man, was probably also the first person to think of a solid as a stack of infinitely thin discs. He was honestly troubled by the paradox that creates (S002).
And in Proposition 14 he compares two infinite collections. Reviel Netz, Ken Saito, and Natalie Tchernetska read the palimpsest with modern imaging. They found Archimedes calling two infinite sets "equal in multitude" and setting up a proportion between infinite aggregates (S003). Disputed: This is contested territory.Disputed The specialists who found it hedge hard: the key step is implicit, and the text may carry an interpolation. And the argument, in their own word, "does remain strange". Popular accounts routinely overstate it into "Archimedes had actual infinity". The honest classroom framing: the people who found the exciting thing are the ones telling you to calm down about it. That is what expertise sounds like (S003).
How the book survived, which is a better story than the mathematics
In April 1229 a scribe named Ioannes Myronas scraped the ink off a tenth-century copy of Archimedes. He wrote a prayer book on the recycled parchment (Disputed: the colophon date is read as either 14 or 29 April 1229Disputed) (S004, S006). Parchment was expensive and Archimedes was not. He rotated the pages ninety degrees and folded them in half. So a single Archimedes page is now split across two prayer-book leaves, sideways and upside down relative to the prayers (S004).
The book survived because it was destroyed. As a prayer book it was worth keeping in a monastery for seven hundred years.
Then, at some point after 1938, a forger painted gold-leaf Byzantine evangelist portraits over four of the pages. The forger did it to raise the sale price (S006). It sold at Christie's in 1998 for two million dollars (Disputed: sources give 28 or 29 OctoberDisputed) to an anonymous buyer. The buyer immediately handed it to the Walters Art Museum, paid for a decade of imaging, and put the results online for free (S006).
To read through the forged gold, researchers used X-ray fluorescence at the Stanford Synchrotron Radiation Lightsource to detect the iron in the original ink. Vandalism defeated by a particle accelerator.
Before any of that, somebody had to take the book apart. Abigail Quandt, senior conservator at the Walters, spent four years doing it. She unbound the prayer book leaf by leaf and stabilized flaking ink with a gelatin solution. She also made, in the museum's own words, "innumerable repairs with Japanese paper" (S004, S006). The project eventually ran to over eighty scientists and scholars (S004). The synchrotron is the part that gets written about. None of it could have run without her.
And the detail that belongs in front of students: the first imaging attempt failed. The technology was impressive and the output was useless. It stayed that way until the classicists and the physicists argued with each other about what they were looking for (S005). Interdisciplinary work is not a slogan in this story. It is the reason the text was read.
Three manuscripts of Archimedes existed in the Middle Ages. One vanished in 1311, one in the 1550s, and the third was scraped clean and prayed over (S005). That is the entire margin by which we still have him.
Zeno, and a question that outlived civilizations
Zeno of Elea's paradoxes are usually taught as a puzzle about infinite series, which handles Achilles and the tortoise and misses the interesting one. The Arrow paradox is not about series at all. It asks whether an instant can contain motion. You cannot answer that without the idea that velocity is defined at a point by a limit (S007).
Zeno posed it around 450 BCE. The mathematics that answers it arrives in the 1820s. A question can outlive dozens of civilizations. Zeno was not even trying to do mathematics: per Plato he was defending Parmenides, and the paradoxes were a debating weapon (S007).
Did Greek mathematics "stop short"?
The standard line is that the Greeks had the pieces and never assembled them. That framing is contested, and the contest is worth showing students, because it is the moment history of mathematics stops being a list of dates.
In 1975 Sabetai Unguru argued that reading Greek geometry as disguised algebra is an anachronism that destroys what the Greeks were doing. Three of the most eminent mathematicians of the century, B. L. van der Waerden, Hans Freudenthal, and André Weil, wrote furious replies in the same journal. Weil titled his "Who betrayed Euclid?" (S491). Fifty years later the argument is still running.
Victor Katz's 2024 survey lays out the whole bibliography and the argument about whether the question is even well posed (S491).
Before you read on: van der Waerden looked at Greek geometry and saw algebra in disguise. Unguru looked at the same pages and said that reading destroys what is there. They are not disagreeing about the mathematics. What are they disagreeing about?
What the fight is really about▸
Whether the notation is the thinking, or just clothing on it.
Van der Waerden's position: a Greek proposition about rectangles on a line and the identity are the same result, written two ways. Translate one into the other and nothing is lost.
Unguru's position: translating it loses the thing you were trying to study. The Greeks reasoned about magnitudes you can construct, not about symbols you can manipulate. A history that rewrites them into symbols has described the historian instead of the Greeks.
You cannot settle this by rechecking the mathematics. That is why three of the century's most eminent mathematicians could write furious replies and change nobody's mind (S491).
So the short version: asking why the Greeks did not invent calculus may be like asking why Shakespeare did not write a screenplay.
Archimedes told Eratosthenes how he found his results, as opposed to how he published them. What was the method?
Show the answer▸
He balanced the slices on an imaginary lever. He cut a figure into infinitely thin sections and weighed them mechanically to find the answer. Then he built the rigorous geometric proof afterwards. He told Eratosthenes that publishing only the polished proof wastes everyone's time (S002).
You meet that lever again in Integrals 3.5, where mass, density and moments look like an application of the integral. They are not an application. They are where it came from.
Chapter 3
Elsewhere: the parts most textbooks skip
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
Calculus is usually taught as a European relay race. It was not. Three other traditions got a long way, and in two cases they got further, earlier, than anything in Europe until the 1600s.
China: the principle Cavalieri gets the credit for
Liu Hui (刘徽, third century CE) wrote a commentary on the Nine Chapters on the Mathematical Art in 263 CE. It includes a circle-cutting algorithm: inscribe a polygon, keep doubling the sides, and squeeze π. His instruction, "divide the circle until it cannot be divided any more," is either a limit or a contradiction, and he knew it (S020). Which one you see depends on how charitable you are.
Then Liu Hui published his own failure. He rejected the sphere formula in the Nine Chapters. He built the right object to attack instead: the mouhe fanggai. That is the "box-lid" solid common to two cylinders inscribed in a cube at right angles. He saw correctly that the sphere is to that solid as π is to 4. Then he could not compute the box-lid's volume, said so explicitly in print, and left the problem open. He closed with a short verse titled "The geometer's frustration":
Though the diminution increases, / It doesn't quite fit... I dare to let the doubtful points stand, / Waiting for one who can expound them. (S023, Donald Wagner's translation, pp. 59 to 79)
Two and a half centuries later, somebody answered. Zu Gengzhi, son of Zu Chongzhi, finished the problem. He used the principle now stated as: if two solids have equal cross-sectional areas at every height, they have equal volumes. In the original:
疊棋成立積 / 緣幂勢既同 / 則積不容異 "If blocks are piled up to form volumes, / And corresponding areas are equal, / Then the volumes cannot be unequal." (S023, Donald Wagner's translation, pp. 59 to 79)
That is Cavalieri's principle, about eleven hundred years before Cavalieri. The chain runs Liu Hui (263) to Zu Chongzhi (429 to 501) to Zu Gengzhi (c. 450 to 520) to Li Chunfeng (602 to 670). Li Chunfeng wrote it down so it survived (S023).
Zu Chongzhi also gave bounds on π that stood for roughly nine hundred years:
Both bounds check out, and agrees with π to six decimal places (Appendix E.2). It is also the best rational approximation to π with a denominator under about sixteen thousand. And it is trivially memorable: write 113355, split it down the middle, put the big half over the small half.
The ending is bleak and useful. Zu Chongzhi's book was dropped from the imperial curriculum for being too hard, and then the only copy rotted. A civilization lost its best mathematics because of an examination-board decision (S021).
The vocabulary is worth a minute of class time on its own. Chinese solid geometry names shapes after things you could point at in a field. A prism is a "moat-wall", a pyramid is a "male horse", a tetrahedron is a "turtle's foreleg" (S020). Compare the Greek habit of naming after abstractions.
India: the first sine table, written at twenty-three
Āryabhaṭa I (476 to Disputed: c. 550Disputed) finished the Āryabhaṭīya in 499 CE, aged 23. It is 118 verses long, and it contains the first known table of sines: twenty-four entries to the quadrant, at intervals of 3°45′. He also introduced the versine, , into trigonometry (S059).
It gives a value for π as well:
and the verse carrying it is an instruction rather than a formula. Add four to a hundred, multiply by eight, add sixty-two thousand, and that is the circumference of a circle of diameter twenty thousand (S059). The book is built to be recited, which is why it survived.
The most important thing in it is one adjective. Āryabhaṭa calls that value āsanna, approximate (S059). A thousand years later Nīlakaṇṭha wrote pages of commentary on why that single word had to be there. He built an argument out of it that the ratio can never be given exactly (S050, §9). When you write an approximation sign instead of an equals sign, you are repeating a piece of care that is fifteen hundred years old.
India: instantaneous velocity, because of the Moon
Bhāskara II (1114 to 1185) needed instantaneous velocity for a religious reason. The Moon moves too fast for a daily average to predict when a tithi, a lunar day, will end. So in the Siddhānta Śiromaṇi (c. 1150) he calls daily average motion "only a rough or approximate rate". He introduces tatkālika gati, instantaneous motion, because the Moon "varies from moment to moment". He also uses the Rcosine as the rate of change of the Rsine (S050, pp. 20 to 23).
The derivative enters Indian mathematics for the same reason it enters Newton's. Somebody needed to know where a moving body is right now, and an average was not good enough.
He also states, in Golādhyāya 4.39, that where the velocity correction vanishes the equation of the center is extremal (S050). This is often described as Rolle's theorem. It is not. It is Fermat's rule for a maximum.
Kerala: infinite series, two centuries before Newton
Mādhava of Saṅgamagrāma (Disputed: c. 1340 to c. 1425Disputed) is the most important mathematician in this chapter. He left no surviving mathematical book. We know him the way we know Socrates: through students, and students of students. They quote his verses and say "as Mādhava said" (S055). The lineage is named and traceable: Mādhava, Parameśvara, Dāmodara, Nīlakaṇṭha, with Jyeṣṭhadeva also a student of Dāmodara.
Two names in that chain do work worth separating out. Nārāyaṇa Paṇḍita (fl. c. 1350), in the Gaṇitakaumudī, gave the general formula for repeated summations, vāra-saṅkalita. These are sums of sums of sums, to any depth you like (S050, §4). That is not a curiosity. It is the machinery a power series runs on, and it was in place before the series arrived.
Parameśvara (c. 1380 to c. 1460), Mādhava's own pupil, spent fifty years observing eclipses. He used the results to correct the models he had inherited (S050). A tradition described from the outside as pure computation was checking itself against the sky for half a century.
Mādhava had the series we call Leibniz's:
and the series we call Newton's for sine and cosine, and the one we call Gregory's for arctangent. He also knew this series is uselessly slow, and did something about it. Running it out to ten thousand terms buys three correct decimal places (Appendix E.4):
| Terms | Value | Error |
|---|---|---|
| 10 | 3.0418396189 | 9.98 × 10⁻² |
| 100 | 3.1315929036 | 1.00 × 10⁻² |
| 1,000 | 3.1405926538 | 1.00 × 10⁻³ |
| 10,000 | 3.1414926536 | 1.00 × 10⁻⁴ |
So Mādhava built correction terms, and the Yuktibhāṣā states an exactness criterion for them, namely that the correction function should satisfy (S050). That is error analysis, not guessing.
His π value is recorded as a poem. In the bhūta-saṅkhyā word-numeral system, numbers are written as words for things that come in known quantities. Such things include gods, eyes, elephants, snakes, fire, the three qualities, the Vedas, the lunar mansions. Decoded, the string is 2827433388233, and the value is:
Eleven correct decimal places (verified, Appendix E.3). The system exists because verses are easier to memorize and harder to corrupt in copying than digits are. Indian mathematics was, for centuries, sung (S050).
Nīlakaṇṭha Somayāji (born 14 June 1444; Disputed: death unsettled: sources give "after 1501", c. 1550, and 1545Disputed) dates his own work by eclipses. He watched them on 6 March 1467 and 28 July 1501 (S056). A student with a laptop and a modern ephemeris can check those. An ephemeris is a table of where the sun, moon and planets sit in the sky, date by date. That makes a five-hundred-year-old Sanskrit text verifiable in a homework assignment.
Jyeṣṭhadeva's Yuktibhāṣā (Disputed: c. 1530Disputed) is written in Malayalam prose, not Sanskrit verse, and it explains why. That is a deliberate pedagogical choice: the text was written to be understood rather than to be authoritative (S050).
The one documented link between these two chapters
Here is a fact worth putting in front of students, because it is concrete where most transmission talk is speculative. The Kerala derivation of the sine series uses ibn al-Haytham's summation identity. Katz traces it precisely. In the step where the derivation passes from one line to the next, "the Indians used ibn al-Haytham's equation for the case k = 1". They then sketch a proof of the general result themselves, for any k. It shows the sum of the kth powers of the first n integers is approximately /(k+1) (S061, p. 172).
And immediately: "Although the Indian mathematicians did not refer to ibn al-Haytham or any other predecessor" (S061, p. 172). The mathematics is shared. The citation is absent. Katz's own reading is that "there seems a good chance that transmission to India did occur" (S061, p. 174).
Baghdad to Kerala and Kerala to Europe are two different arguments, and they do not stand or fall together. They rest on two different bodies of evidence, and the first is on much firmer ground than the second (S061).
The transmission question, and how to teach a live dispute
Charles Whish, a young East India Company civil servant in Malabar, read the manuscripts. He realized the local astronomers had infinite series for π before Newton's birth. He died in 1833, and his paper appeared in London after his death (S051). It is dated Disputed: 1834 or 1835 in different sourcesDisputed. Nobody cared for about a hundred and twenty years.
Whether Kerala mathematics reached Europe through Jesuit missionaries is an open question among working historians. Dennis Almeida and George Gheverghese Joseph argue for transmission on grounds of motive, opportunity, and resemblance. The Jesuits ran a scientific network from Rome to Beijing, they were in Kerala, and they were collecting calendrical material. And the series show up in Europe soon after (S051). Skeptics point out that no document has been produced.
The motive has a name attached to it. Christopher Clavius (1538 to 1612) taught Matteo Ricci at the Collegio Romano and then headed the commission that produced the Gregorian calendar reform of 1582. Ricci reached Goa in 1578 carrying explicit instructions to investigate Indian science. Other Jesuits with real mathematical training followed him. One was Antonio Rubino, who reported in 1610 that European timekeeping tables were inaccurate. Another was Johann Schreck, a student of Viète, whose 1618 letter of observations was meant for Kepler (S051).
So Europe had two expensive and urgent problems in exactly that window: calendar drift, and longitude at sea. Kerala had exactly the calendrical and trigonometric expertise both problems needed. That is what a motive looks like, and a motive is still not a document.
This is a dispute, not a fact. Two competent people looking at identical evidence and disagreeing is the whole lesson (S052).
Before you read on: motive, opportunity, and resemblance are all present. What single thing would settle it?
The answer, and why nobody can produce it▸
A document. One Jesuit letter, inventory, or translation that carries a Kerala series to Europe. Nobody has one.
Notice what that means about the three arguments in favor. Motive, opportunity, and resemblance are exactly the evidence you would expect to find whether or not the transmission happened. That is because the Jesuits really did run a network from Rome to Beijing, and they really were in Kerala. And two people can find the same series independently. Evidence that looks the same under both stories cannot tell you which story is true.
That is the difference between a case and a proof, and it is worth more than any settled result in this chapter.
One more thing belongs here, because it is about how histories get written. Carl Boyer's 1949 history dismissed Indian calculus in two pages, at the exact moment the Yuktibhāṣā was going to press in Kerala (S050). Historiography has a supply-chain problem.
The Islamic world
Ibn al-Haytham (Alhazen, 965 to Disputed: 1039Disputed) needed the sum of fourth powers to find the volume of a paraboloid. That is because a paraboloid's cross-sectional radius squared is linear in height. Squaring the area term pushes you to degree four (S054, S061). Nobody wanted Σi⁴ for its own sake; a solid shape demanded it.
The machinery is one identity, proved by induction:
He states it only for n = 4 and k = 1, 2, 3, and proves each case by induction on n. But the argument generalizes immediately (S061, p. 166). I checked it for k = 1 through 4 (Appendix E.13). He stopped at the fourth power because he needed nothing higher. What he needed it for was this: rotate the parabola x = ky² about the line x = kb² and slice the solid into discs. The volume comes out at exactly eight fifteenths of the circumscribing cylinder (S061, p. 168). That ratio is exact, and I verified it symbolically (Appendix E.14).
He worked out the sum of fourth powers exactly, every term of it, and proved it by induction. You would write and throw the rest away without a second thought. Name the one thing he did not have that lets you do that.
I have a guess▸
A limit. Divide by , let grow, and every term except dies, which is . That move needs a limit concept. He did not have one, so he needed the formula exact, down to the last term (S054).
If the exact formula looked like showing off, turn it around. It is the price of not having limits, and you get to skip it because somebody else paid.
And his proof is a Greek-style double reductio. He brackets the paraboloid between eight fifteenths of the cylinder and eight fifteenths of the cylinder-less-its-top-slice. Then he makes the top slice as small as he likes (S061, p. 168). Six hundred years later, Fermat and Roberval do the same thing with the same inequality. Katz shows the two arguments side by side (S061, pp. 163 to 164).
The formula then travels without its owner. It turns up in Morocco in the work of ibn Haydur (d. 1413) and ibn Ghazi (1437 to 1514). It also turns up in Samarkand in al-Kāshī's Calculator's Key. Katz is blunt about the limit of the evidence. He writes: "we do not know, however, how these mathematicians learned of the formula or for what purpose they used it" (S061, p. 169).
Ibn al-Haytham's biography is the best cautionary tale in the history of engineering. He promised the caliph al-Hakim he could control the Nile, traveled to Aswan, and realized the project was impossible with available technology. And then, knowing what al-Hakim did to people who disappointed him, he spent roughly a decade faking insanity. He used the decade to write the Optics, the book that established experiment as the arbiter of physical theory (S058). Modern optics exists because a scientist was frightened of his boss. The Aswan High Dam was later built roughly where he stood.
Thābit ibn Qurra (836 to 901) computed volumes using unequal partitions (S057). Every calculus student meets Riemann sums with equal-width strips. Thābit saw that unequal widths can be the right tool, a thousand years before Riemann formalized the general partition. His religious background matters historically: he belonged to a pagan, Syriac-speaking minority who had kept reading Greek. The House of Wisdom's translation program depended on that community. The transmission of Greek mathematics to Europe runs through a community that was neither Greek, nor Muslim, nor Christian (S057).
Sharaf al-Dīn al-Ṭūsī (c. 1135 to 1213) found the maximum of a cubic about 450 years before Fermat. He used the discriminant condition D ≥ 0 to ask when a solution exists rather than what it is (S053). Disputed: Whether he "had the derivative" is disputed: Roshdi Rashed says yes, Jan Hogendijk says the evidence does not support it.Disputed
Before you read on: what would al-Ṭūsī have had to write down for you to say he "had" the derivative?
What the two sides are arguing about▸
Not the mathematics. Rashed and Hogendijk agree on what is on the page. They disagree about what counts as having a concept.
Rashed's standard is the procedure: al-Ṭūsī computes the quantity we would compute and gets the answer we would get.
Hogendijk's standard is the object: to have the derivative you have to treat it as a thing in its own right. It is something you can name, differentiate again, and reason about apart from the cubic that produced it.
You will cross that exact line in Derivatives 1.4. There, a derivative stops being something you do to a curve at one point and becomes a function in its own right. Your course walks over the boundary these two historians are arguing about, in one section.
al-Kāshī (d. 1429) stated his error budget as "no more than the width of a horse's hair on the circumference of the universe". That is a fifteenth-century engineer's way of saying sixteen decimal places. It is also a much better hook for a teenager than the digits are (S063). One correction for the textbooks: al-Kāshī did not invent decimal fractions. al-Uqlīdisī did, around 950 (S063).
The word "sine" is a mistranslation, and we never fixed it
This is the best language-and-mathematics crossover available, and it takes three minutes to teach.
- Sanskrit ardha-jyā, "half-bowstring". A chord of a circle looks like a drawn bowstring; the arc is the bow. Shortened to jyā.
- Borrowed into Arabic as jiba, a meaningless loanword.
- Arabic script omits short vowels, so jiba is written identically to jaib, a real Arabic word meaning a fold, bay, or bosom.
- A twelfth-century translator in Toledo read the real word and rendered it into Latin as sinus.
- Edmund Gunter abbreviated it to "sin" in 1624.
(S060; Disputed: the translator is either Robert of Chester or Gerard of Cremona, and sources disagree.Disputed)
Every time a student writes "sin", they are writing a Sanskrit word for archery equipment. The word arrives mangled through Arabic, mistranslated into Latin, and abbreviated by a Jacobean clergyman.
How to close this chapter honestly
The temptation with a chapter like this is to swing from "Europe invented calculus" to "Europe stole calculus". Both are wrong, and the standard citation says so plainly. Katz, having laid out everything above, ends:
There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today. (S061, pp. 173 to 174)
And then, in the next breath, the honest caveat:
But what we do not know is whether the immediate predecessors of Newton and Leibniz, including in particular Fermat and Roberval, learned of some of the ideas of the Islamic or Indian mathematicians through sources of which we are not now aware. (S061, p. 174)
That is the register to write this chapter in. Islamic and Indian mathematicians had large pieces of the calculus, centuries early, and used them for the specific problems they cared about. What they did not do, on the evidence we have, is unify them. Katz's diagnosis: "There were apparently only specific cases in which these ideas were needed" (S061, p. 173).
"Sin" is a Sanskrit word that got mangled twice on the way to your calculator. What did the Sanskrit word mean?
Show the answer▸
Half a bowstring. Ardha-jyā, shortened to jyā, because a chord looks like a drawn bowstring and the arc is the bow. It went into Arabic as jiba, a meaningless loanword written identically to jaib, a fold or a bay. A twelfth-century translator in Toledo read the real word and put sinus into Latin (S060).
That is the word you use all through Derivatives 2.2. It is the one piece of notation in your course that means nothing at all. Nobody ever fixed it, so every "sin" you write is archery equipment.
Chapter 4
Medieval Europe: somebody invents the graph
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The Oxford Calculators
At Merton College, Oxford, in the 1330s and 1340s, a group of logicians produced the most important kinematic result of the Middle Ages. They did it inside a logic textbook. Kinematic means described through motion: a curve traced by a moving point, an area swept by a moving line.
The mean speed theorem concerns a body moving with uniformly changing speed. It says that body covers the same distance as a body moving steadily at the average of its starting and ending speeds. William Heytesbury states it in 1335, in a book about sophismata, the trick sentences logic students were trained to dismantle (S101). The single most important kinematic result of the Middle Ages arrives as a rule for settling a brainteaser. Physics teachers do not look in chapter six of a logic textbook.
The Calculators' method, the wildly artificial thought experiment, gets defended in one line. That line licenses the whole later tradition of frictionless planes and massless ropes. The scenario does not have to be physically real, only conceivable, and "for purposes of the sophisma, that is enough" (S102).
Merton College gets the mean speed theorem, and the line you will read in most histories is that Merton discovered it and proved it. Name the half of that Merton did not supply.
I have a guess▸
The proof. Heytesbury states the theorem in 1335 and never proves it. The proof belongs to Nicole Oresme, in Paris, with a picture. "Stated at Merton, proved at Paris" is the accurate line. Clifford Truesdell's often-quoted claim that it was discovered and proved at Merton overreaches (S101, S104).
If you assumed the people who stated it also proved it, you are with Truesdell. His version is the one that gets quoted. Stating and proving are two jobs, and this is the chapter where they get done in two countries.
The Calculators also died the way medieval people died. Thomas Bradwardine was made Archbishop of Canterbury on 10 July 1349, in the middle of the Black Death. He was dead of plague by 26 August, a primacy of about seven weeks (S103). John Dumbleton also died in 1349. Dumbleton is the most under-documented person in this whole document: for a long time no usable source could be found for him at all. Edith Sylla's DSB entry turned him up: a ten-part Summa surviving in more than twenty manuscripts. Nearly forty percent of it is about living things and minds (S480). He was a bestseller, not a footnote.
Richard Kilvington got the group started and Bradwardine founded it (S480).
Oresme draws the picture
Nicole Oresme (c. 1320 to 1382), in Tractatus de configurationibus qualitatum et motuum, does something nobody had done. He draws time along the bottom, speed up the side, and then says the thing that makes calculus possible. The area of the figure is the total effect.
Marshall Clagett, whose critical edition is the source read here, calls chapter III.vii "the most significant chapter of the work, historically speaking" (S107).
This is a Riemann-integral idea in the 1350s, and a student can reproduce the entire proof with a ruler in ninety seconds. Cut the triangle at the halfway height, rotate the top piece, get the rectangle. Done.
Oresme also proved the harmonic series diverges, by grouping terms into blocks that each sum to at least one half:
The bound holds at every power of two I tested up to 2¹⁴ (Appendix E.5). The series diverges, and it does so at a speed that surprises people: to pass 10 you need more than 12,000 terms.
He was also a king's man. Charles V paid him to translate Aristotle into French, and the by-product was a French scientific vocabulary that had not existed before. He argued at length and brilliantly that no available argument disproves the Earth's daily rotation. Then he said he believed it stood still anyway, because Scripture said so (S100). His argument against astrology is mathematical: if celestial ratios are almost certainly irrational, no exact conjunction ever repeats, so astrological prediction fails in principle (S108). That is one of the earliest uses of "most probable" as an argument in European science.
Oresme put time along the bottom and speed up the side. Say what he claimed the area of that figure measures.
Show the answer▸
The total effect. Distance, when the thing up the side is speed (S107). Clagett calls the chapter it sits in the most significant of the work. The proof takes ninety seconds with a ruler: cut the triangle at half height, rotate the top piece, get the rectangle.
That sentence is the theorem behind Integrals 1.5, the Net Change Theorem. It was written around 1350, roughly three hundred years before anyone had a sign for an integral.
Chapter 5
The seventeenth century: everybody gets close
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
By 1600 the pieces existed in a dozen countries. Between 1615 and 1670 a group of Europeans, most of whom were not professional mathematicians, assembled almost all of them. This chapter is the run-up, and here is the honest summary. Newton and Leibniz were the last two people to arrive at a party that had been going for fifty years.
1615: Kepler gets annoyed about wine
Johannes Kepler's 1615 Nova stereometria doliorum vinariorum is the best origin story in the subject, and the primary source is better than the legend.
Kepler had remarried and moved to Linz. Wine came up the Danube by barge. The merchant shoved a measuring rod diagonally into each barrel and named a price for every cask alike. Kepler's own words for it were sine discrimine, sine respectu figurae, sine ratiocinatione vel calculo. In English: without distinction, without regard to shape, without reasoning or calculation (S159, preface). So Kepler wrote a book computing the volumes of about ninety solids of revolution to check whether the merchant was cheating him.
The method: treat a solid as made up of infinitely many infinitely thin discs, sum them, and then hedge. Margaret Baron's summary quotes Kepler's own hedge word:
Kepler regarded solid bodies as being made up, as it were (veluti), of "infinitely" many "infinitely" small cones or "infinitely" thin disks. (S151)
Veluti, "as it were", is the whole seventeenth century in one adverb. Nobody would say a solid is made of infinitely thin discs. Everybody was willing to say it is made of them as it were, and then do the sum.
The physics hook is stronger still. Kepler's second law is not "planets speed up near the Sun". It is "equal areas in equal times", which is a statement that area measures elapsed time (S152). To use it you add up infinitely many thin triangular slivers of an ellipse. That is an integral, in 1609, sixty-six years before Leibniz wrote the sign for it. And the raw data was thirty years of naked-eye observation by Tycho Brahe, a Danish nobleman with a metal nose.
The human context matters and is documented. Between 1611 and 1615 Kepler's patron abdicated, his six-year-old son died, and his wife died. He was forced to move cities, and he had his wife's body reburied with the child. He remarried for the sake of the children. A few years later he was defending his mother against a witchcraft charge (S154). That is the life in which an integration technique got invented.
1635: Cavalieri, and a fight about the Eucharist
Bonaventura Cavalieri's Geometria indivisibilibus continuorum (1635) is where the principle Zu Gengzhi had in the fifth century enters European mathematics under a new name. The new name is indivisibles: the idea that an area is made of lines and a solid of planes. It is Proposition II.4, and the name "Cavalieri's principle" is not his (S135).
He sat on the manuscript for eight years. He wrote to Galileo more than once. In effect, he said: please do not make me commit to what my own method seems to say (S156). And he denied the atomistic reading in writing, twice, on 2 October 1634 and 28 June 1639 (S135, quoted by Kirsti Andersen). Paul Guldin's objection was blunt:
Between one infinity and another there is never a proportion or ratio. (S156, Guldin, Centrobaryca vol. 4 p. 342, quoted in Cavalieri's Exercitationes p. 201)
Cavalieri's reply was that collections of lines are "finite with respect to extension (in spatio)" though infinite in number (S156, Exercitationes p. 202).
There is a live argument about why the Jesuits suppressed indivisibles. Amir Alexander's popular book Infinitesimal argues it was about hierarchy and the Euclidean model of ordered knowledge. David Sherry argues it was about the Eucharist. If bread's quantity can survive without bread's substance, then quantity had better not reduce to a pile of dimensionless slivers (S132, S133). Disputed: The two appeared in the same journal issue, both with evidence, and the dispute is unresolved.Disputed Popular writers cite Alexander's claim freely; specialists contest it. Say so when you use it.
1636: Fermat, and a word that arrived by accident
Pierre de Fermat was a criminal-court judge in Toulouse who did mathematics in his own time. He mostly refused to publish, sending results to friends in letters, sometimes as challenges (S144).
His method of maxima and minima works like this. To maximize a quantity, perturb the variable by a small amount e and set the two expressions "adequal". Simplify, divide by e, then suppress the remaining e terms. For the classic case of splitting a length a into two parts with the largest product:
I re-derived this symbolically and it solves to a/2 (Appendix E.6).
The word "adequality" is a translation accident. Diophantus, in third-century Alexandria, coined the Greek παρισότης for "approximately equal". Bachet rendered it into Latin morpheme for morpheme as adaequalitas. Fermat picked up the Latin word and built the first general method of maxima and minima on it (S130). Fermat's own French phrase for the procedure was "une comparaison feinte ou une adégalité", a feigned comparison, or an adequality (S130, p. 6).
Disputed: What Fermat meant by it is genuinely disputed and has been for a century. Herbert Breger reads it one way; André Weil, Michael Mahoney, and Mikhail Katz read it anotherDisputed (S130). The one thing the primary text settles: the last step is a suppression of the e-terms, not setting e = 0 (S130). That difference is exactly the crack that Berkeley would drive a wedge into a century later.
Descartes read the result and decided Fermat had got lucky. The two of them feuded about it.
1637: Descartes supplies the enabling technology
La Géométrie (1637) is an appendix. Descartes wrote it as a worked example of the philosophical method in the Discourse, not as the main event. He deliberately made it hard because he wanted readers to have to work (S148).
It is also the single most important enabling step in this chapter. Once a curve is an equation, "find the tangent" becomes an algebra problem you can hand to anybody (S148). Everything after 1637 is downstream of that.
Descartes also gave us x, y, z for unknowns and a, b, c for knowns. He never argued for the convention or even mentioned that he was introducing one. He simply started doing it, and 389 years later every algebra textbook on Earth still does (S450). You will meet a story that he picked the last letters because a printer's type case was running short of them. Nobody has ever found a source for it, so it stays out of this book (S450).
1643 to 1670: the rest of the field
Evangelista Torricelli got about three months with Galileo before the old man died. He turned that into the barometer and the acute hyperbolic solid: an infinitely long solid with finite volume. He had it by 1641 and published it in the 1644 Opera geometrica (S131). That same 1644 volume carried the first printed article on the cycloid. The article brought a letter from Roberval charging plagiarism. Disputed: Cajori attributed Torricelli's death in 1647 to the charge, a claim about a man's state of mind and not a medical recordDisputed (S064, p. 314). His own statement of the hyperbolic solid:
An acute hyperbolic solid, infinitely long ... is equal to that right cylinder. (S157)
Torricelli's solid goes on forever and holds a finite volume. Your textbook probably adds that you could fill it with paint and never paint the outside. Name the half of that story Torricelli never proved.
I have a guess▸
The infinite surface area. He had the finite volume by 1641 and printed it in 1644. The painter's paradox is a nineteenth and twentieth-century overlay on a seventeenth-century theorem (S157). The surface area is infinite and it is a fine exercise. Do not put it in his mouth.
If you were sure the paint was the point, so is nearly every textbook in print. What shocked 1644 was smaller and stranger, and you are about to read it.
The shock in 1644 was not "infinite paint on a finite volume". That framing came later. The shock was simpler: a solid that goes on forever and fits in a cup. Hobbes could not accept it. Barrow said Aristotle had been proved partly wrong. That is a near-universal textbook error today. If a textbook prints the painter's paradox as Torricelli's own, it is repeating a two-century-old embellishment.
Galileo tried to find the area under a cycloid in 1599 by weighing it. He cut an arch and a generating circle from the same material and put them on a balance. He found the arch came out at about three times the circle. Then he abandoned the result, because he believed the ratio was incommensurable, not a ratio of whole numbers (S064, p. 310). It is exactly three (Appendix E.15). An experiment handed him a correct, simple, rational answer and he distrusted it for being too neat. Torricelli reports the episode in 1644, and the historian Moritz Cantor credits Galileo with making the curve well known and with naming it.
Gilles Personne de Roberval got the quadrature around 1634. He did not publish it until it appeared posthumously in the Traité des Indivisibles of 1693 (S064). The institutional reason is clean, and Whitman states it exactly. Roberval won the Chair of Ramus at the Collège Royal in 1634. That chair "automatically became vacant every three years, to be filled again by open competition. As the incumbent set the questions it seems plausible that Roberval should conceal his methods" (S064, pp. 310 to 311). Secrecy was not vanity, it was tenure (S137). Evelyn Walker's verdict, which Whitman quotes: "the accident of occupying this chair caused Roberval to lose credit for many of his discoveries" (S064, p. 311).
Blaise Pascal, having renounced mathematics for God, was told by his doctors to think about something lighter. He thought about the cycloid. His 1658 anonymous contest is the period's best set-piece. It had cash prizes and a pen name (Amos Dettonville, an anagram of Louis de Montalte, the pen name from the Provincial Letters). It had judges, deadlines, angry losers, and a winner declared: nobody (S153). Wallis and Lalouvère both submitted and both were faulted. Pascal then published his own solutions, and inside them sat the Traité des sinus du quart de cercle. Leibniz would later say its characteristic triangle switched on a light in his head.
Disputed: The toothache story says Pascal thought about the cycloid because a toothache kept him awake and the pain stopped when he concentrated. It rests on his sister Gilberte,Disputed and the Internet Encyclopedia of Philosophy explicitly calls her claims questionable (S161, S143). Whitman shows the transmission route: he attributes the story to W. W. Rouse Ball. He adds that Pascal took the relief as "a divine intimation to proceed with the problem". By this account Pascal then "worked incessantly at it for eight days" (S064, p. 314). So the chain is Gilberte, then Ball, then everyone. Present it as a family anecdote reported by Ball.
And the contest produced a result that belongs in every integral calculus course. While it was running, Christopher Wren sent Pascal his rectification of the cycloid, with no proof. Roberval, shown it, is said to have proved it on sight and claimed to have known it for years. He had not published that either. Wallis printed it as Wren's the following year in Tractatus duo (S064, p. 314).
The result is startling and a student can check it:
A curved length exactly equal to a straight one, and no π anywhere in it (verified, Appendix E.16). In 1658 that was close to shocking: curved lengths were widely believed not to be expressible by straight ones at all. The word for finding it, rectification, means "straightening".
John Wallis introduced ∞ in Arithmetica Infinitorum (1656), and the detail worth teaching is that its very first job was to be a denominator:
the height of each one of which is 1/∞ of the whole height, or an infinitely small aliquot part. (S139, Wallis's Latin, translated)
Wallis is worth knowing outside the mathematics. He got his career from cryptography: he broke a Royalist cipher in 1642 and Parliament noticed him. He spent twenty years exchanging insulting pamphlets with Thomas Hobbes about whether the circle can be squared. They had titles like School Discipline for not saying his Lessons Aright (S145).
James Gregory is the great near-miss of British mathematics. In Geometriae Pars Universalis (1668) he proved the inverse relationship between tangents and areas, two years before Barrow. MacTutor calls it "the first attempt to write a systematic text-book on what we should call the calculus" (S136). He had the Taylor series forty years early in a private letter. He designed a reflecting telescope he could not build. At 36 he had a stroke while pointing a telescope at Jupiter for his students (S136).
Isaac Barrow stated the inverse relationship geometrically in Lectiones Geometricae (1670). He resigned the Lucasian chair to his 26-year-old student in 1669 and posted that student's manuscript to London in the same year (S184). How much Barrow had, and what he passed to Newton, is the sharpest unresolved question in this chapter. J. M. Child, translating Barrow in 1916, opens his preface with a flat declaration:
ISAAC BARROW was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication. (S158, Preface p. viii)
D. T. Whiteside called Barrow "a thoroughly competent university don" and no more. Four distinguished scholars have looked at the same book and the same marginal notes in Leibniz's hand and reached four different verdicts (S138).
And there is a good explanation for why they diverge, which is more useful than the dispute itself. Mordechai Feingold's 1993 reinterpretation makes the case that Barrow was a weapon in the priority war, and that the scholarship inherited the damage. Augustus De Morgan saw it in 1835 and said it best:
And here they made of Barrow a sort of retrenched position, on which to fall back in case of defeat, affirming that if the method was not Newton's, it could not belong to Leibnitz, because Barrow had a claim of discovery prior to that of both. This gave a fictitious importance to Barrow's interesting and elegant method. (S073, quoting De Morgan, Penny Cyclopaedia, 1835, vol. 3, p. 508)
Barrow was the Newtonians' fallback trench. And then, Feingold argues, the twentieth-century reaction against Child overcorrected. Newton scholars and Leibniz scholars formed what he calls a "tacit alliance" that emptied Barrow of significance (S073, p. 312).
The detail that surprises everyone: the suggestion that Leibniz owed something to Barrow came from Leibniz's own circle. Ehrenfried Walter von Tschirnhaus, his confidant, wrote to him in 1678 and 1679. He admitted he could not see much in Leibniz's calculus that was new beyond what Barrow had published. He called Barrow's version "much more readily intelligible" than Leibniz's. Tschirnhaus then published in the Acta Eruditorum in 1682 and 1683, claimed the calculus for himself, and credited Barrow's influence. That credit was used against Leibniz decades later (S073, p. 325). Jakob Bernoulli's "Specimen calculi differentialis" of January 1691 pointed at the same similarities. That is why Leibniz's own references to Barrow are, in Feingold's words, "few and far between, and invariably defensive or deprecatory" (S073, p. 325).
Feingold sets his own limits, and they are as sharp as his argument. He explicitly declines Child's thesis. His aim is not "to argue (in the manner of Child) that... he is, in fact, the true inventor of the calculus" (S073, p. 312). And his conclusion: "Due recognition of Barrow shifts no credit for the invention of the calculus away from Newton and Leibniz" (S073, p. 338).
Disputed: The question of what exactly Barrow gave Newton stays open.Disputed But it is now open with a reason, which is a better thing to hand a class than a shrug. The evidence is finite and public, and the people reading it were arguing about something else.
1647 to 1649: the logarithm turns out to be an area, and it happens during a fight
Grégoire de Saint-Vincent's Opus geometricum (1647) runs to ten books and more than 1,200 folio pages. It contains a false claim to have squared the circle. A 22-year-old Dutchman, Christiaan Huygens, found the fatal flaw in the headline claim. Buried in Book VI, on the hyperbola, is something the author valued less and we value more.
The book's own table of contents flags it. Under Book VI, Part III, the indexer wrote:
Proprietates admirandae de superficiebus inter hyperbolam, asymptoton unam, et duas parallelas alteri asymptoto interiacentibus... "Admirable properties concerning the surfaces lying between the hyperbola, one asymptote, and two parallels to the other asymptote..." (S067, Elenchus Materiarum, listing propositions 125 to 129)
And immediately after it, the theorem itself: lines parallel to one asymptote that cut off equal areas are all in continued proportion (S067).
Here is why that matters, and why it is worth ten minutes of class time. Equal areas correspond to abscissas in geometric progression. Multiply the x-values by a fixed factor and you add a fixed amount of area. That is exactly what a logarithm does to numbers, and it is the whole content of the result:
Products become sums. Verified in Appendix E.17.
The person who said so out loud was answering an attack. Marin Mersenne posed a challenge problem: given any three magnitudes and the logarithms of two of them, find the logarithm of the third by geometry. The title page of the reply records the sting in the challenge. The sting was that Saint-Vincent's quadrature of the circle "abeat in illud necdum solutum Problema", collapses into this still-unsolved problem (S066). Mersenne was going after Saint-Vincent.
Alphonse Antonio de Sarasa, Saint-Vincent's pupil, answered it in 1649. Partway through, he stops and addresses the reader directly:
Sed quorsum haec, inquies, ambages? ... ad Logarithmos duco. "But to what end, you will say, all this roundabout? ... I am leading to logarithms." (S066, printed p. 8)
He takes magnitudes in continued proportion whose logarithms are 6, 7, 8, 9, 10. These numbers, in his phrase, "always exceed each other by the same excess, according to logarithmic doctrine". He builds them as lines against a hyperbola, shows the areas between them are all equal, and concludes:
Unde loco numerorum 6, 7, 8, 9, 10, 11, &c. qui erant Logarithmi magnitudinum O, P, Q, R, S, T, assumere poterimus quantitates hyperbolicas X G, H G, I G, K G, L G, M G.
"Hence, in place of the numbers 6, 7, 8, 9, 10, 11, etc., which were the logarithms of the magnitudes O, P, Q, R, S, T, we shall be able to take the hyperbolic quantities XG, HG, IG, KG, LG, MG." (S066, printed p. 8; the Latin is normalized from a long-s scan)
In place of the logarithms, take the areas. That sentence is the birth of the logarithm as an area, and it appears inside a defense of somebody else's book.
"Natural logarithm" was therefore coined for a curve, not a number. The logarithm is natural because it falls out of the area under the plainest hyperbola there is, xy = 1. It comes with no base chosen by anybody (S155).
Two footnotes for honesty. Sarasa's marginal references send the reader to the Opus geometricum, Book VI on the hyperbola, at propositions in the high 120s. The individual digits are not legible with confidence in a 1649 long-s scan. So cite the range 125 to 130 rather than a single number (S066, S067). And the copy of the Opus geometricum read here contains the front matter, the full contents, and Books I to V. Book VI itself was not in the volume, so its proposition texts have not been read (S067).
Pietro Mengoli posed the Basel problem in Bologna in 1650 (S142). Johann Hudde was differentiating polynomials algebraically in 1657 and later ran the city of Amsterdam for thirty years (S147). Nicholas Mercator's 1668 series for log(1+x) is the first infinite series for a logarithm ever printed. It is what pushed Newton to write up De analysi to establish priority (S149). Christiaan Huygens demolished Saint-Vincent's false proof. He built the first pendulum clock that kept time on a cycloid rather than a circle. Then he personally taught the young Leibniz the mathematics he needed (S150).
The scoreboard in 1669
By the time Newton was 26, Europe had coordinate geometry. Europe also had a general method for maxima and minima, and a general method for tangents. It had quadrature of , the area under a hyperbola as a logarithm, and infinite series for logarithms and for π. It also had Cavalieri's principle and two independent statements of the inverse relationship between tangents and areas.
What it did not have was a single symbolic system that made all of that one subject with one set of rules. That is what came next, twice, independently.
Sarasa's sentence of 1649 changes what a logarithm is. What does he say you can take in place of one?
Show the answer▸
The areas. His words: "in place of the numbers 6, 7, 8, 9, 10, 11... which were the logarithms of the magnitudes... we shall be able to take the hyperbolic quantities" (S066). Multiply x by a fixed factor and you add a fixed amount of area under xy = 1, so products turn into sums.
That is the whole of the result you will use in Integrals 2.4, and it is why the natural logarithm is called natural. Nobody chose the base. The curve did.
Chapter 6
Newton
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The dates, and why they are confusing
| Event | Old Style (England) | New Style (Europe) |
|---|---|---|
| Born | 25 December 1642 | 4 January 1643 |
| Baptized | 1 January 1642/3 | 11 January 1643 |
| Trinity closed for plague | 8 August 1665 | 18 August 1665 |
| Back in Cambridge | 20 March 1665/6 | 30 March 1666 |
| Back for good | 22 April 1667 | 2 May 1667 |
| Barrow sends De analysi to Collins | 31 July 1669 | 10 August 1669 |
| "Shoulders of Giants" letter to Hooke | 5 February 1675/6 | 15 February 1676 |
| Epistola posterior | 24 October 1676 | 3 November 1676 |
| Died | 20 March 1726/7 | 31 March 1727 |
| Buried, Westminster Abbey | 28 March 1727 | 8 April 1727 |
(S200, and the full table with sources is in the master timeline.) His tomb reads MDCCXXVI because the English civil year began on 25 March.
The plague years, corrected
The story everyone knows: Cambridge closed for plague, Newton went home to Woolsthorpe, and in eighteen months invented calculus, optics, and gravitation.
The geography is wrong, and Whiteside proved it from the Trinity commons accounts and Newton's own Fitzwilliam pocket-book. Newton was in Cambridge from September 1664 to June 1665, back on 20 March 1666, and back again from 22 April 1667. Whiteside's conclusion:
The conclusion seems inescapable that Newton wrote most of them while in Cambridge... rather than in seclusion in Lincolnshire. (S190, MP 1: 8 n.20)
And the summer of 1665 was not spent at Woolsthorpe either. He was at Humphrey Babington's rectory at Boothby Pagnell, which had a library (S190, MP 1: 8 n.21).
Newton's own famous recollection, written decades later, is at CUL Add. 3968.41, f. 85r. In it he says he was "in the prime of my age for invention" (S190, MP 1: 152 n.25). Whiteside spent his life on these manuscripts and had every reason to be skeptical. Yet he acquits him of lying. The weaknesses in Newton's chronology, he writes, are "lapses in memory... rather than the product of deliberate distortion" (S190, MP 1: 152).
He also, aged 22, working from almost nothing, computed logarithms to fifty-two figures by hand for fun. Whiteside checked the arithmetic: correct to fifty decimal places (S190).
The apple
Stukeley's 1752 Memoirs records the conversation, under apple trees in a garden, tea in hand. The text has Newton "in a comtemplative mood" when the notion of gravitation "was occasion'd by the fall of an apple" (S180, f. 15r).
He saw it fall. It did not hit him. No eighteenth-century source has a head strike (S180). The setting of the telling mirrors the setting of the event, which is a gift for a textbook. Here is the scene: an old man and a younger admirer, under apple trees. And there is Newton saying this is exactly where I was sitting in my head sixty years ago.
The manuscripts, and a forty-two-year delay
The Waste Book (CUL MS Add. 4004) is where the calculus first got written down (S198). Newton did his gravity calculations on the back of a legal document, because he was a poor student and parchment was expensive (S199). In the same notebook: the calculus, the colors of light, the laws of motion, drawings of dissected eyes, and a scheme for dividing the octave. He was 23 (S199).
The notebook was his stepfather's. Barnabas Smith married Newton's mother when Newton was three and sent the boy away to live with his grandmother. Newton took Smith's blank notebook years later and filled it (S198).
The October 1666 tract (CUL Add. 3958.3) is the hardest dated evidence for the anni mirabiles, the "miracle years". It is also the one piece that does not depend on Newton's memory. He wrote it in English, not Latin, because he was writing for himself, then put it in a drawer. It was not printed until 1962 (S183).
De analysi is the paper that shows Newton had the method first. Barrow posted it to London within weeks of Newton finishing it in 1669. Guess the year it reached print.
I have a guess▸
1711, forty-two years later, and only because by then the priority war made printing it a weapon (S184). He was 26 when he wrote it and 68 when it appeared.
If you guessed some year in the 1670s, you assumed a man who has nearly been scooped will run to the press. He did the opposite for four decades, and Chapter 8 is the bill.
De analysi (1669) went to Barrow, who sent it to John Collins in London. It was not published until 1711, forty-two years later, and only because the priority war made publication a weapon (S184). Newton was 26 when he wrote it and 68 when it appeared.
Methodus fluxionum was written in 1670 to 1671 and published in 1736, nine years after he died. It appeared in English before it came out in Latin, because John Colson wanted it read. Colson's subtitle promises "a perpetual comment upon the whole work... in order to make this treatise a compleat institution for the use of learners". The first calculus textbook in English is a dead man's manuscript with a tutor's notes glued on (S185).
De quadratura curvarum (written 1691 to 1693) was published in 1704 as an appendix to the Opticks. If you own a 1704 Opticks, you own Newton's calculus by accident (S187).
He was publishing the whole time. He was just not printing.
The forty-two-year silence is not silence. It is a different distribution system, and Niccolo Guicciardini has reconstructed it.
The name for it is scribal publication: you circulate a text in handwritten copies instead of putting it on a press (S204, p. 460). It was a normal thing to do in Restoration England. The historian Harold Love worked out why people still chose it a century and a half after printing arrived. Guicciardini quotes him:
Interesting in the choice of scribal publication . . . was the idea that the power to be gained from the text was dependent upon possession of it being denied to others. (Love 1993, pp. 183-184, quoted at S204, p. 460)
A printed book cannot choose its readers. A manuscript can, and Newton chose. The men allowed to see the mathematical manuscripts at Cambridge were John Collins, John Craig, and Edmond Halley. So were David Gregory and Nicolas Fatio de Duillier (S204, p. 459). Other papers went to the Royal Society, or into the University Library as Lucasian Lectures. Outside that inner ring sat "a larger circle of philomaths", amateur lovers of mathematics. They got the results second hand from the acolytes, "sometimes in mutilated form" (S204, p. 462).
Mutilated is the exact word, and the mutilation was deliberate. William Jones taught mathematics for a living and held a large stock of Newton manuscripts. James Wilson says Jones "was want to curtail or otherwise disguise the papers [of Newton] he communicated to his scholars, that none might make out a compleat book" (S204, p. 461). Handing out a damaged copy kept Jones's scholars dependent on him and stopped a pirate printer from assembling the whole thing.
It failed anyway, because copies get copied. Collins made at least two transcripts of De analysi, kept one, and sent the second to Wallis in 1677. When Wallis died it passed to David Gregory (S204, p. 460). Jones transcribed De methodis around 1710, and Wilson copied Jones around 1720. Samuel Horsley built the collected Opera omnia of 1779 to 1785 on those copies rather than on Newton's originals (S204, p. 460). The first collected Newton is an edition of copies of copies. Colson's 1736 translation runs off the Jones transcript too (S204, p. 460). That is the same chain that put Jones's tidied notation into the printed text (S192).
By December 1720 Newton had lost track of where his own papers were, and Wilson wrote to tell him:
I imagine, when you sent any of your Friends your papers, the person they got to transcribe them, took a double copy, which is a frequent practice, in order to make profit by it. So that they are in different hands. (S204, p. 461)
And then Newton died, and the sorting took six days. Between 20 and 26 May 1727 Thomas Pellet drew up an inventory of the papers. It ran to eighty-one items, and he judged five of them fit to be printed (S494).
Sit with that ratio. Newton spent fifty years choosing who was allowed to read his mathematics, in the most careful, most deliberate way anybody has ever managed it. Within weeks of his funeral one man with a list did the choosing for him. In under a week, that man passed five things out of eighty-one.
The point is not that Pellet was careless, and this book does not have the evidence to say he was. The point is structural. An archive becomes a canon at the moment somebody sorts it. The sorting is almost always faster, and done by someone less famous, than the making.
Hired scribes were running off second copies and selling them. Meanwhile the originals were wearing out. In 1691 Halley and Joseph Raphson went to look at the De methodis manuscript at Cambridge. They found it "very much worn by having been lent out" (S204, p. 461). A treatise nobody was allowed to buy had been handled until it started falling apart.
The reason underneath: he stopped believing in his own method
Guicciardini splits Newton's working life in two. An early analytical period, then a mature synthetic period that begins in the 1670s (S204, p. 455).
Early Newton built on three things. One was Wallis's habit of accepting a formula once it worked for enough particular cases. The others were a free hand with infinitesimal "moments" and Descartes's move of writing a curve as an equation. Newton's own name for the package was the new analysis (S204, p. 463).
Then he turned on it. He wanted certainty rather than plausibility, and he knew the standing of infinitesimals was disputed (S204, p. 463). At the same time he was deep in Pappus and trying to restore lost books of Apollonius. He decided the ancients had the better style. On Descartes's solution of the Pappus four-line problem, in the late 1670s:
To be sure, their [the Ancients'] method is more elegant by far than the Cartesian one. For he [Descartes] achieved the result by an algebraic calculus which, when transposed into words (following the practice of the Ancients in their writings), would prove to be so tedious and entangled as to provoke nausea, nor might it be understood. (Newton, translated by Westfall, quoted at S204, p. 464)
Guicciardini's conclusion is that Newton came to regard the new analysis as "not a mathematical language fit for publication" (S204, p. 455). He had judged his own best work unprintable. Right after finishing De methodis he drafted an addendum in synthetic form. He reworked it about 1680 as the Geometria curvilinea: the calculus redone in geometry, with limits in place of infinitesimals (S204, p. 463).
And the silence is not a gap in his mathematics. Whiteside's volumes show Newton working on mathematics without interruption until 1696 (S204, p. 459). He was not idle and he had not lost interest. He had decided that what he had was not fit to print.
What changed his mind: somebody nearly scooped him
Craig went to Cambridge in 1684 or 1685. There he heard that Wallis was about to publish a summary of Newton's method for squaring curves. He was allowed to transcribe part of De methodis and of the Epistola posterior (S204). Disputed: The year is disputed: Guicciardini gives 1684 in one place and 1685 in another, in the same article. See Appendix F.Disputed
Craig went home to Scotland and told Archibald Pitcairne and David Gregory what he had seen. Gregory reconstructed Newton's "prime theorem" on series quadratures from the description. The theorem appeared in print in Pitcairne's Solutio problematis de historicis seu inventoribus at Edinburgh in 1688. Newton's answer was to start writing the treatise that became De quadratura (S204, p. 465).
That is the pattern for everything after. Guicciardini names three forces that pushed Newton to the press (S204, pp. 465, 468). One was priority, over Gregory first and Leibniz above all. Another was his new standing as the undisputed leader of British mathematics after the Principia. Finally came a change in what counted as publishing at all, once scientific journals existed. By then the Continental calculus was being built in the Acta Eruditorum and the Paris Mémoires.
Even then he needed pushing. Arithmetica universalis went to press anonymously in 1707 because Newton wanted his Cambridge colleagues' votes in the 1705 parliamentary election. Gregory recorded that Newton "has not seen a sheet of it" and might "buy up the copyes" if the result displeased him (S204, p. 467).
And he edited on the way to the printer. In the 1704 De quadratura, infinite parva became admodum parva. In the 1711 De analysi, Jones changed esse infinite parvam to in infinitum diminui & evanescere, "most probably after Newton's instruction" (S204, p. 466). A man in his sixties went back into manuscripts he had written in his twenties and took the infinitesimals out. He did it so that the young man's mathematics would agree with the old man's doctrine.
The notation nobody used in 1666
Newton's fluxions are rates: if a quantity x flows in time, its fluxion is the rate of flow, and its fluent is the quantity itself.
The dot over the letter is from late 1691, not 1666. The 1666 tract uses p, q, r. The 1671 autograph uses l, m, n, r. The dots in the printed 1736 text are William Jones's 1710 tidying-up, which Colson then translated (S192, MP 3: 73 n.86). Reproductions of "Newton's 1666 notation" that you will meet in textbooks are reproducing Jones (S192).
The binomial series, which is where he really started
Newton's first great result was extending the binomial theorem to fractional and negative exponents, which he got by interpolating from Wallis. For a square root:
Twelve terms at x = 0.2 gets 10 correct decimal places (Appendix E.8).
The Principia, and a claim you should stop repeating
The received story: Newton discovered the Principia results with calculus and then translated them into classical geometry to make them respectable.
That claim traces to a single source, and the source is Newton, writing anonymously about himself in Philosophical Transactions 29 no. 342, p. 206 (S189). Whiteside contextualizes it: Johann Bernoulli said the same thing about his own work. Whiteside also finds the Principia built natively from limit-ratios. The one genuine exception is the solid of least resistance (S191, MP 6: 24 n.74 and MP 6: 26).
Three things now narrow this. First, E. A. Fellmann's 1988 survey of how Continental mathematicians read the Principia never states the received claim at all. The survey independently corroborates only Whiteside's single exception, using Whiteside's own phrase "Newton's suppressed analysis" for the solid of least resistance (S070, p. 24). A clean negative from a specialist survey is worth something.
Second, and more usefully, the argument exists and I can tell you exactly where it is. I. Bernard Cohen's Guide to Newton's Principia, which accompanies the standard modern translation, has a section titled "Methods of Proof versus Method of Discovery; The 'New Analysis' and Newton's Allegations about How the Principia Was Produced; The Use of Fluxions in the Principia", at Guide p. 122 (S072). The word "Allegations" is Cohen's, and it tells you he treats Newton's 1715 account as a claim to be assessed rather than a fact. Related sections sit at Guide pp. 114, 129, 316, and 345. The last, titled "An Example of the Calculus or of Fluxions in Geometric Form", concedes the received story for one named lemma.
Third, Guicciardini narrows it from the other side, and the narrowing cuts both ways. He states that "obvious traces of the use of integration techniques" run through many Principia demonstrations and that Newton "did not give the reader any detail on these highly algorithmic techniques" (S204, p. 465). David Gregory and Roger Cotes, the man editing the second edition, could not finish proofs that turn on quadrature. Newton told them privately how to apply his integral tables (S204, p. 466). So something was withheld, and more than Whiteside's one exception: the editor of the Principia could not complete its proofs from the printed page.
Now read the rest of it. Guicciardini also says the mathematics Newton put on the page is the geometric fluxional calculus of the Geometria curvilinea (S204, p. 465). Newton built that calculus on limits and not on infinitesimals. That is Whiteside's account of how the book was made, not Newton's 1715 one. Hiding your quadrature working is not the same thing as writing the book in calculus and then translating it into geometry. This source establishes the first and does not establish the second (S204). Guicciardini calls the size of the hidden part an open research question. He points at his own Reading the Principia, pp. 179 to 186, for the detail (S204, p. 466). See Appendix F.
Disputed: The question is still open here, because the copy of Cohen obtained contains the Guide's contents and not its body.Disputed But it is no longer a vague gap: Guide pp. 114 to 117, 122 to 127, and 345 to 347 would settle it. They are still the highest-value request. See Appendix L.
Two other Principia facts worth having. Newton anticipated Berkeley's objection almost word for word in 1687, fifty years before Berkeley made it, and answered it (S186). Students enjoy discovering that the famous heckle was pre-refuted. And its reputation for impossibility is, in Whiteside's word, "ill-deserved" (S191, MP 6: 25 n.78). He quotes Todhunter's estimate that "probably a dozen pages would supply the necessities of a student who wished to master even the Principia". A manuscript at King's College records Cambridge students muttering as Newton walked past. There goes, they said, "the man who has writt a book that neither he nor any one else understands" (S191).
Halley paid for the printing out of his own pocket. When Hooke claimed priority on the inverse-square law, Newton threatened to withhold Book III, the part containing the system of the world. Halley had to talk him down (S203). Newton's view was that Hooke had done nothing and left "the drudgery of calculations" to others (S203).
Before you read on: Hooke suggested the inverse-square law and could not derive the orbit from it. Newton derived the orbit. How would you split the credit, and what would change your answer?
Why this one has no clean answer, and what that tells you▸
Take Newton's side and you are saying an idea is worth nothing until somebody makes it work. That standard erases most of the people in Chapter 11 of this book. They did enormous amounts of the drudgery under somebody else's name.
Take Hooke's side and you are saying the first person to say a thing owns it. That standard hands the calculus to Fermat, Barrow, and about six others in Chapter 5. It is also the standard Newton himself used when he claimed priority over Leibniz on the strength of unpublished manuscripts.
Neither man applied his own standard consistently, and that is the point. Watch what happens in Chapter 8 when Newton needs the other rule.
One lemma, and what it shows about how he worked
Here is a small, concrete case that beats a hundred pages of argument about method.
Lemma 12 of Book 1 is the one Newton needs about conjugate diameters of an ellipse. In De motu he writes only "Patet ex Conicis", "it is clear from the Conics". In the 1687 Principia he writes "Constat utrumque ex Conicis". He proves nothing, and points at Apollonius (S076).
William Whiston later turned that into a legend. Newton, he wrote, "saw it by intuition" and "used it before it had ever been demonstrated by any one" (1749). It is not true. The theorem had been demonstrated by Apollonius, by Saint-Vincent in 1647, and by La Hire. And Andrea Del Centina and Alessandra Fiocca found something better. Newton's own copy of De Witt and van Schooten came from Barrow's library. In it, Newton wrote a three-line proof in his own hand as an unpublished "Corollary 6", using Euclid III.36 (S076).
Two details make it evidence rather than an anecdote. Lemma 12 as printed in 1687 is defective, missing the conjugate-diameter condition, and Newton fixed the wording only in 1713. His private marginal corollary carries the identical defect, which is how Del Centina and Fiocca date the two together. And Barrow's library catalog, drafted on Newton's advice, held two copies of Borelli's Apollonius of 1661 alongside Saint-Vincent's 1647 Opus geometricum (S076).
So the pattern is: he had the proof, in Euclidean form, and printed a pointer instead. Del Centina and Fiocca say nothing at all about calculus versus geometry; the words calculus, fluxions, and analysis do not appear in their paper. What this settles is Whiston's legend and one lemma, and it settles those cleanly. They also disagree with Whiteside on a related point, holding that Newton had studied Apollonius more closely than Whiteside allowed (S076).
The rest of the man
Alchemy. John Maynard Keynes bought the alchemical manuscripts at auction in 1936 and told the Royal Society that Newton was "the last of the magicians" (S193). Keynes wrote "at least 100,000 words"; the widely repeated "million words" figure is modern cataloguing wrongly attributed to him (S193). Keynes was in the middle of a world war, designing the post-war financial system. Yet he spent his spare hours buying back and reading a dead alchemist's notebooks, and he called the auction an impiety.
Theology. Newton was an anti-Trinitarian at a college named Trinity, which would have cost him his fellowship, his knighthood, and possibly his liberty. The Twelve Articles (Keynes MS 8) are twelve short sentences on one sheet of paper that he wrote and never showed anybody (S195). In 1675 he obtained a royal dispensation exempting the Lucasian Professor from taking holy orders (S194). The dispensation is reported in several places and nobody working on this book saw the original document. So treat it as very likely rather than as settled. Appendix L.4 says what would settle it.
The Mint. Newton prosecuted counterfeiters, and William Chaloner is the best story in the Newton file for a teenager. Chaloner had counterfeited coin on an industrial scale. He petitioned Parliament twice claiming he could fix the Mint's security while accusing the Mint of incompetence. And he publicly humiliated Newton. Newton built the case. Chaloner, in prison, wrote to him (TNA MINT 15/17/205):
O no body can save me but you.
I shall be murdered unless you save me. (S196, two separate lines of the same letter, TNA MINT 15/17/205)
Coining was high treason, and the sentence was not simple hanging. Newton kept the letter. It is still in the file (S196).
The dot over the letter is the famous piece of Newton's notation, and it is not his 1666 work. Who put it into the printed text?
Show the answer▸
William Jones, in 1710. The dots themselves date from late 1691; the 1666 tract uses p, q, r and the 1671 autograph uses l, m, n, r. Colson then translated Jones's tidied version, which is why most reproductions of "Newton's 1666 notation" are reproducing Jones (S192).
You meet the fluxion itself in Derivatives 1.4. That is where a derivative stops being something you do at a point and becomes a function. The dot you see in reprints of his 1666 tract belongs to a later editor, not to the man who did the mathematics.
Chapter 7
Leibniz
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The notebooks are dated to the day
This is the historian's gift in this story. Newton's chronology depends partly on Newton's memory. Leibniz's depends on dated manuscripts, and you can watch him invent the notation over about three weeks in Paris in the autumn of 1675.
26 October 1675. He writes out a formula so long and so clogged with nested omn. omn. omn. (for omnia, "all of them") that Cajori's comment is simply: "The last equation given above forcibly exhibits the necessity of a simplified notation" (S451).
29 October 1675. Three days later:
It will be useful to write ∫ for omn., so that ∫l = omn. l, or the sum of the l's. (S230, Child p. 80, translating the manuscript Analyseos tetragonisticae pars secunda)
The integral sign is a letter S, written long, standing for summa. It replaced a word because Leibniz thought writing "omn." out every time wasted ink (S238).
On the same sheet, and this is the detail almost nobody knows, d starts life as a divisor, not a multiplier:
Let l = ya/d; then just as ∫ will increase, so d will diminish the dimensions. But ∫ means a sum, and d a difference. (S230, Child p. 82)
And further down the same page, in the same session:
These are sufficiently new and notable, since they will lead to a new calculus. (S230, Child p. 82)
He knew.
Leibniz has d working as a difference on single quantities. He knows dx. He knows dy. Now he wants the difference of a product. Write down the obvious guess for d(xy).
I have a guess▸
He guessed d(xy) = dx · dy (S230). He tested it against one example, the example agreed with him, and he believed it for about an afternoon. Child's marginal note, written in 1920 about a page from 1675: "Leibniz, as a logician, should have known better than to trust a single example."
If that is what you wrote, you are exactly where the inventor of the notation was on 11 November 1675. Read on and watch him catch himself on the same page.
11 November 1675. dx becomes a multiplier (S230, Child p. 91). And on the same day he gets the product rule wrong, in writing:
Let us now examine whether dx dy is the same thing as d(xy)... But you get the same thing if you work out d(xy) in a straightforward manner. (S230, Child p. 100)
He tested it on one example, got the right answer by accident, and believed it. Then he argued himself out of it in the same afternoon's writing:
Hence it must be concluded that d(vψ) is not the same as dv·dψ, and d(v/ψ) is not the same as dv/dψ. (S230, Child p. 102)
This is the single best teaching artifact in the whole history of calculus. Here is why the guess fails, worked with the numbers I checked (Appendix E.9):
The guess is smaller than the actual change by a factor of 8001. That is because it keeps only the second-order term and throws away the two first-order ones. Leibniz left both the error and the refutation on the page. Very few discoverers do.
The analytic derivation comes on 21 November 1675 (S230).
Where the ideas came from
Huygens set him a test. In Paris in 1672, Leibniz was a 26-year-old diplomat who claimed an interest in mathematics. Huygens asked him to sum the reciprocals of the triangular numbers. Leibniz noticed that every term is a difference, that the sum telescopes, and that the answer is exactly 2:
Huygens was impressed (S237). Everything else follows from that afternoon. The principle Leibniz extracted, in Richard Arthur's words, is that "the sum of the differences is the difference between the first term and the last". It becomes "the basis for the fundamental theorem of the calculus: the sum (integral) of the differentials equals the difference of the sums" (S237). Pascal's triangle adds downwards; Leibniz's harmonic triangle (its entries are reciprocals, and each is the sum of the two below it) subtracts downwards. Sums and differences, the two operations he would later call ∫ and d, are the whole architecture already in 1672.
Pascal's characteristic triangle. Reading the Traité des sinus du quart de cercle, Leibniz later wrote that subito magna lux oborta est, "a great light suddenly appeared" (S232). It came not from a falling apple but from a lemma about the area of a sphere.
1684: seven pages that started a subject
Nova Methodus pro Maximis et Minimis appeared in Acta Eruditorum, October 1684, pp. 467 to 473. It came with plate Tabula XII facing p. 466, signed "per G.G.L." (S231, verified against the 1684 scan).
Three facts about it are worth teaching:
- It gives no proofs at all. Contemporaries complained. Jacob and Johann Bernoulli had to reverse-engineer it, and then built most of the early calculus from it (S231).
- It had typographical errors, and one of them inverts the meaning of a sentence about the shape of a curve. The most consequential seven pages in the history of analysis went out with the concavity and the convexity swapped (S231). You will often see the paper called six pages long. It runs pp. 467 to 473, which is seven numbered pages, and it stops part way down the last one, which is where the shorter count comes from (S231). The 1929 translator quietly fixed the printer's errors and left a footnote about it (S232).
- It shares its issue with an anatomical review of the structure of the ear (S231).
The ∫ sign reached print two years later, in De geometria recondita (1686). It came almost in passing, inside a squabble about whether the circle can be squared (S233). Its own subtitle misprints the page number of Nova methodus (S233). The founding documents of calculus are full of typos.
Why the notation won
Leibniz's notation encodes the structure of the operations, and Newton's does not.
- dy/dx looks like a ratio and behaves like one. The chain rule becomes visual cancellation. Substitution in an integral becomes bookkeeping.
- ∫ f(x) dx carries the "of what" in the dx. Change variables and the dx tells you what to do.
- ∫ and d announce themselves as inverses. Leibniz says so directly in 1686: "for as powers and roots in common calculation, so with us sums and differences or ∫ and d, are reciprocals" (S232, p. 625, Evelyn Walker's translation).
And then the joke that turned into a field. In a 1697 letter to Wallis, Leibniz noticed an analogy. If ∫ and d behave like roots and powers, and powers can be one-half, then so, "usefully", can d. He wrote y in his own hand (S241). Fractional calculus, now used in viscoelasticity, anomalous diffusion, and control theory, traces its birthday to two speculative lines in a letter. In the same letter he proposed the neutral name analysis infinitesimalis for the shared subject, which is essentially the name it still has (S241).
The man
Law degree at 20. Disputed: AllegedlyDisputed refused a doctorate at Leipzig over his age, Disputed: and the story that the dean's wife was responsible is unverifiedDisputed (S235). He built a stepped-reckoner calculating machine and demonstrated it at the Royal Society in 1673. He worked out binary arithmetic and published it in 1703. For decades he tried to drain the Harz silver mines with windmills, and every scheme failed. The man who invented a notation that outlived empires could not move water uphill (S235).
His employer became King of England. Leibniz asked to come. He was told to stay in Hanover and finish the family history of the House of Brunswick. He never finished it, and he never saw England again (S236). Nine volumes of documents, no book: the most famous unfinished homework assignment in European history.
Two funerals
State this precisely, because the temptation to dramatize it is strong and unnecessary.
Leibniz died at Hanover on 14 November 1716. On the evidence available he was buried with almost nobody in attendance. The standard account is that his secretary was effectively the only mourner (S235, S200). The Royal Society, of which he had been a Fellow for forty-three years, marked his death with nothing. Leibniz was a foreign member of the Académie des Sciences in Paris. It had Fontenelle deliver a formal éloge (Disputed: that document is real but was not read in full here; do not attribute wording to itDisputed).
Newton died on 20 March 1726/7. His body lay in state in the Jerusalem Chamber at Westminster. He was buried in the Abbey nave on 28 March. Two dukes, three earls, and the Lord Chancellor bore the pall. His grave lies under a monument by the King's own architect (S200).
Eleven years, eight hundred miles, and two funerals.
Huygens handed a 26-year-old diplomat a list of fractions and asked for the total. What did Leibniz notice about every single term?
Show the answer▸
Every term is a difference. So the sum collapses and the answer is exactly 2. Leibniz took a principle out of that afternoon. Richard Arthur states it: "the sum of the differences is the difference between the first term and the last" (S237).
That is the sentence underneath Integrals 1.4, the Fundamental Theorem. Sums and differences are inverse, which is why ∫ and d are, and he had the architecture before he had either symbol.
Chapter 8
The war
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
How it started
Not with a bang. With a subordinate clause.
- 1699. Nicolas Fatio de Duillier publishes an accusation. (His exact wording was never obtained for this book, so you will not find it quoted here.)
- September to October 1708. John Keill, writing in Philosophical Transactions 26 no. 317 about centripetal force, drops in an aside. The Latin, recovered verbatim: the method was published by Leibniz "mutatis nomine & notationis modo", with the name and the manner of notation changed (S189). The accusation is a throwaway line in a paper about something else.
- 4 March 1711 (NS). Leibniz complains to Hans Sloane and asks the Royal Society to make Keill retract publicly. Keill prefers to defend it. Leibniz writes again on 29 December 1711 (S283).
- March 1712. The Royal Society, presided over by Newton, appoints a committee. Nobody tells Leibniz it exists until after it has reported (S288).
- April 1712. It reports.
What the fight put into print
The quarrel was ugly, and it produced a library.
Almost everything Newton wrote about series and fluxions in the 1660s and 1670s reached print only after the dispute began in 1699 (S204, p. 456). Newton finished De analysi in 1669. It came out in 1711 in William Jones's collection, with the fight at its peak (S204, p. 457). Guicciardini puts it plainly: "The printing of Newton's mathematical works took place indeed in the context of the priority dispute" (S204, p. 465).
The fight also fixed what Newton said his own calculus was for. In the anonymous 1715 Account he described the algorithm as a tool and dismissed Leibniz's method. The tool was for "not demonstrating but only investigating a Proposition, for making dispatch"; Leibniz's method "is only for finding it out" (S204, p. 467). He was demoting the algorithm to a search device and reserving proof for geometry. That is the doctrine he had been pressing on his followers since the 1670s (S204). Notice what that move does inside a priority argument. If the algorithm is only a shortcut, then whoever printed one first has won a smaller prize.
The committee
De Morgan named all eleven members (S288, pp. 27 to 29). Five were working mathematicians: Halley, Jones, De Moivre, Machin, and Brook Taylor. Five were Royal Society men without a stake in the mathematics: Robartes, Hill, Burnet, Aston, and Arbuthnot. The eleventh was Bonet, the Prussian minister to London, and his presence is the detail worth stopping on.
Bonet is the entire basis of the anonymous Account's boast that it was "composed of Gentlemen of several Nations". Four were Newton's mathematician friends, one was Keill's friend, two were Newton's intimate personal friends, and one was a foreign diplomat.
The names were never officially published with the report, which is why they are hard to find (S288).
The conflict of interest, documented
This is the most damning verified fact in the whole affair, and it needs no embellishment.
Newton was President of the Royal Society. He appointed the committee that adjudicated his own priority dispute. He wrote much of its report himself: a 24,000-word holograph draft survives at CUL MS Add. 3968, ff. 67r to 96v (S282). The Commercium Epistolicum was dated 1712 and circulated in January 1713.
Then, in January to February 1714/15, he published an anonymous review of his own anonymous report. It ran in his own Society's journal (Philosophical Transactions 29 no. 342, pp. 173 to 224). And in that review he wrote:
But no Man is a Witness in his own Cause. A Judge would be very unjust, and act contrary to the Laws of all Nations, who should admit any Man to be a Witness in his own Cause. (S189, Wilkins ed. p. 13)
He wrote that sentence, anonymously, about a report he had written, reviewing his own case.
The report's governing principle is stated on pages 1 to 2: "Second inventors have no right" (S284). That single sentence explains why the quarrel could not be settled amicably. Under it, being second is being nothing, even if you got there on your own.
The date on the review is itself a calendar joke. Its issue is numbered for January and February 1714 in England and 1715 on the Continent. The two men could not agree what year they were fighting in (S189).
The anagrams
In the Epistola posterior of 24 October 1676, Newton concealed his method in a letter-count cipher:
6accdae13eff7i3l9n4o4qrr4s8t12vx
It decodes to Data æquatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa. In English: given an equation involving any number of flowing quantities, to find the fluxions, and conversely (S188).
Newton sent that string to Leibniz in October 1676 instead of the method itself. Leibniz was a working mathematician with a good head for ciphers and every reason to want what was inside it. Say what he could get out of it.
I have a guess▸
Nothing. It is a letter-count, not a code: six a's, two c's, thirteen e's. You cannot decrypt it, you can only check it once somebody tells you the answer (S188). It proves to a future reader what you knew and when, and conveys nothing at all to a present one.
If you took it for a puzzle to be solved, that is the natural reading. It is also the wrong one. Newton's own reason for sending it is almost mild: he did not want to have to change his plans if somebody else got there first.
It is not a code. It is a letter-count, down to a single d and a single æ among the rest (S188). It is a seventeenth-century hash commitment, or a sealed envelope lodged with a solicitor (S285).
The crux, resolved
Did Leibniz see De analysi before he invented his own calculus?
He saw it in October 1676, not 1673. Gerhardt found his two folios of extracts. He copied the series and root-approximation material. He skipped the tangent and quadrature material, because he already had that (S230, Child pp. 167 to 170). By October 1676 the ∫ and the d were a year old and dated.
The modern verdict
Stated plainly, and each part supported:
- Newton got there first. The October 1666 tract is hard dated evidence (S183).
- Leibniz published first, in October 1684 (S231).
- They worked independently. The dated Paris notebooks show the invention happening, including a wrong turn on 11 November 1675 that nobody copying Newton would make (S230).
- Leibniz's notation was better, and it won everywhere outside Britain (S232, S241).
- No credible evidence supports plagiarism in either direction.
One caveat you are entitled to. Three standard modern books on this fight stayed out of reach: A. Rupert Hall's Philosophers at War, Domenico Bertoloni Meli's Equivalence and Priority, and Guicciardini's Reading the Principia. One Guicciardini article is in hand (S204), and it backs the crux above from the manuscript side. It has Leibniz reading and transcribing parts of Collins's first De analysi transcript on the 1676 London visit. The article also explains the forty-year delay in printing as Newton's own policy rather than as anything Leibniz did. The five points above still rest mainly on the primary documents read here. Appendix L.5 names the three books and what each would add.
What it cost
British mathematics isolated itself for roughly a century, teaching dots while the Continent taught d's and pulled ahead.
The rescue came from undergraduates. In 1812 Charles Babbage, John Herschel, and George Peacock founded the Analytical Society at Cambridge to campaign for Leibniz notation. Babbage's suggested title for their manifesto, recalled in his 1864 Passages from the Life of a Philosopher, pp. 29 to 30:
The Principles of pure D-ism in opposition to the Dot-age of the University (S239)
Both barbs land: d-ism sounds like deism, and dot-age sounds like dotage, senility. The Society was accused of infidelity, and it was accusing Cambridge of senility. The title was suggested and never used. Instead, the volume appeared as Memoirs of the Analytical Society in 1813. Babbage is recalling the joke fifty years later (S239).
Fourteen years after being called a young infidel, Babbage was given Newton's own chair.
Britain spent about a century teaching dots and falling behind. In 1812 three Cambridge undergraduates founded a society to fix it. What were they campaigning for?
Show the answer▸
Leibniz's notation. The d's. Babbage's suggested title for their manifesto was "The Principles of pure D-ism in opposition to the Dot-age of the University". That title accuses Cambridge of senility and gets the authors accused of infidelity in one line. The title was never used (S239).
You can see what they were fighting for in Integrals 2.2. There, the dx in ∫f(x)dx is the thing that tells you what to do when you change variables. Notation is not decoration. It decides which moves look obvious.
Chapter 9
The eighteenth century: attacked, defended, and finally taught
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The 1700s are where calculus stopped being a discovery and became a subject. It got a textbook, it got applied to everything, and it got attacked on grounds that were completely correct.
The Bernoullis, the most dysfunctional family in mathematics
Jacob and Johann Bernoulli reverse-engineered the calculus out of Leibniz's proof-free 1684 paper. Then they built most of the early subject from it (S231).
The word "integral" was negotiated between three men who could not stand each other. Jacob prints it, Johann claims it, and Leibniz has to be talked out of his own preferred name, calculus summatorius (S324). The vocabulary of calculus was hammered out in letters between people who were feuding.
The brachistochrone, 1696. Johann posed the problem of the curve of fastest descent partly to embarrass his brother Jacob, with whom he was by then openly at war. Jacob solved it, and solved it better. Johann then spent years insisting his own solution was superior. The family feud produced the calculus of variations as a by-product (S330).
The famous line about Newton's anonymous solution needs care. Its Latin is "tanquam ex ungue leonem", "as the lion is known by his claw". This book verified it in David Brewster's 1831 Life of Newton, p. 194 (S321):
although that of Newton was anonymous, yet Bernouilli recognised in it his powerful mind, "tanquam," says he, "ex ungue leonem," as the lion is known by his claw. (S321, Brewster 1831, p. 194; spelling and italics as printed)
This is a rare, fully checkable example of how a quotation gets manufactured. A nineteenth-century biographer renders a French sentence into Latin and adds "says he". The Latin version then outlives the original by two centuries (S321). The Latin belongs to Brewster, writing in 1831, so the line is his and not Bernoulli's (S321). Disputed: Bernoulli's own 1697 printing appears to have been in French, and nobody working on this book could retrieve it. So the French reading rests on a citation trail rather than on the page (S337).Disputed
Also debunked here: the story that Newton solved it in twelve hours on returning from the Mint. Brewster's Mint detail belongs to Leibniz's 1716 problem, not the 1697 one (S321).
Jacob's tombstone. He loved the logarithmic spiral because it reproduces itself under scaling. He asked for one on his grave with the motto eadem mutata resurgo, "though changed, I rise again the same". The stonemason carved an Archimedean spiral (S328, S329, confirmed by two independent sources). The difference is a good two-minute exercise:
William Ashworth's dry moral: if you want your mathematics on your tombstone, supervise the mason (S329).
And Johann versus his own son. Daniel Bernoulli published Hydrodynamica. Johann's competing Hydraulica is titled "...Anno 1732", but Johann first mentions it to Euler in October 1738, months after Daniel's book appeared (S333). Read the sequence to a class: son publishes. Two months later the father tells a former student about a book that will turn out to be dated six years earlier. The student receives both in the same post. Johann had earlier thrown Daniel out of the house after they tied for the 1734 Paris Academy prize (S333).
L'Hôpital's rule is Johann Bernoulli's
The first calculus textbook, Analyse des Infiniment Petits (1696), was published anonymously by the Marquis de l'Hôpital. Everyone knew who wrote it. The man who wrote most of the mathematics was in Groningen, under contract not to say so (S316).
The contract is documented: 300 pounds a year from 1 January 1694, offered in a letter of 17 March 1694. It bought both exclusivity and silence (S315). Bernoulli's audition piece at their first meeting in Malebranche's salon was his unpublished general formula for the radius of curvature. He produced it and l'Hôpital hired him on the spot for four lessons a week (S315). L'Hôpital did not always pay in full (S315).
The complication that makes it a better story than "he stole it" sits in l'Hôpital's preface. The preface does acknowledge the Bernoullis ("je me suis servi sans façon de leurs découvertes", I have freely made use of their discoveries). Yet the body of the book credits half a dozen people by name and never Bernoulli (S316). Both things are true, and that is the story.
Euler
Leonhard Euler (1707 to 1783) is pronounced OY-ler, /ˈɔɪlɐ/, and never YOO-ler.
The Basel problem, 1734/35, made him famous at 28 because everybody had tried and failed since Mengoli posed it in 1650. His argument treats sin(x)/x as an infinite-degree polynomial and factors it by its roots:
I verified the coefficient extraction symbolically and the sum numerically (Appendix E.11).
This argument is not rigorous, and that is the point of including it. A university teaching module on the Basel problem (S322) puts it well in a footnote. Euler "was confident we could factor an infinite degree polynomial in the same way that we could factor a finite degree polynomial, and proceeded accordingly" (S322). Justifying moves like this is exactly the job the nineteenth century had to do. And Euler was not guessing blind: he had already computed the sum numerically and recognized 1.644934... as π²/6. The factorization explained a number he had already seen (S322).
Two corrections your textbook probably needs:
- Euler's 1735 Basel paper (E41) uses
p, not π. He adopts π at Introductio §126 in 1748 (S318, S325). - Euler's identity is not in the Introductio. §138 gives only . Euler's
idates from 1777 and was published in 1794 (S318, S325).
Notation he standardized or introduced: f(x), e, Σ, Δ, i, and π by adoption. In 1755, Δ and Σ arrive in one book, in one sentence, as explicit twins (S451). The letter e began life in a paper about cannon. The letter i waited seventeen years between being written and being printed. And π was a Welsh mathematician's abbreviation that a Swiss genius made compulsory (S325).
He went blind around 1771, in his sixties, dictated to his sons and secretaries, and doubled his lifetime output afterwards. The remark that blindness would mean fewer distractions is quoted everywhere and nobody has found where he said it. So you will not see it in quotation marks here (S326). His papers were still being published from the backlog for 43 years after his death. That is why he needed his own catalog, the E-numbers (S323).
The Diderot anecdote is fabricated, and the fabrication is fully traceable. The story has Euler silencing Diderot at the Russian court with ", therefore God exists". It collapses under Ronald Gillings' 1954 analysis. The chain: Thiébault reports an unnamed mathematician, does not ridicule Diderot's competence, and explicitly disclaims the story. De Morgan in 1872 names Euler and adds "algebra was Hebrew". Cajori repeats De Morgan in 1919; Bell says "Chinese" in 1937; Hogben says "Arabic" in 1951. Nobody went back to the source for 138 years. And Gillings' killer detail: Thiébault printed , and De Morgan miscopied the as an (S313, S314).
Before you read on: five writers repeated this story over 138 years and not one of them checked. What could any of them have done in an afternoon that would have caught it?
What would have caught it▸
Open Thiébault. The source is a printed book that was sitting in libraries the whole time.
Anyone who opened it would have found three things at once. The mathematician is unnamed, Thiébault explicitly disclaims the story, and the formula reads . De Morgan miscopied that z as an n in 1872, and every later teller copied De Morgan (S313, S314).
The anecdote survived because it is a good story and checking is boring. Each writer trusted the previous writer, which felt like four independent confirmations and was one unchecked claim repeated four times.
You will meet the same structure whenever a number gets quoted with no source attached.
The "cyclops" jibe is real, and it is not what people think. Frederick the Great wrote it to Voltaire on 25 January 1778: "the Cyclop Euler calculated the effort of the wheels... Vanity of Vanities! Vanity of mathematics." But Michael Eckert shows Euler spent about a month on the Sanssouci fountain job in autumn 1749. Euler correctly diagnosed the oversized pump, warned Frederick in writing, and was ignored (S342). One letter, twenty-nine years later, from a king still sore about a fountain, is not a standing nickname.
Maria Gaetana Agnesi
Pronounced ah-NYEH-zee, /aɲˈɲeːzi/.
Instituzioni analitiche ad uso della gioventù italiana (1748) is the first surviving mathematics book written by a woman. It is also the first textbook to treat differential and integral calculus together.
Why the book looks the way it does is the interesting part. Massimo Mazzotti's argument is that Agnesi set out to show the newest calculus could be understood in purely geometrical terms. Her aim was that modern analysis would "enrich, not undermine" the traditional framework of religious and metaphysical knowledge (S077, p. 675). That is a deliberate intellectual strategy, not timidity. She held that the differential was equivalent to the Newtonian fluxion and insisted on the priority of geometrical evidence. She also left out the mechanical and hydraulic applications that mattered most to working practitioners.
Three things about it a teacher will recognize immediately:
- She wrote in Italian, not Latin, though Latin "is believed by some to be more convenient to these matters". The result was "the first complete systematic presentation of the Italian terminology for the concepts of calculus" (S077, p. 675). She was building a vocabulary, not just a book.
- She introduced signed numbers through debits and credits (S077, p. 675). In 1748.
- She took "unusual care... in explaining every single step of mathematical reasoning" (S077, p. 675). Volume 1 covers algebraic equations and Cartesian geometry; Volume 2 covers differential calculus, integral calculus, and differential equations, which she called "the inverse method of tangents" after the problem they came from.
Her father put a printing press on the ground floor of the palazzo so she could stand over the typographers. These were typographers "who had never before worked with the symbols of differential and integral calculus" (S077, p. 680). That is why the presses were in the house: nobody in Milan could set a d correctly.
The Bologna chair, with the documents. Benedict XIV wrote to her on 10 September 1750 informing her of appointment to a lettura. The university diploma is dated 5 October 1750 and names the post Cattedra di Pubblico Lettore di Matematica. She never went to Bologna to take it up (S077, n. 55, citing Biblioteca Ambrosiana O.202.Sup).
Mazzotti's larger claim is worth putting in front of students because it cuts against the expected story. The space that let a woman become a legitimate scientific author in the 1740s was opened by reformist Catholicism under Benedict XIV. Mazzotti adds that "neither more conservative traditional Catholicism nor the radical culture of the Enlightenment seems to have offered women comparable opportunities" (S077, p. 658). After that pontificate the space closed. As he puts it, "it would be a long time before a woman would again be offered a chair at a European university."
The "witch" is a mistranslation, and the chain is now fully sourced:
- Guido Grandi coined the term in 1718: Latin versoria, the rope that turns a sail, from vertere, to turn (S077, n. 44).
- The Italian for that rope is la versiera.
- Agnesi writes, at Volume 1, page 381: "equazione alla curva da descriversi, che dicesi la Versiera", "the equation of the curve to be described, which is called la Versiera" (S077, n. 44; S319).
- John Colson translated it into English, and John Hellins edited and published it in 1801. At page 222 the phrase reads "which is vulgarly called the Witch" (S077, n. 44).
- Colson's synopsis went further and made it hers: "which she calls the Witch" (S320). He mistranslated and misattributed in one stroke.
Two details are worth getting right. Colson died in 1760, so he translated before then. But the printed book is 1801 and Hellins edited it, which is why both names belong on it (S077, n. 44). And Mazzotti hedges the mechanism: the rendering happened "perhaps because of an interesting confusion" with avversiera, she-devil. "Probably" is as far as the evidence goes, and "because" goes further than Mazzotti does (S077).
Why was the curve in the book at all? It "had attracted little attention in earlier treatises precisely because it could not be associated with any relevant mechanical or physical application. Its interest lay exclusively in its remarkable metric properties" (S077).
The irony is exact: the same John Colson annotated Newton's Method of Fluxions and was thanked by Maclaurin for defending Newton against Berkeley (S311, S320).
And she did far more than one curve. Antonella Cupillari's survey of the Instituzioni is the corrective to a century of reducing her to the witch (S075). One example, ready to teach: at page 361 Agnesi takes , finds both asymptotes, and plots points. Then she establishes concavity with no calculus at all, by comparing the height of a chord against the height of the curve at the midpoint. She reduces that comparison to the inequality . That is midpoint convexity, argued by hand in 1748, and it agrees with the sign of the second derivative.
Her later life, precisely, because the loose version undersells it. She negotiated conditions with her father: dress simply, attend San Nazaro at will, and give up balls and the theater. Another was to volunteer at the Ospedale Maggiore caring for poor and infirm women. Her father died in March 1752. In 1771 Archbishop Pozzobonelli asked her to direct the female department of the newly opened Pio Albergo Trivulzio, which eventually held about 450 patients. In 1783, having renounced everything she owned, she moved in. She died of pneumonia in January 1799, and because war was again ravaging northern Italy, Milanese authorities barely noticed. She was buried in a mass grave outside Porta Romana (S077).
She held a documented university chair and has no grave.
Émilie du Châtelet
Her translation of Newton's Principia into French, with commentary, appeared in 1759. It is to this day the only complete French translation of that work (S069). Her addition beyond translation is substantive: the Exposition abrégée du système du monde commentary (S332).
She had to be taught the mathematics, and the record of who taught her survives. Maupertuis started her around 1733 on Nicolas Guisnée's Application de l'algèbre à la géométrie. He had worked through the same text under Nicole and Bernoulli. She found it dry and told him so (S068). Then Alexis-Claude Clairaut took over and pushed her through equations of the third and fourth degree, multiple unknowns, and operations on radicals. Zinsser makes a point about why that mattered. It was exactly the equipment needed to follow Maupertuis's project of turning the geometric arguments of the Principia into algebraic ones (S068). When she started her own commentary in the late 1740s, "it was Clairaut who met with her and checked her calculations." He was, in Zinsser's words, "the mentor who took her aspirations as a géomètre seriously and helped her to progress beyond that of an amateur", and from their first lessons he had found her "altogether remarkable" (S068).
She argued the Leibnizian side of the vis viva dispute: Newtonians measured "force" by mv, Leibnizians by mv². Both were right about different things, momentum and kinetic energy, and it took a century to see it. Du Châtelet was pointing at what we now call energy (S332, S068). Her Institutions de Physique was written as a physics textbook for her thirteen-year-old son. Its last chapter is an open attack on Dortous de Mairan's 1728 Académie memoir on forces vives. That attack appeared just as Mairan was about to become the Académie's perpetual secretary (S068).
Two of her three major works went out without her name on them. The 1738 essay on the nature and propagation of fire was submitted anonymously to the Académie prize competition. That was because the judges would not have taken a submission known to be by a woman seriously. It did not win, and the Académie published it anyway. The Institutions de physique also appeared anonymously, in 1740, with a frontispiece carrying portraits of Newton, Descartes, and Leibniz (S069). The Académie royale des sciences admitted no women at all, except as spectators at its two public meetings a year (S069).
She was 42, pregnant, and knew what that meant. So she "rushed on with her translation and commentary, hoping to finish the work by the time her child was born", resolving "to sequester myself absolutely" (S069). She was right about the danger. She died in 1749.
The book exists because of Clairaut. It came out ten years after her death because he pressed the publisher Laurent François Prault to bring it out. He also arranged for the illustrations to be finished (S068). That gap is usually reported as neglect. It has a name attached to it, and the name is her mathematics tutor.
Berkeley's The Analyst, 1734
George Berkeley, Bishop of Cloyne, published The Analyst; or, a Discourse Addressed to an Infidel Mathematician. The infidel is generally taken to be Edmond Halley. And there is a documented occasion. Halley had told the dying Dr Samuel Garth that "the doctrines of Christianity are incomprehensible, and the religion itself an imposture" (S310, per Stock).
The famous passage, at Section 35. Note the punctuation, because almost everyone gets it wrong:
And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? (S310, S. 35, body text)
It is a chain of questions, not a statement. The full stops and the quotation marks that circulate come from later editors' summaries (S310). Berkeley closes the book with 66 Queries, questions rather than assertions, which is a rhetorical trick worth showing students directly.
Fermat's method, and Newton's after it, does two things in a row. It divides by a small quantity e, then deletes the e that is left over. Berkeley put a bishop's finger on it in 1734. Name the step that is illegal.
I have a guess▸
Neither of them, on its own. It is the pair. Dividing by e requires e ≠ 0. Deleting e requires e = 0. Each step is fine and you are not allowed both. That is why Berkeley could call the results "the ghosts of departed quantities" and be mathematically right (S130, S310).
If you picked one of the two, that is the move the trap is built on. Everybody from Fermat to Newton made it and got correct answers for a hundred years. That is exactly what made the objection so hard to answer.
His motive was theological and his mathematics was correct. That combination is the whole lesson. He is pointing at the step Fermat had fudged in 1636 and Newton had papered over in 1687. You divide by an increment, which requires it to be nonzero. Then you discard it, which requires it to be zero. Berkeley is not wrong.
The responses: James Jurin's Geometry No Friend to Infidelity, a reply from Benjamin Robins, and Colin Maclaurin's Treatise of Fluxions (1742). Maclaurin's book was a serious eight-year attempt at rigor. Maclaurin says so in his own preface:
A Letter published in the Year 1734, under the Title of the Analyst, first gave Occasion to the ensuing Treatise... The Author of that Piece had represented the Method of Fluxions as founded on false Reasoning, and full of Mysteries. (S311, Preface p. i)
Maclaurin also concedes the substantive point: "the Supposition of an infinitely little Magnitude being too bold a Postulate" (S311).
Both Berkeley and Jurin published anonymously or pseudonymously and then attacked each other for hiding (S310).
Treat this episode as a hinge, not a footnote. Everything in Chapter 10 is a response to it.
Lagrange tries to fix it with algebra
Joseph-Louis Lagrange's Théorie des fonctions analytiques (1797) has a subtitle that is the whole argument. It offers the principles of differential calculus, freed from any consideration of infinitesimals, vanishing quantities, limits, or fluxions. It then reduces them to the algebraic analysis of finite quantities (S317). A book whose subtitle is essentially "no infinitesimals, no limits, no fluxions, just algebra" is a direct public answer to a bishop's pamphlet.
His method: assume every function equals its Taylor series, and define the derivative as a coefficient in it. Articles 16 and 17 give the vocabulary and the notation we still use (S317). Here is Lagrange's French, then our own translation:
si, pour plus de simplicité et d'uniformité, on dénote par f'x la première fonction dérivée de fx, par f''x la première fonction dérivée de f'x... Nous appellerons la fonction fx, fonction primitive... et nous appellerons celles-ci, fonctions dérivées.
"If, for greater simplicity and uniformity, we denote by f'x the first derived function of fx, by f''x the first derived function of f'x..." "We shall call fx the primitive function... and we shall call these the derived functions."
That is the birth of the word "derivative" and of the prime notation.
And the honest thing to tell students is that Lagrange's fix did not work, because not every function is the sum of its Taylor series. Cauchy later produced the counterexample:
The function is smooth everywhere, its Taylor series at the origin is identically zero, and the function is not (S317).
Taylor and Maclaurin did not invent their series
- Brook Taylor, Methodus Incrementorum (1715). Gregory, Leibniz, and Johann Bernoulli had it earlier (S312). Taylor's notation is so bad that James Gibson refused to reproduce it: "The notation used by Taylor would make too heavy demands on the printer to justify me in reproducing it here" (S312).
- Colin Maclaurin. James Stirling published the series in Lineae Tertii Ordinis (1717), p. 32, twenty-five years earlier. Stirling said in 1730 that it was Taylor's, and Maclaurin credited Taylor too (S312).
Gibson's remark is worth quoting to a class. "It is, from our present standpoint, strange to see how near Newton in particular came to Taylor's Theorem and yet did not attain to it" (S312).
At page 361 of her 1748 textbook Agnesi settles where a curve bends, with no calculus at all. What is she comparing?
Show the answer▸
The height of a chord against the height of the curve at the midpoint. She takes , finds both asymptotes, plots points, and reduces the whole question to the inequality (S075). That is midpoint convexity, argued by hand, and it agrees with the sign of the second derivative.
Derivatives 2.8 hands you the same verdict in one line. Her page is what the job took before the test existed, which is the best reason there is to trust the test.
Chapter 10
The nineteenth century: making it true
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
Berkeley's objection stood unanswered for about ninety years. This chapter is the answer, and it took four countries and most of the century.
The pressure came from heat
Joseph Fourier's Théorie analytique de la chaleur (1822) claimed that any function has a trigonometric series. It was not fully true, and it was not properly provable, and it is the single most productive wrong-sounding statement of the century. Forcing it to make sense produced Dirichlet's definition of function, Riemann's integral, Cantor's set theory, and Lebesgue's measure (S383). The whole of modern analysis is downstream of a problem about heat.
There is an institutional cause too. The École Polytechnique was founded by the French revolutionary government to train engineers. Rigorous calculus is partly a by-product of the Revolution's need for them (S360, p. 189).
Fourier himself was arrested twice during the Terror, went to Egypt with Napoleon, and governed a province. He also believed heat was good for you and kept his rooms unbearably hot (S383).
Bolzano, silenced
Bernard Bolzano published Rein analytischer Beweis in Prague in 1817. The title is the whole argument. "Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation". The word doing the work is rein, purely: no pictures, no motion, no geometry (S363).
A curve starts below the axis and ends above it, so it has to cross somewhere. Every mathematician before 1817 agreed, and so does everybody you will ever ask. Bolzano called his paper Purely analytic proof, and the word doing the work is purely. Name what he said was wrong with the standard argument.
I have a guess▸
It is a picture, not a proof. Using a fact about geometry to establish a fact about numbers imports applied mathematics into pure, and he would not have it (S363, S387).
If "it obviously crosses" felt like enough, you are in the company of everyone who came before him. The discomfort you are feeling now is the whole of the nineteenth century, and this chapter is what it produced.
His objection is one a fifteen-year-old feels immediately: a picture is not a proof, however obvious it looks (S363, S387). That distinction is worth a whole lesson.
He got the IVT, a working definition of continuity, and the Bolzano-Weierstrass theorem, all before Cauchy.
Then the state destroyed his career. He was a Catholic priest, appointed to a chair of religious studies created to inoculate students against the French Revolution. He preached pacifism, social justice, and the absurdity of national hatred. Finally the Emperor personally fired him in Disputed: December 1819 or January 1820Disputed and forbade him to publish (S370). His mail was opened. He was tried by his own Church, told to recant, and refused (S371). He kept writing for twenty-nine more years, mostly into a drawer.
Around 1830 he constructed a continuous, nowhere-differentiable function, roughly forty years before Weierstrass. Be precise about what he proved: he showed the non-differentiability points are dense, with a flawed continuity argument. He did not complete the theorem (S384, S386). The manuscript sat in the National Library in Vienna until Martin Jašek found it in 1920. That was a hundred years after Bolzano was silenced and two years after the Habsburg empire that silenced him ceased to exist. Bohemia had become Czechoslovakia, and Czech mathematicians went looking for their own forgotten genius (S386). Six weeks after Jašek's announcement, Rychlík and Jarník independently proved the theorem Bolzano could not finish. The Functionenlehre was printed in 1930.
Cauchy, and a fight that is still running
Augustin-Louis Cauchy's Cours d'Analyse (1821) and Résumé (1823) are where the modern definitions arrive. He defined the derivative by a limit and gave the first real definition of the definite integral as a limit of sums. And he proved the fundamental theorem in something close to the modern way.
He also explained the integral sign to his students by telling them it is a letter S that has been stretched, S for somme (S362). Two centuries later students still draw it without being told why.
His examples of a limit are charmingly old-fashioned: an irrational number is the limit of fractions approximating it. And "in geometry, the surface of the circle is the limit toward which the surfaces of inscribed polygons converge as the number of their sides grows" (S361, p. 4). The most modern definition in the book is illustrated with Archimedes.
And here is the live dispute, which you should teach as a dispute.
Cauchy's 1821 definition of continuity is stated in infinitesimal language. In his own words: "an infinitely small increment of the variable always produces an infinitely small increment of the function itself" (S361, p. 35). No epsilons. And his 1823 preface states his program as reconciling rigor "with the simplicity that results from the direct consideration of infinitely small quantities" (S362).
On page 131 of the Cours he states, with complete confidence, that a convergent series of continuous functions has a continuous sum. That is false as stated, because it needs uniform convergence, which nobody had yet. Niels Henrik Abel published the counterexample in 1826, citing the page number:
Dans l'ouvrage cité de M. Cauchy on trouve (page 131) le théorème suivant...
"In the work of M. Cauchy cited above one finds, on page 131, the following theorem..." (S364, p. 71, footnote. English translated here, not quoted from a published translation.)
His counterexample is . You can watch the error being caught in real time, in print, on facing pages. Abel was 23 or 24. He corrected the most famous analyst in Europe in a footnote, and in the main text he called the book indispensable. It is a masterclass in how to disagree (S364).
Disputed: The dispute: was Cauchy simply wrong, or was he reasoning coherently with infinitesimals in a framework that later got vindicated?Disputed Judith Grabiner, Umberto Bottazzini, Craig Fraser, and Gert Schubring say the theorem is false. Detlef Laugwitz, Mikhail Katz, and others argue for a coherent infinitesimal reading (S367, S368). Both sides were read in full for this book and the question is not settled. One of the papers is titled as a direct riposte to Grabiner's article title, thirty years later (S368). Historians of mathematics subtweet each other in journal titles.
Cauchy the man: born in Paris five weeks after the storming of the Bastille. He was on the losing side of every subsequent French revolution. He followed Charles X into exile in 1830 rather than swear a loyalty oath, giving up his salary, his chairs, and his country (S372).
The Galois myth does not survive the documents. The version students meet, a genius destroyed by a jealous establishment with his life's work binned by Cauchy, is not what they show. Galois's lost manuscript was the February 1830 Grand Prize submission, held by Fourier, who died that May (S383). Cauchy's documented 1829 role was to referee it and advise resubmission (S380). What the documents show is worse and more ordinary. There was a bad-tempered teenager, a referee who died, a second referee who wanted clearer writing, and a pointless duel. That is a better lesson than the myth.
Abel, and a quotation that is a translation of a translation
Abel's "divergent series are the invention of the devil" is not what Abel wrote, and the phrase has been traced back to its source (S490).
The letter is to Bernt Holmboe, from Berlin, not Paris, dated 16 January 1826. (The 1902 editors argue it was written on the 14th and dated when sent.) Abel's original Dano-Norwegian, at p. 16:
Divergente Rækker ere i det Hele noget Fandensskab, og det er en Skam at man vover at grunde nogen Demonstration derpaa.
Here is a direct translation. "Divergent series are, on the whole, a piece of devilry, and it is a disgrace that anyone should dare to found any demonstration upon them."
The French translation in the same 1902 volume, by P. G. la Chesnais, reads "une invention du diable". "Invention of the devil" is la Chesnais's phrase, not Abel's (noget Fandensskab means "some devilry"), and the English version drops Abel's hedge i det Hele, "on the whole" (S490).
The human framing is good too: Abel is 23, on a traveling scholarship, writing home to his old schoolmaster. In the same run of letters he is being fitted for new clothes so as to be complet when he reaches Paris. He is also going to the opera and playing cards. The sentence that ended up on a thousand lecture-room walls was a young man's aside in a chatty letter. He softened it immediately (S490).
Abel died on 6 April 1829, aged 26, of tuberculosis, two days before a letter arrived from Crelle announcing a professorship in Berlin (S364).
His line about Gauss is also worth having: "He is like the fox, who effaces his tracks in the sand with his tail" (S385). The line is about Gauss's habit of removing every trace of how he found things.
Weierstrass
Karl Weierstrass taught gymnasium for years, including gymnastics and handwriting. He published almost nothing for over a decade, and then produced one paper that made him famous across Europe within months (S373). Disputed: Sources say he became a professor at 39, though he was 40 or 41, and give his teaching span as 14 years with some variation.Disputed He is the strongest available answer to a student who thinks it is too late for them.
The ε-δ definition reached its final form through his Berlin lectures. Many of them were published only through students' notes, which is why "who wrote it first" is genuinely tangled (S373). He built on Bolzano and Cauchy. The ε is literally the "e" of erreur, and the δ of différence (S360). The symbol of mathematical precision started life as the symbol of being wrong.
The monster function, 1872. Continuous everywhere, differentiable nowhere. Get three details right:
- Weierstrass presented it on 18 July 1872 and never published it. Paul du Bois-Reymond published it, in 1875 (S386).
- He opened by giving the credit to Riemann (c. 1861) (S386).
- Charles Cellérier is a third independent discoverer (c. 1860, published 1890) (S386).
Hermite's reaction is verified from the primary source, Correspondance vol. II p. 318, letter 374, Paris, 20 May 1893:
Je me détourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n'ont point de dérivées. "I turn away with dread and horror from this lamentable plague of continuous functions that have no derivatives." (S365)
The standard English version drops the word "continuous", which destroys the point. And the better line, in the same letter, is the one nobody quotes. "L'Analyse retire d'une main ce qu'elle donne de l'autre", "Analysis takes back with one hand what it gives with the other" (S365). Hermite wrote it at 70, over seven years before he died, to a friend who had under two years to live. In the same letter he cheerfully says that when it is thundery he does nothing but daydream or sleep.
The ending is the best part. Sixty years later, Banach and Mazurkiewicz proved that almost every continuous function is nowhere differentiable (S388). Hermite was not turning away from a plague. He was turning away from the general case.
And there is an internal irony worth showing a class. The standard proof that Weierstrass's monster is continuous uses the Weierstrass M-test and the theorem that a uniform limit of continuous functions is continuous. The tool that tames the monster is the repair of the exact error Cauchy made in 1821 (S389).
Riemann, Dedekind, Cantor, Lebesgue
Bernhard Riemann defined his integral in his 1854 Habilitationsschrift on trigonometric series. He did so because he could not even state his question until he had said what an integral is (S391). He was 27. It was not published until 1867, after he died at 39. His definition uses arbitrary partitions and arbitrary tags, and it ends:
If it does not have this property, then ∫f(x)dx has no meaning. (S391, Riemann's Werke; translated from the German here, so treat the wording as a rendering, not a set text)
What students learn as "the Riemann integral" is usually Darboux's 1875 reformulation with upper and lower sums (S391, and the notation file).
At the same Habilitation, the candidate proposes three topics and the examiner picks. Gauss, against custom, picked the third, the one on the foundations of geometry that Riemann had prepared least. The result is the mathematics Einstein needed sixty years later (S375). A job interview produced the toolkit of general relativity.
Richard Dedekind gives the best classroom hook in the whole century. A young professor about to teach calculus for the first time realizes he cannot prove the most basic fact about limits without drawing a picture. He is embarrassed and resolves to fix it. He fixed it on a specific Wednesday:
I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Durège with whom I had a long and lively discussion. (S366, Beman's translation, p. 2)
Then he did not publish for fourteen years, because "the presentation did not seem altogether simple, and further, the theory itself had little promise" (S366).
Charles Méray published a construction of the reals in 1869, before both Dedekind (1872) and Cantor (S378). He is usually skipped, and he is a subtler "who got robbed" case than Bolzano: nobody suppressed him. He published in French, in a normal journal, in a country whose mathematical culture had no appetite for the question. He was just in the wrong place (S378).
Georg Cantor built the actual infinite into mathematics and called infinitesimals "the cholera-bacilli of mathematics" (S369). His 1888 manifesto sentence is the fault line running to constructivism: whether we can decide something is irrelevant to whether it is true (S382). Kronecker, and later Brouwer and Bishop, thought exactly the opposite. The romantic story that Kronecker single-handedly drove Cantor mad does not survive checking. MacTutor dates the first breakdown to 1884 and presents the illness as recurrent and independent, while documenting the professional obstruction as real (S376).
Henri Lebesgue's 1902 thesis Intégrale, longueur, aire answered the question the Riemann integral could not. What do you do with a function that is discontinuous everywhere? His coin analogy has Riemann counting coins in the order you pull them out of your pocket, while Lebesgue sorts them into denominations first. That analogy reaches us in paraphrase. Nobody here traced the exact words back to Lebesgue, so take the analogy as his idea rather than as his sentence (S377). Two founders of modern French analysis, Lebesgue and Borel, fell out permanently over war work in 1914 to 1918 (S377). Mathematics is done by people who stop speaking to each other.
The coda: Berkeley answered, 232 years later
In 1966 Abraham Robinson published Non-standard Analysis, giving infinitesimals a rigorous foundation through model theory. Berkeley's ghosts got a birth certificate (S381).
The fix that makes them legal is a single extra idea: the standard part. Take the ratio, then round to the nearest real number. H. Jerome Keisler's 1976 textbook puts the payoff in one line:
Leibniz was on the right track, but 300 years too soon! (S440)
Errett Bishop attacked it on constructivist grounds (S390). Disputed: His 1977 review is quoted all over the internet, and the review itself was blocked on every attempt to reach it here. So the sentences you will meet carry no source anyone has checked (S395, S390).Disputed The argument is the nineteenth century's again in twentieth-century dress. Kronecker told Cantor mathematics must be built from the integers; Bishop told Robinson a proof must carry numerical content. Same objection, ninety years apart, aimed at the two things nineteenth-century rigor made respectable: the actual infinite and the infinitely small (S390).
And the loop closes on Chapter 2. Archimedes had both methods in 250 BCE. He found his answers with infinitesimals and published them with something like exhaustion, because he did not trust the first method in print. Two thousand two hundred years later, that is still what most mathematicians do (S440).
Weierstrass presented a function in 1872 that is continuous everywhere. What can it not do, anywhere at all?
Show the answer▸
Have a derivative. Continuous everywhere, differentiable nowhere. Hermite turned away "with dread and horror from this lamentable plague of continuous functions that have no derivatives" (S365). The standard English version drops the word "continuous", which destroys the point.
Derivatives 1.6 trains you to read a derivative off the shape of a graph. This function is the edge of that skill. Sixty years later Banach and Mazurkiewicz proved almost every continuous function is one of these (S388). So the well-behaved curves in your course are the rare case.
Chapter 11
The people history skipped
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
This chapter is not an appendix to the story. Several of these people did work that is in the standard curriculum under somebody else's name. The reasons they are missing are documented rather than vague.
Sophie Germain
Ranked by what a sixteen-year-old will remember:
- Her parents confiscated her candles, her fire, and her clothes to stop her studying at night. They gave up when they found her asleep in the library at dawn with the ink frozen solid in the inkwell (S410).
- She got into mathematics because she read that Archimedes was killed by a Roman soldier for refusing to stop doing geometry, and decided anything worth dying over was worth doing (S410).
- She corresponded with Gauss for three years as "Monsieur LeBlanc". She revealed herself only because Napoleon's army occupied Gauss's town and she was afraid he would die the way Archimedes did, so she sent a French general to check on a man she had never met (S410).
- The astronomer Lalande told her to put down Laplace and read his Astronomy for Ladies, a book with not one equation in it. He apologized in writing on 4 November 1797. She never forgave him (S410).
- She won the Paris Academy prize for elastic plate theory (Disputed: 1815 or 1816 in different sourcesDisputed). Seventy-two names are carved around the first stage of the Eiffel Tower, a structure that stands up because of elasticity theory. Hers is not one of them (S410).
Mary Somerville
Mechanism of the Heavens (1831) was her translation-and-expansion of Laplace, and it went into use at Cambridge. George Peacock wrote to her on 14 February 1832 about its adoption there (S412). Disputed: Whether the adoption was formal and required, or a strong recommendation, is not settled by the source. Use Peacock's own wording rather than the phrase "required text".Disputed
- She was asked to translate a book that, by the commissioner's own estimate, fewer than twenty people in England understood. She agreed on condition that if she failed, the manuscript would go in the fire (S412).
- Poisson told her husband, at a dinner in Paris where the whole table was congratulating her, that there were "not twenty men in France" who could read her book (S412).
- Her mathematics books were left to Girton, the first Cambridge college for women, a college that did not exist when she was refused an education (S412).
Her best line for a nervous student: do not confuse the mathematics required to discover something with the mathematics required to understand it (S413). A student intimidated by calculus is, in Somerville's diagnosis, making exactly that mistake.
The usual story about the word "scientist" is wrong in two places. That word first appears in print in an anonymous 1834 Quarterly Review article now attributed to William Whewell. The article reviews On the Connexion of the Physical Sciences, not Mechanism of the Heavens. Whewell does not claim the coinage; he credits "some ingenious gentleman" (S411, S442). So "Whewell coined the word" is a sentence this book will not write. The word appears nowhere in Somerville's memoir. What is true and better: the word covering every chemist, geologist, and astrophysicist on Earth began life in a book review. It came as a joke the reviewer thought had not caught on, in the same paragraph as the rejected alternative "nature-poker". And the book under review was written by a woman who had never been allowed to attend a university (S411).
Whewell was the nineteenth century's naming service. When Faraday needed words for the parts of an electrolysis cell, Whewell supplied anode, cathode, and ion (S442).
Ada Lovelace
Note G of her 1843 translation-and-notes contains an algorithm for computing Bernoulli numbers.
- She states in print that she chose the harder Bernoulli formula on purpose, to show the machine off (S415).
- The Analytical Engine was never built. The algorithm in Note G was, for over a century, a program with no computer (S415).
- The Engine's control system came from the silk-weaving industry: punched cards from the Jacquard loom. Her sentence about the engine weaving algebraical patterns as the loom weaves flowers and leaves is the origin of the whole "software as fabric" family of metaphors (S415).
Disputed: What she did and did not do is a live scholarly dispute.Disputed Present both sides; do not pick one. And note the connection: Somerville was Lovelace's mentor and the person who brought her to Babbage's house (S416).
Sofia Kovalevskaya
The obstacle was never the mathematics. It was a registrar (S374). Universities would not admit her, so Weierstrass taught her privately. She solved a partial differential equations problem well enough to have her name permanently attached to Cauchy's, and she was not allowed in the building.
The Cauchy-Kovalevskaya theorem is the only major theorem in the classical canon named jointly for a French royalist man and a Russian radical woman. Their births fell sixty-one years apart, and they never met (S374).
She took her doctorate from Göttingen in 1874 in absentia, won the Prix Bordin, and became a professor at Stockholm in 1889. Disputed: Sources disagree on her Berlin arrival date, the Prix Bordin year, and the exact "first in northern Europe" formulation.Disputed She also wrote a novel and a memoir good enough to be translated and sold on their literary merits. Her Swedish circle knew her as both (S417).
Grace Chisholm Young
This is the strongest single artifact in the whole credit theme. A letter from her husband, William Young, in plain words:
The fact is that our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. No. Mine the laurels now and the knowledge. Yours the knowledge only. Everything under my name now, and later when the loaves and fishes are no more procurable in that way, everything or much under your name. At present you cannot undertake a public career. You have your children. I can and do. (S419)
Read the middle of that letter again. The arrangement is explicitly temporary: his name now, hers later, once the loaves and fishes stop coming that way. Guess how many papers it took before they switched.
I have a guess▸
They never switched. Two hundred and twenty papers later it stood exactly as he had written it (S419).
If you took him at his word, so did she. And she was the one with everything riding on it. When somebody asks how work ends up credited to the wrong person, this is the answer in the participants' own handwriting.
Emmy Noether
For four years she lectured at Göttingen under Hilbert's name because she was not allowed to hold a post. Students went to "Hilbert's" course and found Noether at the board (S418).
Noether's theorem is why a physics student can say why energy is conserved. The answer: because the laws do not change from one moment to the next (S418).
Two stories about her turn up everywhere and neither one could be verified here. They are Hilbert's retort to the faculty senate, "this is a university, not a bath-house", and Einstein's New York Times letter about her. Both are in Appendix L.3, which is where this book keeps the things it could not stand behind (S418).
Mary Cartwright
Chaos theory was found because radio engineers complained. During the Second World War the Radio Research Board circulated a plea for help with valve amplifiers whose output went strange. Cartwright and Littlewood took the van der Pol equation seriously and found solutions that never settle down (S420). That is chaotic dynamics, twenty years before Lorenz's weather model and thirty before anyone called it chaos.
She left school teaching because she was not allowed to experiment with how she taught. Then she spent her research life on equations nobody could predict (S420).
Julia Robinson
Hilbert's Tenth Problem asked for a procedure to decide whether any polynomial equation in whole numbers has a whole-number solution. Robinson spent decades reducing the question to one missing ingredient. In 1970 a 22-year-old in Leningrad, Yuri Matiyasevich, supplied it, using Fibonacci numbers. The answer: no such procedure exists. Robinson wrote to him at once; they became friends and collaborators across the Cold War (S421).
Katherine Johnson and the NASA computers
The strongest single fact: in 1962 the most advanced computer in the United States produced a trajectory. John Glenn refused to fly until a human being had done the same arithmetic by hand and agreed with it (S422). NASA's own words record his request to "get the girl". That is what checking your work means at 17,500 miles per hour.
Start with the person who is almost never named. Dorothy Vaughan (1910 to 2008) ran the West Area Computing section at Langley, a section segregated by law. It was Vaughan who put both Katherine Johnson and Mary Jackson where their famous work happened. Johnson got her assignment to the Flight Research Division within two weeks of arriving in 1953. Jackson was hired into the section under Vaughan in April 1951 (S422, S425).
That is worth saying plainly: neither of the two careers below starts without her. She is also the one gap in this chapter that this book could not close from primary sources. NASA's own biography page for Vaughan could not be located. So this book documents her here through the Johnson and Jackson pages rather than her own (S422, S425). The woman who assigned the work is harder to find than the women she assigned it to.
NASA TN D-233 (Skopinski and Johnson, 1960), read in full for this book, runs the problem backwards. Everyone imagines rocket science as "we launch, where does it land?" Johnson and Skopinski wrote the reverse. Pick the spot on Earth where you want the capsule to come down, and count the orbits. The equations then hand you the compass bearing the rocket must point at when the engines cut out (S423). Her own summary:
You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off. (S423, quoted by MacTutor)
Her own pick for her best work was not the famous one. She said it was the Apollo rendezvous: getting the lunar module to find the command module again in orbit around the Moon (S422). And she got author credit on that report only because her co-author wanted to move to Houston. The co-author told their supervisor, who did not like women, that she should finish it. The reason: she had done most of it anyway (S422, S424).
"There were no textbooks, so we had to write them." That was her line about the era (S424). A sixteen-year-old learning calculus from a textbook should know this. The people who worked out how to put a human in orbit did it without one.
A correction, and it matters because the claim is everywhere. The widely repeated statement that Johnson used "Euler's method" for the Friendship 7 calculations is absent from NASA's biography and from MacTutor. It is also absent from TN D-233 itself, which uses iterative correction of Kepler equations plus partial-derivative sensitivity analysis (S422, S423). Three independent sources are silent on it. No source read for this book supports it.
Gladys West computed the geoid. Every phone map on Earth silently uses a model of the planet's shape. That shape is not a sphere and not even an ellipsoid, but a lumpy surface. It is lumpy because gravity is stronger over mountain ranges and weaker over trenches, so "sea level" is not level. West's team computed that surface on a 1960s mainframe, one correction at a time (S426).
She did it for 42 years and never mentioned it to her friends. She said she "just thought it was my work". She found out it mattered when she was 87 (S426).
Mary Jackson took a demotion on purpose. After nearly twenty years as an engineer, she looked at the promotion statistics for women at Langley and concluded the ceiling was structural. Then she took a lower-graded job whose entire purpose was to change hiring and promotion for the next generation (S425). That is a harder kind of courage than films usually show.
Seki Takakazu, and Japanese calculus
Seki Takakazu (関 孝和) was born in the same year as Newton (Disputed: on the usual dating of 1642, which some sources dispute in favor of 1637Disputed). His birthplace lay on the other side of a sealed border. He ended his career as an accounts examiner for a military government, exactly as Newton ended his career running the Royal Mint. Neither ever heard of the other (S429).
- He got into mathematics because a servant in the house noticed a nine-year-old was good at it.
- He had determinants ten years before Leibniz and the Bernoulli numbers before Bernoulli, and because Japan was closed under sakoku, none of it reached Europe. Every one of those results had to be discovered a second time (S429).
- His tombstone calls him "The Arithmetical Sage".
Takebe Katahiro, his pupil, computed π to more than forty decimal places in 1722. He did it using a convergence-acceleration trick that Western numerical analysis rediscovered in 1927 (Richardson) and 1955 (Romberg) (S430). He also wrote a book whose stated purpose was to explain how to do research. It offers twelve worked examples of investigation rather than twelve theorems. That makes it one of the earliest self-conscious accounts of mathematical method anywhere (S430).
And the misattribution runs in the other direction too. The enri, the "circle principle" that made Japanese calculus possible, was credited to Seki for two hundred years and recently went back to Takebe (S430). Misattribution is not a Western monopoly.
Srinivasa Ramanujan
- Before Hardy, he wrote to three English mathematicians. One replied without understanding him and two did not reply at all. The most famous letter in the history of mathematics was the fourth attempt (S431).
- He learned what mathematics looks like on the page from a crammer's revision handbook of about 5,000 unproved formulae, which is why the world spent a century proving his statements for him (S431).
- He could not get into university because he was brilliant at mathematics and neglected everything else. The examination system that failed him is the direct ancestor of the one his readers are sitting (S431).
- Divergent series, the thing Cauchy, Abel, and Weierstrass spent a century declaring illegitimate, were the thing he was best at. Two decades after his death Hardy wrote Divergent Series and made the subject respectable again (S431).
One person left out, and why
Wang Zhenyi (王贞仪, 1768 to 1797) appears in many popular lists of women in mathematics. She is not in this document for one reason, stated openly: and more decisive, her work is Qing calendrical astronomy, not calculus (S449). Including her would be padding.
Noether's theorem lets you say why energy is conserved, instead of only that it is. What property of the laws makes it true?
Show the answer▸
They do not change from one moment to the next. Symmetry in time gives you conservation of energy (S418). She worked at Göttingen for four years lecturing under Hilbert's name, because she was not allowed a post of her own. Students who signed up for his course found her at the board.
Integrals 3.6 has you computing work as an integral of force. The number that comes out is only worth having because energy is conserved. Her theorem is what sits under the exercise.
Chapter 12
Calculus now
The people in this chapter
Faces where a face survives. Every name links to its full entry in Appendix A.
The chain rule is what trains artificial intelligence
This is the strongest "why does this matter to me" hook in the document for a reader in 2026. Two peer-reviewed surveys back what follows (S436, S437).
Every chatbot, image generator, and recommendation system on Earth is trained by running the chain rule backwards. The rule taught in about week three of a derivatives course:
(Appendix E.18.) That rule is not an exam technique. It is the algorithm that trains artificial intelligence. It is called backpropagation, it is reverse-mode automatic differentiation, and it is the chain rule.
A modern network has one output, the loss, and inputs numbering in the billions. Training it means getting the derivative of that one output with respect to every one of those billions. Say how many passes through the network the backward direction costs.
I have a guess▸
One. Full stop. About two to three times the cost of a single forward evaluation. It does not matter whether the network holds a thousand parameters or a trillion (S436).
If you guessed one pass per input, that is the honest arithmetic for the forward direction, and it is why nobody trains anything that way. Every model you have used was trained in the direction that costs one pass.
Why it has to run backwards is a counting argument a student can check. A modern network has one output (the loss) and inputs numbering in the billions (the parameters). Forward mode costs one pass per input. Reverse mode costs one pass, full stop, at roughly two to three times the cost of a single forward evaluation, regardless of parameter count (S436).
The history is a lesson in how credit works:
- 1970: Seppo Linnainmaa, a Finnish master's student, publishes the first description of reverse mode. He was not thinking about learning machines at all. He was worried about rounding errors. The paper is titled "Taylor expansion of the accumulated rounding error" (S436, S438). Griewank records the idea arriving "on a sunny afternoon in a Copenhagen park", and then sitting unread, in Finnish, until Griewank telephoned him from Argonne nearly twenty years later (S438).
- 1980: Speelpenning's Illinois thesis gives the first genuinely automatic implementation (S436). Disputed: His first name is disputed between sources: Griewank (2012) writes "Bernt", Baydin et al. (2018) write "Bert".Disputed
- 1986: Rumelhart, Hinton, and Williams popularize it. 1986 was popularization, not invention (S436, S437).
- It was reinvented at least four times by people who did not know about each other, in two research communities that overlapped heavily and still failed to read each other's papers (S436). Griewank cheerfully admits he rediscovered it himself in 1987 (S438).
And in 1969 Minsky and Papert published a book about what single-layer networks cannot do. The tools to fix the problem had already been published. The field mostly stopped for a decade anyway (S437). Jürgen Schmidhuber calls this "surprising in hindsight".
Before you read on: the mathematics that fixed the problem was already published when the book came out. Why did the field stop anyway?
Why a correct book can still stop a field▸
Perceptrons was not wrong. Single-layer networks cannot compute XOR, and Minsky and Papert proved it. The book was right and the conclusion people drew from it was not.
The result was about one layer. Readers heard it as being about neural networks. The tools for more than one layer existed, in work that had already been published and that almost nobody in the field had read (S437).
So the field spent a decade acting on a true theorem plus a false generalization. The generalization traveled because the theorem was rigorous and the book was well written.
A correct proof of a narrow claim is the most persuasive way to spread a wide one. That is worth remembering every time you see a study quoted one sentence at a time.
The line from Leibniz to ChatGPT is five steps and unbroken. Leibniz writes the chain rule in 1676. Euler builds the calculus of variations in 1744. Kelley and Bryson use it to steer rockets around 1960. Linnainmaa writes it down as an algorithm about rounding error in 1970. And in 2026 it trains every large language model (S437). Five steps, three hundred and fifty years, one rule.
Two more modern threads worth a paragraph each
Laurent Schwartz and distributions. Physicists had been using Dirac's delta: an object that is zero everywhere except one point, infinite there, with area 1. No such function exists. They used it anyway because it worked. Schwartz's answer was not to ban it but to enlarge the definition of what can be differentiated until the delta became a respectable citizen. That is the exact opposite of the nineteenth-century strategy of tightening definitions until the monsters were excluded (S433).
He worked it out lying in the dark, in bed, in occupied France, under a false name. Then he ran next door in the middle of the night to tell his neighbor. The neighbor's reply is the best one-line summary of the theory anyone has produced: "Now we'll never again have functions without derivatives" (S433). He later lost his professorship for saying the French Army tortured people, and was proved right.
Bourbaki. The most influential mathematics textbook of the twentieth century was written by a man who did not exist. That man, "Nicolas Bourbaki", was a group of young French mathematicians annoyed at the calculus textbook they had to teach from. Their solution was to rewrite mathematics from the ground up. Their style was definition then theorem then proof, with no diagrams, no motivation, and no history. That style is why so many university textbooks feel the way they do (S434). It is also why a book like this one has to argue for its existence.
And the statistics connection: the Radon-Nikodym derivative is why "probability density" means anything. Every time a statistics course writes p(x) dx, the reason that expression is legitimate is a theorem finished in 1930. And Stone-Weierstrass is why it is not crazy to believe a complicated function can be approximated by simpler ones. That is the entire premise of machine learning (S435).
How calculus reached teenagers
This thread is partially sourced, and the gaps are named.
Verified. The College Board states plainly that "The Advanced Placement Program was established in 1955" (S448, Access to Excellence, 2001, p. 1). In 1976 the entire AP Program, every subject nationwide, was about 100,000 examinations taken by roughly 76,000 students. Calculus AB and BC were already both in place by 1977 (S493). In 2024, Calculus AB drew 278,657 exams and Calculus BC drew 148,191, a total of 426,848. That was out of 5,744,259 exams taken by 3,079,134 students (S446, cross-checked against S447, whose per-score counts sum exactly).
| Year | AP Calculus exams | Total AP exams, all subjects |
|---|---|---|
| 1976 | not sourced | 100,000 |
| 2004 | not sourced | 1,887,770 |
| 2024 | 426,848 | 5,744,259 |
A nice statistical aside for a class: the BC exam is harder and has the better score distribution, because its students are self-selected. That is a concrete illustration of selection effects (S447).
Not sourced, and flagged as such. Seven claims about the Advanced Placement program could not be traced to a document this book could read:
- The Ford Foundation and Kenyon Plan origins.
- The 1952 report.
- The 1954 pilot.
- The 1956 first exams.
- The figure of 1,229 candidates.
- The year AP Calculus specifically began.
- When and why the AB/BC split appeared.
Rothschild (1999) and Schneider (2009) are both non-open-access, AP Central has no history page, and ERIC holds the histories without full text. These are blanks, not estimates, and a number you cannot cite is worse than an empty space.
The best primary source in this thread is public domain and from 1901. John Perry, addressing the British Association at Glasgow on 14 September 1901, complained that:
Some boys of ten years of age study the methods of the differential calculus... (S492, p. 19)
His needling comparison is the most quotable line in the whole schools-and-calculus thread. In it, ten-year-olds handle differentiation, while nineteen-year-olds "working at mathematics" all their lives cannot easily grasp what a derivative is (S492). Silvanus P. Thompson, who spoke at that same meeting, published Calculus Made Easy nine years later. Its famous line, "What one fool can do, another can", is the same argument in a book for the general reader (S492).
And the first calculus textbook in English (1704) belongs here, uncomfortably. Charles Hayes's A Treatise of Fluxions was written by a director of the Royal African Company, a slave-trading corporation (S439). That is a fact, and it belongs in a chapter about who gets to write the books. Its first sentence defines an infinitely small quantity, and its first proof shows there are infinitely small quantities inside the infinitely small quantities. Thirty years later Berkeley would call these the ghosts of departed quantities. Two hundred and sixty-two years later Robinson would prove Hayes had been entitled to them all along (S439).
Hayes's own warning, "this Doctrine may seem hard to most Readers at first", is the oldest sentence in English calculus pedagogy (S439).
Every statistics course writes p(x) dx, and nobody stops to say why that expression is allowed to mean anything. Which theorem makes it legitimate?
Show the answer▸
The Radon-Nikodym derivative, finished in 1930 (S435). It is what turns a probability measure into a density you can integrate. That move is behind every sentence that says the area under the curve is the probability.
Integrals 3.7 is where you meet that sentence. The reason it is not a fudge is a piece of measure theory. That piece landed two and a half centuries after Leibniz wrote the integral sign.
The master timeline: every date, in one place
327 dated events across ten eras, merged from 255 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.
Part II
How to read the timeline
Every dated event from every source, merged into one sequence and deduplicated. 327 entries.
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 tag is not settled: the date, the attribution, or both. The reason is stated underneath the event in small text. Appendix F gives both sides in full. You will meet the tag often, since 58 of this book's 129 logged disagreements are still open. That is the honest state of the evidence, not a gap in the research.
Newton's dates carry both Old Style and New Style, because England kept the Julian calendar until 1752 and began its year on 25 March. The Style Key explains what that does to a February date.
Two overlapping pairs were merged while the timeline was assembled. Newton's anonymous 1715 Account appeared in the log twice, and his death and burial appeared both as one row and as two. Ten duplicate people records were merged and their identifiers remapped. A later consistency pass found twelve more duplicated events, one of them logged three times (the 29 October 1675 integral sign). The pass merged each into a single row carrying every source the duplicates had between them. Nothing was deleted.
Timeline
Antiquity to 500 CE
10 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| c. 490 BCE Disputed |
Zeno of Elea born; his paradoxes of motion and plurality state the problems limits and measure theory would later address Dispute: Date inferred from the dramatic setting of Plato's Parmenides, not from a document |
Zeno of Elea | Elea, southern Italy | S007 |
| c. 460 BCE Disputed |
Democritus born; he asserts that a cone is one third of its cylinder, treating a solid as a stack of indivisible slices Dispute: Known only from Archimedes' Method (recovered 1906) and Plutarch, De Comm. Not. adv. Stoicos xxxix.3 |
Democritus | Abdera, Thrace | S002 |
| c. 370 BCE Disputed |
Eudoxus makes the method of exhaustion rigorous and defines equal ratios (Euclid, Elements V, Def. 5) Dispute: All of Eudoxus' own works are lost; dated from his floruit |
Eudoxus | Cnidus / Athens | S001, S008 |
| c. 250 BCE Disputed |
Archimedes writes The Method of Mechanical Theorems to Eratosthenes, explaining how he discovers theorems by weighing infinitesimal slices Dispute: Heath's approximate dating |
Archimedes | Syracuse | S002 |
| c. 250 BCE | Archimedes proves a parabolic segment is of the inscribed triangle (Quadrature of the Parabola) and brackets pi between 3+ and 3+ using a 96-gon | Archimedes | Syracuse | S001 |
| 212 BCE | Archimedes killed in the sack of Syracuse; he had asked for a sphere in a cylinder on his tomb | Archimedes | Syracuse | S001 |
| 263 CE | Liu Hui completes his commentary on the Nine Chapters, supplying proofs, cutting the circle to a 3072-gon for pi, and admitting he cannot find the sphere's volume | Liu Hui | Kingdom of Wei, China | S020, S022, S023 |
| 462 CE | Zu Chongzhi proposes the Daming calendar, the first Chinese calendar to allow for precession of the equinoxes | Zu Chongzhi | Jiankang (Nanjing) | S021 |
| late 5th c. CE | Zu Gengzhi states the equal-cross-sections principle and derives the volume of a sphere, about 1100 years before Cavalieri | Zu Gengzhi | Liang dynasty China | S023 |
| 499 CE | Aryabhata completes the Aryabhatiya at age 23, containing the first known sine table and pi = , expressly called approximate | Aryabhata I | Kusumapura, India | S050, S059 |
Timeline
500 to 1400: the wider world and the medieval schools
17 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| c. 500 CE | Aryabhata is the first to write down explicitly the formula for the sum of cubes, according to Katz; the formula for the sum of squares had been stated by Archimedes around 250 BCE | - | India | S061 |
| c. 628 CE | Brahmagupta's Brahmasphuta-siddhanta gives a second-order interpolation formula for Rsines and the arithmetic of zero | Brahmagupta | Rajasthan, India | S050 |
| c. 870 CE Disputed |
Thabit ibn Qurra computes areas and volumes of parabolas and paraboloids by upper and lower sums, dividing the interval into unequal parts Dispute: Dated from his floruit; specific treatise not identified in the source read |
Thabit ibn Qurra | Baghdad | S057 |
| c. 1000 CE Disputed |
Ibn al-Haytham sums fourth powers and uses it to find the volume of a paraboloid as of the circumscribed cylinder Dispute: Dated to his mature Cairo period; exact year of the treatise not established here |
Ibn al-Haytham | Cairo | S054, S058 |
| c. 932 CE | Munjala's Laghumanasa gives the first correct formula for true daily planetary motion using the Rcosine as rate of change of the Rsine | Munjala | India | S050 |
| c. 1000 to 1030 | Ibn al-Haytham derives the sums of integral powers through the fourth and uses them to show that the paraboloid formed by rotating about is exactly of its circumscribing cylinder, by a Greek-style double reductio argument | - | Cairo | S061 |
| c. 1150 CE | Bhaskara II's Siddhanta-siromani distinguishes average from instantaneous velocity (tatkalika gati) and states that the equation of the center is extremal where the velocity correction vanishes | Bhaskara II | India | S050 |
| c. 1209 Disputed |
Sharaf al-Din al-Tusi's Treatise on Equations classifies cubics into 25 types and locates the maximum of at Dispute: Whether this involves a derivative is disputed between Rashed and Hogendijk |
Sharaf al-Din al-Tusi | Persia | S053 |
| 14 or 29 April 1229 Disputed |
The scribe Ioannes Myronas completes the prayer book written over the erased text of Archimedes' Method Dispute: Walters gives 14 April 1229; Smithsonian, reporting the X-ray fluorescence reading of the colophon, gives 29 April 1229 |
Ioannes Myronas | Jerusalem | S004, S006 |
| before 1325 | Kilvington's Sophismata, the earliest calculatory treatment of motion at Oxford | Richard Kilvington | Oxford | S102 |
| 1328 | Bradwardine's Tractatus de proportionibus velocitatum in motibus states his rule relating velocity to the force/resistance ratio | Thomas Bradwardine | Merton College, Oxford | S103, S104 |
| 1335 | Heytesbury's Regulae solvendi sophismata states the mean speed theorem (ch. VI, De tribus praedicamentis) without proving it | William Heytesbury | Merton College, Oxford | S101 |
| c. 1340-1350 | Swineshead's Liber calculationum, sixteen treatises, printed at Venice in 1520 | Richard Swineshead | Oxford | S106 |
| 26 August 1349 | Bradwardine dies of plague at Lambeth, about seven weeks after being consecrated Archbishop of Canterbury | Thomas Bradwardine | Lambeth, London | S103 |
| c. 1350 Disputed |
Oresme proves the divergence of the harmonic series in Questiones super geometriam Euclidis by grouping terms into blocks each exceeding Dispute: Work is dated only loosely by the sources read |
Nicole Oresme | Paris | S100 |
| c. 1350s Disputed |
Oresme's Tractatus de configurationibus qualitatum et motuum; III.vii proves the mean speed theorem by area under a velocity graph Dispute: No source read gives a firm composition year; SEP (S100) and Caroti (S108) both decline to date it |
Nicole Oresme | Paris | S100, S107, S108 |
| 11 July 1382 | Nicole Oresme dies as bishop of Lisieux | Nicole Oresme | Lisieux, Normandy | S100 |
Timeline
1400 to 1637: Kerala, print, and the algebraic turn
15 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| c. 1400 Disputed |
Madhava of Sangamagrama obtains infinite series for pi, sine, cosine and arctangent, with end-correction (antya-samskara) terms Dispute: Madhava's own works are lost; attribution rests on quotations by Nilakantha and Sankara Variyar |
Madhava of Sangamagrama | Sangamagrama, Kerala | S050, S055 |
| c. 1502 | Nilakantha's Aryabhatiya-bhasya argues that the circumference and diameter of a circle cannot both be integers in any unit - an argument for incommensurability | Nilakantha Somayaji | Kerala | S050, S056 |
| c. 1530 Disputed |
Jyeshthadeva's Yuktibhasa sets out, in Malayalam prose, demonstrations of the Kerala series including the derivation of the error terms Dispute: S050 gives c. 1530; MacTutor gives c. 1550 |
Jyeshthadeva | Kerala | S050, S055 |
| c. 1530 | The Tantrasangraha-vyakhya states the sine and cosine power series in Sanskrit verse; the Yuktibhasa, in Malayalam, gives the derivation and credits the series to Madhava. The derivation uses ibn al-Haytham's summation identity for the case k = 1, without naming him | - | Kerala | S061 |
| 1578 | Matteo Ricci arrives in Goa with instructions to investigate Indian science - the opportunity in the Kerala transmission argument | Matteo Ricci | Goa, India | S051 |
| 1599 | Galileo attempts the quadrature of the cycloid by weighing it: he cuts an arch and a generating circle from the same material, balances them, finds the arch about three times the circle, and abandons the result believing the ratio incommensurable. The true ratio is exactly 3 | - | Padua / Florence | S064 |
| 1609 | Kepler's Astronomia Nova gives the ellipse law and the area law; time in orbit is identified with area swept out | Johannes Kepler | Prague | S152, S154 |
| 1613 Disputed |
At his second wedding in Linz Kepler watches a wine merchant gauge a barrel with a rod through the bung-hole and objects to the method Dispute: MacTutor (S154) and Cardil (S134) tie the incident to the 1613 wedding; the 1911 Britannica (S151) frames it around the abundant vintage of 1612 |
Johannes Kepler | Linz, Austria | S134, S154, S151 |
| 1615 | Kepler's Nova stereometria doliorum vinariorum, printed at Linz by Johannes Plancus; volumes of some ninety solids of revolution by summing infinitely thin slices | Johannes Kepler | Linz, Austria | S134, S151, S159 |
| 15 December 1621 | Cavalieri first puts the problem of ratios between infinite collections to Galileo | Bonaventura Cavalieri | Bologna/Florence | S156 |
| 1624 | Edmund Gunter introduces the abbreviation 'sin' | - | England | S060 |
| 2 October 1634 | Cavalieri writes to Galileo: 'I absolutely do not declare to compose the continuum by indivisibles' | Bonaventura Cavalieri | Bologna | S156 |
| c. 1634 | Roberval effects the quadrature of the cycloid using a companion curve and Cavalieri's theorem; he does not publish, and the proof appears only in the Traite des Indivisibles of 1693 | - | Paris | S064 |
| 1635 | Cavalieri's Geometria indivisibilibus continuorum nova quadam ratione promota published at Bologna, about 700 pages, eight years after the manuscript was finished | Bonaventura Cavalieri | Bologna | S135, S156 |
| 1636 | Fermat's Methodus ad disquirendam maximam et minimam begins circulating in Paris through Mersenne | Pierre de Fermat | Toulouse/Paris | S144, S130 |
Timeline
1637 to 1665: the race begins
27 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 1637 | Descartes' La Geometrie published as an appendix to the Discours de la methode | Rene Descartes | Leiden | S148 |
| 1638 | Fermat communicates his method to Descartes on receiving the Geometrie; the quarrel over maxima, minima and tangents begins | Pierre de Fermat, Rene Descartes | Paris | S144, S151 |
| August 1638 | Roberval, Fermat and Descartes each give Mersenne a different method of drawing a tangent to the cycloid; the ensuing quarrel turns partly on the fact that a tangent as the limiting position of a secant was not yet accepted | - | Paris | S064 |
| 1641 | Torricelli obtains the acute hyperbolic solid: an infinitely long solid of revolution with finite volume | Evangelista Torricelli | Florence | S131, S157 |
| 25 December 1642 (Old Style, Julian, England) = 4 January 1643 (New Style, Gregorian) | Isaac Newton born, Woolsthorpe Manor, Colsterworth, Lincolnshire; posthumous child, father died c. 3 months earlier | Isaac Newton | Woolsthorpe Manor, Lincolnshire, England | S180, S181, S200 |
| 1 January 1642/3 (OS) = 11 January 1643 (NS) | Newton baptised at Colsterworth | Isaac Newton | Colsterworth, Lincolnshire | S180 |
| 1643 | Torricelli's barometer experiment, performed by Viviani | Evangelista Torricelli | Florence | S131 |
| October 1643 | Torricelli writes to Roberval on the cycloid, opening the priority quarrel | Evangelista Torricelli, Gilles Personne de Roberval | Florence/Paris | S131, S137 |
| 1644 | Torricelli's Opera geometrica, containing De solido hyperbolico acuto, published | Evangelista Torricelli | Florence | S131, S157 |
| 1644 Disputed |
Torricelli publishes his quadrature of the cycloid and a tangent method, the earliest printed article on the curve. Roberval writes charging plagiarism; Cajori attributes Torricelli's death in 1647 to the charge Dispute: The claim that the plagiarism charge caused Torricelli's death is Cajori's, reported by Whitman, not a medical record |
- | Florence | S064 |
| 1 July 1646 (Gregorian) = 21 June 1646 (Julian) | Leibniz born at Leipzig, Saxony | Gottfried Wilhelm Leibniz | Leipzig | S235, S236 |
| 1647 | Saint-Vincent publishes the Opus geometricum, ten books in over 1,200 folio pages, containing a false claim to have squared the circle and, in Book VI on the hyperbola, the propositions that make the area under a hyperbola behave as a logarithm | Gregoire de Saint-Vincent | Antwerp | S067, S141, S155 |
| 1647 | Cavalieri's Exercitationes geometricae sex, containing Exercitatio 3, his full reply to Guldin; he dies 30 November | Bonaventura Cavalieri, Paul Guldin | Bologna | S156, S135 |
| 1649 | Sarasa answers a challenge problem of Mersenne's, posed partly to attack his teacher's quadrature of the circle, and in the scholium states that hyperbolic areas can be taken in place of logarithms: the first identification of the area under a hyperbola with a logarithm | Alphonse Antonio de Sarasa, Gregoire de Saint-Vincent | Antwerp | S066, S141, S155 |
| 1650 | Mengoli's Novae quadraturae arithmeticae proves the harmonic series diverges and poses the Basel problem he cannot solve | Pietro Mengoli | Bologna | S142 |
| 1651 | Huygens exposes the flaw in Saint-Vincent's claimed circle quadrature at Book X Prop. 39, p. 1121 | Christiaan Huygens, Gregoire de Saint-Vincent | The Hague | S141, S150 |
| 1655 | John Wallis introduces the lemniscate for infinity in De sectionibus conicis, and uses it first as a DENOMINATOR: each of infinitely many strips has height 1/infinity of the whole | John Wallis | Oxford | S139, S145, S451 |
| 1656 Disputed |
Wallis's Arithmetica Infinitorum gives the infinite product for pi by interpolation Dispute: MacTutor (S145) gives 1656; the 1911 Britannica (S151) and Miller (S139) allow 1655 |
John Wallis | Oxford | S145, S151, S139 |
| 1657 | Hudde's rule published in van Schooten's Exercitationes mathematicae; fuller treatment in the 1659 Latin Geometria of Descartes | Johann Hudde | Leiden/Amsterdam | S147 |
| 1658 | Pascal's anonymous cycloid prize challenge to the mathematicians of Europe; Wallis, Huygens, Wren and Fermat respond and none wins | Blaise Pascal, John Wallis, Christiaan Huygens | Paris | S153, S143 |
| 1658 | Christopher Wren sends Pascal his rectification of the cycloid, without proof, during Pascal's contest. One arch has length exactly 8a, four times the diameter of the generating circle: a curved length equal to a straight one, with no pi in it. Wallis publishes it as Wren's the following year in Tractatus duo | - | London / Paris | S064 |
| 1659 | Lettres de A. Dettonville, containing the Traite des sinus du quart de cercle and the characteristic triangle | Blaise Pascal | Paris | S143 |
| 5 June 1661 (OS) = 15 June 1661 (NS) Disputed |
Newton admitted to Trinity College, Cambridge, as a subsizar Dispute: Conduitt's draft memoir (Keynes Ms. 129.02, f. 1v) writes 'the 5th of Iune in 1660'. Standard scholarship gives 1661. See Appendix F. |
Isaac Newton | Trinity College, Cambridge | S181 |
| 1663 | Leibniz takes his bachelor's degree at Leipzig aged 17 (thesis De Principio Individui); summer at Jena under Erhard Weigel | Gottfried Wilhelm Leibniz | Leipzig / Jena | S235 |
Timeline
1665 to 1687: Newton and Leibniz
66 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 23 May 1665 (OS) = 2 June 1665 (NS) | Pocket-book entry proves Newton still resident in Cambridge | Isaac Newton | Cambridge | S190 |
| 8 August 1665 (OS) = 18 August 1665 (NS) | Trinity College dismissed on account of the plague | - | Trinity College, Cambridge | S190 |
| January 1664/5 (OS) = January 1665 (NS) | Newton graduated B.A. | Isaac Newton | Cambridge | S190 |
| June 1665 (OS) | Newton leaves Cambridge because of plague | Isaac Newton | Cambridge | S190 |
| summer 1665 (OS) | Newton computes the area of the hyperbola (i.e. a logarithm) to 'two & fifty figures' at Boothby Pagnell, Lincolnshire | Isaac Newton, Humphrey Babington | Boothby Pagnell, Lincolnshire | S190, S199 |
| 20 March 1665/6 (OS) = 30 March 1666 (NS) | Newton travels back to Cambridge (recorded in the Fitzwilliam notebook) | Isaac Newton | Cambridge | S190 |
| October 1666 (OS) | Newton dates his first systematic tract on fluxions, 'To resolve Problems by Motion' | Isaac Newton | Lincolnshire (probably) | S183, S199 |
| c.1666 Disputed |
Leibniz refused the doctorate in law at Leipzig Dispute: Reason unclear. MacTutor gives two: (a) his youth plus a quota on places, so he should wait a year; (b) a story that the Dean's wife persuaded the Dean to argue against him. No primary source obtained. |
Gottfried Wilhelm Leibniz | Leipzig | S235 |
| mid-June 1666 (OS) | Newton leaves Cambridge again | Isaac Newton | Cambridge | S190 |
| 1667 | James Gregory's Vera circuli et hyperbolae quadratura published at Padua, using convergent infinite series | James Gregory | Padua | S136 |
| 22 April 1667 (OS) = 2 May 1667 (NS) | Newton returns to Cambridge for good | Isaac Newton | Cambridge | S190 |
| February 1667 | Doctorate in law at Altdorf (De Casibus Perplexis); Leibniz declines the professorship offered | Gottfried Wilhelm Leibniz | Altdorf | S235, S236 |
| 1668 | Gregory's Geometriae pars universalis - the first systematic calculus textbook - proves that differentiation and integration are inverse operations | James Gregory | Padua/London | S136 |
| 1668 | Mercator's Logarithmotechnia prints the series and the term logarithmus naturalis | Nicolaus Mercator | London | S149, S155 |
| 1669 | Barrow resigns the Lucasian chair and Newton succeeds him | Isaac Barrow | Cambridge | S140, S158 |
| 31 July 1669 (OS) = 10 August 1669 (NS) | Isaac Barrow sends De analysi to John Collins in London | Isaac Newton, Isaac Barrow | Cambridge to London | S184 |
| 1670 | Barrow's Lectiones Geometricae; Lecture X Prop. 11 states the inverse relation between tangents and areas in geometric form | Isaac Barrow | Cambridge | S140, S138, S158 |
| 15 February 1671 | Gregory's letter to John Collins contains what is now called Taylor's series, over forty years before Taylor | James Gregory | St Andrews | S136 |
| 1670-1671 | Newton writes De methodus serierum et fluxionum (the 'Method of Fluxions') | Isaac Newton | Cambridge | S192, S185 |
| 1672 Disputed |
Sluse's general tangent rule published in the Philosophical Transactions Dispute: MacTutor (S146) gives 1672; the date 1673 is also widely cited but was not confirmed by any source read |
Rene-Francois de Sluse | Liege/London | S146 |
| 1672 | Leibniz arrives in Paris on a diplomatic mission for the Elector of Mainz; stays until October 1676 | Gottfried Wilhelm Leibniz | Paris | S230, S236 |
| September 1672 | Huygens sets Leibniz the problem of summing the reciprocals of the triangular numbers; Leibniz solves it (sum = 2) by telescoping differences | Gottfried Wilhelm Leibniz, Christiaan Huygens | Paris | S237 |
| 15 February 1673 | Royal Society meeting (Leibniz absent) at which Hooke criticizes the calculating machine | Gottfried Wilhelm Leibniz | London | S235 |
| 1673 | Leibniz acquires Barrow's Lectiones Geometricae in London; annotations suggest he read it 1674-1676 | Isaac Barrow | London | S138 |
| 1673 | Huygens's Horologium oscillatorium proves the cycloid tautochronous and introduces evolutes and involutes | Christiaan Huygens | Paris | S150 |
| 9 April 1673 (Old Style) Disputed |
Leibniz elected Fellow of the Royal Society Dispute: MacTutor gives 9 April 1673; 19 April 1673 is also widely printed. Royal Society register not consulted. |
Gottfried Wilhelm Leibniz | London | S235 |
| January-February 1673 | First London visit: Leibniz demonstrates a model of his stepped-reckoner calculating machine at the Royal Society; meets Boyle, Hooke, Pell, Oldenburg | Gottfried Wilhelm Leibniz, Henry Oldenburg | London | S230, S235 |
| 11 November 1675 | Leibniz moves d from denominator to multiplier: "Idem est dx et x/d". dx, dy and the quotient dx/dy all date from this manuscript | Gottfried Wilhelm Leibniz | Paris | S451, S139 |
| 11 November 1675 (Gregorian; = 1 November Julian) | Manuscript Methodi tangentium inversae exempla: dx takes its modern position (multiplier, not divisor); Leibniz asserts d(xy)=dx.dy and d(x/y)=dx/dy, then refutes both later in the same paper | Gottfried Wilhelm Leibniz | Paris | S230, S238 |
| 21 November 1675 | Leibniz derives the product rule analytically, and records that "nothing new can be deduced from it, because we had already obtained it" | Gottfried Wilhelm Leibniz | Paris | S230 |
| 26 October 1675 | Leibniz, still writing omn. for "all the", produces a formula so tangled with nested sums that a new notation becomes unavoidable | Gottfried Wilhelm Leibniz | Paris | S451 |
| 29 October 1675 (Gregorian; = 19 October Julian) | Leibniz writes the integral sign for the first time, in the manuscript Analyseos tetragonisticae pars secunda: "It will be useful to write [integral sign] for omn." On the same sheet d appears, but as a divisor (l = ya/d) | Gottfried Wilhelm Leibniz | Paris | S139, S230, S238, S451 |
| 13 June 1676 (OS) = 23 June 1676 (NS) | Newton's Epistola prior written for Leibniz and sent through Oldenburg | Isaac Newton, Henry Oldenburg, Gottfried Wilhelm Leibniz | Cambridge / London | S188, S189 |
| 18 November 1676 (OS) = 28 November 1676 (NS) | Leibniz writes to Oldenburg from Amsterdam that Slusius's tangent method "is not yet at its summit", his first reference to the tangent problem in that correspondence | Gottfried Wilhelm Leibniz, Henry Oldenburg | Amsterdam | S230 |
| 24 October 1676 (OS) = 3 November 1676 (NS) | Newton's Epistola posterior to Oldenburg, for transmission to Leibniz: the binomial series by interpolation from Wallis, and the two anagrams concealing his method | Isaac Newton, Henry Oldenburg, Gottfried Wilhelm Leibniz | Cambridge / London | S188, S189, S285 |
| 5 February 1675/6 (OS) = 15 February 1676 (NS) Disputed |
Newton to Hooke: 'If I have seen further it is by standing on the sholders of Giants' Dispute: Frequently mis-dated to 1675 because of the English civil year. Full text not yet verified by this agent (see S197). |
Isaac Newton, Robert Hooke | Cambridge to London | S197 |
| December 1676 | Leibniz enters the service of Duke Johann Friedrich of Brunswick-Luneburg at Hanover as librarian and Court Councillor | Gottfried Wilhelm Leibniz | Hanover | S235, S236 |
| October 1676 | Second London visit, about one week: Leibniz reads Collins's papers and makes extracts headed "Excerpta ex tractatu Newtoni de Analysi per aequationes numero terminorum infinitas". He copies the series and equation-solving material, not the tangent/quadrature material | Gottfried Wilhelm Leibniz, John Collins | London | S230 |
| 21 June 1677 | Leibniz's reply to Newton, the first appearance of dy in the correspondence with the English | Gottfried Wilhelm Leibniz | Hanover | S189 |
| from 1678 | Leibniz begins the Harz mountains project to drain the silver mines with windmills; the schemes run for years and all fail | Gottfried Wilhelm Leibniz | Harz mountains | S235 |
| 24 November 1679 (OS) = 4 December 1679 (NS) | Hooke writes to Newton proposing the compounding of tangential motion with a central attraction | Robert Hooke, Isaac Newton | London to Cambridge | S203 |
| 1683 | Seki Takakazu is the first to study determinants - ten years before Leibniz, and in a more general form | Seki Takakazu | Edo, Japan | S429 |
| October 1684 | Nova methodus pro maximis et minimis published in Acta Eruditorum, pp. 467-473; the first publication of the calculus. Signed only "per G.G.L."; no proofs; contains printer's errors including concavity/convexity interchanged | Gottfried Wilhelm Leibniz, Otto Mencke | Leipzig | S231, S232 |
| 1686 | The integral sign reaches print (Acta eruditorum, pp. 297, 299) with its descender cut off, looking like a small s | Gottfried Wilhelm Leibniz | Leipzig | S451 |
| 20 June 1686 (OS) = 30 June 1686 (NS) | Newton writes to Halley about Hooke's claim to the inverse-square law; threatens to suppress Book III of the Principia | Isaac Newton, Edmond Halley, Robert Hooke | Cambridge to London | S203 |
| June 1686 | De geometria recondita et analysi indivisibilium atque infinitorum, Acta Eruditorum p. 292 onward; the first appearance of the integral sign in print | Gottfried Wilhelm Leibniz | Leipzig | S233, S232 |
| July 1687 Disputed |
Philosophiae Naturalis Principia Mathematica published, London Dispute: Precise publication day not verified by this agent; Halley paid for the printing. Imprimatur dated 5 July 1686. |
Isaac Newton, Edmond Halley | London | S186, S191 |
| November 1687 - June 1690 | Leibniz's archive journey through Bavaria, Austria and Italy researching the history of the House of Brunswick | Gottfried Wilhelm Leibniz | Central Europe / Italy | S235 |
| 12 February 1669/1670 (OS) = 22 February 1670 (NS) | Collins writes to James Gregory that Newton will probably let De analysi be printed with Barrow's Lectiones geometricae; the plan comes to nothing | Isaac Newton, John Collins, James Gregory, Isaac Barrow | London | S204 |
| 24 December 1670 (OS) = 3 January 1671 (NS) | Collins tells James Gregory that the Cambridge mathematics lecturer must either print nine lectures a year or deposit them in the public library, where copies could be transcribed | John Collins, James Gregory | London | S204 |
| 25 May 1672 (OS) = 4 June 1672 (NS) | Newton tells Collins he will not print his optical lectures: he will not enjoy 'my former serene liberty' until he has done with the press | Isaac Newton, John Collins | Cambridge | S204 |
| 1674 | Newton gives John Flamsteed a set of notes on algebra, one of the earliest documented loans of a Newton mathematical manuscript outside the Cambridge circle | Isaac Newton, John Flamsteed | England | S204 |
| 29 June 1675 (OS) = 9 July 1675 (NS) | Collins reports to James Gregory that 'Mr Newton intends not to publish anything' and will only deposit his lectures in the public library | Isaac Newton, John Collins, James Gregory | London | S204 |
| 19 October 1675 (OS) = 29 October 1675 (NS) | Collins tells James Gregory that Newton and Barrow are beginning to think mathematical speculations 'nice and dry, if not somewhat barren' | Isaac Newton, John Collins, James Gregory, Isaac Barrow | London | S204 |
| 5 September 1676 (OS) = 15 September 1676 (NS) | Newton tells Collins that the discourse on infinite equations he wrote about five years earlier has never been read out and he keeps it by him | Isaac Newton, John Collins | Cambridge | S204 |
| 26 October 1676 (OS) = 5 November 1676 (NS) | Newton instructs Oldenburg: 'Pray let none of my mathematical papers be printed wthout my special licence' | Isaac Newton, Henry Oldenburg | Cambridge | S204 |
| 1677 | Collins makes a second transcript of De analysi and sends it to John Wallis; after Wallis's death it passes to David Gregory | John Collins, John Wallis, David Gregory | England | S204 |
| c. 1680 Disputed |
Newton reworks the synthetic addendum to De methodis as Geometria curvilinea, the geometric version of the fluxional calculus later used in the Principia Dispute: Guicciardini hedges: he dates the Geometria curvilinea 'c. 1680' and places the synthetic addendum 'just after completing the De methodis' (1671), giving no year for either |
Isaac Newton | Cambridge | S204 |
| 1684 | Newton's Lucasian lectures on algebra are deposited in the Cambridge University Library and become public in principle; they are printed in 1707 as Arithmetica universalis | Isaac Newton | Cambridge | S204 |
| 9 June 1684 (OS) = 19 June 1684 (NS) | David Gregory sends Newton his Exercitatio geometrica de dimensione figurarum, which contains series quadrature theorems already in De analysi and which Gregory credits to his uncle James's papers | David Gregory, Isaac Newton, James Gregory | Edinburgh to Cambridge | S204 |
| 1684 or 1685 Disputed |
John Craig visits Newton at Cambridge and is allowed to transcribe part of De methodis and of the Epistola posterior to Leibniz Dispute: Guicciardini dates this visit 1684 at pp. 460 and 462 and 1685 at p. 465, in the same article. The 1685 passage carries the fuller account (Wallis's 1683 Proposal, Craig's letter, the meeting, the report to Pitcairne and David Gregory); the 1684 statement cites Whiteside, Mathematical Papers vol. 7, pp. 3-4 |
John Craig, Isaac Newton | Cambridge | S204 |
| 1685 | Newton communicates his 'prime theorem' on quadratures to John Craig; excerpts from his letters and manuscripts appear in print in Wallis's Treatise of Algebra | Isaac Newton, John Craig, John Wallis | England | S204 |
Timeline
1688 to 1734: publication, textbooks, and the war
63 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 9 October 1690 | Huygens submits his catenary solution to Leibniz as an encrypted anagram, to claim priority without disclosing the answer | Christiaan Huygens | The Hague | S334 |
| May 1690 | Jacob Bernoulli prints 'Ergo et horum Integralia aequantur' in Acta Eruditorum p. 218 - the first appearance in print of 'integral' in its modern mathematical sense - and in the same issue sets the funicularia (hanging-chain) challenge | Jacob Bernoulli | Leipzig / Basel | S324, S334 |
| 1691 | Rolle states his theorem, about polynomials, in a book on solving equations - nine years before he begins campaigning against the calculus | Michel Rolle | Paris | S459, S453 |
| June 1691 | Acta Eruditorum prints the catenary solutions of Leibniz, Huygens and Johann Bernoulli together; the hanging chain is a catenary, not the parabola Galileo supposed | Johann Bernoulli, Christiaan Huygens | Leipzig | S334 |
| late 1691 | Newton introduces the dotted-letter (fluxion) notation | Isaac Newton | Cambridge/London | S192 |
| 1691-1693 | Newton composes Tractatus de quadratura curvarum | Isaac Newton | Cambridge | S187 |
| 1693 | Newton's fluxional dots reach print for the first time, in the Latin edition of Wallis Algebra | Isaac Newton, John Wallis | Oxford | S451 |
| 17 March 1694 | L'Hopital writes letter no. 20 to Johann Bernoulli offering a retainer of 300 pounds a year, backdated to 1 January, in exchange for Bernoulli's discoveries and his silence about them | Guillaume de l'Hopital, Johann Bernoulli | Paris / Basel | S315 |
| 30 September 1695 Disputed |
Leibniz replies to l'Hopital's question about a derivative of order one-half; the conventional birthday of fractional calculus Dispute: The letter was NOT obtained in full text. The verified attestation of Leibniz writing a half-order differential is his letter to Wallis of 1697. |
Gottfried Wilhelm Leibniz | Hanover | S242, S241 |
| 1696 | Newton appointed Warden of the Royal Mint | Isaac Newton | London | S194 |
| 1696 | L'Hopital publishes Analyse des infiniment petits anonymously in Paris: the first calculus textbook. Section IX Prop. I (art. 163) contains what is now called L'Hopital's rule, discovered by Johann Bernoulli | Guillaume de l'Hopital, Johann Bernoulli | Paris | S316, S315 |
| June 1696 | Johann Bernoulli poses the brachistochrone problem in Acta Eruditorum, allowing six months; Leibniz asks for an extension | Johann Bernoulli | Groningen / Leipzig | S330, S321 |
| 1697 | In a letter to John Wallis, Leibniz writes y and the general , and proposes "analysis infinitesimalis" as the common name for his method and Newton's | Gottfried Wilhelm Leibniz | Hanover | S241 |
| January 1697 | Newton's solution of the brachistochrone appears anonymously in Philosophical Transactions, sent via Charles Montagu; the curve is the cycloid | Charles Montagu | London | S330, S321 |
| June 1697 Disputed |
Johann Bernoulli, writing in Basnage's Histoire des ouvrages des scavans, says he recognizes Newton behind the anonymous solution 'comme ex ungue Leonem' Dispute: The 1697 original could NOT be retrieved. The familiar Latin form 'tanquam ex ungue leonem' is verified only as far back as Brewster 1831 p. 194 |
Johann Bernoulli | Rotterdam | S337, S321 |
| 1699 | Fatio de Duillier, in Lineae brevissimi descensus investigatio geometrica duplex, publicly claims Newton as first inventor and casts Leibniz as at best a second | Nicolas Fatio de Duillier | London | S284 |
| 22 March 1698/9 (OS) = 1 April 1699 (NS) Disputed |
William Chaloner executed at Tyburn Dispute: Date obtained only from tertiary sources; NOT verified against the Old Bailey Proceedings or a Newgate calendar. See Appendix F. |
William Chaloner | Tyburn, London | S196 |
| early March 1698/9 (OS) = mid-March 1699 (NS) | William Chaloner writes to Newton from prison: 'O no body can save me but you' | William Chaloner, Isaac Newton | Newgate, London | S196 |
| December 1701 - 1702 | Newton MP for Cambridge University (second term) | Isaac Newton | Westminster | S201 |
| 1703 (volume published 1705) Disputed |
Explication de l'arithmetique binaire, Histoire de l'Academie Royale des Sciences, Annee 1703, Memoires pp. 85-89 (Histoire section pp. 58-63), connecting binary to the hexagrams of Fuxi Dispute: Full text not obtained; the pagination comes from dealer bibliographies (tier 4) and has never been checked against the Gallica scan. |
Gottfried Wilhelm Leibniz | Paris | S236 |
| 1704 | Tractatus de quadratura curvarum published as an appendix to Opticks | Isaac Newton | London | S187 |
| 1704 | Charles Hayes publishes A Treatise of Fluxions, the first book in English on the calculus, founded openly on actual infinitesimals | Charles Hayes | London | S439 |
| 16 August 1705 | Jacob Bernoulli dies at Basel, aged 50 years 7 months; his tomb is later carved with an Archimedean spiral instead of the logarithmic spiral he asked for, under the motto Eadem mutata resurgo | Jacob Bernoulli | Basel | S328, S329 |
| January 1705 | Anonymous review in Acta Eruditorum of Newton's De quadratura says Newton "substituted fluxions for differences", read in London as a plagiarism charge against Newton, and the proximate trigger of the whole dispute | Gottfried Wilhelm Leibniz | Leipzig | S189 |
| 1706 | William Jones uses the symbol pi for the ratio of circumference to diameter, on p. 263 of Synopsis palmariorum mathesios, without comment | William Jones | London | S325, S451, S462 |
| September-October 1708 | John Keill, in Philosophical Transactions 26(317), accuses Leibniz of publishing Newton's arithmetic of fluxions "mutatis nomine et notationis modo" | John Keill | London | S189, S283 |
| 1710 | The 20-volume Taisei sankei is completed; none of it written by Seki, volumes 1-12 by Takebe Katahiro and 13-20 by Takebe Kataaki. It contains the enri, Japan's nearest equivalent of definite integration | Takebe Katahiro, Takebe Kataaki, Seki Takakazu | Edo, Japan | S430 |
| 1711 | De analysi first printed, in William Jones's Analysis per quantitatum series, fluxiones, ac differentias | Isaac Newton, William Jones | London | S184 |
| 29 December 1711 | Leibniz's second letter to Sloane: he declines to say how he came by the method, calls Keill a novice, and asks Newton to speak | Gottfried Wilhelm Leibniz, Hans Sloane | Hanover / London | S189 |
| 4 March 1711 (New Style) | Leibniz complains to Hans Sloane and asks the Royal Society to make Keill publicly retract | Gottfried Wilhelm Leibniz, Hans Sloane | Hanover / London | S189 |
| 1712 (committee appointed March, reported April; book dated 1712, circulated January 1713) Disputed |
The Royal Society appoints a committee to examine the old letters and papers; its report and the documents are published as the Commercium Epistolicum. Leibniz is never asked for his account. Dispute: Committee membership and the exact appointment/report dates were NOT verified from the Royal Society Journal Book. Conventional list: Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robartes, Bonet, de Moivre, Aston, Brook Taylor; appointed 6 March 1711/12, reported 24 April 1712. |
Gottfried Wilhelm Leibniz | London | S189, S282, S284 |
| 26 July 1712 | Brook Taylor communicates his theorem to John Machin by letter, three years before publishing it | Brook Taylor | England | S312 |
| 1713 Disputed |
Charta volans ("flying sheet"), an anonymous single-sheet rebuttal circulated on the Continent for Leibniz, quoting an unnamed "eminent mathematician" (Johann Bernoulli, also anonymous) Dispute: Not verified in full text. The date is usually given as 29 July 1713, quoting a Bernoulli letter of 7 June 1713, and nothing here rests on the sheet itself or on a scholarly edition of it. |
Gottfried Wilhelm Leibniz | Continental Europe | S284 |
| 1715 | Taylor publishes Methodus Incrementorum Directa et Inversa; the theorem now named after him is Prop. VII Cor. 2, p. 23 | Brook Taylor | London | S312 |
| January-February 1714/15 (OS) = 1715 (NS) | Newton publishes, anonymously and in his own Society's journal, "An Account of the Book entituled Commercium Epistolicum" in Phil. Trans. 29 no. 342, pp. 173-224: a review of the report he had himself written | Isaac Newton | London | S189, S191, S282 |
| November 1715 | Leibniz opens the correspondence with Samuel Clarke, routed through Caroline, Princess of Wales; five papers each, ending with Clarke's fifth reply of 29 October 1716 | Gottfried Wilhelm Leibniz, Samuel Clarke, Caroline of Ansbach | Hanover / London | S240 |
| 14 November 1716 | Leibniz dies at Hanover. On the standard account his secretary was effectively the only mourner | Gottfried Wilhelm Leibniz | Hanover | S235, S236 |
| July 1716 | Johann Bernoulli publishes an anonymous attack on Taylor, accusing him of plagiarism over centers of oscillation; the exchange runs to 1719 | Johann Bernoulli, Brook Taylor | Basel / London | S331 |
| 1717 | Clarke publishes the Leibniz-Clarke correspondence in his own translation, after Leibniz is dead and cannot reply | Samuel Clarke | London | S240 |
| 1717 | Stirling publishes, in Lineae Tertii Ordinis Neutonianae p. 32, the expansion that will be named after Maclaurin twenty-five years later, and uses it as we now use it | James Stirling | Oxford | S312 |
| 1718 | Guido Grandi names the curve versoria (Latin, the rope that turns a sail) and la versiera in Italian | Guido Grandi | Pisa | S327 |
| 1722 | Takebe Katahiro publishes the Tetsujutsu Sankei, containing his series for arc length in terms of the sagitta and a convergence-acceleration method equivalent to Romberg's; chapter 9 gives pi to over 40 decimal places | Takebe Katahiro | Edo, Japan | S430 |
| 1727-1728 | Euler introduces the symbol e for the base of natural logarithms in the manuscript Meditatio in experimenta explosione tormentorum nuper instituta, a paper on the firing of cannon; it reaches print in the Mechanica of 1736 | Leonhard Euler | St Petersburg | S325 |
| 20 March 1726/7 (OS) = 31 March 1727 (NS) | Newton dies at Kensington in the early hours; his body lies in state in the Jerusalem Chamber at Westminster | Isaac Newton | Kensington, London | S200 |
| 28 March 1727 (OS) = 8 April 1727 (NS) | Newton buried in Westminster Abbey, the pall borne by two dukes, three earls and the Lord Chancellor | Isaac Newton | Westminster Abbey, London | S200 |
| August 1730 | Newton's monument completed by Michael Rysbrack to a design by William Kent; unveiled the following year | Isaac Newton | Westminster Abbey, London | S200 |
| 'Anno 1732' (as printed on the title) Disputed |
The year printed in the title of Johann Bernoulli's Hydraulica; the work is not mentioned to Euler until October 1738 and does not appear in print until 1742/1743 Dispute: Craik's reference gives Opera omnia (Bousquet) 1742; Tou and others give 1743 for the appearance. The chronology of the Euler correspondence is what makes the antedating claim serious |
Johann Bernoulli | Basel | S333, S336, S341 |
| 1688 | David Gregory's reconstruction of Newton's prime theorem on series quadratures is printed in Archibald Pitcairne's Solutio problematis de historicis seu inventoribus, Edinburgh; Newton answers by starting the treatise that becomes De quadratura | David Gregory, Archibald Pitcairne, Isaac Newton | Edinburgh | S204 |
| 1691 | Edmond Halley and Joseph Raphson examine the original De methodis manuscript at Cambridge and find it 'very much worn by having been lent out' | Edmond Halley, Joseph Raphson, Isaac Newton | Cambridge | S204 |
| 1693 | Extracts from De quadratura are printed in Wallis's Opera mathematica, vol. 2 | Isaac Newton, John Wallis | Oxford | S204 |
| 1694 | David Gregory visits Cambridge and transcribes Newton mathematical manuscripts; his summary treatise Methodus fluxionum then circulates in copies held by Jones and Keill | David Gregory, Isaac Newton, William Jones, John Keill | Cambridge | S204 |
| 7 September 1695 (OS) = 17 September 1695 (NS) | Halley writes to Newton that he still has De quadratura for transcription and that nobody has seen it or will without Newton's directions | Edmond Halley, Isaac Newton | London | S204 |
| late 1690s Disputed |
The manuscript of Enumeratio linearum tertii ordinis is shown to David Gregory; it is printed in 1704 with the Opticks Dispute: Guicciardini hedges: 'in the late 1690s', with no year |
Isaac Newton, David Gregory | Cambridge | S204 |
| 1702 | Newton's 'classical scholia', after restricted manuscript circulation, are printed in David Gregory's Astronomiae physicae et geometricae elementa | Isaac Newton, David Gregory | Oxford | S204 |
| 1704 | In the printed De quadratura Newton replaces 'infinite parva' with 'admodum parva', editing the infinitesimals out of his own early text | Isaac Newton | London | S204 |
| 1707 | Arithmetica universalis published anonymously at Cambridge; Newton let it go to press to secure his colleagues' support in the 1705 parliamentary election and had not seen a sheet of it | Isaac Newton, David Gregory | Cambridge | S204 |
| c. 1710 Disputed |
William Jones transcribes De methodis; Colson's 1736 English translation descends from this copy, not from Newton's original Dispute: Guicciardini hedges: the transcript was made by William Jones 'in about 1710' |
William Jones, Isaac Newton, John Colson | London | S204 |
| 1711 Disputed |
In the printed De analysi William Jones changes 'esse infinite parvam' to 'in infinitum diminui & evanescere', most probably on Newton's instruction Dispute: Guicciardini hedges the attribution, not the year: Jones made the change 'most probably after Newton's instruction'. The instruction itself is not documented |
William Jones, Isaac Newton | London | S204 |
| December 1720 (OS) | James Wilson writes to Newton reporting incorrectly copied transcripts of the 1666 and 1671 treatises circulating in at least two private hands, and guesses the copyists took double copies to sell | James Wilson, Isaac Newton | London | S204 |
| c. 1720 Disputed |
James Wilson makes a secondary transcript of De methodis from Jones's copy; the transcript is now lost Dispute: Guicciardini hedges: Wilson's secondary transcript was made 'in ca. 1720', and the transcript is lost, so it cannot be dated from the object |
James Wilson, William Jones | London | S204 |
Timeline
1734 to 1799: attack, repair, and the Enlightenment
33 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 1734 | Berkeley publishes The Analyst; or A Discourse Addressed to an Infidel Mathematician, in London and Dublin, in two non-identical texts | George Berkeley | London and Dublin | S310 |
| 1734 | Jurin replies as 'Philalethes Cantabrigiensis' with Geometry No Friend to Infidelity; Robins, Walton, Pemberton, Bayes and others join in over the following decade | James Jurin, Benjamin Robins | London | S310, S311 |
| 1735 | Euler presents his solution of the Basel problem to the St Petersburg Academy: . He is 28. Published as E41 in 1740 | Leonhard Euler | St Petersburg | S322, S323 |
| 1736 | The Method of Fluxions and Infinite Series published in English, trans. John Colson | Isaac Newton, John Colson | London | S185, S192 |
| 1738 | Du Chatelet submits her essay on the nature and propagation of fire to the Academie royale des sciences prize competition anonymously, because a submission known to be by a woman would not have been taken seriously. It does not win, and the Academie publishes it anyway | - | Paris | S069 |
| April/May 1738 | Daniel Bernoulli's Hydrodynamica appears in print at Strasbourg; in October his father first mentions a rival Hydraulica to Euler | Daniel Bernoulli, Johann Bernoulli | Strasbourg / Basel | S333 |
| 1740 | Emilie du Chatelet publishes Institutions de physique anonymously in Paris, its frontispiece carrying portraits of Newton, Descartes and Leibniz; its last chapter openly attacks Mairan's 1728 memoire on forces vives, and the 1742 Amsterdam edition carries her name | Emilie du Chatelet | Paris | S068, S069, S332 |
| 1740 | Simpson publishes the iteration , the form in which "Newton's method" is taught | Thomas Simpson | London | S462 |
| 1742 | Maclaurin publishes A Treatise of Fluxions, 2 vols., Edinburgh; the Preface states that Berkeley's Analyst 'first gave Occasion to the ensuing Treatise' | Colin Maclaurin, George Berkeley | Edinburgh | S311 |
| 1744 | Euler publishes Methodus inveniendi (E65), founding the calculus of variations and giving the first statement of the principle of least action | Leonhard Euler | Lausanne and Geneva | S323 |
| 1748 | Euler publishes Introductio in analysin infinitorum (E101): analysis becomes the study of functions. S.126 adopts pi; S.138 gives | Leonhard Euler | Lausanne | S318, S325 |
| 1748 | Agnesi publishes Instituzioni analitiche ad uso della gioventu italiana in Milan: the first surviving mathematics book by a woman and the first textbook to cover differential and integral calculus together | Maria Gaetana Agnesi | Milan | S319, S327 |
| 10 September 1749 | Emilie du Chatelet dies at Luneville, six days after childbirth, having raced to finish her Principia translation | Emilie du Chatelet | Luneville, Lorraine | S332 |
| 26 September 1750 | Pope Benedict XIV writes to Agnesi offering her the chair of mathematics at Bologna; she accepts the honor and, on Truesdell's account, never goes | Maria Gaetana Agnesi | Rome / Bologna | S327, S340 |
| 1755 | Euler's Institutiones calculi differentialis (E212) published at St Petersburg; written in 1748 | Leonhard Euler | St Petersburg | S323 |
| 1755 | Euler introduces Delta for difference and Sigma for sum in one book, explicitly as a matched pair | Leonhard Euler | St Petersburg | S451 |
| 1759 | Du Chatelet's complete French Principia, with her Exposition abregee du systeme du monde, appears in Paris ten years after her death, still the only complete French translation, because Clairaut pressed the publisher Prault to bring it out and arranged for the illustrations to be finished | Emilie du Chatelet, Alexis-Claude Clairaut | Paris | S068, S332 |
| c. 1760 | John Colson, translating Agnesi into English, renders la versiera as 'the Witch'; published posthumously in 1801 | John Colson, Maria Gaetana Agnesi | Cambridge | S320, S319 |
| 1766 | Euler returns to St Petersburg at Catherine II's invitation, aged 59; he goes on to produce almost half his life's output there, most of it after losing his sight | Leonhard Euler, Frederick II of Prussia | Berlin to St Petersburg | S326 |
| 1768-1770 | Euler's Institutiones calculi integralis published in three volumes at St Petersburg | Leonhard Euler | St Petersburg | S323 |
| 1770 | Condorcet first uses the 'curly d' partial-derivative symbol; Legendre gives it its modern form in 1786 | Nicolas de Condorcet, Adrien-Marie Legendre | Paris | S325 |
| 1771 Disputed |
Euler undergoes cataract surgery; sight is briefly restored, then lost entirely. He dictates for the remaining twelve years and his output increases Dispute: MacTutor records Calinger's view that the standard chronology of Euler's sight loss (1738 strain, 1740 one eye) is unreliable |
Leonhard Euler | St Petersburg | S326 |
| 1772 | Lagrange uses 'fonction derivee' in the Nouveaux Memoires of the Berlin Academy - the origin of the English word 'derivative' | Joseph-Louis Lagrange | Berlin | S324 |
| 1773-1774 Disputed |
Diderot visits St Petersburg at Catherine II's invitation; the alleged confrontation with Euler is placed here by the anecdote and is unsupported by any Russian source Dispute: Thiebault, the only source, does not name Euler and declines to vouch for the story |
Denis Diderot, Leonhard Euler | St Petersburg | S313, S314, S339 |
| 5 May 1777 | Euler presents 'De formulis differentialibus angularibus', introducing the letter for ; published only in 1794 | Leonhard Euler | St Petersburg | S325 |
| 1786 | Legendre publishes the modern partial-derivative symbol - and abandons it the following year | Adrien-Marie Legendre | Paris | S451, S139 |
| 1786 | Simon L'Huilier writes "lim." for the first time, in the essay that won the Berlin Academy prize on the nature of infinity | Simon L'Huilier | Berlin/Geneva | S451, S139 |
| c. 1795 | Sophie Germain submits observations on Lagrange's Ecole Polytechnique analysis course under the name of a real former student, M LeBlanc | Sophie Germain | Paris | S410 |
| 1797 | Lagrange publishes Theorie des fonctions analytiques, founding the calculus on power series 'freed from every consideration of infinitely small or vanishing quantities, of limits or of fluxions', and introducing f'x, f''x and the primitive/derived pair, "pour plus de simplicite et d'uniformite" | Joseph-Louis Lagrange | Paris | S317, S325, S451 |
| 1779-1785 | Samuel Horsley uses both the Jones and the Wilson transcripts for his edition of Newton's Opera omnia | Samuel Horsley, William Jones, James Wilson | London | S204 |
Timeline
1800 to 1872: the age of rigor
48 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 1800 | Arbogast introduces D for derivation and for its inverse | Louis Arbogast | Strasbourg | S451, S139 |
| 1806 | Ampere publishes an attempted justification that continuous functions are differentiable except at isolated points - the 'theorem' Weierstrass would refute | - | Paris | S386 |
| 1806 | Fearing Gauss would die as Archimedes did during the French occupation of Braunschweig, Germain sends a family-friend general to protect him, and is forced to reveal that she is LeBlanc | Sophie Germain | Braunschweig and Paris | S410 |
| 21 December 1807 | Fourier presents 'On the Propagation of Heat in Solid Bodies' to the Paris Institute; referees Lagrange, Laplace, Monge, Lacroix | Fourier | Paris | S383 |
| 1811 | Fourier wins the Institute prize, with a citation saying the work 'still leaves something to be desired on the score of generality and even rigour' | Fourier | Paris | S383 |
| 1812 | Babbage, Herschel, Peacock and others found the Analytical Society at Cambridge to replace Newton's dot notation with Leibniz's d | Charles Babbage, John Herschel, George Peacock | Cambridge | S239 |
| 1813 | Memoirs of the Analytical Society published (Cambridge, 4to), written entirely by Herschel and Babbage. Babbage had suggested the title "The Principles of pure D-ism in opposition to the Dot-age of the University"; it was not used | Charles Babbage, John Herschel | Cambridge | S239 |
| 1815 Disputed |
Germain wins the Institut de France prize for elastic surfaces - a medal of one kilogram of gold - at the third attempt; she does not attend the ceremony Dispute: Very widely printed as 1816. MacTutor says the re-opened contest of 1815. See Appendix F. |
Sophie Germain | Paris | S410 |
| 1817 | Bolzano publishes Rein analytischer Beweis: arithmetic definition of continuity, proof of the intermediate value theorem, the convergence criterion later named after Cauchy | Bolzano | Prague | S363, S371, S387, S360 |
| December 1819 Disputed |
Bolzano suspended from his chair after Austrian government pressure; house arrest, mail censorship and a publication ban follow Dispute: MacTutor (S371) dates the suspension to December 1819; the Stanford Encyclopedia (S370) says Emperor Franz removed him from the professorship in January 1820. Probable resolution: suspended Dec 1819, formally removed Jan 1820. |
Bolzano | Prague | S371, S370 |
| 1821 | Cauchy publishes the Cours d'analyse: verbal definitions of limit, infinitesimal, continuity and convergent series, plus the false Theorem I on sums of continuous functions (pp. 131-132) | Cauchy | Paris | S361, S360 |
| 1822 | Fourier publishes Theorie analytique de la chaleur, whose claim about arbitrary functions breaks the existing concept of 'function' | Fourier | Paris | S383 |
| 1822 | Fourier prints the limits of integration on the integral sign | Fourier | Paris | S451, S139 |
| 1823 | Cauchy publishes the Resume: the definite integral defined as a limit of sums and its existence proved; the preface states his aim of reconciling rigor with 'the direct consideration of infinitely small quantities' | Cauchy | Paris | S362 |
| 1823 | Sarrus introduces the evaluation bar with limits attached | Pierre Frederic Sarrus | Strasbourg | S451, S139 |
| 16 January 1826 Disputed |
Abel writes to Holmboe from Berlin that divergent series are 'noget Fandensskab' (a piece of devilry) and that it is a shame to base any demonstration on them; the 1902 French translation renders this 'une invention du diable', which is where the English 'invention of the devil' comes from Dispute: The 1902 editors note the letter is clearly dated 16 January 1826 'mais est certainement du 14' - Abel says he will see Die schoene Muellerin 'tomorrow', while letter VII of the same nominal date says he saw it 'yesterday'; the date was probably added when the letter was sent |
Abel | Berlin | S490 |
| 1826 | Abel publishes his binomial-series memoir with the footnote counterexample to Cauchy's Cours d'analyse p. 131 | Abel, Cauchy | Berlin (Crelle's Journal) | S364 |
| 27 March 1827 | Lord Brougham writes to Dr Somerville asking that Mrs Somerville write up Laplace's Mecanique Celeste for the Society for the Diffusion of Useful Knowledge | Mary Somerville | London | S412 |
| 25 May and 1 June 1829 | Galois submits articles on the algebraic solution of equations to the Academy; Cauchy is assigned as referee | Galois, Cauchy | Paris | S380 |
| 29 June 1829 | Cauchy finally reports on Abel's 1826 memoir; the report is described as hasty, nasty and superficial | Cauchy, Abel | Paris | S372 |
| 6 April 1829 | Abel dies aged 26; Cauchy has still not reported on the memoir Abel submitted in 1826 | Abel, Cauchy | Froland, Norway | S372 |
| February 1830 | Galois submits his Grand Prize memoir to Fourier; Fourier dies on 16 May 1830 and the manuscript is never found | Galois, Fourier | Paris | S380, S383 |
| September 1830 | Cauchy leaves Paris after the July Revolution; he will not swear the oath of allegiance and loses all his positions | Cauchy | Paris to Switzerland | S372 |
| around 1830 Disputed |
Bolzano writes the Functionenlehre, containing a continuous nowhere-differentiable function built by a geometric construction Dispute: S386 dates the manuscript 1830; MacTutor (S384) says 'before 1834'. |
Bolzano | Prague | S386, S384 |
| 1831 | Brewster's Life of Sir Isaac Newton (p. 194) puts the Latin 'tanquam ex ungue leonem' into Johann Bernoulli's mouth, creating the version quoted ever since | David Brewster, Johann Bernoulli | London | S321, S337 |
| 1831 | Mechanism of the Heavens published by John Murray in an edition of 750 copies | Mary Somerville | London | S412, S413 |
| 14 February 1832 | George Peacock writes that he and Whewell "have already taken steps to introduce it into the course of our studies at Cambridge"; the whole edition then sells chiefly at Cambridge | Mary Somerville, William Whewell | Cambridge | S412 |
| 1833 | Cauchy goes to Prague to tutor the grandson of the exiled Charles X | Cauchy | Prague | S372 |
| March 1834 | The word "scientist" appears in print for the first time, in Whewell's anonymous review of Somerville's On the Connexion of the Physical Sciences, Quarterly Review vol. LI no. CI, pp. 54-68 - attributed by the reviewer to "some ingenious gentleman" and reported as not generally palatable | William Whewell, Mary Somerville | London | S411 |
| 1834-35 Disputed |
Charles Whish's paper appears in Transactions of the Royal Asiatic Society after his death. It is the first European account of the Kerala infinite series for pi. Dispute: Almeida & Joseph give 1835 vol. 3; the paper is very commonly cited as 1834 |
Charles Whish | London / Malabar | S051, S052 |
| 1837 | Dirichlet states the modern definition of a function as an arbitrary rule of assignment | Dirichlet | Berlin | S379 |
| 1841 | Jacobi reintroduces the curly d for partial derivatives, unaware that Legendre had published it 55 years before; adoption takes another half-century | Carl Gustav Jacob Jacobi | Konigsberg | S451, S139 |
| 1841 | Weierstrass writes absolute value with vertical bars, in a paper not printed at the time | Weierstrass | Munster | S451 |
| 1843 | Lovelace publishes her translation of Menabrea with Notes A-G; Note G contains a looping, recurrence-based algorithm for the Bernoulli numbers with a trace table for the case n=4 | Ada Lovelace, Luigi Menabrea | London | S415 |
| 10 June 1854 | Riemann delivers the Habilitationsvortrag 'Ueber die Hypothesen welche der Geometrie zu Grunde liegen'; Gauss praises it with rare enthusiasm | Riemann | Goettingen | S375 |
| 1854 | Riemann's Habilitationsschrift on trigonometric series defines the definite integral by arbitrary partitions and arbitrary sample points | Riemann | Goettingen | S391, S375 |
| 31 March 1854 | Koenigsberg awards Weierstrass an honorary doctorate after the Abelian functions paper, ending fourteen years of schoolteaching | Weierstrass | Koenigsberg | S373 |
| October 1856 Disputed |
Weierstrass takes up a professorship at the University of Berlin, aged 40-41 Dispute: Commonly reported as 'professor at 39'; MacTutor's dating makes him 40 turning 41. See Appendix F. |
Weierstrass | Berlin | S373 |
| 24 November 1858 | Dedekind, preparing his first calculus lectures at Zurich, finds the definition of continuity by cuts, and tells his friend Durege days later | Dedekind | Zurich | S366, S382 |
| 1859 | Bunyakovsky publishes the integral form of what will be called the Cauchy-Schwarz inequality - in the year he retires | Viktor Bunyakovsky | St Petersburg | S457, S458 |
| around 1861 | Riemann suggests a counterexample to Ampere's theorem in his lectures, per Weierstrass; the proof was never written down | Riemann, Weierstrass | Goettingen | S386 |
| 1869 | Meray publishes the first coherent rigorous theory of the irrational numbers to appear in print, three years before Dedekind, Cantor and Heine | Meray | France | S378 |
| 7 November 1870 | Maxwell writes to Tait from Glenlair proposing names for the results of the nabla operator: slope, convergence, twist/turn/twirl/curl, concentration | James Clerk Maxwell, Peter Guthrie Tait | Glenlair, Dumfriesshire | S454 |
| c. 1870 | William Robertson Smith suggests the name "nabla" to Tait, from the Assyrian harp the inverted delta resembles | William Robertson Smith, Peter Guthrie Tait | Scotland | S454 |
| 18 July 1872 | Weierstrass presents his continuous nowhere-differentiable function to the Royal Academy of Science in Berlin, crediting Riemann with the idea | Weierstrass | Berlin | S386 |
| 1872 | Dedekind (cuts), Cantor (fundamental sequences) and Heine all publish constructions of the real numbers | Dedekind, Cantor | Germany | S366, S369, S378 |
| 1872 | De Morgan's A Budget of Paradoxes names Euler in the Diderot story, adds 'algebra was Hebrew to Diderot', and miscopies Thiebault's z as n | Augustus De Morgan | London | S313, S314 |
| 20 March 1872 | Dedekind writes the preface to Stetigkeit und irrationale Zahlen; Cantor's Math. Annalen vol. 5 paper arrives that same day | Dedekind, Cantor | Braunschweig | S366 |
Timeline
1873 to 1913: measure, set theory, and the modern integral
21 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 1874 | Cantor proves the real numbers are not countable while the algebraic numbers are | Cantor | Halle | S376 |
| 1874 | Kovalevskaya is awarded her doctorate by Goettingen in absentia, having been barred from enrolling at Berlin | Kovalevskaya, Weierstrass | Goettingen | S374 |
| 1875 | Weierstrass's nowhere-differentiable function is published for the first time - by Paul du Bois-Reymond, in Borchardt's Journal | Weierstrass, du Bois-Reymond | Heidelberg / Berlin | S386 |
| 1875 | Darboux publishes the upper-and-lower-sum construction of the integral that textbooks call the Riemann integral | Gaston Darboux | Paris | S453 |
| 1877 | Kronecker treats Cantor's dimension paper with suspicion and delays publication until Dedekind intervenes; Cantor never submits to Crelle again | Cantor, Kronecker, Dedekind | Berlin / Halle | S376 |
| May 1884 | Cantor's first recorded attack of depression | Cantor | Halle | S376 |
| 1885 | Mittag-Leffler persuades Cantor to withdraw a paper already in proof, saying it is 'about one hundred years too soon' | Cantor | Stockholm | S376 |
| 1889 | Peano introduces the set-membership symbol in Arithmetices principia, as an abbreviation for the Latin 'est' | Peano | Turin | S393 |
| June 1889 | Kovalevskaya becomes full professor at Stockholm, having been extraordinary professor since June 1884 | Kovalevskaya | Stockholm | S374 |
| 1890 | Cellerier's independently discovered nowhere-differentiable function is published, posthumously, thirty years after its construction | Cellerier | Geneva | S386 |
| 1891 | Cantor gives the diagonal argument | Cantor | Halle | S376 |
| 20 May 1893 | Hermite writes to Stieltjes that he turns away 'avec effroi et horreur' from the lamentable plague of continuous functions with no derivatives | Hermite, Stieltjes | Paris | S365, S388 |
| 1895 | Grace Chisholm is awarded a Gottingen doctorate under Klein - twenty-one years after Kovalevskaya could receive one only in absentia | Grace Chisholm Young | Gottingen | S419 |
| 14 September 1901 | John Perry opens the British Association discussion on the teaching of mathematics at Glasgow, arguing that the methods of the calculus are 'easily learnt by all young boys' and should be taught in ordinary schools - the manifesto of the Perry movement | - | Glasgow | S492 |
| 29 April 1901 | Lebesgue announces his integral in a Comptes Rendus note, 'Sur une generalisation de l'integrale definie' | Lebesgue | Paris | S377 |
| 1902 | Lebesgue's thesis Integrale, longueur, aire is presented at Paris and published in the Annali di Matematica, Milan | Lebesgue | Paris / Milan | S377 |
| 1905 | J. G. Leathem introduces the arrow for passage to a limit | John Gaston Leathem | Cambridge | S451, S139 |
| 1906 | Heiberg identifies and transcribes Archimedes' Method from a palimpsest in Constantinople, after following up an 1899 catalog notice | Johan Ludvig Heiberg | Constantinople | S002, S006 |
| 1906 | The Theory of Sets of Points published under both Youngs' names; the other 219 joint papers carry William Young's name alone, by written agreement | Grace Chisholm Young, William Henry Young | Geneva | S419 |
| 1908 | Hardy's Course of Pure Mathematics and Bromwich's Infinite Series make the arrow standard; Hardy credits Leathem and Bromwich in his Preface | G. H. Hardy, Thomas Bromwich | Cambridge | S451, S139 |
| January 1913 | Ramanujan writes to Hardy - his fourth attempt, after Hill misunderstood his divergent series and Hobson and Baker did not reply | Srinivasa Ramanujan | Madras and Cambridge | S431 |
Timeline
1914 to the present: vindication and the machine age
40 entries.
| When | What happened | People | Places | Sources |
|---|---|---|---|---|
| 1916 | Hardy shows that ab >= 1 suffices for the Weierstrass function to be nowhere differentiable, weakening Weierstrass's own condition | Weierstrass | Cambridge | S389, S388 |
| 1918 | Noether proves that to every continuous symmetry of a system there corresponds a conservation law | Emmy Noether | Gottingen | S418 |
| 1919 | Noether is finally granted the habilitation after a four-year fight by Hilbert and Klein; until then she had lectured under Hilbert's name | Emmy Noether | Gottingen | S418 |
| 1920 | Martin Jasek finds Bolzano's Functionenlehre manuscript in the National Library, Vienna | Jasek, Bolzano | Vienna | S386 |
| 3 December 1921 | Jasek gives the first lecture reporting on the rediscovered Functionenlehre | Jasek | Prague | S386 |
| 1922 | Schafheitlin finds and publishes, in the Basel public library, Part I of Johann Bernoulli's 1691-92 Course - proving that L'Hopital's Analyse was Bernoulli's content in L'Hopital's exposition | Johann Bernoulli, Guillaume de l'Hopital | Basel | S315 |
| 3 February 1922 | Rychlik presents the correct proof that Bolzano's function is continuous and nowhere differentiable; Jarnik proves it independently at the same time | Rychlik, Jarnik, Bolzano | Prague | S386 |
| 1930 Disputed |
Bolzano's Functionenlehre is published for the first time, edited by Rychlik, in Prague, about a century after it was written Dispute: S386's abstract says 1922, its body and MacTutor (S384) say 1930. Resolution: the proofs were published in 1922, the manuscript in 1930. See Appendix F. |
Bolzano, Rychlik | Prague | S384, S386 |
| 1930 | Nikodym completes the Radon-Nikodym theorem, the abstract form of "one measure has a density with respect to another" | Otton Nikodym, Johann Radon | Poland | S435 |
| 1931 | Banach and Mazurkiewicz independently prove that in the space of continuous functions, the nowhere-differentiable ones are generic | - | Poland | S388 |
| 1934 | Skolem constructs the hyperintegers by the ultraproduct method and proves the analogue of the Transfer Principle - the machinery of nonstandard analysis, 27 years before its application | Thoralf Skolem | Norway | S440 |
| 1934-1935 | The Bourbaki group forms in Paris, initially to write a better analysis textbook than the one they were teaching from | Nicolas Bourbaki, Henri Cartan | Paris | S434 |
| 1936 Disputed |
Sotheby dispersal of the non-scientific Newton papers; J. M. Keynes reassembles about half Dispute: Keynes says 'dispersed in the auction room in 1936'; the sale is usually given as 13-14 July 1936, which this agent did not verify. |
John Maynard Keynes | London | S193 |
| 1937 | Stone generalises the Weierstrass approximation theorem to compact Hausdorff spaces | Marshall Stone | USA | S435 |
| 1942 | B. H. Brown publishes 'The Euler-Diderot Anecdote' in the American Mathematical Monthly, the first proper debunking; Gillings extends it in 1954 | - | Hanover, New Hampshire | S313, S314 |
| 1944 | Laurent Schwartz conceives the theory of distributions during a night of insomnia in occupied France, and runs next door to tell Henri Cartan | Laurent Schwartz, Henri Cartan | Nancy or Grenoble, France | S433 |
| July 1946 | 'Newton, the Man' read at the Royal Society tercentenary by Geoffrey Keynes; J. M. Keynes had died in April 1946 | John Maynard Keynes | London | S193 |
| 1955 Disputed |
The College Board dates the establishment of the Advanced Placement Program to this year ('The Advanced Placement Program was established in 1955') Dispute: This is the College Board's own institutional date. Other accounts date the pilot examinations to 1954 (run by the School and College Study of Admission with Advanced Standing) and the first examinations under the College Board to 1956. Neither was verified in this project - see S444 |
- | United States | S448 |
| 1958 | Mary Jackson becomes NASA's first black female engineer, having needed the City of Hampton's permission to attend the required classes in a segregated high school | Mary Jackson | Hampton, Virginia | S425 |
| 1960 | NASA TN D-233 published: given a chosen landing point and a number of orbits, solve for the launch azimuth. First author credit for a woman in Langley's Flight Research Division | Katherine Johnson, Ted Skopinski | Langley Field, Virginia | S423, S422, S424 |
| 1961 | Robinson introduces nonstandard analysis | Abraham Robinson | USA | S381 |
| February 1962 | Before Friendship 7, John Glenn asks engineers to "get the girl" to re-run by hand, on a desktop mechanical calculator, the same equations programmed into the IBM computers: "If she says they are good, then I am ready to go" | Katherine Johnson | Cape Canaveral and Langley | S422 |
| 1964 | Wengert essentially discovers forward-mode automatic differentiation | Robert Wengert | USA | S436 |
| 1966 | Robinson publishes Non-standard Analysis, giving infinitesimals that obey the same laws as ordinary numbers, as Leibniz had claimed | Abraham Robinson | USA | S381 |
| 1970 | Linnainmaa's master's thesis gives the first published description of the reverse mode of automatic differentiation - the algorithm now called backpropagation - with no reference to neural networks; the 1976 BIT paper is titled "Taylor expansion of the accumulated rounding error" | Seppo Linnainmaa | Helsinki | S436, S437 |
| 1970 Disputed |
Seppo Linnainmaa conceives the reverse mode of differentiation - the ancestor of backpropagation - 'on a sunny afternoon in a Copenhagen park'; G. M. Ostrovskii and co-workers had used the same technique about five years earlier, and it was independently rediscovered repeatedly into the 1980s Dispute: Griewank's whole point is that there is no single inventor: Ostrovskii et al. c. 1965 (published 1971), Linnainmaa 1970, Speelpenning 1980, Werbos 1982, Baur-Strassen 1983, Griewank himself 1987 |
- | Copenhagen / Helsinki | S438 |
| 1974 | Bishop's lecture at the Workshop on the Evolution of Modern Mathematics, Boston; the phrase 'debasement of meaning' is in the draft but not in the audio of the lecture as delivered | Bishop, Abraham Robinson | Boston | S390 |
| 1974 | Werbos's thesis Beyond Regression casts the reverse mode in formal discrete-time terms | Paul Werbos | Harvard, USA | S436, S437 |
| 1975 | Sabetai Unguru publishes 'On the need to rewrite the history of Greek mathematics' (Archive for History of Exact Sciences 15, 67-114), denying that Greek geometry is algebra in disguise; van der Waerden (1976), Freudenthal (1977) and Weil (1978) reply in the same journal - the debate that governs whether 'why did the Greeks not invent calculus?' is even a well-posed question | - | - | S491 |
| 1976 | Keisler publishes Elementary Calculus: An Approach Using Infinitesimals, the first calculus textbook built on Robinson's hyperreals; Bishop reviews it hostilely in the Bulletin of the AMS in 1977 | H. Jerome Keisler | Wisconsin, USA | S440, S441 |
| 1976 | About 76,000 students sat nearly 100,000 Advanced Placement examinations in the United States, all subjects combined - against 3,079,134 students and 5,744,259 examinations in 2024 | - | United States | S493, S446 |
| late 1970s to early 1980s | Gladys West leads a five-person team computing the geoid on an IBM 7030 Stretch - the Earth-shape model that satellite positioning is measured against | Gladys West | Dahlgren, Virginia | S426 |
| 1977 | Bishop reviews Keisler's Elementary Calculus in the Bulletin of the AMS | Bishop | USA | S395, S390 |
| by 1977 Disputed |
AP Calculus already consisted of two examinations, 'calculus AB' and 'calculus BC'; the booklet lists 'Calculus (two courses)' among fourteen AP subjects Dispute: This fixes only an upper bound: the AB/BC split predates 1977. The year it was introduced, and the meaning of the letters, remain unsourced - see S444 |
- | United States | S493 |
| 1980 | Speelpenning gives the first genuinely automatic implementation of reverse-mode AD | Speelpenning | Illinois, USA | S436 |
| 1986 | Rumelhart, Hinton and Williams popularise backpropagation; the PDP group had not read Parker, and Parker had not read Werbos | - | USA | S436, S437 |
| 1997 | Grabiner's 'Was Newton's Calculus a Dead End?' argues that Maclaurin's Treatise transmitted improved Newtonian calculus to the Continent | Colin Maclaurin | Claremont, California | S335 |
| 28 or 29 October 1998 Disputed |
The Archimedes Palimpsest is sold at Christie's for $2 million to an anonymous collector, who deposits it at the Walters Art Museum Dispute: Walters gives 28 October 1998; the project website gives 29 October 1998 |
William Noel | New York | S004, S005, S006 |
| 2002 | Netz, Saito and Tchernetska publish a new reading of Method Proposition 14 in which Archimedes calls two infinite collections 'equal in multitude' | Reviel Netz, Archimedes | Stanford / Osaka / Cambridge | S003 |
| 2024 | 426,848 AP Calculus examinations taken in one year in the United States - 278,657 Calculus AB and 148,191 Calculus BC - out of 5,744,259 AP examinations sat by 3,079,134 students | - | United States | S446, S447 |
Reference: the registers everything else is built on
Twelve appendices. 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. If a word stopped you, start with the Glossary. The rest: people in A, symbols in B, how to say a name in C, where each story fits your course in I.
- Look up any person, symbol, term, or source used anywhere in this book.
- Check a worked calculation for yourself, since every one was re-derived in code.
- See exactly where two sources disagree, and whether the disagreement is settled.
- Find where a story belongs in your Derivatives or Integrals course.
- Tell the difference between a story that checks out and one that does not.
Appendix A
People register
212 people, sorted by birth year, each with life dates, pronunciation, region, contribution, and sources. Life dates in italics are not settled.
| Name | Lived | Pronunciation: IPA above, respelling below |
Region | Why they are here | Sources |
|---|---|---|---|---|---|
 Zeno of Elea Not from life |
490 BCE to 430 BCE Disputed |
/ˈziːnoʊ əv ˈɛliə/ ZEE-noh of EL-ee-uh |
Elea, southern Italy (Greek) | Framed the paradoxes of motion and plurality that state, 2300 years early, the problems limits and measure theory were built to solveAppears in Chapter 2, and in the timeline under Antiquity to 500 CE 3 times. | S007 |
| Antiphon | 480 BCE to 411 BCE Disputed |
/ˈæntɪfɒn/ AN-tih-fon |
Athens (Greek) | Inscribed polygons of ever more sides in a circle - the germ of the method of exhaustionAppears in Chapter 2 3 times. | S008 |
| Democritus | 460 BCE to 370 BCE Disputed |
/dɪˈmɒkrɪtəs/ dih-MOK-rih-tus |
Abdera, Thrace (Greek) | First asserted that a cone is one third of its cylinder, apparently by treating solids as stacks of indivisible slicesAppears in Chapter 2 3 times, and in the timeline under Antiquity to 500 CE twice. | S002, S008 |
| Eudoxus | 408 BCE to 355 BCE Disputed |
/juːˈdɒksəs/ yoo-DOK-sus |
Cnidus, Asia Minor (Greek) | Made the method of exhaustion rigorous and defined equal ratios in a way that survives as Euclid Book VAppears in Chapter 2 7 times, and in the timeline under Antiquity to 500 CE 3 times. | S001, S002, S008 |
 Archimedes Not from life |
287 BCE to 212 BCE | /ˌɑːrkɪˈmiːdiːz/ ar-kih-MEE-deez |
Syracuse, Sicily (Greek) | Squared the parabola, bounded pi with a 96-gon, and left a mechanical 'method' of discovery recovered only in 1906Appears in Chapter 2 24 times, Chapter 10 3 times and Chapter 11 3 times, and in the timeline under Antiquity to 500 CE 7 times, 500 to 1400 twice, 1800 to 1872, 1873 to 1913 and 1914 to the present 3 times. | S001, S002, S003 |
| Liu Hui | 220 to 280 Disputed |
/ljoʊ xweɪ/ lyoh HWAY |
Kingdom of Wei, China | Proved the results of the Nine Chapters in his 263 CE commentary, cut the circle to a 3072-gon for pi, and published his failure on the sphereAppears in Chapter 3 3 times, and in the timeline under Antiquity to 500 CE twice. | S020, S022, S023 |
| Zu Chongzhi | 429 to 501 | /tsù ʈʂʰʊ́ŋ ʈʂɻ̩́/ dzoo chong-JRR |
Jiankang (Nanjing), Liu Song China | Bounded pi between 3.1415926 and 3.1415927 and gave , unmatched for roughly 900 yearsAppears in Chapter 3 6 times, and in the timeline under Antiquity to 500 CE twice. | S021, S023 |
| Zu Gengzhi | 450 to 520 Disputed |
/tsu kɤŋ ʈʂɻ̩/ dzoo gung-JRR |
Liang dynasty China | Stated the equal-cross-sections principle and used it to derive the volume of a sphere, about 1100 years before CavalieriAppears in Chapter 3 3 times and Chapter 5, and in the timeline under Antiquity to 500 CE twice. | S021, S023 |
| Aryabhata I | 476 to 550 | /ˈaːrjɐbʱɐʈɐ/ AAR-yuh-bhuh-tuh |
Kusumapura (Pataliputra), India | Wrote the Aryabhatiya at 23, containing the first known table of sines and a pi value he explicitly labeled approximateAppears in Chapter 3 4 times, and in the timeline under Antiquity to 500 CE twice and 500 to 1400. | S050, S059, S060 |
| Brahmagupta | 598 to 668 | /ˈbrɐɦmɐɡuptɐ/ BRUH-muh-GOOP-tuh |
Rajasthan, India | Gave a second-order interpolation formula for sine tables and the first systematic arithmetic of zeroIn the timeline 500 to 1400 twice. | S050 |
 Li Chunfeng Not from life |
602 to 670 | /li tʂʰwən fɤŋ/ lee chwun-FUNG |
Tang dynasty China | His commentary on the Nine Chapters is the only reason Zu Gengzhi's sphere proof survivesAppears in Chapter 3 3 times. | S023 |
| al-Khwarizmi | 780 to 850 Disputed |
Arabic [al xwaːrizmiː] al-KHWAA-riz-mee (kh as in Scottish loch) |
Khwarazm and Baghdad | His name became "algorithm" and his book title al-jabr became "algebra"Named in the registers, not in the story or the timeline. | S463 |
| Thabit ibn Qurra | 836 to 901 | /ˈθaːbɪt ɪbn ˈqurra/ THAA-bit ib-n KOOR-ruh |
Harran and Baghdad | Computed areas and volumes of parabolas and paraboloids by upper and lower sums, dividing the interval into unequal partsAppears in Chapter 3, and in the timeline under 500 to 1400 twice. | S057 |
 Ibn al-Haytham Not from life |
965 to 1040 Disputed |
/ˈɪbn æl ˈhaɪθæm/ IB-n al-HY-tham |
Basra and Cairo | Summed fourth powers and used it to find the volume of a paraboloid as of its cylinder - a Riemann-sum computation around 1000 CEAppears in Chapter 3 8 times, and in the timeline under 500 to 1400 3 times and 1400 to 1637. | S054, S058 |
| Bhaskara II | 1114 to 1185 | /ˈbʱaːskɐrɐ/ BHAAS-kuh-ruh |
Bijjada Bida, India | Distinguished average from instantaneous velocity (tatkalika gati) around 1150, using the Rcosine as the rate of change of the Rsine (the sine and cosine scaled by the radius R)Appears in Chapter 3, and in the timeline under 500 to 1400 twice. | S050 |
| Sharaf al-Din al-Tusi | 1135 to 1213 Disputed |
/ˈʃarafʔadˈdiːn atˈtuːsiː/ SHA-raf ad-DEEN at-TOO-see |
Tus, Persia | Found the maximum of the cubic at and gave the condition for a positive root, around 1209Appears in Chapter 3 3 times, and in the timeline under 500 to 1400 twice. | S053 |
| Thomas Bradwardine | 1295 to 1349 Disputed |
/ˈbrædwərdiːn/ BRAD-wer-deen |
Oxford / Canterbury (England) | Bradwardine's rule: velocity grows arithmetically as the force-to-resistance ratio grows geometrically - an implicitly logarithmic law of motion in 1328Appears in Chapter 4 twice, and in the timeline under 500 to 1400 4 times. | S103, S104, S100 |
| Richard Kilvington | 1302 to 1361 Disputed |
/ˈkɪlvɪŋtən/ KIL-ving-tuhn |
Oxford (England) | Wrote the Sophismata before 1325, using calculatory techniques on motion and instantaneous speed, and prompted Bradwardine's ruleAppears in Chapter 4, and in the timeline under 500 to 1400 twice. | S102 |
| William Heytesbury | 1313 to 1372 Disputed |
/ˈheɪtsbəri/ HAYTS-bur-ee |
Oxford (England) | Gave the first clear statement of the mean speed theorem, in a 1335 logic textbook about how to solve sophismsAppears in Chapter 4 5 times, and in the timeline under 500 to 1400 twice. | S101, S104 |
 Nicole Oresme Not from life |
1320 to 1382 Disputed |
/ɔˈʁɛm/ oh-REM |
Normandy / Paris (France) | Proved the mean speed theorem geometrically by showing the area under a velocity graph is the distance traveled, and proved the harmonic series divergesAppears in Chapter 4 12 times, and in the timeline under 500 to 1400 6 times. | S100, S107, S108 |
| Madhava of Sangamagrama | 1340 to 1425 Disputed |
/ˈmaːdʱɐʋɐ/ MAAD-huh-vuh |
Sangamagrama, near Cochin, Kerala | Found infinite series for pi, sine, cosine and arctangent around 1400, with error terms, roughly 250 years before Newton and LeibnizAppears in Chapter 3, and in the timeline under 1400 to 1637 3 times. | S050, S051, S055 |
| Narayana Pandita | 1340 to 1400 Disputed |
/naːˈraːjɐɳɐ ˈpɐɳɖitɐ/ naa-RAA-yuh-nuh PUN-dih-tuh |
India | Gave the general formula for repeated summations (vara-sankalita), the combinatorial engine the Kerala series neededAppears in Chapter 3. | S050 |
| Parameshvara | 1380 to 1460 Disputed |
/pɐrɐˈmeːʃʋɐrɐ/ puh-ruh-MAY-shvuh-ruh |
Kerala, India | Madhava's pupil; sustained a fifty-year program of eclipse observation used to correct the modelsAppears in Chapter 3 5 times. | S050 |
| Nilakantha Somayaji | 1444 to 1550 Disputed |
/ˈniːlɐkɐɳʈʰɐ soːmɐˈjaːdʒiː/ NEE-luh-KUN-tuh soh-muh-YAA-jee |
Trikkantiyur, Kerala | Wrote the Tantrasamgraha and the Aryabhatiya-bhasya, argued that pi is incommensurable with the diameter, and summed infinite geometric seriesAppears in Chapter 3, and in the timeline under 1400 to 1637. | S050, S056 |
| Jyeshthadeva | 1500 to 1575 Disputed |
/ˈdʒjeːʂʈʰɐdeːʋɐ/ JYESHT-huh-DAY-vuh |
Kerala, India | Wrote the Yuktibhasa in Malayalam prose, setting out demonstrations (upapatti) of the Kerala series including the error termsAppears in Chapter 3 5 times, and in the timeline under 1400 to 1637 twice. | S050, S051 |
| Sankara Variyar | 1500 to 1560 Disputed |
/ˈʃɐŋkɐrɐ ˈʋaːrijɐr/ SHUN-kuh-ruh VAA-ree-yar |
Kerala, India | His commentaries Kriyakramakari and Yukti-dipika preserve the verses in which Madhava's series are statedIn the timeline 1400 to 1637. | S050 |
| Christopher Clavius | 1538 to 1612 | /ˈklɑːviəs/ KLAH-vee-us |
Germany / Rome | Taught Ricci at the Collegio Romano and headed the 1582 Gregorian calendar reform - the motive in the transmission argumentAppears in Chapter 3. | S051 |
| Matteo Ricci | 1552 to 1610 | /matˈtɛːo ˈrittʃi/ mat-TAY-oh REE-chee |
Italy / Goa / China | Mathematically trained Jesuit who reached Goa in 1578 with instructions to investigate Indian science - a central figure in the Kerala transmission argumentAppears in Chapter 3 3 times, and in the timeline under 1400 to 1637 twice. | S051 |
 Johannes Kepler |
1571 to 1630 | /joˈhanəs ˈkɛplɐ/ yo-HAH-nes KEP-luh |
Wuerttemberg / Prague / Linz | Computed volumes of 90-odd solids of revolution by summing infinitely thin slices (Nova stereometria, 1615), and made time proportional to swept area in his second lawAppears in Chapter 3, Chapter 5 18 times and Chapter 11 3 times, and in the timeline under 1400 to 1637 6 times. | S134, S151, S152, S154, S159 |
| Paul Guldin | 1577 to 1643 | /ˈpaʊl ˈɡʊldiːn/ POWL GOOL-deen |
Switzerland / Vienna / Graz | The leading Jesuit critic of indivisibles, arguing that there is never a ratio between one infinity and anotherAppears in Chapter 5 twice, and in the timeline under 1637 to 1665 twice. | S135, S156, S132 |
| Claude Gaspard Bachet | 1581 to 1638 | /klod ɡaspaʁ baʃˈɛ/ klohd gas-PAR bash-AY |
Bourg-en-Bresse (France) | His Latin Diophantus turned the Greek parisotes into adaequalitas by a morpheme-for-morpheme calque, handing Fermat the word 'adequality'Appears in Chapter 5. | S130 |
 Gregoire de Saint-Vincent |
1584 to 1667 | /ɡʁeɡwaʁ də sɛ̃ vɛ̃ˈsɑ̃/ greg-WAHR duh san van-SAHN |
Bruges / Ghent (Spanish Netherlands) | Proved (Opus geometricum, 1647) that equal ratios of abscissae under a hyperbola cut off equal areas - the functional equation of the logarithm, found geometricallyAppears in Chapter 5 8 times and Chapter 6 3 times, and in the timeline under 1637 to 1665 5 times. | S141, S151, S155 |
 Rene Descartes |
1596 to 1650 | /ʁəne deˈkaʁt/ ruh-NAY deh-KART |
Touraine (France) / Netherlands / Stockholm | La Geometrie (1637) turned curves into equations, making tangents and areas problems you can compute rather than constructAppears in Chapter 5 7 times, Chapter 6 4 times and Chapter 9, and in the timeline under 1637 to 1665 6 times and 1734 to 1799. | S148, S144, S151 |
 Bonaventura Cavalieri Not from life |
1598 to 1647 | /kavaˈljɛːri/ kah-vah-LYEH-ree |
Milan / Bologna (Italy) | Built the method of indivisibles (Geometria, 1635) and stated as Prop. II.4 what is now called Cavalieri's principleAppears in Chapter 3 4 times and Chapter 5 9 times, and in the timeline under Antiquity to 500 CE, 1400 to 1637 7 times and 1637 to 1665 twice. | S135, S156, S132, S151 |
 Pierre de Fermat |
1601 to 1665 Disputed |
/pjɛʁ də fɛʁˈma/ pyair duh fair-MAH |
Beaumont-de-Lomagne / Toulouse (France) | Invented the method of adequality for maxima, minima and tangents (c.1636) and quadratured y= by cutting the axis in geometric progressionAppears in Chapter 3 5 times, Chapter 5 12 times, Chapter 6 and Chapter 9 4 times, and in the timeline under 1400 to 1637 twice and 1637 to 1665 4 times. | S130, S144, S151 |
| Gilles Personne de Roberval | 1602 to 1675 | /ʒil pɛʁˈsɔn də ʁɔbɛʁˈval/ zheel pair-SUN duh roh-bair-VAL |
Paris (France) | Quadratured the cycloid before 1636 and found tangents by composing the velocities of the generating motionsAppears in Chapter 3 twice and Chapter 5 9 times, and in the timeline under 1400 to 1637 and 1637 to 1665 4 times. | S137, S131, S151 |
 Evangelista Torricelli |
1608 to 1647 | /torriˈtʃɛlli/ tor-ree-CHEL-lee |
Faenza / Florence (Italy) | Showed in 1641 (published 1644) that an infinitely long solid of revolution can have finite volume - the acute hyperbolic solidAppears in Chapter 5 9 times, and in the timeline under 1637 to 1665 11 times. | S131, S157 |
| Humphrey Babington | 1615 to 1691 Disputed |
/ˈhʌmfri ˈbæbɪŋtən/ HUM-free BAB-ing-tuhn |
England | Senior fellow of Trinity and rector of Boothby Pagnell, who housed Newton during the plague of 1665 and probably secured his place at Trinity.Appears in Chapter 6, and in the timeline under 1665 to 1687. | S190, S180 |
 John Wallis |
1616 to 1703 | /ˈwɒlɪs/ WOL-iss |
Kent / Oxford (England) | Arithmetica Infinitorum (1656) gave the infinite product for pi by interpolation, and De sectionibus conicis (1655) introduced the symbol infinityAppears in Chapter 5, and in the timeline under 1637 to 1665 4 times, 1665 to 1687 3 times and 1688 to 1734 3 times. | S145, S139, S151, S155 |
 Alphonse Antonio de Sarasa |
1618 to 1667 | /alˈfons anˈtonjo de saˈɾasa/ al-FONS an-TOH-nee-oh duh sah-RAH-sah |
Spanish Netherlands | Recognized in 1649 that Saint-Vincent's hyperbolic areas behave as logarithms, joining the hyperbola to the logarithm for the first time, in a tract written to defend his teacher against a challenge problem of Mersenne'sAppears in Chapter 5 7 times, and in the timeline under 1637 to 1665 twice. | S066, S141, S155 |
| Henry Oldenburg | c.1619 to 1677 Disputed |
German [ˈɔldn̩ˌbʊʁk]; English /ˈoʊldənbɜːrɡ/ OLE-den-boork (German) / OLE-den-berg (English) |
Bremen / London | First Secretary of the Royal Society; the postal channel through which Newton and Leibniz communicated in 1676, including the anagram letters.In the timeline 1665 to 1687 10 times. | S230, S188, S285 |
| Nicolaus Mercator | 1620 to 1687 | /mɛrˈkaːtɔr/ mer-KAH-tor |
Holstein / London / Paris | Logarithmotechnia (1668) printed the first infinite series for a logarithm, Appears in Chapter 5 3 times, and in the timeline under 1665 to 1687 twice. | S149, S155 |
| Rene-Francois de Sluse | 1622 to 1685 | /ʁəne fʁɑ̃swa də slyz/ ruh-NAY frahn-SWAH duh SLOOZ |
Visé / Liege (Prince-Bishopric of Liege) | Published a general rule for tangents to algebraic curves in the Philosophical Transactions, effectively implicit differentiation without calculusIn the timeline 1665 to 1687 twice. | S146 |
 Blaise Pascal |
1623 to 1662 | /blɛz pasˈkal/ BLEZ pahs-KAL |
Clermont-Ferrand / Paris (France) | The Traite des sinus du quart de cercle (1659) contains the characteristic triangle, the figure Leibniz later credited with opening up his calculusAppears in Chapter 5 10 times and Chapter 7 8 times, and in the timeline under 1637 to 1665 5 times. | S143, S153, S137 |
| John Collins | 1625 to 1683 | /dʒɒn ˈkɒlɪnz/ JON KOL-inz |
England | Kept the archive of English mathematical letters that Leibniz was shown in October 1676 and that became the Commercium Epistolicum in 1713.Appears in Chapter 6 8 times and Chapter 8, and in the timeline under 1665 to 1687 18 times. | S230, S189, S282 |
| Pietro Mengoli | 1626 to 1686 | /ˈpjɛːtro ˈmɛŋɡoli/ PYEH-troh MEN-goh-lee |
Bologna (Italy) | Proved the harmonic series diverges (1650), summed reciprocals of triangular numbers, and posed the Basel problem he could not solveAppears in Chapter 5 3 times and Chapter 9, and in the timeline under 1637 to 1665 twice. | S142, S155 |
 Johann Hudde |
1628 to 1704 | /ˈjoːhɑn ˈhʏdə/ YOH-hahn HUD-duh |
Amsterdam (Netherlands) | Hudde's rule (1657/1659) multiplies polynomial coefficients by an arithmetic progression to find repeated roots and extrema - algebraic differentiation before the calculusAppears in Chapter 5 4 times, and in the timeline under 1637 to 1665 twice. | S147 |
 Christiaan Huygens |
1629 to 1695 | Dutch [ˈkrɪstijaːn ˈhœyɣəns]; anglicised /ˈhaɪɡənz/ Dutch: KRIS-tee-yahn HOY-khuns (the g is a throaty kh). Anglicised: HY-genz. |
The Hague (Netherlands) / Paris | Horologium oscillatorium (1673) proved the cycloid tautochronous and introduced evolutes and involutes; he taught Leibniz his mathematics from 1672Appears in Chapter 5 3 times and Chapter 7 6 times, and in the timeline under 1637 to 1665 4 times, 1665 to 1687 4 times and 1688 to 1734 4 times. | S150, S141, S235, S237, S334, S315 |
 Isaac Barrow |
1630 to 1677 | /ˈaɪzək ˈbæroʊ/ EYE-zuk BARR-oh |
London / Cambridge (England) | Lectiones Geometricae (1670), Lecture X Prop. 11, gives the geometric form of the fundamental theorem of calculusAppears in Chapter 5 23 times and Chapter 6 9 times, and in the timeline under 1665 to 1687 12 times. | S140, S138, S158, S151 |
 Christopher Wren |
1632 to 1723 | /ˈkrɪstəfə rɛn/ KRIS-tuh-fer REN |
England | Rectified the cycloid in 1658, showing one arch has length exactly four times the diameter of its generating circleAppears in Chapter 5 6 times, and in the timeline under 1637 to 1665 3 times. | S064 |
 Robert Hooke Not from life |
1635 to 1703 | /ˈrɒbərt hʊk/ ROB-ert HUUK |
England | Curator of Experiments at the Royal Society; proposed to Newton in 1679 the decomposition of orbital motion into tangential motion plus central attraction, and claimed priority for the inverse-square law.Appears in Chapter 6 6 times, and in the timeline under 1665 to 1687 8 times. | S203, S197 |
 James Gregory |
1638 to 1675 | /ˈdʒeɪmz ˈɡrɛɡəri/ JAYMZ GREG-uh-ree |
Aberdeenshire / St Andrews / Edinburgh (Scotland) | Geometriae pars universalis (1668), the first systematic calculus textbook, proved that differentiation and integration are inverse operationsAppears in Chapter 5, and in the timeline under 1665 to 1687 13 times. | S136 |
 Isaac Newton |
1642 to 1727 | /ˈaɪzək ˈnjuːtən/ EYE-zuhk NEW-tuhn |
England | Invented the calculus of fluxions and fluents in 1664-66, founded classical mechanics in the Principia (1687), and spent more of his life on alchemy and anti-Trinitarian theology than on either.Appears in Chapter 1 3 times, Chapter 3 7 times, Chapter 5 14 times, Chapter 6 97 times, Chapter 7 5 times, Chapter 8 28 times, Chapter 9 13 times and Chapter 11 3 times, and in the timeline under 1637 to 1665 6 times, 1665 to 1687 58 times, 1688 to 1734 55 times, 1734 to 1799 4 times, 1800 to 1872 twice and 1914 to the present 3 times. | S180, S183, S184, S185, S186, S187, S188, S189, S190, S191, S192, S195, S196, S200 |
| Seki Takakazu | 1642 to 1708 Disputed |
Japanese [seki takakazɯ] SEH-kee tah-kah-KAH-tsoo |
Edo Japan (Kozuke and Edo) | First study of determinants (1683), ten years before Leibniz and in more general form; the Bernoulli numbers before Jacob BernoulliAppears in Chapter 11 3 times, and in the timeline under 1665 to 1687 twice and 1688 to 1734. | S463, S429, S430 |
| Otto Mencke | 1644 to 1707 | German [ˈɔto ˈmɛŋkə] OT-oh MENK-uh |
Leipzig | Founded the Acta Eruditorum at Leipzig in 1682 and edited it: the journal that printed the first calculus paper in 1684.In the timeline 1665 to 1687. | S231, S233 |
 Gottfried Wilhelm Leibniz |
1646 to 1716 | German [ˈlaɪpnɪts]; usual English /ˈlaɪbnɪts/ or /ˈlaɪpnɪts/ German: LYPE-nits (the b devoices to p, as in Leib). Common English: LYBE-nitz. Both are in use; the German is the one he answered to. |
Saxony / Hanover (Holy Roman Empire) | Invented the differential and integral calculus independently of Newton, and gave it the notation (dy/dx and integral-sign f(x)dx) the world still uses.Appears in Chapter 1 3 times, Chapter 3 4 times, Chapter 5 19 times, Chapter 6 3 times, Chapter 7 31 times, Chapter 8 23 times, Chapter 9 6 times, Chapter 10, Chapter 11 4 times and Chapter 12 8 times, and in the timeline under 1637 to 1665 4 times, 1665 to 1687 48 times, 1688 to 1734 25 times, 1734 to 1799, 1800 to 1872 and 1914 to the present. | S230, S231, S232, S233, S235, S236, S237, S241 |
| William Chaloner | 1650 to 1699 Disputed |
/ˈwɪljəm ˈtʃælənər/ WILL-yuhm CHAL-uh-ner |
England | The most audacious coiner in London, who twice petitioned Parliament claiming he could secure the Mint, and was caught, convicted and hanged through Isaac Newton's personal prosecution.Appears in Chapter 6 3 times, and in the timeline under 1688 to 1734 4 times. | S196 |
| Michel Rolle | 1652 to 1719 | French [miˈʃɛl ʁɔl] mee-SHEL ROLL |
France | Stated the theorem now named after him (1691) and then spent 1700-1705 attacking the infinitesimal calculus as "a collection of ingenious fallacies"Appears in Chapter 3, and in the timeline under 1688 to 1734 twice. | S459, S453 |
 Jacob Bernoulli |
1655 to 1705 | /ˈjaːkɔp bɛʁˈnʊli/ YAH-kop bair-NOO-lee |
Basel, Switzerland | Put the word 'integral' into print (May 1690), set the catenary challenge, founded the calculus of variations with his brachistochrone method, and wrote Ars ConjectandiIn the timeline 1688 to 1734 4 times. | S324, S328, S329, S330, S334, S322 |
 Hans Sloane |
1660 to 1753 | /hænz sloʊn/ HANZ SLOHN |
Ireland / London | Secretary of the Royal Society; the recipient of Leibniz's two formal complaints (4 March 1711 NS and 29 December 1711) demanding that Keill retract.Appears in Chapter 8, and in the timeline under 1688 to 1734 4 times. | S189 |
| Charles Montagu | 1661 to 1715 | /ˈmɒntəɡjuː/ MON-tuh-gyoo |
England | President of the Royal Society in 1697; Newton sent him the anonymous brachistochrone solution for publicationIn the timeline 1688 to 1734 twice. | S330, S321 |
 Guillaume de l'Hopital |
1661 to 1704 | /ɡijom də lopiˈtal/ gee-YOHM duh loh-pee-TAL |
Paris, France | Published the first calculus textbook, Analyse des infiniment petits (1696), containing the rule for that Johann Bernoulli had sold himAppears in Chapter 9 4 times, and in the timeline under 1688 to 1734 3 times and 1914 to the present. | S315, S316, S330 |
| Takebe Kataaki | 1661 to 1716 | /takebe kataaki/ TAH-keh-beh kah-tah-AH-kee |
Edo Japan | Wrote volumes 13-20 of the 20-volume Taisei sankei (1710), completed after Seki's deathIn the timeline 1688 to 1734 twice. | S430 |
| Nicolas Fatio de Duillier | 1664 to 1753 | French [nikɔla faˈtjo də dɥiˈlje] nee-koh-LAH fa-TYOH duh dwee-LYAY |
Geneva / London | In his 1699 Lineae brevissimi descensus investigatio geometrica duplex he first put in print the claim that Newton was first inventor and Leibniz at best a second; the opening move of the dispute.Appears in Chapter 6 and Chapter 8, and in the timeline under 1688 to 1734 twice. | S284 |
| Takebe Katahiro | 1664 to 1739 | /takebe kataçiɾo/ TAH-keh-beh kah-tah-HEE-roh |
Edo Japan | Series for in the sagitta - in modern terms the Taylor series of arcsin-squared - and a convergence-acceleration method equivalent to Romberg's, used to get pi to 40+ places in 1722Appears in Chapter 11 twice, and in the timeline under 1688 to 1734 4 times. | S430, S429 |
| John Harris | 1666 to 1719 Disputed |
/dʒɒn ˈhærɪs/ JON HARR-iss |
England | Compiler of the Lexicon Technicum (1704-1710), the first English technical encyclopedia, and translator of Newton's De quadratura curvarum into English in 1710.Named in the registers, not in the story or the timeline. | S187 |
 Johann Bernoulli |
1667 to 1748 | /ˈjoːhan bɛʁˈnʊli/ YOH-hahn bair-NOO-lee |
Basel and Groningen | Taught the calculus to L'Hopital under a contract of exclusivity, posed the brachistochrone in 1696, named the integral calculus, and taught EulerAppears in Chapter 6, Chapter 7 and Chapter 9 4 times, and in the timeline under 1688 to 1734 15 times, 1734 to 1799, 1800 to 1872 twice and 1914 to the present twice. | S315, S316, S324, S330, S331, S333, S337 |
 Guido Grandi |
1671 to 1742 | /ˈɡwiːdo ˈɡrandi/ GWEE-doh GRAHN-dee |
Pisa, Italy | Named the curve versoria (Latin, the rope that turns a sail) and la versiera (Italian) in 1718, the name Agnesi used and Colson mistranslatedAppears in Chapter 9 4 times, and in the timeline under 1688 to 1734 twice. | S327 |
| John Keill | 1671 to 1721 | /kiːl/ KEEL |
Scotland / Oxford | Wrote the September-October 1708 sentence in Philosophical Transactions accusing Leibniz of republishing Newton's method "with the name and manner of notation changed", the shot that started the war.Appears in Chapter 8 7 times, and in the timeline under 1688 to 1734 6 times. | S189, S283 |
| Samuel Clarke | 1675 to 1729 | /ˈsæmjuəl klɑːrk/ SAM-yoo-ul KLARK |
England | Newton's proxy in the 1715-16 correspondence with Leibniz on space, time and God; published the exchange in 1717 after Leibniz's death.In the timeline 1688 to 1734 5 times. | S240, S189 |
 William Jones |
1675 to 1749 | /ˈwɪljəm dʒoʊnz/ WIL-yum JOHNZ |
Wales and London | First to use the symbol pi for the circumference-to-diameter ratio, in Synopsis palmariorum mathesios (1706), forty-two years before Euler made it standardAppears in Chapter 6 22 times and Chapter 8 5 times, and in the timeline under 1688 to 1734 15 times and 1734 to 1799 twice. | S325, S451, S462 |
 Charles Hayes |
1678 to 1760 | /tʃɑːlz heɪz/ charlz HAYZ |
London, England | Wrote A Treatise of Fluxions (1704), the first book in English on the calculusAppears in Chapter 12 4 times, and in the timeline under 1688 to 1734 twice. | S439 |
 Jean-Jacques Dortous de Mairan |
1678 to 1771 | /ʒɑ̃ ʒak dɔʁtu də mɛˈʁɑ̃/ zhahn zhahk dor-TOO duh meh-RAHN |
Paris, France | Dedicatee of Euler's Introductio and du Chatelet's opponent in the vis viva disputeAppears in Chapter 9 3 times, and in the timeline under 1734 to 1799. | S318, S332 |
 John Colson |
1680 to 1760 | /dʒɒn ˈkoʊlsən/ JON KOHL-suhn |
England | Lucasian Professor at Cambridge; annotated Newton's Method of Fluxions (1736) and translated Agnesi into English, coining 'the Witch of Agnesi' by misreading versiera as avversieraAppears in Chapter 6 10 times and Chapter 9 8 times, and in the timeline under 1688 to 1734 twice and 1734 to 1799 4 times. | S185, S192, S320, S311, S327 |
| Caroline of Ansbach | 1683 to 1737 | German [ˈkaʁoliːnə fɔn ˈansbax]; English /ˈkærəlaɪn/ KA-roh-lee-nuh fon AHNS-bakh (German); KARR-uh-line (English) |
Ansbach / Hanover / London | Leibniz's former pupil; as Princess of Wales she carried every paper of the Leibniz-Clarke correspondence between the two men.In the timeline 1688 to 1734. | S240 |
 James Jurin |
1684 to 1750 | /ˈdʒʊərɪn/ JOOR-in |
England | Replied to Berkeley as 'Philalethes Cantabrigiensis' in Geometry No Friend to Infidelity (1734)Appears in Chapter 9 3 times, and in the timeline under 1734 to 1799 twice. | S310, S311 |
 Brook Taylor |
1685 to 1731 | /brʊk ˈteɪlər/ BRUUK TAY-lur |
England | Methodus Incrementorum Directa et Inversa (1715), Prop. VII Cor. 2: the series named after him, which he had by July 1712Appears in Chapter 5, Chapter 8 4 times, Chapter 9 15 times and Chapter 12, and in the timeline under 1665 to 1687 twice, 1688 to 1734 7 times and 1914 to the present. | S312, S331 |
 George Berkeley |
1685 to 1753 | /ˈbɑːrkli/ BARK-lee (not BURK-lee) |
Ireland and England | Wrote The Analyst (1734), showing that the calculus rested on reasoning nobody could defend, and thereby set the agenda for a century of foundational workAppears in Chapter 5, Chapter 6 3 times, Chapter 9 11 times, Chapter 10 4 times and Chapter 12, and in the timeline under 1734 to 1799 4 times. | S310, S311, S335 |
 William Stukeley |
1687 to 1765 | /ˈwɪljəm ˈstjuːkli/ WILL-yuhm STEW-klee |
England | Knew Newton personally and wrote the 1752 Memoirs that preserve the only first-hand record of Newton telling the apple story himself.Appears in Chapter 6. | S180, S190 |
| John Conduitt | 1688 to 1737 | /dʒɒn ˈkɒndɪt/ JON KON-dit |
England | Newton's half-niece's husband and his successor as Master of the Mint; collected the recollections that became the basis of every Newton biography.In the timeline 1637 to 1665. | S181, S182, S191 |
 James Stirling |
1692 to 1770 | /ˈstɜːrlɪŋ/ STUR-ling |
Scotland | Published in Lineae Tertii Ordinis (1717, p. 32) the expansion now universally called Maclaurin's series, twenty-five years before Maclaurin's FluxionsAppears in Chapter 9 3 times, and in the timeline under 1688 to 1734 twice. | S312, S311, S453 |
| Andrew Motte | 1696 to 1734 Disputed |
/ˈændruː mɒt/ AN-droo MOT |
England | Made the 1729 English translation of the Principia that almost every English reader has used ever since.Named in the registers, not in the story or the timeline. | S186 |
 Colin Maclaurin |
1698 to 1746 | /məˈklɔːrɪn/ muh-KLOR-in |
Scotland | Answered Berkeley with A Treatise of Fluxions (1742), the century's most serious attempt at rigorous foundations, and refused to use infinitesimals at allAppears in Chapter 9 9 times, and in the timeline under 1688 to 1734, 1734 to 1799 twice and 1914 to the present twice. | S311, S312, S335 |
 Daniel Bernoulli |
1700 to 1782 | /ˈdaːni̯eːl bɛʁˈnʊli/ DAH-nee-el bair-NOO-lee |
Basel and St Petersburg | Wrote Hydrodynamica (1738), founding fluid dynamics, and was rewarded by his father with a rival book carrying an earlier dateAppears in Chapter 9, and in the timeline under 1734 to 1799 twice. | S333, S336, S341, S326 |
 Emilie du Chatelet |
1706 to 1749 | /emiˈli dy ʃatˈlɛ/ ay-mee-LEE dew shat-LAY |
Paris and Cirey, France | Made the only complete French translation of Newton's Principia, with her own analytical commentary, and argued the Leibnizian measure of force that became kinetic energyAppears in Chapter 9 3 times, and in the timeline under 1734 to 1799 7 times. | S332 |
| Benjamin Robins | 1707 to 1751 | /ˈrɒbɪnz/ ROB-inz |
England | Wrote A Discourse Concerning ... Newton's Methods of Fluxions (1735), the most mathematically careful reply to BerkeleyAppears in Chapter 9, and in the timeline under 1734 to 1799 twice. | S310, S311 |
 Leonhard Euler |
1707 to 1783 | /ˈleːɔnhaʁt ˈɔɪlɐ/ LAY-on-hart OY-ler (never YOO-ler) |
Basel, St Petersburg, Berlin | Solved the Basel problem at 28, made analysis a science of functions, standardised f(x), e, pi and i, and produced almost half his work after going blindAppears in Chapter 9 17 times, Chapter 11 and Chapter 12 4 times, and in the timeline under 1688 to 1734 4 times, 1734 to 1799 23 times and 1800 to 1872. | S318, S322, S323, S325, S326, S313, S314, S333 |
| Thomas Simpson | 1710 to 1761 | /ˈtɒməs ˈsɪmpsən/ TOM-us SIMP-sun |
UK | The rule named after him is Newton's (and Kepler's, and Cavalieri's, and Gregory's); the method named after Newton and Raphson is, in the form taught, his (1740)In the timeline 1734 to 1799 twice. | S462, S453 |
| Frederick II of Prussia | 1712 to 1786 | /ˈfriːdrɪç/ (Friedrich) FREED-rikh |
Berlin, Prussia | Brought Euler to Berlin in 1741 and undervalued him for twenty-five years; B. H. Brown, who traced the anecdote in 1942, suggests he or his courtiers invented the Diderot storyIn the timeline 1734 to 1799. | S313, S314, S326 |
 Alexis-Claude Clairaut |
1713 to 1765 | /alɛksi klod klɛˈʁo/ ah-lek-SEE klohd kleh-ROH |
Paris, France | Du Chatelet's mathematics mentor: he checked the calculations in her Principia commentary and, ten years after her death, saw the complete translation through the press in 1759Appears in Chapter 9 4 times, and in the timeline under 1734 to 1799 twice. | S068, S332 |
 Denis Diderot |
1713 to 1784 | /dəni didˈʁo/ duh-NEE dee-duh-ROH |
Paris and St Petersburg | Editor of the Encyclopedie and author of five creditable mathematical memoirs; the alleged victim of an anecdote that never happenedAppears in Chapter 9 6 times, and in the timeline under 1734 to 1799 twice and 1800 to 1872 twice. | S313, S314 |
| Maria Gaetana Agnesi | 1718 to 1799 | /maˈriːa ɡaeˈtaːna aɲˈɲeːzi/ mah-REE-ah gah-eh-TAH-nah ah-NYEH-zee |
Milan, Italy | Wrote Instituzioni analitiche (1748), the first surviving mathematics book by a woman and the first textbook to treat differential and integral calculus together; first woman appointed to a chair of mathematics (Bologna, 1750)Appears in Chapter 9 11 times, and in the timeline under 1734 to 1799 6 times. | S319, S320, S327, S340 |
| John Landen | 1719 to 1790 | /ˈdʒɒn ˈlændən/ JON LAN-dun |
UK | Invented an entire rival notation for the derivative in 1758 that nobody ever usedNamed in the registers, not in the story or the timeline. | S451 |
 Dieudonne Thiebault |
1733 to 1807 | /djødɔˈne tjeˈbo/ dyuh-doh-NAY tyay-BOH |
Berlin and Paris | Sole original source of the Euler-Diderot anecdote, which he told without naming Euler and expressly declined to vouch forAppears in Chapter 9 10 times, and in the timeline under 1734 to 1799 and 1800 to 1872. | S313, S314, S339 |
 Joseph-Louis Lagrange |
1736 to 1813 | /ʒozɛf lwi laˈɡʁɑ̃ʒ/ zhoh-ZEF lwee lah-GRAHNZH |
Turin, Berlin, Paris | Tried to found the calculus on power series with no infinitesimals and no limits (1797), and in doing so gave us the word 'derivative' and the notation f'(x)Appears in Chapter 9 5 times, and in the timeline under 1734 to 1799 5 times and 1800 to 1872. | S317, S324, S325 |
| Nicolas de Condorcet | 1743 to 1794 | /nikɔla də kɔ̃dɔʁˈsɛ/ nee-koh-LAH duh kohn-dor-SAY |
Paris, France | First user of the 'curly d' partial-derivative symbol, 1770; also the Enlightenment's leading advocate of women's educationIn the timeline 1734 to 1799 twice. | S325 |
 Pierre-Simon Laplace |
1749 to 1827 | French [pjɛʁ simɔ̃ laˈplas] pyair see-MON lah-PLASS |
France | His name is on the Laplacian, the operator Maxwell wanted to call "concentration", and on the Laplace transformAppears in Chapter 11 3 times, and in the timeline under 1800 to 1872 twice. | S451, S453 |
| Simon L'Huilier | 1750 to 1840 | French [simɔ̃ lɥiˈlje] see-MON lwee-LYAY |
Geneva, Switzerland | First person to write "lim." for a limit (1786), in the prize essay the Berlin Academy set on the nature of infinityIn the timeline 1734 to 1799 twice. | S451, S139 |
| Adrien-Marie Legendre | 1752 to 1833 | /adʁijɛ̃ maʁi ləˈʒɑ̃dʁ/ ah-dree-AN mah-REE luh-ZHAHNDR |
Paris, France | First used the partial-derivative symbol in the modern form in 1786, in a memoir on maxima and minima in the calculus of variationsIn the timeline 1734 to 1799 4 times and 1800 to 1872. | S325 |
| Louis Arbogast | 1759 to 1803 | French [lwi aʁbɔˈɡast] lwee ar-boh-GAST |
Alsace, France | Introduced D as the operator of differentiation and for its inverse, in Du calcul des derivations (1800)In the timeline 1800 to 1872 twice. | S451, S139, S462 |
 Fourier |
1768 to 1830 | /fuʁˈje/ foor-YAY |
Auxerre and Paris, France | His 1822 Theorie analytique de la chaleur claimed that arbitrary functions have trigonometric expansions, which broke the existing idea of 'function' and forced the rigorisation of analysis.Appears in Chapter 10 8 times, and in the timeline under 1800 to 1872 11 times. | S383, S391, S379 |
 Sophie Germain Not from life |
1776 to 1831 | /sɔ.fi ʒɛʁ.mɛ̃/ so-FEE zher-MAN |
Paris, France | Won the Institut de France elastic-surfaces prize (third round, 1815) and produced the most important result on Fermat's Last Theorem between 1738 and KummerAppears in Chapter 11, and in the timeline under 1734 to 1799 twice and 1800 to 1872 4 times. | S410 |
 Mary Somerville |
1780 to 1872 | /ˈsʌməvɪl/ SUM-uh-vil |
Scotland and England | Turned a commission to translate Laplace into Mechanism of the Heavens (1831), which Whewell and Peacock introduced into the Cambridge course of studyAppears in Chapter 11 5 times, and in the timeline under 1800 to 1872 7 times. | S412, S413, S414, S411 |
 Bolzano |
1781 to 1848 | /bolˈtsaːno/ bol-TSAH-noh |
Prague, Bohemia (Habsburg Empire) | Proved the intermediate value theorem and defined continuity arithmetically in 1817, four years before Cauchy, and built a continuous nowhere-differentiable function around 1830.Appears in Chapter 10 10 times, and in the timeline under 1800 to 1872 6 times and 1914 to the present 6 times. | S363, S370, S371, S384, S386, S387, S369, S360 |
 David Brewster |
1781 to 1868 | /ˈdeɪvɪd ˈbruːstər/ DAY-vid BROO-stur |
Scotland | In The Life of Sir Isaac Newton (1831, p. 194) rendered Bernoulli's remark as the Latin 'tanquam ex ungue leonem', the form now universally quotedAppears in Chapter 9 5 times, and in the timeline under 1688 to 1734 and 1800 to 1872 twice. | S321, S337 |
 Cauchy |
1789 to 1857 | /koˈʃi/ koh-SHEE |
Paris, France | Defined limit, continuity, derivative, convergence and (in 1823) the definite integral as a limit of sums, and proved the fundamental theorem in something close to the modern way.Appears in Chapter 9 4 times, Chapter 10 18 times and Chapter 11 3 times, and in the timeline under 1800 to 1872 17 times. | S361, S362, S360, S367, S368, S372, S369 |
 Charles Babbage |
1791 to 1871 | /ˈbæbɪdʒ/ BAB-ij |
England | Co-founded the Analytical Society at Cambridge (1812) to replace Newton's dots with Leibniz's d, and coined "pure D-ism in opposition to the Dot-age of the University".Appears in Chapter 8 6 times and Chapter 11, and in the timeline under 1800 to 1872 5 times. | S239 |
| George Peacock | 1791 to 1858 | /ˈpiːkɒk/ PEE-kok |
England | Analytical Society founder member; as a Cambridge examiner he put Leibnizian notation into the Tripos, which is what made the change stick.Appears in Chapter 8 and Chapter 11 twice, and in the timeline under 1800 to 1872 3 times. | S239 |
 John Herschel |
1792 to 1871 | /ˈhɜːrʃəl/ HUR-shul |
England | Co-founder of the Analytical Society; with Babbage wrote the whole of its 1813 Memoirs.Appears in Chapter 8, and in the timeline under 1800 to 1872 4 times. | S239 |
| Charles Whish | 1794 to 1833 | /wɪʃ/ WISH |
Britain / Malabar, India | East India Company civil servant whose paper, published in 1834-35 after his death, gave the first European account of the Kerala infinite seriesAppears in Chapter 3, and in the timeline under 1800 to 1872 twice. | S051, S052 |
 William Whewell |
1794 to 1866 | /ˈhjuːəl/ HYOO-uhl |
Cambridge, England | Wrote the 1834 review in which the word "scientist" first appears in print; co-adopter of Somerville's book into the Cambridge courseAppears in Chapter 11 6 times, and in the timeline under 1800 to 1872 4 times. | S411, S412, S442 |
| Pierre Frederic Sarrus | 1798 to 1861 | French [pjɛʁ fʁedeˈʁik saˈʁys] pyair fray-day-REEK sa-ROOS |
France | Introduced the evaluation bar [F(x)] with limits, in 1823In the timeline 1800 to 1872 twice. | S451, S139 |
 Abel |
1802 to 1829 | /ˈɑːbl̩/ AH-bel |
Norway | Published in 1826 the counterexample sin(x) - ()sin(2x) + ()sin(3x) - ... that broke Cauchy's theorem on sums of continuous functions, and rigorised the binomial series.Appears in Chapter 10 22 times and Chapter 11, and in the timeline under 1800 to 1872 10 times. | S364, S385, S372 |
| Carl Gustav Jacob Jacobi | 1804 to 1851 | German [ˈkaʁl ˈɡʊstaf ˈjaːkɔp jaˈkoːbi] karl GOOS-tahf YAH-kop yah-KOH-bee |
Prussia (Potsdam, Konigsberg, Berlin) | Revived the curly d for partial derivatives in 1841 and made it stickIn the timeline 1800 to 1872 twice. | S451, S139 |
| Viktor Bunyakovsky | 1804 to 1889 | Russian [bʊnʲɪˈkofskʲɪj] boo-nyih-KOFF-skee |
Russian Empire (St Petersburg) | Published the integral form of the Cauchy-Schwarz inequality in 1859, twenty-six years before SchwarzIn the timeline 1800 to 1872 twice. | S457, S458 |
| Dirichlet | 1805 to 1859 | /diʁiˈkleː/ dee-ree-KLAY |
Rhineland, Berlin and Goettingen | Gave in 1837 the modern definition of a function as an arbitrary rule of assignment, and the first rigorous convergence theorem for Fourier series.Appears in Chapter 10 3 times, and in the timeline under 1800 to 1872 twice. | S379, S391, S383 |
| William Rowan Hamilton | 1805 to 1865 | /ˈwɪljəm ˈroʊən ˈhæmɪltən/ WIL-yum ROH-un HAM-il-tun |
Ireland | Introduced the vector differential operator, first in the modern orientation and then rotated to avoid clashing with other uses of the symbolNamed in the registers, not in the story or the timeline. | S139, S451 |
 Augustus De Morgan |
1806 to 1871 | /ˈɔːɡəstəs də ˈmɔːrɡən/ AW-gus-tus duh MOR-gun |
England | In A Budget of Paradoxes (1872) named Euler, added 'algebra was Hebrew to Diderot', and miscopied Thiebault's z as n, creating the version everyone repeatsAppears in Chapter 5 3 times, Chapter 8 3 times and Chapter 9 9 times, and in the timeline under 1800 to 1872 twice. | S313, S314 |
| Luigi Menabrea | 1809 to 1896 | /luˈiːdʒi me.naˈbrɛː.a/ loo-EE-jee meh-nah-BREH-ah |
Piedmont, Italy | Wrote the 1842 Sketch of the Analytical Engine that Lovelace translated and annotatedIn the timeline 1800 to 1872 twice. | S415 |
 Galois |
1811 to 1832 | /ɡalˈwa/ gal-WAH |
Paris, France | He appears here only to correct the record. The memoir that vanished was the February 1830 Grand Prize submission, which was with Fourier, who died on 16 May 1830.Appears in Chapter 10 3 times, and in the timeline under 1800 to 1872 4 times. | S380, S372, S383 |
| James Joseph Sylvester | 1814 to 1897 | /ˈsɪlvɪstər/ SIL-vis-ter |
UK and USA | Coined the word "matrix" in 1850, meaning the womb from which determinants are bornNamed in the registers, not in the story or the timeline. | S453 |
 Ada Lovelace |
1815 to 1852 | /ˈeɪdə ˈlʌvleɪs/ AY-duh LUV-layss |
London, England | Note G of the 1843 Sketch: a looping, recurrence-based algorithm for the Bernoulli numbers with a full trace table, plus the first published speculation on symbolic differentiation by machineAppears in Chapter 11 3 times, and in the timeline under 1800 to 1872 twice. | S415, S416 |
 Weierstrass |
1815 to 1897 | /ˈvaɪ.ɐʃtʁaːs/ VY-er-shtrahss |
Westphalia and Berlin, Prussia | Settled the epsilon-delta definitions in his Berlin lectures and presented on 18 July 1872 a function continuous everywhere and differentiable nowhere.Appears in Chapter 10 13 times and Chapter 11 3 times, and in the timeline under 1800 to 1872 11 times, 1873 to 1913 3 times and 1914 to the present 4 times. | S373, S369, S386, S388, S389, S360 |
 Cellerier |
1818 to 1889 | /selerˈje/ sel-ay-RYAY |
Geneva, Switzerland | Independently constructed a continuous nowhere-differentiable function around 1860; published posthumously in 1890.Appears in Chapter 10 4 times, and in the timeline under 1873 to 1913 twice. | S386 |
 Hermite |
1822 to 1901 | /ɛʁˈmit/ air-MEET |
Paris, France | Wrote to Stieltjes on 20 May 1893 that he turned away 'avec effroi et horreur' from the lamentable plague of continuous functions with no derivatives.Appears in Chapter 10 5 times, and in the timeline under 1873 to 1913 twice. | S365, S388 |
| Kronecker | 1823 to 1891 | /ˈkʁoːnɛkɐ/ KROH-nek-er |
Berlin, Prussia | Opposed Cantor's transfinite mathematics on constructivist grounds and delayed publication of his 1877 dimension paper.Appears in Chapter 10 3 times, and in the timeline under 1873 to 1913 twice. | S376, S390 |
 Riemann |
1826 to 1866 | /ˈʁiːman/ REE-mahn |
Hanover, Germany | Defined the integral by arbitrary partitions and arbitrary sample points in his 1854 Habilitationsschrift on trigonometric series, and reinvented geometry in his 1854 Habilitation lecture.Appears in Chapter 3 3 times and Chapter 10 18 times, and in the timeline under 1800 to 1872 7 times and 1873 to 1913. | S391, S375, S386 |
 Dedekind |
1831 to 1916 | /ˈdeːdəkɪnt/ DAY-duh-kint |
Braunschweig and Zurich | Constructed the real numbers by cuts, an idea he dates precisely to 24 November 1858 while preparing his first calculus lectures at Zurich.Appears in Chapter 10 6 times, and in the timeline under 1800 to 1872 7 times and 1873 to 1913 twice. | S366, S382, S369 |
| James Clerk Maxwell | 1831 to 1879 | /ˈdʒeɪmz ˈklɑːrk ˈmækswɛl/ (Clerk is pronounced CLARK) JAYMZ KLARK MAKS-wel |
Scotland | Coined curl, convergence (now divergence), slope (now gradient) and concentration (the Laplacian), in one letter to Tait on 7 November 1870In the timeline 1800 to 1872 twice. | S454, S453 |
| Peter Guthrie Tait | 1831 to 1901 | /ˈpiːtər ˈɡʌθri teɪt/ PEE-ter GUTH-ree TAYT |
Scotland | Developed and named the nabla operator; established the inverted-delta symbol in his Quaternions (1867)In the timeline 1800 to 1872 4 times. | S454, S139 |
| du Bois-Reymond | 1831 to 1889 | /dy bwa ʁɛˈmɔ̃/ dyoo bwah ray-MOHN |
Germany | Published Weierstrass's nowhere-differentiable function in 1875, three years after Weierstrass presented it and never published it himself.Appears in Chapter 10 3 times, and in the timeline under 1873 to 1913 twice. | S386, S368 |
| Meray | 1835 to 1911 | /meˈʁɛ/ may-RAY |
Dijon, France | Published in 1869 the first coherent and rigorous theory of irrational numbers to appear in print, three years before Dedekind, Cantor and Heine.Appears in Chapter 10, and in the timeline under 1800 to 1872 twice. | S378 |
 Gaston Darboux |
1842 to 1917 | French [ɡasˈtɔ̃ daʁˈbu] gas-TON dar-BOO |
France | His 1875 memoir gave the upper-and-lower-sum construction of the integral that every textbook calls the Riemann integralAppears in Chapter 10 5 times, and in the timeline under 1873 to 1913 twice. | S453 |
| Hermann Schwarz | 1843 to 1921 | German [ˈhɛʁman ʃvaʁts] HAIR-mahn SHVARTS |
Germany (Berlin, Gottingen) | His 1885 paper is the source of the proof of the inequality now taughtNamed in the registers, not in the story or the timeline. | S458 |
 Cantor |
1845 to 1918 | /ˈkantoːɐ/ KAN-tor |
Halle, Germany | Built the reals from fundamental sequences in 1872, proved the reals uncountable in 1874, and gave the diagonal argument in 1891.Appears in Chapter 5 and Chapter 10 9 times, and in the timeline under 1800 to 1872 5 times and 1873 to 1913 11 times. | S376, S369, S366 |
| William Robertson Smith | 1846 to 1894 | /ˈrɒbərtsən smɪθ/ ROB-ert-sun SMITH |
Scotland | Suggested the name "nabla" to Tait, after the Assyrian harp the symbol resemblesAppears in Chapter 6 twice, and in the timeline under 1800 to 1872 twice. | S454, S453 |
| Horace Lamb | 1849 to 1934 | /ˈhɒrɪs læm/ HORR-iss LAM |
UK | Coined "gradient" in 1897 - for the slope of a plane curve, not for the vectorNamed in the registers, not in the story or the timeline. | S453 |
 Kovalevskaya |
1850 to 1891 | /kəvɐˈlʲefskəjə/ kuh-vuh-LYEF-skuh-yuh |
Russia and Sweden | Weierstrass's private pupil because Berlin would not enrol a woman; the Cauchy-Kovalevskaya theorem; doctorate from Goettingen in absentia in 1874; full professor at Stockholm in June 1889.Appears in Chapter 11, and in the timeline under 1873 to 1913 5 times. | S374, S373 |
| Henri Poincare | 1854 to 1912 | French [ʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe] zhool on-REE pwan-ka-RAY |
France | Named Fuchsian and Kleinian functions after other people, over Klein's objections, and defended the choice in printNamed in the registers, not in the story or the timeline. | S453 |
| Johan Ludvig Heiberg | 1854 to 1928 | /ˈjoːan ˈluðvi ˈhɑjbɐɡ/ YOH-ahn LOOD-vee HY-berg |
Denmark | Followed up an 1899 catalog notice, traveled to Constantinople in 1906, and read Archimedes' lost Method out of a prayer bookAppears in Chapter 2 3 times, and in the timeline under 1873 to 1913 twice. | S002, S003, S006 |
| Stieltjes | 1856 to 1894 | /ˈstiːltjəs/ STEELT-yuhs |
Netherlands and Toulouse, France | Recipient of Hermite's famous letter; his integral generalises Riemann's by integrating against an arbitrary increasing function.In the timeline 1873 to 1913 twice. | S365 |
| Peano | 1858 to 1932 | /peˈaːno/ peh-AH-noh |
Turin, Italy | Introduced the set-membership symbol in 1889 as an abbreviation for the Latin 'est', plus union, intersection and the existential quantifier.In the timeline 1873 to 1913 twice. | S393 |
| Florian Cajori | 1859 to 1930 | anglicised /kəˈdʒɔːri/; Swiss-Italian [kaˈdʒoːri] kuh-JOR-ee (as he was called in America) |
Graubunden, Switzerland; then USA | Wrote the two-volume History of Mathematical Notations (1928-29), still the only place most calculus symbols have ever been traced to their first sentenceAppears in Chapter 5, Chapter 7 and Chapter 9 3 times, and in the timeline under 1637 to 1665 twice. | S450, S451, S456 |
| William Henry Young | 1863 to 1942 | /ˈwɪljəm jʌŋ/ WIL-yum YUNG |
England and Switzerland | Co-author with Grace Chisholm Young of 220 papers and of The Theory of Sets of PointsIn the timeline 1873 to 1913. | S419 |
| Grace Chisholm Young | 1868 to 1944 | /ɡreɪs ˈtʃɪzəm jʌŋ/ grayss CHIZ-uhm YUNG |
England, Germany, Switzerland | Gottingen doctorate 1895; co-author of The Theory of Sets of Points (1906); independent work on infinite derivatives, 1914-1916Appears in Chapter 11, and in the timeline under 1873 to 1913 twice. | S419 |
| J. M. Child | 1871 to 1960 Disputed |
/tʃaɪld/ CHYLD |
England | Translated Leibniz's Paris manuscripts into English (1920), making the dated notebooks readable outside Germany.Appears in Chapter 5. | S230 |
| John Gaston Leathem | 1871 to 1923 | /ˈdʒɒn ˈɡæstən ˈliːðəm/ JON GAS-tun LEE-thum |
Ireland and Cambridge, UK | Introduced the arrow for "tends to" in 1905, in a tract on volume and surface integralsIn the timeline 1873 to 1913 3 times. | S451, S139 |
| Lebesgue | 1875 to 1941 | /ləˈbɛɡ/ luh-BEG |
France | Announced his integral in a Comptes Rendus note of 29 April 1901 and set it out in his 1902 thesis Integrale, longueur, aire.Appears in Chapter 10 8 times, and in the timeline under 1873 to 1913 4 times. | S377 |
| Thomas Bromwich | 1875 to 1929 | /ˈbrɒmɪdʒ/ (the w is silent, as in Greenwich) BROM-ij |
UK | With Hardy, spread Leathem's arrow notation in 1908; Hardy's Preface credits him by nameIn the timeline 1873 to 1913 3 times. | S451, S139 |
| G. H. Hardy | 1877 to 1947 | /ˈɡɒdfri ˈhærəld ˈhɑːrdi/ GOD-free HARR-uld HAR-dee |
UK | His Course of Pure Mathematics (1908) made the arrow standard and argued, on teaching grounds, that x = infinity should never be writtenAppears in Chapter 11 twice, and in the timeline under 1873 to 1913 4 times and 1914 to the present. | S451, S139 |
| Jasek | 1879 to 1945 | /ˈjaʃɛk/ YAH-shek |
Bohemia, Czechoslovakia | Found Bolzano's Functionenlehre manuscript in the National Library in Vienna in 1920 and announced it in a lecture on 3 December 1921.Appears in Chapter 10 4 times, and in the timeline under 1914 to the present 4 times. | S386 |
 Emmy Noether |
1882 to 1935 | German [ˈnøːtɐ] NUR-ter (round the lips as if for "oo") |
Erlangen and Gottingen, Germany; Bryn Mawr, USA | Noether's theorem (1918): every continuous symmetry of a physical system corresponds to a conservation lawAppears in Chapter 11 9 times, and in the timeline under 1914 to the present 4 times. | S463, S418 |
 John Maynard Keynes |
1883 to 1946 | /dʒɒn ˈmeɪnərd keɪnz/ JON MAY-nerd KAYNZ |
England | Bought back about half the Newton papers dispersed at auction in 1936, read the alchemy, and called Newton 'the last of the magicians' in a 1946 lecture read posthumously.Appears in Chapter 6 5 times, and in the timeline under 1637 to 1665 and 1914 to the present 6 times. | S193 |
| Rychlik | 1885 to 1968 | /ˈrɪxliːk/ RIKH-leek |
Prague, Czechoslovakia | Proved in February 1922 that Bolzano's function is continuous and nowhere differentiable, and edited its first publication in 1930.Appears in Chapter 10, and in the timeline under 1914 to the present 4 times. | S386, S384 |
| Johann Radon | 1887 to 1956 | /ˈjoːhan ˈʁaːdɔn/ YOH-hahn RAH-don |
Austria | Radon measures (1913) and the Radon transformIn the timeline 1914 to the present. | S435 |
 Otton Nikodym |
1887 to 1974 | /ˈɔtɔn ɲiˈkɔdɨm/ OT-on nyih-KOH-dim |
Poland and USA | Completed the Radon-Nikodym theorem, 1930In the timeline 1914 to the present twice. | S435 |
 Srinivasa Ramanujan |
1887 to 1920 | /ˈʃriːnɪˌvɑːsə rɑːˈmɑːnʊdʒən/ SHREE-nih-VAH-suh rah-MAH-nuu-jun |
Tamil Nadu, India, and Cambridge, England | Divergent series, continued fractions, elliptic and modular functions; Euler's constant to 15 places at sixteenAppears in Chapter 11, and in the timeline under 1873 to 1913 twice. | S431 |
| Thoralf Skolem | 1887 to 1963 | /ˈtuːrɑlf ˈskuːlɛm/ TOO-rahlf SKOO-lem |
Norway | 1934 ultraproduct construction of the hyperintegers, with the analogue of the Transfer PrincipleIn the timeline 1914 to the present twice. | S440 |
| Jarnik | 1897 to 1970 | /ˈjarɲiːk/ YAR-nyeek |
Prague, Czechoslovakia | Independently and simultaneously with Rychlik proved in 1922 that Bolzano's function is nowhere differentiable.Appears in Chapter 10, and in the timeline under 1914 to the present twice. | S386 |
| Mary Cartwright | 1900 to 1998 | /ˈkɑːtraɪt/ KART-ryte |
England | With Littlewood, found chaotic solutions of the van der Pol equation in the 1940s, decades before the word "chaos" existed for themAppears in Chapter 11 twice. | S420 |
| Marshall Stone | 1903 to 1989 | /ˈmɑːʃəl stoʊn/ MAR-shul STOHN |
USA | Stone-Weierstrass theorem (1937), generalising Weierstrass's polynomial approximation to any compact Hausdorff spaceIn the timeline 1914 to the present. | S435 |
| Henri Cartan | 1904 to 2008 | /ɑ̃ˈʁi kaʁˈtɑ̃/ ahn-REE kar-TAHN |
France | Bourbaki founder; the first person Schwartz told about distributions, in the middle of the nightIn the timeline 1914 to the present 3 times. | S433, S434 |
| Dorothy Vaughan | 1910 to 2008 | /ˈdɒrəθi vɔːn/ DOR-uh-thee VAWN |
Virginia, USA | Supervisor of the segregated West Area Computing section at NACA Langley; assigned both Jackson and Johnson to the work that made them famousAppears in Chapter 11 9 times. | S422, S425 |
| Robert K. Merton | 1910 to 2003 | /ˈmɜːrtən/ MUR-tun |
USA | The scholar Stigler's 1980 paper was written to honor; Stigler is commonly said to have credited the law of eponymy to Merton, a claim I could not verifyAppears in Chapter 4 11 times, and in the timeline under 500 to 1400 twice. | S460 |
| Marjorie Lee Browne | 1914 to 1979 | /ˈmɑːrdʒəri liː braʊn/ MAR-juh-ree lee BROWN |
USA | Michigan doctorate completed 1949 (conferred February 1950); built mathematics teaching for black students at North Carolina Central UniversityNamed in the registers, not in the story or the timeline. | S428 |
| Laurent Schwartz | 1915 to 2002 | French [loˈʁɑ̃ ʃvaʁts] loh-RAHN SHVARTS |
Paris, Nancy, France | Theory of distributions (conceived 1944, published 1948-1951): made the Dirac delta a legitimate mathematical object and every distribution infinitely differentiableAppears in Chapter 12 4 times, and in the timeline under 1914 to the present twice. | S463, S433 |
| Abraham Robinson | 1918 to 1974 | /ˈrɒbɪnsən/ ROB-in-sun |
Germany, Israel, UK, USA | Introduced nonstandard analysis in 1961 and in the 1966 book, giving infinitesimals obeying the same laws as ordinary numbers, as Leibniz had claimed.Appears in Chapter 10, and in the timeline under 1914 to the present 3 times. | S381, S367, S368, S390 |
 Katherine Johnson |
1918 to 2020 | /ˈkæθrɪn ˈdʒɒnsən/ KATH-rin JON-sun |
West Virginia and Virginia, USA | Co-authored NASA TN D-233 (1960), the report solving for launch azimuth from a chosen landing point; hand-checked the Friendship 7 trajectory at John Glenn's requestAppears in Chapter 11 13 times, and in the timeline under 1914 to the present twice. | S422, S423, S424 |
 Julia Robinson |
1919 to 1985 | /ˈdʒuːliə ˈrɒbɪnsən/ JOO-lee-uh ROB-in-sun |
USA | Reduced Hilbert's Tenth Problem to a single missing hypothesis, supplied by Matiyasevich in 1970Appears in Chapter 11. | S421 |
 Mary Jackson |
1921 to 2005 | /ˈmɛəri ˈdʒæksən/ MAIR-ee JAK-sun |
Hampton, Virginia, USA | NASA's first black female engineer, 1958; a dozen or so reports on the boundary layerAppears in Chapter 11 8 times, and in the timeline under 1914 to the present twice. | S425 |
| Evelyn Boyd Granville | 1924 to 2023 | /ˈɛvlɪn bɔɪd ˈɡrænvɪl/ EV-lin boyd GRAN-vil |
USA | Yale doctorate 1949 in functional analysis; IBM programmer on NASA contracts; later a leader in teacher educationNamed in the registers, not in the story or the timeline. | S427 |
| Bishop | 1928 to 1983 | /ˈɛrɪt ˈbɪʃəp/ ERR-it BISH-up |
USA | Objected that nonstandard analysis, especially in calculus teaching, risked 'a debasement of meaning' because its objects carry no numerical content.Appears in Chapter 9 and Chapter 10 3 times, and in the timeline under 1914 to the present 5 times. | S390, S395 |
 Gladys West | 1930 to ? | /ˈɡlædɪs wɛst/ GLAD-iss WEST |
Virginia, USA | Led the team that computed the geoid, the Earth's true gravitational shape, on which satellite positioning dependsAppears in Chapter 11 8 times, and in the timeline under 1914 to the present twice. | S426 |
| D. T. Whiteside | 1932 to 2008 | /ˈdɛrək ˈwaɪtsaɪd/ DERR-uhk WYTE-syde |
England | Edited all eight volumes of The Mathematical Papers of Isaac Newton (1967-1981), the critical edition that made real Newton scholarship possible and dismantled several popular myths.Appears in Chapter 5 and Chapter 6 14 times, and in the timeline under 1665 to 1687. | S190, S191, S192 |
| Laugwitz | 1932 to 2000 | /ˈlaʊkvɪts/ LOWK-vits |
Germany | Argued in Historia Mathematica and the Archive for History of Exact Sciences that Cauchy's procedures must be read as infinitesimal mathematics, not paraphrased into epsilon-delta.Appears in Chapter 10. | S367, S368 |
| Michael Nauenberg | 1934 to 2023 | /ˈmaɪkəl ˈnaʊənbɜːrɡ/ MY-kuhl NOW-uhn-berg |
USA | Physicist who reconstructed Newton's orbital methods and defended the mathematical adequacy of Principia Proposition I against Whiteside and Aiton.Named in the registers, not in the story or the timeline. | S202 |
| H. Jerome Keisler | 1936 to ? | /ˈhaʊərd dʒəˈroʊm ˈkaɪslər/ HOW-erd juh-ROHM KYSE-ler |
Wisconsin, USA | Elementary Calculus: An Approach Using Infinitesimals (1976), the first calculus textbook built on Robinson's hyperrealsAppears in Chapter 10, and in the timeline under 1914 to the present 3 times. | S440, S441 |
| Roshdi Rashed | 1936 to ? | /ˈrʊʃdi ˈraːʃɪd/ ROOSH-dee RAA-shid |
Egypt / France | Argues that Sharaf al-Din al-Tusi's method rests on an implicit use of the derivativeAppears in Chapter 3 3 times, and in the timeline under 500 to 1400. | S053 |
| Grabiner | 1938 to ? | /ˈɡreɪbɪnər/ GRAY-bin-er |
USA | Showed that Cauchy's rigor was built from the eighteenth-century algebra of inequalities, and that the epsilon is the 'e' of erreur.Appears in Chapter 1 and Chapter 10 twice, and in the timeline under 1914 to the present. | S360, S367 |
| Stephen M. Stigler | 1941 to ? | /ˈstiːɡlər/ STEEG-ler |
USA | Stated the law of eponymy in 1980Named in the registers, not in the story or the timeline. | S460 |
| Victor J. Katz | 1942 to ? | /ˈvɪktər kæts/ VIK-ter KATS |
USA | Wrote the standard short account of how far Islamic and Indian mathematicians got toward calculus, and traced the summation identity from ibn al-Haytham into the Kerala derivationsAppears in Chapter 2, Chapter 3 7 times, Chapter 5 and Chapter 10, and in the timeline under 500 to 1400. | S061 |
| Seppo Linnainmaa | 1945 to ? | /ˈsepːo ˈlinːɑi̯nˌmɑː/ SEP-poh LIN-nine-mah |
Finland | First published description of the reverse mode of automatic differentiation, in a 1970 master's thesis and a 1976 BIT paper about accumulated rounding errorAppears in Chapter 12 6 times, and in the timeline under 1914 to the present 4 times. | S436, S437, S438 |
| Bottazzini | 1947 to ? | /bottatˈtsiːni/ bot-taht-TSEE-nee |
Italy | Author of The Higher Calculus; argues it was precisely the use of infinitesimals that stopped Cauchy and Abel seeing uniform convergence.Appears in Chapter 10. | S367 |
| Paul Werbos | 1947 to ? | /pɔːl ˈwɜːrboʊs/ pawl WUR-bohss |
USA | 1974 doctoral thesis Beyond Regression, which cast reverse-mode differentiation in formal discrete-time terms; first NN-specific application, 1981In the timeline 1914 to the present 4 times. | S436, S437 |
| Russ | 1947 to ? | /rʌs/ RUSS |
UK | Translator of Bolzano's 1817 paper (Historia Mathematica 7, 1980) and of The Mathematical Works of Bernard Bolzano (OUP, 2004).Named in the registers, not in the story or the timeline. | S387, S360 |
| Jan P. Hogendijk | 1955 to ? | /jɑn ˈhoːɣəndɛik/ yahn HOH-khen-dike |
Netherlands | Argues that a different, non-derivative method lies behind al-Tusi's determination of the maximumAppears in Chapter 3 3 times, and in the timeline under 500 to 1400. | S053 |
| William Noel | 1965 to ? | /ˈwɪljəm noʊˈɛl/ WIL-yum noh-EL |
United Kingdom / United States | Curator who ran the decade-long Archimedes Palimpsest project and put all its data online freeIn the timeline 1914 to the present. | S006 |
| Reviel Netz | 1968 to ? | /rɛviˈɛl nɛts/ reh-vee-EL NETS |
Israel / United States | Led the reading of the recovered palimpsest text and published the Method Proposition 14 argument comparing infinite collectionsAppears in Chapter 2, and in the timeline under 1914 to the present twice. | S003, S006 |
| Abigail Quandt | - | /ˈæbɪɡeɪl kwɒnt/ AB-ih-gayl KWONT |
United States | Spent four years disassembling and stabilising the Archimedes Palimpsest so it could be imaged at allAppears in Chapter 2. | S004, S006 |
 Archibald Pitcairne |
- | /ˈɑːrtʃɪbɔːld pɪtˈkɛərn/ or /ˈpɪtkɛərn/ AR-chih-bawld pit-KAIRN, also PIT-kairn |
Scotland | Friend of Craig and David Gregory whose Solutio problematis de historicis seu inventoribus (Edinburgh, 1688) printed Gregory's reconstruction of Newton's prime theorem on series quadraturesAppears in Chapter 6 3 times, and in the timeline under 1665 to 1687 and 1688 to 1734 twice. | S204 |
| David Gregory | - | /ˈdeɪvɪd ˈɡrɛɡəri/ DAY-vid GREG-uh-ree |
Scotland; Oxford | Scottish acolyte who transcribed Newton manuscripts after his 1694 Cambridge visit, wrote the circulating summary Methodus fluxionum, was shown the Enumeratio in the late 1690s, and printed Newton's 'classical scholia' in his own 1702 bookAppears in Chapter 6 6 times, and in the timeline under 1665 to 1687 5 times and 1688 to 1734 9 times. | S204 |
 Edmond Halley |
- | /ˈhæli/ or /ˈheɪli/; Ronan preferred /ˈhɔːli/ Contested: HAL-ee (rhymes with valley) is the form comet astronomers and 89 percent of Halley families use; HAY-lee is common; HAW-lee is the form his biographer Ronan preferred |
- | One of the small group of experts allowed to see Newton's mathematical manuscripts; examined the worn De methodis at Cambridge with Raphson in 1691 and was given De quadratura to transcribe in 1695 on the condition that no one else saw itAppears in Chapter 6 5 times, Chapter 8 4 times and Chapter 9 3 times, and in the timeline under 1665 to 1687 4 times and 1688 to 1734 5 times. | S204 |
| Evelyn Walker | - Disputed |
/ˈɛvəlɪn ˈwɔːkər/ EV-uh-lin WAW-ker |
New York, USA | Professor at Hunter College; made the standard English translation of Leibniz's 1684 and 1686 calculus papers for Smith's Source Book (1929), and documented the originals' printing errors.Appears in Chapter 5 and Chapter 7. | S232 |
| Ioannes Myronas | - | /joʊˈhænɪs maɪˈroʊnəs/ yoh-HAN-iss my-ROH-nus |
Jerusalem (Byzantine) | The scribe who erased Archimedes' Method in 1229 and wrote a prayer book over it - thereby preserving itAppears in Chapter 2 twice, and in the timeline under 500 to 1400 twice. | S004, S006 |
 James Wilson |
- | /ˈdʒeɪmz ˈwɪlsən/ JAYMZ WIL-sun |
- | Made a secondary transcript of De methodis from Jones's copy about 1720, now lost, and wrote to Newton in December 1720 reporting that badly copied transcripts of the 1666 and 1671 treatises were circulating in private handsAppears in Chapter 6 7 times, and in the timeline under 1688 to 1734 5 times and 1734 to 1799 twice. | S204 |
| Jeff Miller | - | /dʒɛf ˈmɪlər/ JEFF MIL-er |
USA | Built and maintained the Earliest Uses of Symbols and Words pages, which have corrected Cajori in dozens of places since 1994In the timeline 1637 to 1665. | S139, S453 |
| John Craig | - | /dʒɒn kreɪɡ/ JON KRAYG |
Scotland | Scottish mathematician who went to Cambridge after hearing that Wallis meant to publish a summary of Newton's method of squaring curves, transcribed part of De methodis and of the Epistola posterior, and carried the results back to Pitcairne and David GregoryAppears in Chapter 6 3 times and Chapter 10, and in the timeline under 1665 to 1687 5 times. | S204 |
| John Flamsteed | - | - | - | Given a set of notes on algebra by Newton in 1674, one of the earliest documented loans of a Newton mathematical manuscriptIn the timeline 1665 to 1687 twice. | S204 |
| John Perry | - Not known |
/dʒɒn ˈpɛri/ jon PERR-ee |
Britain | Opened the British Association debate of 14 September 1901 that launched the campaign to teach the calculus in ordinary schoolsAppears in Chapter 12, and in the timeline under 1873 to 1913 twice. | S492 |
| Joseph Raphson | - | - | - | Examined the original De methodis manuscript at Cambridge with Halley in 1691 and recorded that it was 'very much worn by having been lent out'Appears in Chapter 6, and in the timeline under 1688 to 1734 twice. | S204 |
| Munjala | - Disputed |
/ˈmuɲdʒaːlɐ/ MOON-jaa-luh |
India | Gave, around 932 CE, the first correct formula for a planet's true daily motion using the Rcosine as a rate of changeIn the timeline 500 to 1400 twice. | S050 |
| Niccolo Guicciardini | - | /nikkoˈlɔ ɡwittʃarˈdini/ nee-koh-LOH gweet-char-DEE-nee |
Italy | Established that Newton ran a deliberate scribal publication strategy for his mathematics, and that the withholding followed from a methodological judgment against the 'new analysis' formed in the 1670sAppears in Chapter 6 9 times and Chapter 8 3 times, and in the timeline under 1665 to 1687 twice and 1688 to 1734 4 times. | S204 |
| Nicolas Bourbaki | - | /ni.kɔ.la buʁ.ba.ki/ nee-koh-LAH boor-BAH-kee |
France | Elements de mathematique (from 1939): rebuilt mathematics in a fixed logical order with no pictures and no historyAppears in Chapter 12 twice, and in the timeline under 1914 to the present twice. | S434 |
| Richard Swineshead | - Disputed |
/ˈswaɪnzhɛd/ SWYNZ-hed |
Oxford (England) | Wrote the Liber calculationum (c.1340-50), sixteen treatises of quantitative analysis of change, earning him the single name 'the Calculator'In the timeline 500 to 1400 twice. | S106 |
| Robert Wengert | - Not known |
/ˈwɛŋɡərt/ WENG-gert |
USA | Forward-mode automatic differentiation, 1964; the evaluation trace is still called a Wengert listIn the timeline 1914 to the present twice. | S436 |
| Roger Cotes | - | /ˈrɒdʒər koʊts/ ROJ-er KOHTS (rhymes with coats) |
- | Editor of the second edition of the Principia (1713); one of the disciples who had to ask Newton directly how to complete proofs depending on the quadrature of curves, and who made the anti-Cartesian intent of Book 1 Sections 4 and 5 explicit in the indexAppears in Chapter 6. | S204 |
| Sabetai Unguru | - Not known |
/saˈbetaɪ uŋˈɡuru/ sa-BET-eye ung-GOO-roo |
Israel / United States | Argued in 1975 that Greek geometry must not be read as algebra in disguise, provoking replies from van der Waerden, Freudenthal and Weil and reframing what it means to ask why the Greeks did not develop calculusAppears in Chapter 2 6 times, and in the timeline under 1914 to the present. | S491 |
 Samuel Horsley |
- | /ˈsæmjuəl ˈhɔːrsli/ SAM-yoo-ul HORS-lee |
- | Editor of Newton's Opera omnia (1779-1785), which rests on the Jones transcript of De methodis and on Wilson's secondary transcript of it rather than on Newton's originalsAppears in Chapter 6 4 times, and in the timeline under 1734 to 1799 twice. | S204 |
| Speelpenning | - Not known |
/bɛrnt ˈspeːlpɛnɪŋ/ bairnt SPAIL-pen-ing |
United States (Illinois) | 1980 PhD: the first fully automatic implementation of reverse mode, transforming programs written in a general-purpose languageAppears in Chapter 12 3 times, and in the timeline under 1914 to the present 3 times. | S436, S438 |
| Stephen D. Snobelen | - | /ˈstiːvən ˈsnoʊbələn/ STEE-vuhn SNOH-buh-luhn |
Canada | Established the detailed picture of Newton's concealed anti-Trinitarianism and of the specific strategies he used to survive as a heretic in office.Named in the registers, not in the story or the timeline. | S194 |
| Ted Skopinski | - Not known |
/skəˈpɪnski/ skuh-PIN-skee |
Virginia, USA | First author of NASA TN D-233Appears in Chapter 11 4 times, and in the timeline under 1914 to the present. | S423, S424 |
| Thomas Pellet | - | - | - | Received an incomplete copy of De methodis from William Jones, an instance of the deliberate practice of passing on curtailed Newton manuscriptsAppears in Chapter 6 7 times. | S204 |
Appendix B
Symbols, words, and eponyms
191 entries: where every symbol and every word on the page in front of you came from, with its first attested use. An entry marked Disputed has a first use or an originator that the sources do not agree on.
Symbols
41 entries.
| Item | First attested |
Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| (the dot as a hint of smallness, pre-Newton) | 1592 | Leonard Digges, Tectonicon p. 15 | English | a prick beside a fraction "betokeneth some small quantitie lesse then" it; two pricks, a little more | Not calculus, but the direct English ancestor of Newton's dot: a mark meaning "and a bit". Mercator used a dot over a letter for an infinitesimal in 1668 | S451 |
| (abbreviation) | 1624 | Edmund Gunter | English | - | Standard abbreviation for the sine function | S060 |
| for variables and for constants | 1637 | Rene Descartes, La geometrie | notation | first letters of the alphabet = known; last letters = unknown | The convention behind every "let x be ..." in mathematics. Descartes introduced it, in Cajori's words, "without comment" | S450 |
| (the infinity sign) | 1655 | John Wallis, De sectionibus conicis, Prop. 1 | unknown; two conjectures | Wallis wrote "esto enim nota numeri infiniti" - let this be the sign of an infinite number. His FIRST use is as a denominator: each of infinitely many strips has height of the whole, an infinitely small part | Infinity. Two conjectures for the shape: the late Roman numeral for 1000 (Wattenbach 1872, followed by Cajori), or the lowercase omega (), last letter of the Greek alphabet. A Roman hand-abacus in the Bibliotheque Nationale has a 1000-column marked with a shape close to the lemniscate | S451, S139, S145 |
| Disputed: (x-dot, the fluxion) Disputed |
1665 (20 May, per Cajori/Rigaud) or 1691 (per Whiteside); 1693 in print | Isaac Newton | notation | a dot over the letter, for the velocity of a flowing quantity | The time derivative. First PRINTED in the Latin edition of Wallis Algebra (De algebra tractatus, Oxford 1693, pp. 390-96). The 1691 entry below carries Whiteside's case against the early dating | S451, S192 |
| 1666-1671 | Isaac Newton | Latin/algebraic convention | the letter o, used for 'an indefinitely small Quantity' | Newton's infinitesimal time-increment. The moment of a fluent is its fluxion multiplied by . The step 'divide by , then set ' is exactly what Berkeley attacked in 1734 | S185, S192, S187 | |
| (integral sign) | 1675 (29 Oct MS); 1686 in print | Gottfried Wilhelm Leibniz | Latin summa | An elongated letter s, the ordinary long s of 17th-century printing, standing for summa ("sum"). It replaced the abbreviation omn. for omnia ("all the ...", as in omnia lineae, "all the lines") | The integral, i.e. the limit of a sum. NOTE the first PRINTED integral sign (Acta eruditorum 1686, pp. 297, 299) had its descender cut off and looked like a small f-shaped s; the modern tall form comes later | S139, S230, S233, S238, S451 |
| (the differential) | 1675 | Gottfried Wilhelm Leibniz | Latin, from differentia | From differentia, "difference". First used as a DIVISOR (Leibniz wrote ) because he thought of as lowering dimension, the inverse of which raises it | The differential, as in and ; the derivative | S139, S230, S238 |
| , | 1675 (11 Nov) | Gottfried Wilhelm Leibniz | Latin differentia | "the difference between two proximate 's"; from 11 November 1675 written as a multiplier rather than . Leibniz: "Idem est et " | An infinitesimal or differential increment of x; in it carries the variable of integration | S230, S238, S231, S451, S139 |
| omn. (omnia) | before 29 Oct 1675 | Gottfried Wilhelm Leibniz | Latin omnia | "all the ..." - Leibniz wrote omn. for "all the 's" | The thing the integral sign replaced. Leibniz abandoned it on 29 Oct 1675 because nested sums produced formulas like omn. omn. omn. that were unreadable | S451, S139 |
| as a divisor () | 1675 (29 Oct) | Gottfried Wilhelm Leibniz | Latin differentia | Leibniz first wrote the differential UNDER a line, as , because the integral raises dimension and lowers it: "ut augebit, ita minuet dimensiones" | Superseded on 11 Nov 1675 by the multiplier form dx. The symbol we write a thousand times was a denominator for its first sixteen days | S451, S139 |
| (built-up quotient) | 1675 (11 Nov) in MS; NOT in the 1684 print | Gottfried Wilhelm Leibniz | Latin | the ratio of the two differentials | The derivative. Cajori §571: the 1684 Acta paper has " ad " and "" but not the built-up fraction | S451 |
| (pricked or dotted letter) | 1691 | Isaac Newton | notation | a dot placed over a letter | The fluxion (derivative) of the fluent. CAUTION: Newton did NOT use this in 1666 (he used p, q, r) or in the 1671 treatise (l, m, n, r). Whiteside dates the dot notation to late 1691. The dots printed in the 1736 English Method of Fluxions were introduced by William Jones's 1710 transcript. The disputed 1665 dating above rests on Cajori and Rigaud | S192, S183, S185 |
| Disputed: (fractional differential) Disputed |
1695/1697 | Gottfried Wilhelm Leibniz | Latin | A differential of order one-half, proposed by analogy with fractional exponents on powers | Fractional (non-integer-order) derivative; the founding notion of fractional calculus | S241, S242 |
| , (higher differentials) | 1695 (numerals); ddy before that | Gottfried Wilhelm Leibniz | Latin | repeat the d | Higher derivatives. L'Hopital 1696 and Fontaine as late as 1764 still wrote in full | S451 |
| 1706 | William Jones, Synopsis palmariorum mathesios, p. 263; standardised by Euler, Introductio S.126, 1748 | Greek periphereia | the first letter of the Greek for "periphery, circumference" | Ratio of circumference to diameter; Euler defines it as the semicircumference of a circle of radius 1. Jones introduced it in a beginners' textbook with no fanfare; Cajori: "It simply came, unheralded." Euler used p in 1734, adopted pi in the Mechanica (1736), and made it universal with the Introductio (1748) | S318, S325, S451, S462 | |
| 1727 or 1728 (MS, printed 1862); 1736 in print | Leonhard Euler, Meditatio in experimenta explosione tormentorum nuper instituta; printed in Mechanica | notation | no attested reason. Not "Euler"; not "exponential" on any evidence he left | Base of natural logarithms, 2.71828...; Euler's manuscript gives '2,7182817', wrong in the last digit. Leibniz had used b for it in letters to Huygens in 1690-91; d'Alembert and others used c into the 1780s | S325, S451 | |
| (finite difference) | 1755 | Leonhard Euler, Institutiones calculi differentialis pp. 3-7 | Greek capital delta, for Latin differentia | the Greek capital of the same letter as Leibniz's Latin d | A finite difference: . Johann Bernoulli had used for a differential coefficient in 1706 | S451 |
| (summation) | 1755 | Leonhard Euler, Institutiones calculi differentialis p. 27 | Greek capital sigma, for Latin summa | the Greek S for "sum", exactly parallel to Leibniz's long Latin s for the integral | Summation. Euler's own sentence: "Quemadmodum ad differentiam denotandam vsi sumus signo , ita summam indicabimus signo " - just as we used for the difference, we shall indicate the sum by . and are twins, born in one book | S451 |
| (Landen's derivative) | 1758 | John Landen, Discourse concerning Residual Analysis, p. 12 | notation | a bracketed pair with a mark between | A dead rival to . Its selling point was typographical: no fraction bar, no letters above normal type height. "Landen had no followers in the use of his symbols" | S451 |
| (curly d, the partial derivative) | 1770 Condorcet; 1786 Legendre (modern form); 1841 Jacobi (revival) | Condorcet, Memoire sur les equations aux difference partielles; Legendre for the modern form, Memoire sur ... le calcul des variations | French/Latin; the letter itself is a cursive d | a cursive/rounded d; it is the same letter as the Cyrillic cursive de | Partial differentiation. Legendre used it once, dropped it the following year, and it lay unused for 55 years until Jacobi picked it up | S325, S451, S139 |
| 1777 (presented 5 May); published 1794 | Euler, De formulis differentialibus angularibus: 'formulam littera i in posterum designabo, ita ut sit ' | Latin imaginarius | - | The imaginary unit; too late for the Introductio, which writes throughout | S325, S318 | |
| 1786 | Simon L'Huilier, Exposition elementaire des principes des calculs superieurs, pp. 24, 31 | Latin limes | abbreviation of Latin limes, a boundary | The limit operator. L'Huilier wrote "" and said it means the same as . The full stop was gradually dropped; Weierstrass writes "lim" without it (1841) | S451, S139 | |
| , (prime notation) | 1797 (prime); 1770 and 1759 for earlier accents | Lagrange, Theorie des fonctions analytiques art. 16 | notation | a stroke over the letter, from the accent | Successive derivatives; introduced, in Lagrange's own words, 'pour plus de simplicite et d'uniformite' | S317, S325, S451, S139 |
| Disputed: Disputed |
1797 in this form | Lagrange (art. 2) writes fx for a simple variable and f(), f(a+bx), f(x,y) for compound arguments; Euler had used f(x) earlier | - | - | Function notation | S317, S318 |
| the inverted rounded E (Gruson's limit) | 1798-1800 | Johann Philipp Gruson, Le calcul d'exposition | notation | an inverted and rounded capital E | A dead rival notation for the limit of . Along with Pasquich's , an example of how many limit notations were tried and lost | S451 |
| (as an operator) | 1800 (Arbogast); used non-operationally by Johann Bernoulli earlier | Louis Francois Antoine Arbogast, Du calcul des derivations | French/Latin derivare | D for "derivation" | The differentiation operator. Arbogast also introduced for the inverse operation, which survives | S451, S139, S462 |
| Disputed: (subscripted operator) Disputed |
1800 | Arbogast (attribution from Julio Gonzalez Cabillon) | notation | D with the variable written below | Partial or total differentiation with respect to x. Cajori shows Arbogast's D but not the subscripted form, so the exact page is unconfirmed | S139, S451 |
| (limits on the integral sign) | 1819-20 (Memoires); 1822 (book) | Joseph Fourier, Theorie analytique de la chaleur, p. 252 (S 231) | French | "Nous designons en general par le signe l'integrale qui commence lorsque la variable equivaut a a, et qui est complete lorsque la variable equivaut a b" | The definite integral. Adopted at once by Plana, Fresnel and Cauchy; Poisson called it "tres-commode". Euler had written the limits in words (ab ... ad) inside brackets | S451, S139 |
| 1821 and 1823 | Augustin-Louis Cauchy, Cours d'analyse (Oeuvres II.3, pp. 49-50, 54); attested for 'erreur' in his 1853 probability work | Greek letter for French erreur | the initial of erreur, error | The arbitrary positive tolerance in a limit argument. The symbol of rigor is named after being wrong | S451, S139, S360, S362 | |
| (in epsilon-delta) | 1823 | Augustin-Louis Cauchy, Resume des lecons sur le calcul infinitesimal, Lecon 7 | Greek letter for French difference | the initial of difference | The input tolerance. Cauchy sometimes used where he elsewhere used ; the fixed division of labor ( for the output, for the input) settles in Weierstrass's Berlin lectures | S451, S139, S360 |
| (the evaluation bar with limits) | 1823 | Pierre Frederic Sarrus, in Gergonne's Annales XIV, p. 197 | French | a vertical rule (or bracket) with the two limits attached | The "plug in the top, subtract the bottom" bar. Used afterwards by Moigno and Cauchy | S451, S139 |
| (the Laplacian written as capital delta) | 1833 | Robert Murphy, Elementary Principles of the Theories of Electricity | notation | the same capital as finite difference, doing an entirely different job | The Laplace operator, also written . Maxwell's own name for it was "concentration" | S139 |
| (lim with the variable underneath) | 1841 and 1854 | Karl Weierstrass | notation | writing below the word lim | Specifying WHAT the variable approaches. Before this, "lim" alone left it to context. W. R. Hamilton used in 1853; Dirksen in Berlin used Gr. for German Grenze in 1832 | S451, S139 |
| (absolute-value bars) | 1841 | Karl Weierstrass, in a paper not printed at the time (Werke I, p. 67) | notation | two vertical rules | Absolute value / modulus. Weierstrass also used single bars for determinants and had to disambiguate in words. The competing name "module" is Argand's and Cauchy's | S451 |
| (matrix, double vertical lines) | 1843 and 1845 | Arthur Cayley | notation | two vertical rules on each side | Matrix delimiters. W. A. Whitworth even used TRIPLE lines in 1866 to signal that three independent determinant equations could be formed | S451 |
| (nabla, the symbol) | 1846-47 (Hamilton, rotated form); 1867 (Tait, modern form) | William Rowan Hamilton; established by P. G. Tait | notation | an inverted capital | The vector differential operator. Hamilton first wrote it in the modern orientation in an unpublished Southampton paper, then ROTATED it because "this more common sign has been so often used with other meanings"; Tait turned it back | S139, S451 |
| (the dotted-equals sign for a limit) | 1880 in print; coined by James E. Oliver | James E. Oliver (Cornell); printed in W. E. Byerly, Elements of the Differential Calculus (1880) p. 7 | notation | an equals sign with a dot above | "Approaches as a limit". The standard American limit arrow before the British arrow crossed the Atlantic. Oliver's own gloss: "Using [it] in the sense Approaches in value to, as a limit" | S451 |
| (from the lunate epsilon , for set membership) | 1889 | Giuseppe Peano, Arithmetices principia nova methodo exposita, Turin | Greek lunate epsilon, standing for the Latin 'est' | 'is' - Peano called it an abbreviation for 'est' | , x is an element of the set A. Peano's Formulaire (1895) lists 158 signs, of which only 26 are purpose-made ideographs | S393, S451 |
| (the arrow in limits) | 1905 | John Gaston Leathem, Volume and Surface Integrals Used in Physics | notation | an arrow, for "tends to" | . Popularised by Bromwich (Infinite Series, 1908) and Hardy (A Course of Pure Mathematics, 1908). Hardy CREDITS Leathem and Bromwich in his Preface; Leathem complained in 1925 that the arrow had been "erroneously attributed to another writer owing to its use, with inadvertent omission of acknowledgment, in an important book published three years later" | S451, S139 |
| (the integral round a closed path) | 1917 | Arnold Sommerfeld, Annalen der Physik | notation | a small circle drawn through the integral sign | Contour integral. Earlier than the 1923 use Cajori knew; E. B. Wilson (1901) put a small circle BELOW the sign instead | S139 |
Words and their origins
101 entries.
| Item | First attested |
Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| asymptote | before 200 BCE (Apollonius, broader sense); 1656 in English | Apollonius of Perga | Greek a- + sym- + ptotos | "not falling together" - i.e. not meeting | A line a curve approaches without meeting. In Apollonius it meant ANY lines that do not meet, in any direction. First English citation 1656, Hobbes: "Asymptotes ... come still nearer and nearer, but never touch" | S453 |
| parisotes (παρισότης) | 250 | Diophantus of Alexandria | Greek | para (beside) + isotes (equality) = near-equality | approximate equality; the ancestor of 'adequality' | S130 |
| sankalita | 499 | Aryabhata I | Sanskrit | summation | Sum of a series; vara-sankalita is repeated (higher-order) summation | S050 |
| asanna | 499 | Aryabhata I | Sanskrit | near, approximate | Aryabhata's own label for his pi value ; Nilakantha built an incommensurability argument on this single word | S050, S059 |
| kuttaka | 499 | Aryabhata I | Sanskrit | pulveriser | The algorithm for integer solutions of linear Diophantine equations, essentially the Euclidean algorithm | S059 |
| khahara | 1150 | Bhaskara II | Sanskrit | having kha (zero) as divisor | A quantity divided by zero, which Bhaskara II says is unaltered by addition or subtraction | S050 |
| latitudo (latitude of a form) | 1300 | Oxford Calculators / Kilvington | Latin | breadth | the intensity or magnitude of a quality, plotted as the ordinate | S100, S101, S102, S107 |
| longitudo | 1355 | Nicole Oresme | Latin | length | the extension of the subject, plotted as the abscissa (for motion, time) | S100, S107 |
| average | c. 1500 in English | - | Old French avarie, ultimately from Arabic awariya, damaged goods | originally a maritime-trade term: the loss on damaged cargo, and then the equitable SHARING of that loss among the merchants | The arithmetic mean. The mathematics came from insurance: "average" means dividing a loss fairly, and only later dividing anything by n | S453 |
| quadrature | classical Latin (a land measure); 1569 in English; used by Newton 1669 | no coiner recorded; Newton uses it in 1669 | Latin quadratura, a squaring | literally "a squaring": in classical Latin, the division of land into squares; in mathematics, the making of a square of equal area | Finding an area under a curve; computing a definite integral. To "square" a figure is to construct a square of the same area, which is why finding areas is called squaring. Newton's De analysi Rule I is a quadrature rule | S184, S187, S453 |
| concave | 1571 in English | Thomas Digges, A Geometricall Practise named Pantometria | Latin concavus (com- + cavus, hollow) | "hollowed out" | Curving inwards. Its opposite, convex, is Latin convexus, "vaulted, arched" | S453 |
| tangent (linea tangens) | 1583 | Thomas Fincke, Geometriae rotundi | Latin tangere | "the touching line" | A line meeting a curve and not crossing it there. In English from 1597 (Blundevil): "two other right lines belonging to a Circle, called lines Tangent, and lines Secant" | S453 |
| secant (secans) | 1583 | Thomas Fincke, Geometriae rotundi | Latin secare | "the cutting line" | A line cutting a curve in two points; in trigonometry, 1/cos. Viete disliked the word for fear of confusion with the geometric sense and proposed Transsinuosa instead | S453 |
| logarithm (logarithmus) | 1614 | John Napier, Mirifici logarithmorum canonis descriptio | Greek logos + arithmos | commonly read as "ratio-number"; but the OED warns that Napier never explained it and logos may simply mean "reckoning" | The exponent to which a base must be raised. Before coining the word Napier called them numeri artificiales, "artificial numbers", as against the numeri naturales they act on | S453 |
| veluti ('as it were') | 1615 | Johannes Kepler | Latin | as it were, just as if | Kepler's qualifier for treating a solid as composed of infinitely many infinitely thin parts | S151 |
| stereometria doliorum (doliometry) | 1615 | Johannes Kepler | Latin, from Greek stereos + metron; dolium = cask | solid-measurement of casks | the measurement of barrel volumes; the title subject of Kepler's 1615 book | S159 |
| cosine | 1620 (co.sinus); 1658 (cosinus) | Edmund Gunter proposed co.sinus; John Newton contracted it to cosinus | Latin sinus complementi | "the sine of the complement" - the sine of what is left when you take the angle from a right angle | cos. Unlike sine, cosine has no Sanskrit or Arabic ancestry: it is a European Latin abbreviation. Earlier European names for it include sinus rectus complementi (Regiomontanus c. 1463) and sinus rectus secundus (Apianus 1541) | S453 |
| parameter | 1631 (Mydorge, for the latus rectum); 1673 onwards in the modern sense (Leibniz) | Claude Mydorge; then Leibniz | Greek para + metron | "measured alongside" - a quantity measured beside the others | A constant that indexes a family of curves or functions. Van Schooten (1659) is probably where Leibniz got it | S453 |
| indivisibilia / indivisibles | 1635 | Bonaventura Cavalieri | Latin | things that cannot be divided | the lines or planes into which a figure is resolved for comparison of magnitudes | S135, S156, S132 |
| adaequalitas / adequality | 1636 | Pierre de Fermat (via Bachet) | Latin, calqued from Greek parisotes | ad- (towards) + aequalitas (equality) = near-equality | the step of setting f(a+e) 'almost equal' to f(a) before dividing by e and suppressing e | S130 |
| inflection point | c. 1638 (written); 1679 (printed) | Pierre de Fermat, appendix to Methodus ad disquirendam maximam et minimam | Latin inflectere | "a bending in" | Where a curve changes from convex to concave. Fermat: "investiganda sunt ex arte puncta inflexionum, in quibus curvatura ex convexa fit concava vel contra" - the points of inflection must be found by art, where the curvature turns from convex to concave or the reverse | S453 |
| fluxion / fluent | 1671; 1676 (concealed), 1693 (in print); 1704 (Hayes brings "fluxion" into English) | Isaac Newton; Charles Hayes brought the word into English in 1704 | Latin fluere, to flow (fluens, fluxio) | a flowing quantity (fluent) and the velocity with which it flows (fluxion) | A fluent is a quantity that varies continuously with the independent variable, what we now call a function of time; its fluxion is the instantaneous rate of change of that fluent, the derivative. Newton's own gloss: "the Velocities by which every Fluent is increased by its generating Motion". Hayes treats "Moments, Differences or Fluxions" as three names for one thing. NOTE the words are older than Newton: Richard Swineshead used fluxus and fluens in the Liber calculationum c. 1350 (Boyer). The word fluxion appears once in the Principia, apparently by oversight | S185, S187, S192, S189, S188, S286, S439, S453 |
| Disputed: calculus (in English) Disputed |
1666 (general sense) | anonymous, Philosophical Transactions vol. 1, p. 369 | English from Latin | "a system or method of calculating" | The English noun predates Newton's and Leibniz's calculus, which is why English long said "the calculus" (i.e. THE method of reckoning) rather than "calculus". I could not pin down in any document when the definite article was dropped | S453, S455 |
| limit (limes) | 1667 (terminus, Gregory of St Vincent); 1687 (Newton, ultimate ratio); 1821 (Cauchy, modern) | Cauchy for the modern definition; Newton 1687 'ultimate ratio'; Gregory of St Vincent used 'terminus' | Latin limes; terminus, a boundary stone | a boundary path between fields; later the fortified frontier of the Roman Empire | The value approached, so as to end up differing from it by as little as one wishes. The Latin word is a path, not a wall: a limes is the track along a boundary. Newton's Principia scholium argues for it with a moving body: the ultimate velocity is "that with which the body arrives at its last place" | S361, S392, S451, S453 |
| convergent and divergent | 1667 | James Gregory, Vera circuli et hyperbolae quadratura | Latin convergere / divergere | to bend or lean together / to bend apart | Of a series, having or not having a limiting sum. Gregory wrote series convergens | S456, S453 |
| logarithmus naturalis (natural logarithm) | 1668 | Nicolaus Mercator | Latin | natural logarithm | logarithm to base ; so named because it arises with no chosen base from the area under | S155 |
| moment | 1671-1704 | Isaac Newton | Latin momentum, contracted from movimentum: a movement, an instant | "a movement", hence a moment of time and a moving weight; in Newton, an infinitely small increment accruing in an infinitely small time | The infinitesimal increment of a fluent: Newton writes the moment of x as x-dot times o, roughly the modern differential dx. Newton, De quadratura curvarum 1704: "Momenta id est incrementa momentanea synchrona" | S185, S453 |
| function (functio) | 1673 | Gottfried Wilhelm Leibniz (non-analytical); Johann Bernoulli 1698 (analytical); Euler 1748 (general) | Latin fungi, functus | the performing of a duty, an office, a task | A rule assigning one output to each input. Leibniz's 1673 use is geometric (a line "performing an office" in a figure), not analytic; Johann Bernoulli 1698 and Euler 1748 make it analytic | S317, S318, S324, S453 |
| variable | 1673 onwards | Gottfried Wilhelm Leibniz | Latin variabilis, from variare | changeable | A quantity that varies. In English as an adjective 1710 (Harris), as a NOUN only in 1816, in the translation of Lacroix - meaning that for a century English speakers said "variable quantity" and never "a variable" | S453 |
| rectification / to rectify | 1658 (the cycloid result, Christopher Wren); 1673 (the word, in English) | Christopher Wren for the cycloid result; William Brouncker, Phil. Trans. VIII, 6150, for the English word | Latin rectus + facere | "to make straight" | Finding the arc length of a curve, i.e. straightening it out. Brouncker: "It was easier to infer, That, if we can Rectifie the one, we may square the other" | S064, S453 |
| algorithm (algorithmus, of the calculus) | 1684 | Gottfried Wilhelm Leibniz | Latin (from Arabic al-Khwarizmi) | Leibniz calls his rules "an algorithm, so to speak, of this calculus": a set of formal rules operable without thinking about the geometry | Algorithm | S232, S231 |
| triangulum characteristicum (characteristic triangle) | 1686 | Gottfried Wilhelm Leibniz | Latin | "the distinguishing/defining triangle": the infinitely small triangle with sides dx, dy and the element of arc | The differential triangle underlying the derivative and arc length | S232, S233 |
| analysis indivisibilium atque infinitorum | 1686 | Gottfried Wilhelm Leibniz | Latin | "analysis of indivisibles and infinites" | Infinitesimal analysis | S233, S232 |
| evanescent / nascent | 1687 | Isaac Newton, Principia Book I Section I | Latin evanescere / nasci | "vanishing" / "being born" | Quantities in the act of shrinking to zero, or of growing from it: a nascent quantity is just beginning to increase from zero, "the first ratio of nascent quantities is that with which they begin to be"; an evanescent quantity is in the act of shrinking to zero, and Newton insists these are "evanescent divisible quantities", not indivisibles. His scholium: "by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish" | S186, S187, S453 |
| Disputed: integral (integralis) Disputed |
1690 | Jacob Bernoulli, Acta Eruditorum May 1690, p. 218 | Latin integer / integralis | whole, entire, untouched | The total accumulated from the parts; and, as a noun, the result of the inverse of differentiation. Leibniz wanted calculus SUMMATORIUS; Johann Bernoulli wanted calculus INTEGRALIS and the capital letter I as the sign. They compromised in 1696: Bernoulli's WORD, Leibniz's SYMBOL | S324, S451, S453 |
| catenary (catenaria) / funicularia | 1690-1691 | Jacob Bernoulli posed it as the funicularia; catenaria from Huygens | Latin catena / funiculus | chain / little rope | The curve of a chain hanging under its own weight; the hyperbolic cosine, not the parabola Galileo supposed | S334 |
| differential (as a noun) | 1690 (10 Sept MS) | Gottfried Wilhelm Leibniz, "Methodus pro differentialibus, ponendo " | Latin differentia | a difference | An infinitesimal increment. In English from 1704, in Harris's Lexicon Technicum | S453 |
| brachistochrone | 1696 | Johann Bernoulli's challenge | Greek brachistos + chronos | shortest time | The curve of quickest descent under gravity; it is the cycloid | S330 |
| normal | c. 1696 in English | Edmund Scarburgh, The English Euclide ("Normal Line") | Latin norma | a carpenter's square; hence "at right angles, according to the square" | Perpendicular to a curve or surface. The word for "normal" in the everyday sense - conforming to a rule - is the same carpenter's square | S453 |
| analysis infinitesimalis | 1697 | Gottfried Wilhelm Leibniz | Latin | "infinitesimal analysis": Leibniz's proposed NEUTRAL name covering both his differential method and Newton's fluxions | Infinitesimal calculus | S241 |
| series | before 1700 as proportio; "series" standard by the 18th century | - | Latin series, from serere, to join, to link | "a row, a chain, a linked succession" - the same root as "assert" and "insert" | A sum of a sequence of terms. Smith: "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century" | S453 |
| exponential | 1704 in English | John Harris, Lexicon Technicum | Latin exponere | "setting out, putting forth" - the exponent is the number "set above" | Of the function . Harris: "Exponential curves are such as partake both of the nature of Algebraick and Transcendent ones" | S453 |
| Disputed: enri (円理) Disputed |
1710 | Takebe Katahiro (long credited to Seki Takakazu) | Japanese | circle principle | The Japanese method most closely corresponding to definite integration | S430 |
| Commercium Epistolicum | 1713 | Royal Society (drafted by Isaac Newton) | Latin | "exchange of letters"; short for Commercium Epistolicum D. Johannis Collinsii et aliorum de analysi promota | The Royal Society's published dossier and report on the priority dispute | S189, S282 |
| Disputed: Charta volans Disputed |
1713 | anonymous (for Leibniz; quoting Johann Bernoulli anonymously) | Latin | "flying sheet": a single loose printed sheet circulated without imprint | The anonymous Continental rebuttal to the Commercium Epistolicum | S284 |
| versoria / la versiera | 1718 | Guido Grandi | Latin versoria, from vertere, to turn | the sheet or rope that turns a sail | The curve ; in modern form | S327, S319 |
| continuous / fonction continue | 1748 (Euler, in the OLD sense); 1817 Bolzano and 1821 Cauchy (modern) | Euler for the old sense; Bolzano's equivalent definition 1817; Cauchy in print, 1821 | Latin continuus, from continere | hanging together, unbroken | WARNING FOR READERS OF OLD TEXTS: Euler's 1748 "continuous curve" means one given by a SINGLE formula. A curve made of two formulas was "discontinuous" even if you could draw it without lifting the pen. The modern sense is the opposite kind of criterion: f is continuous if an infinitely small increment of the variable always produces an infinitely small increment of the function (Cauchy 1821), now epsilon-delta | S361, S363, S369, S387, S453 |
| derivative (fonction derivee) | 1772 (term); 1797 (systematic) | Joseph-Louis Lagrange, Nouveaux Memoires de l'Academie de Berlin; systematised in Theorie des fonctions analytiques 1797 | French deriver, from Latin derivare (de + rivus, a stream) | to draw off water from a stream | The rate of change of a function: literally the function DERIVED from another. The literal image is a channel led off a river, the derived function drawn off from the primitive one | S317, S324, S451, S453 |
| D-ism / Dot-age | 1812 (recalled 1864) | Charles Babbage | English (pun) | "D-ism" puns on deism (Leibniz's d); "Dot-age" puns on dotage/senility (Newton's dot). Babbage's footnote: "Leibnitz indicated fluxions by a d, Newton by a dot." | Slogan of the Analytical Society campaign to adopt Leibnizian notation in Britain | S239 |
| infiniment petit / infinitesimal | 1667/68 (Wallis, 17/27 Feb, letter; printed 1685); 1821 (Cauchy redefines it) | John Wallis, letter to Vincent Leotaud; Cauchy as a defined term in the Cours d'analyse | Latin infinitesimus; French infiniment petit | "the infinite-th" - one part out of infinitely many, formed like decimus, centesimus, millesimus | Vanishingly small. Wallis's sentence: "Dic Angulum Contactus esse, Infinitesimam partem duorum Rectorum" - say the angle of contact is an infinitesimal part of two right angles. Cauchy redefined the term in 1821: for him an infinitesimal is a variable whose successive numerical values fall below any given number, i.e. a variable with limit zero, not a fixed number | S361, S369, S453 |
| serie convergente / convergent series | 1821 | Cauchy (definition by limit of partial sums) | Latin 'convergere', to incline together | to bend or lean together toward a point | a series whose partial sums approach a limit, which is called the sum of the series | S361, S364 |
| solution de continuite (a discontinuity) | 1821 | Cauchy | French, from Latin 'solutio continuitatis', a breaking of the continuity | a breaking-apart of the holding-together; the phrase was medical, for a wound or fracture | a point where a function fails to be continuous | S361 |
| integrale definie / definite integral (as a limit of sums) | 1823 | Cauchy; generalised by Riemann 1854 | Latin 'integer', whole, untouched | making whole; the integral sign is a stretched letter S for 'somme', sum | the limit of sums of the form sum f(x_i)(x_{i+1}-x_i) as the subdivisions shrink | S362, S391 |
| Disputed: epsilon-delta Disputed |
1823 | Cauchy uses the inequalities in proofs; the notation stabilises through Weierstrass's Berlin lectures | Greek letters | - | the quantified inequality definition of limit and continuity; 'who wrote it first' is genuinely tangled because Weierstrass's lectures circulated as students' notes | S360, S373, S369, S368 |
| the first epsilon-delta proof | 1823 | Augustin-Louis Cauchy, Calcul infinitesimal, Lecon 7 (Oeuvres ser. 2 vol. 4, pp. 44-45) | - | Cauchy's proof of what is essentially the mean value theorem for derivatives | The moment the verbal definition of a limit is translated into inequalities. Judith Grabiner: in the 1820s Cauchy did not say what his delta depended on, which is why the pointwise/uniform distinction is missing | S139, S360 |
| scientist | 1834 | Attributed by Whewell to "some ingenious gentleman" at the British Association | English (from Latin scientia, by analogy with artist) | one who does science | A person engaged in systematic study of the natural world | S411 |
| nature-poker / nature-peeper | 1834 | Whewell | English (rendering German Naturforscher) | one who pokes at nature | Rejected alternatives to "scientist", offered half-seriously in the same paragraph | S411 |
| vector | c. 1840 (Hamilton); earlier in "radius vector" in astronomy | William Rowan Hamilton | Latin vector, from vehere, to carry | "a carrier, one who conveys" | A directed magnitude. The word arrives in mathematics out of astronomy's radius vector - the "carrying radius" that sweeps a planet round - and Hamilton kept the noun after dropping the radius | S453 |
| range | 1848 in mathematical use (statistics) | H. Lloyd, Proc. Royal Irish Acad. 4 | Old French rangier, to set in a row | a row, a rank; hence the spread from lowest to highest | The set of outputs of a function. NOTE the function sense and the statistical sense (max minus min) are two different senses of one word, and students conflate them | S453 |
| matrix | 1850 | James Joseph Sylvester, Philosophical Magazine, pp. 363-370 | Latin matrix, from mater | "womb; the thing from which something else is born" | A rectangular array. Sylvester meant it literally: the oblong array "is, as it were, a Matrix out of which we may form various systems of determinants". For him the matrix was the mother of the determinants, not an object in itself; Cayley made it an object in 1855 and 1858 | S453 |
| Menge / set | 1851 | Bolzano, Paradoxien des Unendlichen (posthumous) | German 'Menge', a quantity, a crowd | a crowd, a multitude, an amount | a collection of objects considered as a single object | S371 |
| Disputed: uniform convergence Disputed |
1853 | Concept absent from Cauchy 1821; the missing hypothesis identified through Abel 1826, Bjorling, Seidel, Stokes and Weierstrass; Cauchy adds a condition in 1853 | Latin 'uniformis', of one shape | having one and the same form throughout | convergence at a rate independent of the point; whether Cauchy's 1853 condition IS uniform convergence is disputed | S367, S364, S360 |
| nabla (the name) | c. 1870 | William Robertson Smith, suggesting it to P. G. Tait | Greek nabla, from Hebrew/Phoenician nevel | an ancient Levantine harp of ten or twelve strings | The name of the symbol, chosen by a Semitic-languages scholar purely because the inverted delta looks like a harp. Maxwell thought it a joke and used it in print exactly once: in the dedication of a comic poem, "To the Chief Musician upon Nabla: A Tyndallic Ode" | S454, S453 |
| atled | 1870 | James Clerk Maxwell, letter to Tait, 7 Nov 1870 | English | "delta" spelled backwards | Maxwell's rival name for the nabla operator. It lost | S454 |
| divergence | 1870 (as "convergence", Maxwell); 1892 in the modern sense | James Clerk Maxwell coined the concept-name; the sign was flipped later | Latin divergere | "bending apart" | div of a vector field. Maxwell's own word, in the letter to Tait of 7 Nov 1870, was CONVERGENCE, and modern divergence is minus Maxwell's convergence. Clifford used div u or dv u | S454, S453, S451 |
| curl | 1870 (7 Nov) | James Clerk Maxwell, letter to P. G. Tait | English | a curl, as of hair or a scroll | curl of a vector field. Maxwell's fourth attempt: he tried Twist ("suggests a screw"), then turn or version, then Twirl ("sufficiently racy ... perhaps too dynamical for pure mathematicians"), then "for Cayley's sake I might say Curl (after the fashion of Scroll)". Rival names in other languages survive: Rot (Lorentz, Abraham), Quirl (Wiechert), Vort (Voigt) | S454, S453 |
| concentration | 1870 (7 Nov) | James Clerk Maxwell, letter to P. G. Tait | English | Maxwell's name for the Laplacian, "because it indicates the mode in which the value of the function at a point exceeds the average value of the function in a little spherical surface drawn round it" | A dead name for - but the best one-sentence explanation of what the Laplacian MEANS that anyone has written | S454 |
| Schnitt / Dedekind cut | 1872 | Dedekind (idea found 24 November 1858) | German 'Schnitt', a cut | a cut, a slice | a partition of the rationals into two classes with every member of the first less than every member of the second; each cut is produced by exactly one real number | S366, S369 |
| domain | 1886 | Arthur Cayley, "On Linear Differential Equations", Quart. J. Pure Appl. Math. | Latin dominium, from dominus | "lordship, the land a lord rules" | The set of allowed inputs. Cayley: "points x within the domain of the point a" - originally a region of the complex plane, not a set of inputs | S453 |
| gradient | 1897 (scalar sense); c. 1900-1909 (vector sense) | Horace Lamb, An Elementary Course of Infinitesimal Calculus | Latin gradi, to step | "a stepping, a walking" - hence a slope, as in a railway gradient | Lamb coined it for the slope of a plane curve: "It is convenient to have a name for the property of a curve which is measured by the derived function." The vector sense is later, via H. Weber's 1900 edition of Riemann's lectures. Maxwell had called it SLOPE (1870), then space-variation; Gibbs called it the derivative | S453, S454 |
| Radon-Nikodym derivative | 1930 | Johann Radon and Otton Nikodym | English (Austrian and Polish authors) | the density of one measure with respect to another | : the general form of the Fundamental Theorem of Calculus, and the reason "probability density" is well defined | S435 |
| hyperreal | 1948 | Edwin Hewitt | English | beyond-real | An element of an ordered field properly extending the reals, containing infinitesimals and infinite numbers | S440 |
| distribution (generalised function) | 1948 | Laurent Schwartz | French (distribution) | a distribution of values over test functions | A continuous linear functional on test functions; every distribution has derivatives of all orders, so the Dirac delta becomes legitimate | S433 |
| nonstandard analysis | 1961 | Abraham Robinson | English | analysis over a nonstandard model | Calculus and analysis done rigorously with infinitesimals, via the hyperreals and the Transfer Principle | S440, S381 |
| standard part | 1961 | Abraham Robinson | English | the standard (real) part of a finite hyperreal | The operation that rounds a finite hyperreal to the unique real infinitely close to it - the single step that makes Leibniz's rigorous | S440 |
| Transfer Principle | 1961 | Abraham Robinson (anticipated by Leibniz's law of continuity and by Skolem 1934) | English | transfer of statements between models | Every first-order statement true of the reals is true of the hyperreals and conversely | S440, S381 |
| Wengert list | 1964 | Robert Wengert | English | - | The linear trace of elementary operations in a computation, on which automatic differentiation operates | S436 |
| reverse mode / backpropagation | 1970 | Seppo Linnainmaa (reverse mode); the name backpropagation from the neural network community | English | propagating derivatives backwards | Evaluation of the gradient by traversing the computational graph backwards; the chain rule applied in reverse. Cost is a small constant multiple of one forward evaluation regardless of the number of inputs | S436, S437 |
| Nicodemite | 16th century (Calvin); applied to Newton 1999 | Stephen Snobelen (application) | Greek/Latin via John 3:1-2 | one who follows Nicodemus, who came to Jesus by night | A person who conceals heretical belief behind outward conformity. Snobelen's term for Newton's fifty-year strategy of public Anglican conformity and private anti-Trinitarianism | S194 |
| palimpsest | - | - | Greek (palimpsestos) | scraped again | A manuscript whose original text was erased so the parchment could be reused | S004 |
| upapatti | - | Indian mathematical tradition | Sanskrit | that which establishes, demonstration | Proof or rationale; the Indian tradition's own word for what the Yuktibhasa supplies | S050 |
| del | 20th century | American usage | English | a clipping of delta | The usual American name for the nabla symbol. Cajori (1929) reports "It is often read del" | S451 |
| calculus (the Latin word) | classical Latin | - | Latin, diminutive of calx (limestone) | a little stone, a pebble | The word that names the subject. Lewis and Short give six classical senses, all from "pebble": a pebble; a stone in the bladder or kidneys; a draughtsman in the board game duodecim scripta; a counter on a reckoning board and hence a reckoning; a voting stone (white to acquit, black to condemn); a small weight | S455, S453 |
| calculus (the kidney stone) | 1st century CE | Celsus, De medicina 7.26; Pliny | Latin | the same pebble, lodged in the bladder | A renal calculus. It is NOT a metaphor from mathematics and mathematics is not a metaphor from it: both senses descend independently from "pebble", and both were already current in classical Latin | S455 |
| calculator / calculo | classical and late Latin | - | Latin | one who reckons with pebbles | A teacher of arithmetic. Roman usage split by class: a slave teaching calculation was a calculo, a freeborn one a calculator or numerarius | S453 |
| analysis | Greek; named as a method by Theon of Alexandria (4th c. CE) | Theon of Alexandria (per Kline); revived by Viete 1591 | Greek analysis, from analyein | "a loosening up, an undoing" - untying a knot | Now the branch of mathematics built on limits. Anciently, a method: assume the result and reduce it backwards to something known. Viete pushed "analytic art" as a replacement for "algebra"; he failed to kill algebra but succeeded in planting analysis | S453 |
| maximum and minimum | classical Latin words, mathematical use from Fermat onwards | Latin writers on the calculus | Latin | the neuter superlatives of magnus (great) and parvus/minus (small): "the greatest thing", "the least thing" | The largest and smallest values. They are neuter singular nouns, which is why the plurals are maxima and minima, not maximums | S453 |
| Disputed: sine (the mistranslation chain) Disputed |
Sanskrit 5th c.; Arabic 8th-9th c.; Latin 12th c. | Aryabhata (jya, 499) -> Arabic jiba -> misread as jaib -> Latin sinus | Sanskrit ardha-jya (half-bowstring) > jya > Arabic jiba > written unvowelled j-b > read as jaib (bosom, fold, bay) > Latin sinus (a fold in a toga, a bay) | "bowstring" (Sanskrit jya) becoming "bay, fold, bosom" (Latin sinus) | The ratio opposite/hypotenuse. Latin chorda would have been the correct translation and WAS used by Plato of Tivoli c. 1120; sinus won anyway. WHO made the misreading is genuinely unsettled: Robert of Chester (1145) and Gerard of Cremona (c. 1150) are both named in the literature | S060, S453 |
| mean | Greek theory of means, Pythagorean school | the Pythagoreans, per Heath | Latin medianus, middle | the middle one | Arithmetic, geometric or harmonic mean. Heath: the Pythagoreans had three means, "the arithmetic, the geometric and the subcontrary"; the third was later renamed the harmonic | S453 |
| wasan (和算) | - | - | Japanese | Japanese calculation | The indigenous mathematics of Edo-period Japan, developed under the closed-country policy | S429, S430 |
| trochoid / roulette | 17th century | Roberval used "trochoid"; French writers preferred "roulette" | Greek trochos, a wheel; French rouler, to roll | wheel-curve / little rolling thing | the cycloid and its relatives | S064 |
| philomath | 17th century (usage reported) | Guicciardini, describing Collins and the outer readership | Greek | lover of learning | A non-specialist enthusiast for mathematics; in Newton's case the outer ring of readers who learned his results second hand from the acolytes, sometimes in mutilated form | S204 |
Named theorems, curves, and people
25 entries.
| Item | First attested |
Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| Cavalieri's principle | c. 480 CE (Zu Gengzhi); 1635 (Cavalieri) | Zu Gengzhi | - | - | The Chinese statement is about 1150 years earlier and is quoted verbatim in the Chinese sources | S023, S135 |
| Pascal's triangle | 10th c. Halayudha (India); c. 1000 al-Karaji, c. 1100 Khayyam (Islamic world); 1050-1303 Jia Xian, Yang Hui, Zhu Shijie (China); Tartaglia and Stifel (Europe); 1654/1665 Pascal | many, independently | - | - | Cajori, writing in 1919, already says "known in the West as Pascal's arithmetical triangle" and notes it was in Chu Shih-Chieh (1303), known to Arab mathematicians in the eleventh century, and "before Pascal ... constructed by N. Tartaglia and M. Stifel". Italian still says triangolo di Tartaglia, Iranian usage says Khayyam's triangle, Chinese says Yang Hui's triangle | S456 |
| Alhazen | 1270 | Latin translators of Kitab al-Manazir | Latinisation of Arabic al-Hasan | - | The European name for Ibn al-Haytham; nobody in Basra ever called him that | S058 |
| Calculator (as a name for Richard Swineshead) | 1350 | contemporaries of Swineshead | Latin | one who reckons | the standard medieval nickname for Swineshead; later the name of a machine | S106 |
| Gregory's series | c. 1400 (Madhava); 1671 (Gregory); 1674 (Leibniz) | Madhava of Sangamagrama | - | - | The arctan series, and the case. Miller: "the result had been obtained around three hundred years earlier by the Indian mathematician Madhava. Western mathematicians only learnt of Madhava's contribution in the twentieth century" | S453, S055, S050 |
| Simpson's rule | 1639 Cavalieri; 1615 Kepler (the barrel rule); 1668 James Gregory; 1743 Simpson | Simpson himself said he had it from Newton | - | - | MacTutor: "the numerical method known today as Simpson's rule, although it did appear in his work, was something he learned from Newton as Simpson himself acknowledged". Whittaker and Robinson: "first given (in a geometrical form) by Cavalieri [1639], and later by James Gregory [1668] and by Thomas Simpson [1743]". German still calls it the Keplersche Fassregel, Kepler's barrel rule | S462, S453 |
| Amos Dettonville | 1658 | Blaise Pascal | French | anagram of Louis de Montalte, his Provincial Letters pseudonym | Pascal's pen-name for the cycloid contest and the Traite des sinus | S143 |
| Newton's method | 1669 Newton (algebraic, no derivative); 1690 Raphson; 1740 Simpson (the modern iteration) | Thomas Simpson for the form taught | - | - | MacTutor: "the Newton-Raphson method for solving is, in its present form, due to Simpson ... Newton described an algebraic process for solving polynomial equations which Raphson later improved. The method of approximating the roots did not use the differential calculus. The modern iterative form is due to Simpson, who published it in 1740." There are antecedents in Sharaf al-Din al-Tusi (12th c.) and in Chinese root-extraction (Jia Xian, Qin Jiushao) | S462, S053 |
| Gregory-Newton interpolation formula | 1670 (Gregory, unpublished); 1687 (Newton, in print) | James Gregory | - | - | Whittaker and Robinson (1924): "The formula is often referred to as Newton's formula of interpolation, although it was discovered by James Gregory in 1670." They coined the double-barrelled name to fix it, and it partly stuck | S453 |
| Rolle's theorem | 1691 | Michel Rolle stated it, without proof, in a book on solving polynomial equations | - | - | It was a theorem about polynomials, produced by Rolle's "method of cascades". Yet Rolle led the campaign in the Paris Academy from 1700 to 1705 against the infinitesimal calculus, which he "described as a collection of ingenious fallacies". The theorem underpinning the differential calculus is named after one of its loudest opponents | S459, S453 |
| Mean Value Theorem | 1691 (Rolle, special case); 1806/1823 (Lagrange and Cauchy); Bolzano 1817 | Cauchy for the proof now taught; Lagrange for the statement; Bolzano for the tools | - | - | The name in English is late: "theorem of mean value" 1889, "mean value theorem" 1894. Cauchy's 1823 Lecon 7 proof, the first epsilon-delta proof in history, is a proof of essentially this theorem | S453, S139, S360 |
| L'Hopital's rule | 1696 (published); 1694 (bought) | Johann Bernoulli discovered it; Guillaume de l'Hopital published it | - | - | The rule. L'Hopital paid Bernoulli a retainer for his mathematical discoveries and the right to use them; the rule appears in L'Hopital's Analyse des infiniment petits (1696), the first calculus textbook, which does credit Bernoulli and Leibniz in general terms in its preface | S315, S316, S242 |
| Taylor series | 1712-1715 (Taylor); earlier in Gregory and Johann Bernoulli | Brook Taylor stated it in print (Methodus incrementorum, 1715, Prop. VII Cor. II, p. 23) after communicating it to Machin on 26 July 1712 | - | - | Kline: the result was known to Gregory and Leibniz, and Johann Bernoulli published "practically the same result in 1694". Gibson (1921) concludes Taylor's Prop. XI is essentially Bernoulli's theorem and that "one can not be surprised that Bernoulli felt aggrieved at Taylor's omission of any reference to him" | S312, S453 |
| Maclaurin series | 1717 (Stirling); 1742 (Maclaurin) | James Stirling published it first; Maclaurin himself attributed the general theorem to Taylor | - | - | Stirling, Lineae Tertii Ordinis Neutonianae (1717), p. 32, establishes "the form of Taylor's Theorem now usually called Maclaurin's Theorem", by undetermined coefficients, with worked examples including and - twenty-five years before Maclaurin's Treatise of Fluxions (1742) art. 751, which uses the same method. Stirling in 1730 wrote that "the first to discover this Theorem was Mr Taylor" | S312, S453, S461 |
| Disputed: Stirling's approximation Disputed |
1730 (Stirling); 1730 and earlier (De Moivre) | Abraham de Moivre found the asymptotic form; Stirling supplied the constant | - | - | Stirling's formula appears at Methodus Differentialis (1730), Prop. 28, Example 2, p. 136. De Moivre had the form without identifying C; Stirling identified . Reasonable people split the credit; the name does not | S453 |
| witch of Agnesi | c. 1760 (Colson's mistranslation; printed 1801); the curve was named versoria by Grandi in 1718 | John Colson, translating Agnesi | English, from a misreading of Italian la versiera as l'avversiera | she-devil, adversary, from Latin adversarius | Nothing. It is a mistranslation of a word for a nautical rope, and it has stuck for 265 years. La versiera is a nautical rope, l'avversiera is a she-devil, and Colson read the wrong one. The curve is in Fermat and Grandi before Agnesi | S319, S320, S327 |
| Bolzano-Weierstrass theorem | 1817 (Bolzano); 1860s-70s (Weierstrass lectures) | Bernard Bolzano | - | - | Bolzano proved it in the 1817 Rein analytischer Beweis, as a lemma; the paper was almost unread for fifty years, and Weierstrass reproved it for his Berlin lectures. This is the rare case where BOTH names are on it, and the first one still had to be rediscovered to get there | S363, S370, S371 |
| Cauchy-Schwarz inequality | 1821 Cauchy (sums); 1836 Liouville (implicit); 1859 Bunyakovsky (integrals); 1862 Grassmann; 1885 Schwarz | Viktor Bunyakovsky for the integral form | - | - | Bunyakovsky, "Sur quelques inegalites concernant les integrales ordinaires et les integrales aux differences finies" (Mem. Acad. imp. sci. St-Petersbourg, 1859), inequality (C) on p. 4, WITH the equality case - twenty-six years before Schwarz's 1885 paper (S 15, p. 344), which is the source of the proof now taught. Russian and much European usage correctly says Cauchy-Bunyakovsky-Schwarz | S457, S458 |
| Riemann integral | 1854 (Riemann, Habilitationsschrift); 1875 (Darboux) | Riemann for the definition; Gaston Darboux for the upper-and-lower-sum machinery taught today | - | - | The construction in every modern textbook - upper sums, lower sums, and the criterion that they meet - is Darboux's, from "Memoire sur les fonctions discontinues", Ann. Sci. Ecole Norm. Sup. (2) 4 (1875), 57-112. The name "Riemann integral" is in print by 1907 (Hobson) | S453, S391 |
| Holder's inequality | 1888 | Leonard James Rogers, "An extension of a certain theorem in inequalities", Messenger of Math. 17, 145-150 | - | - | Named after Otto Holder, whose paper is 1889 | S458 |
| Jensen's inequality | 1889 | Otto Holder, Nachr. Gottingen, 38-47 | - | - | Named after Jensen, who pointed out in his own paper that Holder had it first, and was ignored | S458 |
| Fundamental Theorem of Calculus (the name) | 1942 in English under that exact name | Sherwood and Taylor, Calculus (1942), per Miller | - | - | The RESULT belongs to Gregory (Geometriae pars universalis, 1668) and Barrow before Newton and Leibniz, and its modern statement to Cauchy; the NAME is a twentieth-century textbook coinage, about 270 years younger than the theorem | S453, S138, S140 |
| Disputed: Stigler's law of eponymy Disputed |
1980 | Stephen M. Stigler, Transactions of the New York Academy of Sciences | - | "No scientific discovery is named after its original discoverer" | The law this whole section documents. The frequently repeated claim that Stigler credited the law's discovery to the sociologist Robert K. Merton - making the law an instance of itself - could not be verified: the 1980 paper sits behind a paywall, so the Merton attribution stays unconfirmed here | S460 |
| Disputed: Gabriel's horn Disputed |
- | unknown | English | the trumpet of the archangel Gabriel | Torricelli's acute hyperbolic solid; the origin of the name is genuinely unknown | S157 |
| Vandermonde determinant | never discussed by Vandermonde | - | - | - | Lebesgue believed the attribution arose from a MISREADING of Vandermonde's unfamiliar notation. Nobody knows who coined the name; it is in print by 1888 | S453 |
Phrases and quotations that travel
33 entries.
| Item | First attested |
Introduced by | Origin | Literal meaning | What it means now | Sources |
|---|---|---|---|---|---|---|
| theorein vs. apodeiknynai | c. 250 BCE | Archimedes | Greek (θεωρεῖν / ἀποδεικνύναι) | to investigate / to prove | Archimedes' own distinction between a heuristic that suggests a theorem and a demonstration that establishes it | S002 |
| ge yuan shu | 263 | Liu Hui | Chinese (割圓術) | circle-cutting technique | Liu Hui's method of doubling the sides of an inscribed polygon to approach the circle | S020 |
| mouhe fanggai | 263 | Liu Hui | Chinese (牟合方蓋) | box-lid / joined covers | The solid common to two cylinders inscribed at right angles in a cube; Liu Hui saw the sphere is to it as is to 4, but could not compute its volume | S023 |
| yangma / bienao / qiandu | 263 | Liu Hui | Chinese (陽馬 / 鱉臑 / 塹堵) | male horse / turtle's foreleg / moat-wall | A pyramid, a tetrahedron and a right prism; Liu Hui proved the yangma is and the bienao of the qiandu | S020 |
| Zu Geng principle | c. 480 | Zu Gengzhi | Chinese (緣幂勢既同，則積不容異) | if the corresponding areas are the same, the volumes cannot be different | Cavalieri's principle: solids with equal cross-sections at every height have equal volume, stated about 1100 years before Cavalieri | S023 |
| tatkalika gati | 1150 | Bhaskara II | Sanskrit | motion belonging to this instant | Instantaneous velocity | S050 |
| uniformiter difformis | 1335 | William Heytesbury | Latin | uniformly non-uniform | uniformly accelerated motion (constant acceleration) | S101, S107 |
| antya-samskara | c. 1400 | Madhava of Sangamagrama | Sanskrit | end-correction | A remainder term added where an infinite series is truncated, with a derived criterion for exactness | S050 |
| Disputed: method of exhaustion Disputed |
1600s | European mathematicians, not the Greeks | Latin/English coinage describing a Greek method | using up | A double reductio proving a magnitude is neither greater nor less than a target; note the Greek method does not exhaust anything, it brackets | S001, S008 |
| omnes lineae ('all the lines') | 1635 | Bonaventura Cavalieri | Latin | all the lines | the totality of parallel sections of a plane figure, used collectively as a magnitude | S156 |
| ductus plani in planum | 1647 | Gregoire de Saint-Vincent | Latin | the leading of a plane into a plane | Saint-Vincent's construction for generating solids and comparing volumes | S141 |
| new analysis | 1660s-1670s | Newton's own name for his early method | English rendering of Latin usage | the modern analysis, as against ancient geometry | Wallisian induction from particular cases, free use of infinitesimal moments, and Cartesian representation of curves by equations; Newton decided by the late 1670s that it was not a language fit for print | S204 |
| method of fluxions | 1671 | Isaac Newton | English/Latin hybrid | the method of rates of flow | Newton's calculus, as opposed to Leibniz's 'differential calculus'. The two names split the mathematical world for a century | S185, S187 |
| calculus differentialis | 1684 | Gottfried Wilhelm Leibniz, Acta Eruditorum | Latin | "the calculus of differences", literally "the reckoning of differences" | Differential calculus. Before coining it Leibniz called his procedure methodus tangentium directa, "the direct method of tangents" | S231, S232, S453 |
| calculus summatorius | 1684 and 1686 | Gottfried Wilhelm Leibniz | Latin | summing calculus | Superseded name for the integral calculus: Leibniz lost the naming argument to Johann Bernoulli but kept his integral sign, the long s of summa, so his defeated word survives inside the winner's notation | S324, S451, S453 |
| first and last ratios (rationes primae et ultimae) | 1687 | Isaac Newton | Latin | the first and the last ratios | Newton's name for his limit method: the ratio with which quantities begin to be, or with which they vanish. Book I Section I of the Principia carries this title. NOTE ON RENDERING: Motte's 1729 English gives 'first and last ratios', and that phrasing is everywhere; the standard modern translation (Cohen and Whitman, 1999) renders rationes primae et ultimae as 'first and ULTIMATE ratios'. Use whichever suits, but say which edition you are quoting | S186, S187, S072 |
| spira mirabilis / eadem mutata resurgo | 1690s; on the tomb from 1705 | Jacob Bernoulli | Latin | the marvellous spiral / though changed, I rise again the same | The logarithmic (equiangular) spiral, which reproduces itself under scaling - the property the motto celebrates and the tomb's carving fails to show | S328, S329 |
| calculus integralis / integral calculus | 1696 | Johann Bernoulli proposed it; Leibniz adopted it in 1696 in place of his own calculus summatorius | Latin | calculus of the whole | The branch dealing with integration | S324 |
| ghosts of departed quantities | 1734 | Berkeley, The Analyst S.35 | English | - | Berkeley's name for evanescent increments: quantities that are neither finite, nor infinitely small, nor zero | S310 |
| vis viva | 1740s debate | Leibniz's term, defended by du Chatelet against the Newtonian mv | Latin | living force | , twice what we now call kinetic energy; the Newtonians measured force by mv, which is momentum. Both sides were right about different quantities | S332 |
| Disputed: calculus of variations Disputed |
1744 (subject founded); the name is later | Euler, Methodus inveniendi (E65) | Latin variatio | a change, a turning | The calculus of functionals: finding the curve or function that maximises or minimises an integral | S323 |
| why Legendre dropped the curly d | 1787 | (Cajori's explanation) | - | Legendre abandoned his own notation the year after inventing it, and in the 1780s d, D and the rounded d were all being fought over by three different branches of mathematics at once | Cajori §596: "It looked indeed as if the different mathematical architects ... found themselves confronted with the curse of having their sign language confounded" | S451 |
| primitive function (fonction primitive) | 1797 | Lagrange, Theorie des fonctions analytiques art. 17 | French/Latin primitivus | first, original | The function from which a derivative is derived; hence 'primitive' for an antiderivative | S317 |
| fonction prime / seconde / tierce | 1797 | Joseph-Louis Lagrange, Theorie des fonctions analytiques p. 14 | French | "prime function", "second function", "third function" | Lagrange's own names for f prime, f double prime, f triple prime - the origin of saying "f prime" out loud | S451, S317 |
| Disputed: tanquam ex ungue leonem Disputed |
1831 in this Latin form | David Brewster, Life of Sir Isaac Newton p. 194, attributing it to Johann Bernoulli | Latin proverb | as by the claw, the lion | Used for recognising a master by a small sample of work. Bernoulli's 1697 print version appears to have been French with the Latin tag embedded ('comme ex ungue Leonem') | S321, S337 |
| Jacobi's claim to the partial-derivative sign | 1841 | Carl Gustav Jacob Jacobi, De determinantibus functionalibus, Crelle XXII p. 319 | Latin | Jacobi says he preferred the rounded d simply because "an accumulation of parentheses for reading and writing is rather onerous" | Jacobi made no historical search and reinvented Legendre's symbol; Cajori: "the mathematical public, uncritical on matters of priority, is crediting Jacobi with devices first proposed by Legendre and Hamilton". Adoption still took ~50 years: Cayley ignored it in 1857 | S451, S139 |
| matrix, the objection to | 1867 | Charles L. Dodgson (Lewis Carroll), An Elementary Treatise on Determinants | English | Dodgson wanted the word BLOCK instead: "surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities" | A lost terminological argument, by the author of Alice in Wonderland | S453 |
| why Hardy insisted on the arrow | 1908 | G. H. Hardy, A Course of Pure Mathematics, Preface | English | "to write , (as if anything ever were equal to infinity) ... is in the early stages of mathematical training to go out of one's way to encourage incoherence and confusion of thought" | The pedagogical argument that replaced with . It is a teaching argument, made by a textbook writer, and it won | S451 |
| scribal publication | 1993 | Harold Love (the historian's category, applied to Newton by Guicciardini) | English | publishing by handwritten copy | Circulating a text in manuscript copies instead of print, in a culture with established ways of generating, transmitting and selling such copies; its point is that the power in a text depends on possession of it being denied to others | S204 |
| geometric backlash | 1997 | Helena Pycior (quoted by Guicciardini) | English | a swing back toward geometry | The reaction in English seventeenth-century mathematics against reliance on algebraic symbolism, whose main voices were Barrow and Hobbes and whose influence shows in Newton's turn to synthetic methods after 1671 | S204 |
| bhuta-sankhya | - | Indian mathematical tradition | Sanskrit | object-numbers | A system encoding digits as words (eyes = 2, Vedas = 4, lunar mansions = 27), so numbers could be written as memorable verse; Madhava's 11-decimal is recorded this way | S050 |
| Disputed: anni mirabiles / annus mirabilis Disputed |
20th century (historians) | not Newton | Latin | wonderful years / wonderful year | The historians' label for Newton's 1665-1667. NOT Newton's own phrase. Newton's own words were 'in those days I was in the prime of my age for invention' (CUL Add. 3968.41, f. 85r) | S190 |
| calculos subducere | classical Latin | Cicero, De finibus 2.19.60 | Latin | "to draw the pebbles up/away" | "To do the sums, to cast up accounts." The Romans already had the metaphor; Latin also has ad calculos vocare, "to call to the pebbles" = to hold someone to strict account (Cicero, Laelius 16.58) | S455 |
Appendix C
Pronunciation quick reference
The 60 names most often mispronounced in a calculus classroom. IPA gives the native pronunciation; the respelling is what to say out loud in English. The full register of 212 people is Appendix A.
The three worth memorizing first: Euler is OY-ler, not YOO-ler. Cauchy is koh-SHEE. Agnesi is ah-NYEH-zee.
| Name | IPA | Say it like this | Region |
|---|---|---|---|
| Abel | /ˈɑːbl̩/ | AH-bel | Norway |
| Adrien-Marie Legendre | /adʁijɛ̃ maʁi ləˈʒɑ̃dʁ/ | ah-dree-AN mah-REE luh-ZHAHNDR | Paris, France |
| Archimedes | /ˌɑːrkɪˈmiːdiːz/ | ar-kih-MEE-deez | Syracuse, Sicily (Greek) |
| Bhaskara II | /ˈbʱaːskɐrɐ/ | BHAAS-kuh-ruh | Bijjada Bida, India |
| Bolzano | /bolˈtsaːno/ | bol-TSAH-noh | Prague, Bohemia (Habsburg Empire) |
| Bonaventura Cavalieri | /kavaˈljɛːri/ | kah-vah-LYEH-ree | Milan / Bologna (Italy) |
| Cantor | /ˈkantoːɐ/ | KAN-tor | Halle, Germany |
| Carl Gustav Jacob Jacobi | German [ˈkaʁl ˈɡʊstaf ˈjaːkɔp jaˈkoːbi] | karl GOOS-tahf YAH-kop yah-KOH-bee | Prussia (Potsdam, Konigsberg, Berlin) |
| Cauchy | /koˈʃi/ | koh-SHEE | Paris, France |
| Christiaan Huygens | Dutch [ˈkrɪstijaːn ˈhœyɣəns]; anglicised /ˈhaɪɡənz/ | Dutch: KRIS-tee-yahn HOY-khuns (the g is a throaty kh). Anglicised: HY-genz. | The Hague (Netherlands) / Paris |
| Colin Maclaurin | /məˈklɔːrɪn/ | muh-KLOR-in | Scotland |
| Daniel Bernoulli | /ˈdaːni̯eːl bɛʁˈnʊli/ | DAH-nee-el bair-NOO-lee | Basel and St Petersburg |
| Dedekind | /ˈdeːdəkɪnt/ | DAY-duh-kint | Braunschweig and Zurich |
| Dirichlet | /diʁiˈkleː/ | dee-ree-KLAY | Rhineland, Berlin and Goettingen |
| Emilie du Chatelet | /emiˈli dy ʃatˈlɛ/ | ay-mee-LEE dew shat-LAY | Paris and Cirey, France |
| Emmy Noether | German [ˈnøːtɐ] | NUR-ter (round the lips as if for "oo") | Erlangen and Gottingen, Germany; Bryn Mawr, USA |
| Eudoxus | /juːˈdɒksəs/ | yoo-DOK-sus | Cnidus, Asia Minor (Greek) |
| Evangelista Torricelli | /torriˈtʃɛlli/ | tor-ree-CHEL-lee | Faenza / Florence (Italy) |
| Fourier | /fuʁˈje/ | foor-YAY | Auxerre and Paris, France |
| Galois | /ɡalˈwa/ | gal-WAH | Paris, France |
| Gilles Personne de Roberval | /ʒil pɛʁˈsɔn də ʁɔbɛʁˈval/ | zheel pair-SUN duh roh-bair-VAL | Paris (France) |
| Gottfried Wilhelm Leibniz | German [ˈlaɪpnɪts]; usual English /ˈlaɪbnɪts/ or /ˈlaɪpnɪts/ | German: LYPE-nits (the b devoices to p, as in Leib). Common English: LYBE-nitz. Both are in use; the German is the one he answered to. | Saxony / Hanover (Holy Roman Empire) |
| Guillaume de l'Hopital | /ɡijom də lopiˈtal/ | gee-YOHM duh loh-pee-TAL | Paris, France |
| Hermite | /ɛʁˈmit/ | air-MEET | Paris, France |
| Ibn al-Haytham | /ˈɪbn æl ˈhaɪθæm/ | IB-n al-HY-tham | Basra and Cairo |
| Isaac Barrow | /ˈaɪzək ˈbæroʊ/ | EYE-zuk BARR-oh | London / Cambridge (England) |
| Jacob Bernoulli | /ˈjaːkɔp bɛʁˈnʊli/ | YAH-kop bair-NOO-lee | Basel, Switzerland |
| Johann Bernoulli | /ˈjoːhan bɛʁˈnʊli/ | YOH-hahn bair-NOO-lee | Basel and Groningen |
| John Wallis | /ˈwɒlɪs/ | WOL-iss | Kent / Oxford (England) |
| Joseph-Louis Lagrange | /ʒozɛf lwi laˈɡʁɑ̃ʒ/ | zhoh-ZEF lwee lah-GRAHNZH | Turin, Berlin, Paris |
| Jyeshthadeva | /ˈdʒjeːʂʈʰɐdeːʋɐ/ | JYESHT-huh-DAY-vuh | Kerala, India |
| Kovalevskaya | /kəvɐˈlʲefskəjə/ | kuh-vuh-LYEF-skuh-yuh | Russia and Sweden |
| Kronecker | /ˈkʁoːnɛkɐ/ | KROH-nek-er | Berlin, Prussia |
| Laurent Schwartz | French [loˈʁɑ̃ ʃvaʁts] | loh-RAHN SHVARTS | Paris, Nancy, France |
| Lebesgue | /ləˈbɛɡ/ | luh-BEG | France |
| Leonhard Euler | /ˈleːɔnhaʁt ˈɔɪlɐ/ | LAY-on-hart OY-ler (never YOO-ler) | Basel, St Petersburg, Berlin |
| Liu Hui | /ljoʊ xweɪ/ | lyoh HWAY | Kingdom of Wei, China |
| Madhava of Sangamagrama | /ˈmaːdʱɐʋɐ/ | MAAD-huh-vuh | Sangamagrama, near Cochin, Kerala |
| Maria Gaetana Agnesi | /maˈriːa ɡaeˈtaːna aɲˈɲeːzi/ | mah-REE-ah gah-eh-TAH-nah ah-NYEH-zee | Milan, Italy |
| Mary Somerville | /ˈsʌməvɪl/ | SUM-uh-vil | Scotland and England |
| Nicole Oresme | /ɔˈʁɛm/ | oh-REM | Normandy / Paris (France) |
| Nilakantha Somayaji | /ˈniːlɐkɐɳʈʰɐ soːmɐˈjaːdʒiː/ | NEE-luh-KUN-tuh soh-muh-YAA-jee | Trikkantiyur, Kerala |
| Peano | /peˈaːno/ | peh-AH-noh | Turin, Italy |
| Pierre de Fermat | /pjɛʁ də fɛʁˈma/ | pyair duh fair-MAH | Beaumont-de-Lomagne / Toulouse (France) |
| Pierre-Simon Laplace | French [pjɛʁ simɔ̃ laˈplas] | pyair see-MON lah-PLASS | France |
| Rene Descartes | /ʁəne deˈkaʁt/ | ruh-NAY deh-KART | Touraine (France) / Netherlands / Stockholm |
| Riemann | /ˈʁiːman/ | REE-mahn | Hanover, Germany |
| Seki Takakazu | Japanese [seki takakazɯ] | SEH-kee tah-kah-KAH-tsoo | Edo Japan (Kozuke and Edo) |
| Sharaf al-Din al-Tusi | /ˈʃarafʔadˈdiːn atˈtuːsiː/ | SHA-raf ad-DEEN at-TOO-see | Tus, Persia |
| Sophie Germain | /sɔ.fi ʒɛʁ.mɛ̃/ | so-FEE zher-MAN | Paris, France |
| Srinivasa Ramanujan | /ˈʃriːnɪˌvɑːsə rɑːˈmɑːnʊdʒən/ | SHREE-nih-VAH-suh rah-MAH-nuu-jun | Tamil Nadu, India, and Cambridge, England |
| Stieltjes | /ˈstiːltjəs/ | STEELT-yuhs | Netherlands and Toulouse, France |
| Takebe Kataaki | /takebe kataaki/ | TAH-keh-beh kah-tah-AH-kee | Edo Japan |
| Takebe Katahiro | /takebe kataçiɾo/ | TAH-keh-beh kah-tah-HEE-roh | Edo Japan |
| Thabit ibn Qurra | /ˈθaːbɪt ɪbn ˈqurra/ | THAA-bit ib-n KOOR-ruh | Harran and Baghdad |
| Thomas Bradwardine | /ˈbrædwərdiːn/ | BRAD-wer-deen | Oxford / Canterbury (England) |
| Weierstrass | /ˈvaɪ.ɐʃtʁaːs/ | VY-er-shtrahss | Westphalia and Berlin, Prussia |
| William Heytesbury | /ˈheɪtsbəri/ | HAYTS-bur-ee | Oxford (England) |
| Zu Chongzhi | /tsù ʈʂʰʊ́ŋ ʈʂɻ̩́/ | dzoo chong-JRR | Jiankang (Nanjing), Liu Song China |
| Zu Gengzhi | /tsu kɤŋ ʈʂɻ̩/ | dzoo gung-JRR | Liang dynasty China |
Appendix D
Whose name is on it, and who did the work
Stigler's law of eponymy says no scientific discovery is named after its original discoverer. Stephen Stigler proposed it in 1980 and joked that Robert Merton had discovered the law first. The 1980 paper sits behind a paywall, so the Merton joke is unverified here and Appendix L.3 lists it (S460).
Every row below was verified against a source. This table is a ready-made lesson on its own.
| Named for | Who had it first | The evidence | Sources |
|---|---|---|---|
| L'Hopital's rule | Johann Bernoulli | Contract of 17 March 1694: 300 pounds a year from 1 January 1694, buying exclusivity and silence. L'Hopital's preface does thank the Bernoullis generally; the body text never credits him | S315, S316 |
| Maclaurin series | James Stirling, 1717 | Lineae Tertii Ordinis p. 32, twenty-five years early. Stirling said in 1730 it was Taylor's; Maclaurin (art. 751) credited Taylor too | S312, S451 |
| Taylor series | Gregory, Leibniz, Johann Bernoulli (1694) | Gibson 1921. Bernoulli's priority claim was about Prop. XI, not Prop. VII Cor. 2 | S312 |
| Rolle's theorem | Rolle 1691, but unproved and only about polynomials | And Rolle called the calculus "a collection of ingenious fallacies" in print from 1700 to 1705 | S459 |
| Cavalieri's principle | Zu Geng, c. 500 CE | 緣幂勢既同，則積不容異. About 1,100 years earlier. The name is not Cavalieri's either; it is his Prop. II.4 | S023, S135 |
| Riemann integral | Darboux, 1875, for what is taught | Riemann's own definition uses arbitrary partitions and arbitrary tags. The upper and lower sums students learn are Darboux's reformulation | S391 |
| The Fundamental Theorem of Calculus | Gregory 1668, Barrow 1670, Cauchy for the modern statement | The name itself dates only to 1942 (Sherwood and Taylor) | S136, S158, S362 |
| Cauchy-Schwarz inequality | Bunyakovsky, 1859 | Integral form with the equality case, 26 years before Schwarz 1885. Bunyakovsky retired the same year he published it and lived another thirty years watching other names go on it | S457, S458 |
| Holder's inequality | Rogers, 1888 | Holder 1889 | S458 |
| Jensen's inequality | Holder, 1889 | And Jensen said so himself, and was ignored anyway | S458 |
| Simpson's rule | Kepler 1615, Cavalieri 1639, Gregory 1668, Newton | Simpson said so himself | S451, S462 |
| Newton's method | Simpson, 1740, for the iteration taught | Newton's own version was algebraic and derivative-free | S451, S462 |
| Stirling's approximation | De Moivre, disputed | Not settled from the primary sources | S451 |
| The Witch of Agnesi | Curve named by Guido Grandi (1718) as versoria, a sail rope | Agnesi wrote "che dicesi la Versiera", "which is called", at Vol. 1 p. 381. The English "witch" is Colson's translation, edited by John Hellins and published in 1801, p. 222: "which is vulgarly called the Witch". Colson's synopsis went further and made it hers: "which she calls the Witch". Mazzotti offers the confusion with avversiera (she-devil) only as a "perhaps", so it is probable, not proven | S077, S319, S320 |
| The Leibniz series for pi | Madhava, c. 1400 | Roy 1990. Note that the Madhava attribution itself rests on Nilakantha's testimony | S062 |
| Gregory's series | Madhava, c. 1400 | Same | S062 |
| Pascal's triangle | Halayudha, al-Karaji, Khayyam, Jia Xian, Yang Hui, Zhu Shijie | Cajori already wrote "known in the West as Pascal's arithmetical triangle" in 1919. The correction is over a century old | S456 |
| Vandermonde determinant | Nobody | Vandermonde never discussed it | S451 |
| The enri (Japanese circle principle) | Takebe Katahiro, not Seki | Credited to the teacher for two hundred years, handed back to the pupil recently | S430 |
| Backpropagation | Linnainmaa, 1970 | 1986 was popularization. Reinvented at least four times | S436, S437, S438 |
The one who went the other way
Every row above is a name in the wrong place. Here is the opposite, and it is worth knowing that the opposite exists.
In 1882 Henri Poincare (1854 to 1912) published a family of functions he had discovered, and named them after somebody else. He called them fonctions Fuchsiennes and groupes Fuchsiens, after Lazarus Fuchs, in Mathematische Annalen volume 19 (S453). He later used Kleinian for a second family, after Felix Klein (S453).
Klein objected to the Fuchsian name, and Poincare wrote back defending it. The reply appeared in the very next volume of the same journal, volume 20, pages 52 to 53. Poincare's reason was that he wanted to commemorate what Fuchs had discovered (S453).
Read that against the whole table above. Stigler's law describes what usually happens: a name drifts onto the wrong person and nobody corrects it. This is a mathematician arguing, in print, for the right to keep someone else's name on his own work. He did so against the objections of the man whose name it was. It is the rarer thing by a long way, which is why it is here.
Appendix E
Worked mathematics, re-derived and checked in code
I re-derived every result below and checked it numerically or symbolically in Python (sympy, fractions.Fraction, decimal.Decimal at 60-digit precision). Eighteen results, eighteen passes.
One check failed on the first run, and the log keeps the failure rather than hiding it. The test was written wrong: it compared a 400-term partial sum of a geometric series to its exact limit and expected exact equality. The mathematics was right and the test was wrong. The corrected version verifies Archimedes' closed form symbolically and then takes the limit. Both versions are in the log, which is the point of keeping one.
Numeric columns are right-aligned to a consistent number of decimal places. Any column that can go negative shows an explicit sign.
E.1 Archimedes: quadrature of the parabola (c. 250 BCE)
Archimedes proves the finite identity and then argues by contradiction. He never writes the third line.
Check: sympy confirms the closed form of the finite sum equals Archimedes' expression identically, and that both the infinite sum and the limit equal .
E.2 Zu Chongzhi's bounds on π (5th century CE)
Check: both bounds hold; agrees with π to 6 decimal places. It is also the best rational approximation to π with denominator below about 16,000.
E.3 Mādhava's π, decoded from verse (c. 1400)
The bhūta-saṅkhyā word-numerals decode to 2827433388233.
Check: agrees with π to 11 decimal places.
E.4 The Mādhava-Leibniz series, and why it is unusable raw
| Terms | Partial sum | Absolute error |
Correct decimals |
|---|---|---|---|
| 10 | 3.0418396189 | 9.98 × 10⁻² | 0 |
| 100 | 3.1315929036 | 1.00 × 10⁻² | 1 |
| 1000 | 3.1405926538 | 1.00 × 10⁻³ | 2 |
| 10000 | 3.1414926536 | 1.00 × 10⁻⁴ | 3 |
Ten thousand terms buys three decimal places. This is exactly why Mādhava built correction terms, and why the Yuktibhāṣā states an exactness criterion for them rather than guessing.
E.5 Oresme: the harmonic series diverges (c. 1350)
Group the terms into blocks that each sum to at least one half.
| n | H(n) | Oresme's lower bound |
Bound holds |
|---|---|---|---|
| 16 | 3.380729 | 3.000000 | yes |
| 256 | 6.124345 | 5.000000 | yes |
| 4096 | 8.895104 | 7.000000 | yes |
| 16384 | 10.281307 | 8.000000 | yes |
Check: the bound holds at every power of two up to . The series diverges, and slowly: passing 10 takes more than 12,000 terms.
E.6 Fermat's adequality: maximize (c. 1636)
Check: sympy expands the quotient to and solves, at , to .
The pedagogical point: the last step is a suppression, not a substitution of zero. You divide by e, which requires e ≠ 0, and then you delete it, which requires e = 0. That is precisely the crack Berkeley drove a wedge into in 1734.
E.7 Cavalieri and Wallis: the area under on
| n | Riemann sum, N = 200,000 |
1/(n+1) | Absolute error |
|---|---|---|---|
| 1 | 0.500002500 | 0.500000000 | 2.50 × 10⁻⁶ |
| 2 | 0.333335833 | 0.333333333 | 2.50 × 10⁻⁶ |
| 3 | 0.250002500 | 0.250000000 | 2.50 × 10⁻⁶ |
| 4 | 0.200002500 | 0.200000000 | 2.50 × 10⁻⁶ |
E.8 Newton's binomial series (1665)
Check: coefficients confirmed symbolically against sympy's series expansion.
E.9 Leibniz's product-rule error, 11 November 1675
He guessed d(xy) = dx·dy, tested it on one example, and refuted himself the same afternoon.
With x = 3, y = 5, :
| Quantity | Exact | Decimal | Verdict |
|---|---|---|---|
| Actual change in xy | 0.008001000 | the truth | |
| dx times dy | 0.000001000 | Leibniz's first guess: smaller than the actual change by a factor of 8001 | |
| x dy + y dx | 0.008000000 | correct to first order | |
| Actual minus (x dy + y dx) | 0.000001000 | exactly dx dy, the discarded second-order term |
Check: the identity holds exactly in rational arithmetic.
Why this is the best teaching artifact in the subject: Leibniz left both the wrong answer and the right answer on the page. They differ by exactly the term you are entitled to throw away.
E.10 Wallis's product for π (1656)
| Factors | 2 x product | Absolute error |
|---|---|---|
| 20000 | 3.1415533849 | 3.93 × 10⁻⁵ |
E.11 Euler and the Basel problem (1734/35)
Euler treats sin(x)/x as an infinite-degree polynomial and factors it by its roots at x = ±nπ.
Check: sympy extracts the coefficient of as exactly , and simplifies to , matching zeta(2). The partial sum to 200,000 terms gives 1.6449290669, error 5.00 × 10⁻⁶.
The step that is not rigorous is assuming an infinite-degree polynomial factors like a finite one. Making that legitimate is a large part of what Chapter 10 is about.
E.12 Lagrange's program fails: a smooth function that is not its Taylor series
This is why Théorie des fonctions analytiques (1797) could not do what Lagrange wanted.
E.13 Ibn al-Haytham's summation identity (c. 1000)
He states it only for n = 4 and k = 1, 2, 3, proving each by induction on n (S061, p. 166).
Check: verified exactly in integer arithmetic for k = 1, 2, 3, 4 and n = 4, 5, 9. His closed form for the fourth powers also matches the standard one:
E.14 The paraboloid is eight fifteenths of its cylinder (c. 1000)
Rotate the parabola x = ky² about the line x = kb². Slice into discs; the disc at height y has radius kb² - ky².
Check: sympy returns exactly (S061, p. 168).
What makes it a calculus result and not just an answer? He brackets the solid between eight fifteenths of the cylinder and eight fifteenths of the cylinder less its top slice. Then he makes the top slice as small as he likes. That is the same squeeze Fermat and Roberval use in 1636.
E.15 The cycloid's area, which Galileo weighed and threw away (1599)
For one arch of x = a(t - sin t), y = a(1 - cos t):
Check: exactly 3. Galileo's balance gave him "about three times as heavy" in 1599 and he discarded the result, believing the ratio incommensurable (S064, p. 310). The experiment was right and the intuition about it was wrong.
E.16 Wren's rectification of the cycloid (1658)
Check: exactly 8a, confirmed symbolically and numerically (8.00000000000000 for a = 1).
Why it mattered in 1658: a curved length exactly equal to a straight one. The length is four times the diameter of the generating circle, with no π in it. Curved lengths were widely believed not to be expressible by straight ones at all (S064, p. 314).
E.17 Sarasa 1649: hyperbolic areas behave as logarithms
Sarasa's claim is that in place of the logarithms of magnitudes in continued proportion, you may take the areas under a hyperbola. Here is why that works.
Equal ratios give equal areas. Geometric progression in x becomes arithmetic progression in area. That is precisely the behavior of the numbers 6, 7, 8, 9, 10 that Sarasa starts from.
The functional equation, which is what makes it a logarithm rather than merely log-like:
Products become sums. That is the property the whole seventeenth century wanted logarithms for, and here it falls out of an area.
His numerical case, checked with common ratio 2:
| Magnitude | Area from 1 | Difference from the previous row |
|---|---|---|
| 1 | 0.000000 | - |
| 2 | 0.693147 | 0.693147 |
| 4 | 1.386294 | 0.693147 |
| 8 | 2.079442 | 0.693147 |
| 16 | 2.772589 | 0.693147 |
| 32 | 3.465736 | 0.693147 |
Check: every difference is identical, at ln 2 = 0.693147. The magnitudes double; the areas step up by a constant. That is a logarithm, found as an area, in 1649, and stated in a pamphlet defending somebody else's book (S066).
E.18 The chain rule, run backwards, is backpropagation
The cost argument, which is the whole reason reverse mode exists: take a function with m inputs and n outputs. Forward mode costs roughly m passes, and reverse mode costs roughly n. Training a neural network has n = 1 (the loss) and m in the billions. Reverse mode costs one pass regardless of parameter count, at roughly two to three times a single forward evaluation (S436).
Appendix F
Disagreements between sources
129 disagreements logged. 58 were resolved from the sources, 13 partly, and 58 remain open.
The open ones are why the Disputed label exists. Every claim resting on an open conflict, or on the open half of a partly resolved one, carries that label. That holds wherever the claim appears in this document. A labeled claim is never settled fact. Italics do a different job, and the style key at the front of the book says which.
Resolved
58 of 129 logged disagreements were settled from the sources.
| Topic | One source says | Another says | Resolution |
|---|---|---|---|
| Who first stated the cone cylinder theorem | Archimedes credits Eudoxus, naming no one else, in the preface to On the Sphere and Cylinder I (Heath 1897, p. 2) S001 |
Archimedes, in The Method, says 'we should give no small share of the credit to Democritus who was the first to make the assertion with regard to the said figure though he did not prove it' S002 |
Resolved, and the resolution is the interesting part: both are Archimedes. Eudoxus first PROVED it; Democritus first ASSERTED it. The Sphere and Cylinder preface credits the proof; The Method credits the conjecture. Nobody knew about the Democritus attribution until 1906, because The Method was lost. Method: Read both primary prefaces in full; Heath 1912 introductory note pp. 10-11 states the same resolution |
| Who initiated the Oxford Calculators | Kilvington's questions on physics inspired Bradwardine's rule; Kilvington is in the first academic generation but is not credited as founder S102 |
Bradwardine is presented as the originating figure of the Merton tradition S104 |
Resolved in favor of the 'both can be true' reading, now on scholarly authority rather than inference. The Stanford Encyclopedia entry on Heytesbury (E. Jung, rev. 2026 = S101) presents Bradwardine as the earlier founding figure of the Oxford Calculators, with Kilvington as Heytesbury's likely teacher in the first generation, Dumbleton and Roger Swyneshed as contemporaries, and Richard Swineshead as successor; Heytesbury is 'the second generation'. Sylla's Dictionary of Scientific Biography entry on Dumbleton (S480) independently describes Dumbleton as taking up Bradwardine's mathematization program. So Kilvington's physical questions prompted the work, Bradwardine's 1328 rule formalised it, and 'founder' properly attaches to Bradwardine. The Merton College outreach essay (S104) is an outreach page rather than an authority on priority, and nothing here rests on it for that. Method: Re-read SEP Heytesbury in full and read Sylla DSB Dumbleton in full; compared with S102 and S104 |
| Was the mean speed theorem proved at Merton | Truesdell: the kinematic properties 'were discovered and proved by scholars of Merton college' S104 |
Heytesbury 'stated the theorem as a general rule but did not provide a formal mathematical proof' S101 |
The geometric proof that survives is Oresme's (S107), written at Paris and not at Merton. The mean speed theorem was stated at Merton and proved at Paris. Method: primary_text_check |
| Occasion of Kepler's wine-barrel work | The incident occurred at Kepler's 1613 second wedding in Linz; he watched the merchant's gauging rod S134, S154 |
Framed around the abundant vintage of 1612 rather than the wedding; primary preface (S159) confirms both S151, S159 |
Resolved from the primary source: Kepler's own 1615 preface (S159) contains BOTH framings - wine delivered up the Danube to his new house in Linz, AND his duty as a newly married householder to lay in drink. The merchant measured every barrel alike 'sine discrimine, sine respectu figurae, sine ratiocinatione vel calculo'. The two secondary accounts are not in conflict. Method: primary_text_check |
| Pascal and the toothache | Gilberte Perier: the solution came 'during a bout of sleeplessness caused by a toothache' - but the IEP adds 'Gilberte's claims are questionable' S143 |
MacTutor says only that Pascal worked on the cycloid 'while bedridden with pain'; no toothache; Whitman 1943 (S064) attributes the toothache story to W. W. Rouse Ball, not to Gilberte Pascal S153, S064 |
The toothache rests on his sister Gilberte's memoir alone, and a reference work flags her claims as questionable. Whitman (S064) traces the story to W. W. Rouse Ball's account and adds the eight-day detail, so the chain runs Gilberte to Ball to everyone else. It is a family anecdote reported by Ball, not a documented fact. Method: source_comparison |
| Did Cavalieri believe the continuum is composed of indivisibles | Guldin's critique and the standard textbook account assume he did S151, S135 |
Cavalieri to Galileo, 2 Oct 1634: 'I absolutely do not declare to compose the continuum by indivisibles'; and 28 June 1639: 'I have not dared to say that the continuum was composed of these' S156 |
Cavalieri's own letters settle it: he explicitly and repeatedly denied the atomistic reading. Even the 1911 Britannica concedes 'It is possible to contend that Cavalieri did not himself hold the unsound doctrine' Method: primary_text_check |
| Did Torricelli prove the infinite surface area of his solid | Standard textbook claim: a finite volume with infinite surface area, proved by Torricelli in 1643/44 (no scholarly source found for this attribution) |
Torricelli proved the finite volume using indivisibles but did NOT establish the infinite surface area; that came later with integral calculus S157 |
S157 is the only scholarly source here and it is clear: Torricelli proved the finite volume, and the infinite-area painter's paradox framing came later. Method: source_comparison |
| Where the anni mirabiles work was done | Popular and traditional account: Newton did his greatest work alone in seclusion at Woolsthorpe Manor during the plague years 1665-67 S181, S182 (Conduitt, relaying Newton) and general tradition |
Whiteside, from the Fitzwilliam pocket-book and the Trinity commons accounts: Newton was in Cambridge Sept 1664-June 1665, again 20 March-mid-June 1666, and from 22 April 1667. 'The conclusion seems inescapable that Newton wrote most of them while in Cambridge (and hence with full access to university and college libraries) rather than in seclusion in Lincolnshire.' Part of the Lincolnshire time was at Boothby Pagnell, not Woolsthorpe S190 (Whiteside MP 1: 8, notes 20 and 21) |
Whiteside wins on documentary evidence. The plague scattered Newton between Cambridge, Woolsthorpe and Boothby Pagnell, and the October 1666 tract is the one securely dated Lincolnshire product. Method: Compared Newton's/Conduitt's retrospective testimony against contemporaneous financial and travel records as analyzed by Whiteside |
| What the apple did | Popular version: an apple fell on Newton's head No source. I could find no 17th- or 18th-century source for the head-strike |
Stukeley 1752, from Newton's own telling: the notion of gravitation came to him 'occasion'd by the fall of an apple, as he sat in a comtemplative mood' - he saw it fall and asked why it fell perpendicularly. Conduitt independently: 'he took the first hint of it from seeing an apple fall from a tree' S180 (RS MS/142, f. 15r), S181 (Keynes Ms. 129.02, f. 2v) |
Resolved. The apple fell; it did not hit him. Both early witnesses say 'seeing' / 'the fall of an apple ... as he sat', not impact Method: Went to the two earliest manuscript witnesses and read the exact wording |
| The year Newton died | Newton's own monument in Westminster Abbey: 'OBIIT XX MAR MDCCXXVI' - died 20 March 1726 S200 |
Modern reference works: died 20 March 1727 (Old Style) = 31 March 1727 (New Style) S200 (same source, explaining the convention) |
Resolved, and worth teaching. The two dates are the same day. Before 1752 the English civil year began on 25 March, so 20 March still fell in the civil year 1726; the historical year, which begins on 1 January, makes it 1727; and the Gregorian calendar, 11 days ahead by then, makes it 31 March 1727. The full form is 20 March 1726/7 (OS) = 31 March 1727 (NS). Method: Read the inscription and the repository's own explanation of the dating convention |
| Date of the first use of the integral sign | First written 29 October 1675, in Analyseos tetragonisticae pars secunda ("It will be useful to write ∫ for omn.") S230, S238 |
MacTutor: "21 November 1675: wrote manuscript using ∫f(x)dx integral notation for first time; manuscript also contained the product rule for differentiation" S235 |
The symbol dates from 29 October 1675. MacTutor is probably compressing three things into one: the symbol on 29 October 1675, the full form ∫f(x)dx with dx after the sign on 11 November 1675 per Cajori, and the analytic product rule on 21 November 1675. The primary manuscripts in Child settle it. Method: Read the primary manuscript translation (Child pp. 79-83, 90-102, 182) and cross-checked against Cajori via Miller. |
| When Leibniz corrected his wrong product rule | He asserted d(xy)=dx.dy and d(x/y)=dx/dy, and refuted both LATER IN THE SAME MANUSCRIPT of 11 November 1675: "although just above I stated that this was the case, and it appeared to be proved" S230 |
Popular accounts (and the common textbook telling) say he corrected the error "days later", pointing at the 21 November 1675 manuscript S230, S235 |
Both events are real and they are two different events. On 11 November 1675 Leibniz made the error, then recognized it and retracted it on the same sheet. On 21 November 1675 he derived the correct rule analytically for the first time. The pair is worth more than either one alone. Method: Read Child pp. 100-102 (the manuscript) and p. 182 (Child's own chronological summary). |
| Length of Nova Methodus | Six pages common secondary description |
pp. 467-473 of the October 1684 Acta Eruditorum; the article ends part-way down p. 473, where L. Strauss's article begins S231 |
The article runs pp. 467-473 and stops part way down p. 473, where L. Strauss's article begins, which is where the shorter counts come from. 'Six pages' is a rounding rather than an error, and 'six and a half pages' is closer, but the page range is the exact form. Method: Counted from the running heads and the start of the following article in the 1684 volume scan. |
| Whether Leibniz saw Newton's De analysi | Yes, but in OCTOBER 1676, on his second London visit, from Collins's papers; his surviving extracts (found by Gerhardt) copy the series and equation-solving material, not the tangent/quadrature material S230 |
The dispute-era claim (and much popular retelling) that he saw it on the 1673 visit and took the calculus from it S189 |
Resolved: 1676, not 1673, and the content of what he copied shows he took series methods, not the differential calculus, which he already had (the notation is dated to 1675). Method: Read Gerhardt's account of the two folios of extracts in Child's translation, pp. 167-170, against Newton's own narrative in the 1715 Account. |
| The altered date on the 11 November manuscript | The sheet reads "11 November 1673"; the 5 was altered to a 3 in darker ink, apparently by Leibniz S230 |
Child argues the alteration was an honest later misreading by Leibniz of his own phrase superiore anno, and that "The date 1675 is incontestable" S230 |
Resolved on the internal evidence: the paper develops the manuscript of 1 November 1675 directly, so 1675 is the year. The alteration on the sheet is real and Newton's side used it against Leibniz. It is a genuine ambiguity honestly resolved, not a forgery. Method: Read Child pp. 90-91 including Gerhardt's ink observation. |
| Year of Newton's anonymous Account | 1714: Philosophical Transactions no. 342 is for January and February, which concluded the year 1714 in England's Julian calendar S189 |
1715: the same months were the first two months of 1715 on the Continent's Gregorian calendar S189 |
Both years are correct, because the English legal year began on 25 March. The issue is no. 342 (January and February 1714/5), pp. 173-224, and the double year is the honest form of it. Method: Wilkins's editorial note to the TCD edition states both explicitly. |
| Date Newton's De analysi reached Barrow and Collins | Child's text reads "the paper which Newton sent in June, 1699, to Barrow, from whom Collins received it on July 30, 1699" S230 |
The universally accepted dates are June 1669 and 31 July 1669 general scholarship |
Resolved: 1669. The '1699' in the Child text is a typographic or OCR corruption, and Barrow died in 1677, so the sentence as printed cannot be right and no quotation of it appears here. Method: Internal inconsistency: De analysi could not have been sent to Barrow in 1699, since Barrow died in 1677. |
| When the English legal year began | Child, p. 171 n. 82: "the commencement of the legal year in England was altered from May 25 to January 1" S230 |
The English legal year began on 25 MARCH (Lady Day), not 25 May general scholarship / S189 editorial note |
Resolved: Child is wrong; it is 25 March. His worked example (November 18/28 1676) is nonetheless correct. Method: Cross-checked against the Wilkins note on Phil. Trans. no. 342 and the standard rule for double-dating between 1 January and 25 March. |
| The "D-ism / Dot-age" title | Babbage: "I suggested that the most appropriate title would be: The Principles of pure D-ism in opposition to the Dot-age of the University" S239 |
It is often quoted as though it were the actual title of an Analytical Society publication common secondary usage |
Resolved: the title was a suggestion, recalled by Babbage in 1864, and nobody used it. The volume appeared as Memoirs of the Analytical Society (Cambridge, 1813), so the line is Babbage's joke rather than a title page. Method: Read Babbage, Passages from the Life of a Philosopher, pp. 28-30 and his own bibliography of the 1813 Memoirs. |
| Whether Newton wrote the anonymous 1715 Account | Yes: Wilkins's edition states "This Account appeared anonymously, but is known to have been written by Sir Isaac Newton"; Newton's holograph draft survives as CUL MS Add. 3968 ff. 67r-96v (c. 24,000 words), cataloged by the Newton Project S189, S282 |
The article itself carries no author and speaks of Newton in the third person throughout S189 |
Resolved beyond doubt by the surviving manuscript: the holograph draft is in Newton's own hand, CUL MS Add. 3968 ff. 67r-96v. It is the single most damning documented fact in the affair and it needs no hedging. Method: Read the published text in full and checked the Newton Project catalog record for the holograph. |
| Composition and dates of the 1712 Royal Society committee | Conventionally: Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robartes, Bonet, de Moivre, Aston, Brook Taylor; appointed 6 March 1711/12, reported 24 April 1712 common secondary accounts |
Newton, anonymously: "The Committee was numerous and skilful and composed of Gentlemen of several Nations"; he names nobody. Kollerstrom names only Halley, as secretary "for the three months of its existence" S189, S284 |
Resolved for membership. De Morgan, Essays on the Life and Work of Newton (1914), pp. 27-29 (S288), names all eleven: Halley, Jones, De Moivre and Machin (mathematicians and Newton's friends); Brook Taylor (mathematician, friend of Keill); Robartes, Hill, Burnet, Aston and Arbuthnot (non-mathematicians, Aston and Arbuthnot 'intimate personal friends of Newton'); and Bonet, the Prussian minister in London. They were appointed 'at different times in March' 1712 and reported in April 1712. De Morgan states that the names were never officially published with the report, which is exactly why Newton's anonymous Account (S189) could say only 'numerous and skilful and composed of Gentlemen of several Nations', the several nations being Bonet. Still open: the day-level dates conventionally given, appointed 6 March 1711/12 and reported 24 April 1712, are not confirmed by any full text obtained, so the dates read March 1712 and April 1712 until somebody reads the Royal Society Journal Book. De Morgan's verdict: 'the Committee ... were not judges at all'; the members were 'unscrupulous partisans'; Leibniz was given no notice and never invited to produce documents. Method: Read De Morgan 1914 in full text on archive.org (public domain); cross-checked against S189 and S284 |
| Cauchy's death year | Cauchy (1789-1867) S360 (matharticles.com reprint of Grabiner 1983, p. 185) |
Cauchy died 23 May 1857 at Sceaux S372 |
1857 is correct; the 1867 in the reprint is a typographic error and not a scholarly claim Method: cross-check against biography; the error appears only in this reprint |
| Wording of Cauchy's 1823 methodological motto | 'avec la simplicite que produit la consideration directe des quantites infiniment petites' S367 n.11 |
'avec la simplicite qui resulte de la consideration directe des quantites infiniment petites' S362 (1823 first edition, Avertissement) |
The 1823 first edition reading stands. The other wording changes the phrasing and not the meaning, and the primary is what this book quotes. Method: read the primary scan |
| Abel's 'invention of the devil' quotation | 'The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever', commonly attributed to a letter to Holmboe of 16 January 1826 S385 (attributes only to a popular secondary book, with no letter, recipient or date) |
Abel's securely attested printed sentences on divergent series are in the 1826 binomial memoir S364 (Oeuvres 1881, vol. 1, pp. 66-67) |
Resolved. The letter is genuine and the received English wording is a translation of a translation. Abel to Holmboe, dated 16 January 1826, written from Berlin and not Paris, which he reached only in July 1826. Dano-Norwegian original, p. 16: “Divergente Rækker ere i det Hele noget Fandensskab, og det er en Skam at man vover at grunde nogen Demonstration derpaa.” French translation printed in the same 1902 volume by P. G. la Chesnais, p. 16: “Les séries divergentes sont en bloc une invention du diable, et c'est une honte que l'on ose fonder sur elles la moindre démonstration.” So four things follow. ‘Invention of the devil’ is la Chesnais's phrase and not Abel's, who wrote ‘noget Fandensskab’, ‘some devilry’. The English drops Abel's qualifier ‘i det Hele’ and ‘en bloc’, meaning on the whole, which turns a hedge into an absolute. The second half, ‘it is a shame that one dares found any demonstration on them’, is faithful. And the volume's editors argue the letter was written on 14 January and dated 16 January when it was sent. All of that is documented, so the sentence can appear here with the original beside the translation and with the caveat attached (S490). Method: Read both texts verbatim in the 1902 Memorial (archive.org item nielshenrikabelm00hols) through the item's own full-text index (fulltext/inside.php), which returns the surrounding paragraph and the scan leaf; printed page numbers taken from the item's nielshenrikabelm00hols_page_numbers.json (leaf 161 = p. 13 and leaf 164 = p. 16 of the French section; leaf 297 = p. 13 and leaf 300 = p. 16 of the parallel original-language section). Translator identified at leaf 16; the editors' dating note (“mais est certainement du 14”) at leaf 265; ‘lettre à Holmboe datée de Berlin’ at leaf 461. |
| Wording of the Hermite quotation | 'I turn away with fright and horror from this lamentable plague of functions with no derivatives' common English rendering, incl. the project brief |
French original: 'Je me detourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n'ont point de derivees' S365 (Correspondance vol. II, Lettre 374, 20 May 1893, p. 318) |
Primary settles it. 'effroi et horreur' = dread/fright and horror; and the functions are CONTINUOUS functions with no derivatives. English versions that drop 'continuous' destroy the point Method: read the primary French text |
| When Cauchy went into exile with Charles X | Cauchy followed Charles X into exile in 1830 project brief / common accounts |
Cauchy left Paris September 1830 for Switzerland, then Turin, and went to Prague to tutor Charles X's grandson only in 1833 S372 |
MacTutor's staged chronology is the accurate one. The 1830 departure and the 1833 Prague tutorship are different events. Method: biography read |
| Weierstrass's age on becoming professor | Professor at 39, after one paper project brief / common accounts |
Berlin professorship October 1856; born 31 October 1815, so 40 turning 41 S373 |
MacTutor's dating is the accurate one. The '39' probably comes from his age, 38, when the 1854 Abelian functions paper appeared. Method: biography read; arithmetic checked |
| Length of Weierstrass's schoolteaching | 14 years project brief |
Deutsch Krone 1842-1848, Braunsberg from 1848, university post October 1856: roughly 14-16 years depending on what is counted S373 |
'Roughly fourteen years' holds up for 1842 to 1856 in secondary schools. The count runs from 14 to 16 depending on what you include, so 'roughly' is the honest word and a precise number is not. Method: biography read |
| Kovalevskaya's Stockholm chair | The first professorship for a woman in northern Europe, 1889 project brief |
'the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university', June 1889 (extraordinary professor from June 1884) S374 |
MacTutor's careful formulation is the accurate one: Kovalevskaya was 'the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university', in June 1889. She had been an extraordinary professor since June 1884, which the bare '1889' hides. Method: biography read |
| Who first constructed the real numbers | Dedekind and Cantor, 1872 standard textbook accounts |
Meray published 'the earliest coherent and rigorous theory of the irrational numbers to appear in print' in 1869 S378 |
Meray has priority in print, with 'the earliest coherent and rigorous theory of the irrational numbers to appear in print' in 1869, three years before the 1872 work of Dedekind and Cantor. Textbooks routinely leave him out, which is the reason to name him. Method: biography read |
| When Bolzano's function was published | 'published in 1922' S386 abstract and section heading |
'The manuscript Functionenlehre was published in Prague in 1930 (in the Schriften I)' / MacTutor: first published 1930 by Rychlik S386 body, S384 |
Resolved: the PROOFS (Rychlik, Jarnik) were published in 1922; the MANUSCRIPT was published in 1930. S386 is internally inconsistent on this Method: internal comparison plus MacTutor |
| What Bolzano proved about his function | Bolzano exhibited a continuous nowhere-differentiable function S386 abstract |
MacTutor calls it a 'continuous but nowhere monotonic function'; S386's body says Bolzano proved only that the non-differentiability points are DENSE, and his continuity argument was 'not completely correct' S384, S386 body |
Resolved, and the precision matters: Bolzano constructed the function around 1830, and Rychlik and Jarnik proved the full theorem in 1922. The priority claim only holds in that form. Method: internal comparison plus MacTutor |
| First publication of the Weierstrass function | 'Weierstrass ... giving the first published example' in 1872 S389 |
Presented 18 July 1872; first PUBLISHED in 1875 by du Bois-Reymond in Borchardt's Journal S386 |
Resolved: presented 1872, published 1875 by someone else Method: cross-check of two sources |
| The formula in the Euler-Diderot anecdote | S313 |
Thiebault prints , in italics; De Morgan miscopied the z as n and Cajori, Bell, Smith, Sanford and Hogben followed him S314 |
Resolved in favor of S314, which examined the printed text and states explicitly that the letter 'is however z clearly enough and not n'. The famous formula is therefore itself a transcription error inside a fabricated anecdote. Method: Gillings 1954 footnote 7 read in full; note that the underlying Thiebault volume was not independently checked |
| Who is the mathematician in the Diderot anecdote | Euler, named S313 |
Thiebault does not name anyone; he writes 'a Russian philosopher, a learned mathematician and a distinguished member of the Academy'. De Morgan supplied the name Euler S314 |
Resolved. Both articles agree that the naming is De Morgan's addition. Gillings concedes Thiebault's phrase 'seems rather definitely to refer to' Euler, but a strong inference is not a source. Method: Both articles read in full |
| Who first used the word 'integral' in its modern sense | Jacob Bernoulli, Acta Eruditorum, May 1690, p. 218 ('Ergo et horum Integralia aequantur'); the DSB calls this the first use in the present mathematical sense S324 |
Johann Bernoulli also claimed it, and D. E. Smith credits him with 'the use of the term integral in its technical sense in the calculus' S324 |
Resolved by separating two things. The first printed occurrence of the word is Jacob's, in 1690; the naming of the branch, calculus integralis, is Johann's proposal, which Leibniz accepted in 1696 over his own preferred calculus summatorius. Both belong in the story. Method: Read Miller's entries for INTEGRAL and INTEGRAL CALCULUS, including the Cajori and Smith citations, in full |
| Whether the 'ghosts of departed quantities' passage is punctuated as a statement or a question | The Luce & Jessop editorial summary prints 'The velocities of evanescent increments.' with a full stop and encloses the last clause in quotation marks S310 |
Berkeley's body text at S.35 prints 'The velocities of evanescent increments?' with a question mark and no internal quotation marks S310 |
Resolved in favor of the body text, which prints a question mark where the editors' summary prints a full stop. The whole passage is a chain of rhetorical questions, which is the point: Berkeley is interrogating, not asserting. Method: Compared the editors' summary and the printed section within the same critical edition |
| Frederick the Great's alleged 'cyclops' jibe at Euler | Frederick mocked the half-blind Euler behind his back as a 'limited cyclops' (widely repeated in popular biographies) - |
No such remark appears in MacTutor's biography of Euler, in Wikiquote's sourced Frederick II page, or in the Gutenberg text of the Voltaire-Frederick correspondence S326 |
Resolved, and the popular version is wrong in its framing. The jibe is real and locatable: Frederick II to Voltaire, 25 January 1778, writing about the failed Sanssouci fountain, calls him 'the Cyclop Euler' (Eckert's translation, S342): 'I wanted to make a fountain jet in my Garden; the Cyclop Euler calculated the effort of the wheels for raising the water to a basin ... My mill was constructed mathematically, and it could not raise one drop of water to a distance of fifty feet from the basin. Vanity of Vanities! Vanity of mathematics.' So it is one remark in one letter, written in 1778 about events of 1749, and not a standing nickname. It is also unjust: Eckert shows Euler was involved for about a month in autumn 1749, correctly diagnosed that the pump capacity was too large, warned Frederick, and was ignored. The sentence and Eckert's correction belong together. The French wording has not been checked against the Trier Oeuvres de Frederic le Grand, so the English translation is the safe one. Method: Located the letter through Eckert (Deutsches Museum), read in full text; supersedes the earlier negative search |
| Whether L'Hopital concealed his debt to Johann Bernoulli | The 1696 preface names Bernoulli explicitly and invites the Bernoullis to reclaim whatever they please of the contents S316 |
There is not a single acknowledgment to Bernoulli in the body text, although half a dozen other people are credited by name S315 |
Resolved, and the resolution is the interesting part: both are true. L'Hopital made a general acknowledgment in the preface and then never credited Bernoulli for a single specific result. The contract required Bernoulli's silence, so the preface was the most L'Hopital could concede without breaching his own arrangement. It is a case study in how a technically honest statement can still mislead. Method: Read the 1716 preface in the original French and Truesdell's section IV in full |
| Who introduced the arrow in limits | Commonly credited to G. H. Hardy, A Course of Pure Mathematics (1908) (common textbook claim) |
J. G. Leathem introduced it in Volume and Surface Integrals Used in Physics (1905); Hardy's own Preface credits "Mr J. G. Leathem and Mr T. J. IA. Bromwich"; Leathem stated in 1925 that it "has been erroneously attributed to another writer owing to its use, with inadvertent omission of acknowledgment, in an important book published three years later" S451, S139 |
Resolved for Leathem 1905. Hardy popularised it and said so. The "three years later" book is Hardy's. Method: Cajori §636 and Miller quote both Hardy's Preface and Leathem's 1925 Preface in full |
| Who introduced the partial-derivative symbol | Universally credited to Jacobi, 1841 (common textbook claim) |
Condorcet used a rounded d for a partial difference in 1770; Legendre published the modern form in 1786 and abandoned it in 1787; Jacobi reintroduced it in 1841 without any historical check S451, S139 |
Resolved as a three-stage story. Cajori §611: "the mathematical public, uncritical on matters of priority, is crediting Jacobi with devices first proposed by Legendre and Hamilton." Jacobi is the reason we use it; Legendre is the reason it exists. Method: Read Cajori §§596 and 611 in full, including Legendre's and Jacobi's own sentences |
| Date of the epsilon in Cauchy | Existing row T-R365 dates the epsilon of rigor to 1823 S360, S362 |
Cajori §634 and Miller both cite Cauchy, Cours d'analyse (1821), pp. 49, 50, 61, and Oeuvres II.3 p. 54, for epsilon used as an arbitrary small quantity in a limit argument S451, S139 |
Resolved in favor of 1821 for the symbol in use and 1823 for the first full epsilon-delta proof, in Calcul infinitesimal, Lecon 7. Both dates are right about different things. Method: Read Cajori §634 and Miller's Delta and epsilon entry, which quotes the 1821 passage |
| Whether the definite-integral limits are Fourier's 1822 or earlier | Fourier, Theorie analytique de la chaleur (1822), p. 252 S139 |
Cajori §623: "Fourier had used this notation somewhat earlier in the Memoires of the French Academy for 1819-20, in an article of which the early part of his book of 1822 is a reprint" S451 |
Resolved: Fourier in both cases. The first appearance is a paper of 1819-20 and the famous one is the 1822 book, so the notation is Fourier's, 1822, and earlier in 1819-20. Method: Cajori §623 read in full |
| What "continuous" means in an eighteenth-century text | Modern: a function with no jumps, defined by an epsilon-delta or limit condition S361, S363 |
Euler, Introductio vol. 2 (1748): "A continuous curve is one such that its nature can be expressed by a single function of x", and a curve needing different functions on different parts "we call discontinuous" S453 |
Resolved, and it is a trap worth knowing about: the two senses are not merely different, they can disagree outright. A function given by two formulas that agree at the join is discontinuous in Euler's sense and continuous in ours. Method: Miller's CONTINUOUS entry read in full, quoting Euler via Katz p. 524 |
| Whether Whewell coined "scientist" and about whom | The word appears in the 1834 Quarterly Review article, attributed by the anonymous author to "some ingenious gentleman" at the British Association, and reported as "not generally palatable". The book under review is On the Connexion of the Physical Sciences (1834) S411 |
Popular accounts say Whewell coined "scientist" specifically to describe Mary Somerville, in a review of Mechanism of the Heavens (1831) popular accounts, the claim is absent from MacTutor Somerville (S414) and from Somerville's own memoir (S412) |
Resolved against the popular version, from the primary text. The review is of the wrong book, and the reviewer attributes the word to a third party rather than claiming it. What the evidence supports: the word scientist first appears in print in an anonymous 1834 review of Mary Somerville's On the Connexion of the Physical Sciences, written by William Whewell. The word appears nowhere in Somerville's Personal Recollections. Method: Read the 1834 article in full from a scan of the original, and searched the whole of Somerville's memoir for the word |
| Whether Mechanism of the Heavens was a required text at Cambridge | Peacock wrote on 14 February 1832 that he and Whewell "have already taken steps to introduce it into the course of our studies at Cambridge, and I have little doubt that it will immediately become an essential work to those of our students who aspire to the highest places in our examinations"; the whole 750-copy edition then sold chiefly at Cambridge S412 |
Commonly stated flatly as "it became a required textbook at Cambridge" popular accounts |
Resolved to the precise version. Peacock and Whewell took steps to bring the book into the course of study and expected it to be essential for the top Tripos candidates, which is not the same as requiring it of every undergraduate. Peacock's own sentence is sharper than any paraphrase of it. Method: Read the commissioning and adoption correspondence in Somerville's own memoir |
| Attribution of the enri (circle principle) | Modern research leads most historians to attribute the enri to Takebe Katahiro S430 |
For many years it was believed to be Seki's, with only the writing due to Takebe; MacTutor's Seki page still credits Seki with "undocumented calculus discoveries" S429 |
Resolved in favor of Takebe, on the explicit statement of the more detailed page. The enri is not Seki's, and this is a reattribution from master to pupil, which runs the opposite way from the usual direction of these corrections. Method: Compared the two MacTutor pages, which disagree in emphasis |
| Marjorie Lee Browne's doctorate year | 1949 - all requirements completed, and the convention used in her own lifetime S428 |
1950 - the earliest graduation ceremony was February 1950, and from 1999 some references switched to this date S428 |
Resolved as a documented ambiguity with a known cause: she completed the requirements in 1949 and the earliest graduation ceremony was February 1950. The full form is completed 1949, conferred February 1950. Method: MacTutor states both facts and the reason for the discrepancy |
| Whether Rumelhart, Hinton and Williams invented backpropagation | No. It was "brought to fame" by them in 1986; the reverse mode was published by Linnainmaa in 1970, formalised by Werbos in 1974, and implemented automatically by Speelpenning in 1980 S436 |
A very common textbook and press claim that the 1986 paper introduced the algorithm popular accounts |
Resolved. Two independent peer-reviewed surveys say explicitly that 1986 was popularisation. Both note that the PDP group had not read Parker and that Parker had not read Werbos. Method: Read both surveys in full |
| Gladys West: computer model and Hall of Fame year | MacTutor text: "IBM Stretch 7073" and "inducted into the Space and Missiles Pioneers Hall of Fame in 2028" S426 |
The IBM Stretch was the model 7030; the Air Force induction is elsewhere dated 19 December 2018, consistent with the other 2018 honors listed in the same paragraph S426 (its own reference list, Colorado Springs Gazette, 19 December 2018) |
Resolved as typographical errors in the source. The machine is the IBM 7030 Stretch and not the 7073, and the Hall of Fame induction is 2018 and not 2028. Both errors are on the record here so that nobody reintroduces them from the same page. Method: Internal consistency check against the same page's own reference list |
| Who called the cycloid "the Helen of Geometers" | Whitman 1943 was the expected source for the phrase and for its coiner S064 |
Whitman 1943 does not contain the phrase anywhere in its eight pages; Martin 2010 (S161) uses "The Helen of Geometry" as a title with no citation S064, S161 |
Closed as unattributable from the sources available. Whitman's full text was read and searched on 2026-08-14, and the words Helen, Troy, Trojan and discord do not occur in it. The phrase has no owner here, so it carries no name. Method: Full-text read and keyword search of S064, 2026-08-14 |
| Whether Kerala mathematicians used ibn al-Haytham's summation identity | Katz 1995 states that in deriving the sine series the Indian mathematicians used ibn al-Haytham's equation for the case k=1, and sketched a proof of the general result themselves S061 |
Katz also states the Indian mathematicians did not refer to ibn al-Haytham or any other predecessor S061 |
Not a contradiction, and worth stating carefully. The identity is used and the attribution is absent. Katz treats independent rediscovery as likely and says separately that 'there seems a good chance that transmission to India did occur'. So the mathematical link is documented and the transmission is open, and they are two different questions. Method: Read S061 in full, 2026-08-14 |
| Status of du Chatelet's Principia translation | Described in this document as "still the standard French edition" S332 |
Allen 2007 states it is "to this day the only complete French translation of this great work" S069 |
Resolved to the stronger and more precise claim. Allen 2007 says du Chatelet's is 'to this day the only complete French translation of this great work', which is why it is still the standard one. Method: Read S069 in full, 2026-08-14 |
| Whether the Principia geometric style was a barrier to Continental readers | The standard story is that its classical geometry made it hard for Continental analysts to use - |
Fellmann 1988 shows Leibniz had 'not the slightest difficulty in mastering Newton's arguments in detail' (p. 20) and 'did not find it particularly difficult to convert the substance of Newton's text into his own mathematical form' (p. 24). The evidence of limited engagement is about attention, not intelligibility S070 |
The usual framing is wrong. The obstacle was not that Continental mathematicians could not read the Principia. Huygens, the leading Continental figure, preferred geometry himself, and Fellmann attributes his distance from the calculus to 'conservative adherence to a strictly geometrical conception' (p. 14). The analytical reworking has a dated chain: Johann Bernoulli teaching l'Hospital and Varignon in Paris from 1692, Varignon transposing the propositions into Leibnizian symbolism from 1700, then Hermann's Phoronomia in 1716, then Euler. Method: Read Fellmann 1988 in full text, 2026-08-14 |
| Page and date for the versiera passage and the "witch" mistranslation | Cupillari 2014 puts the passage at Instituzioni p. 380, spells the confused word "aversiera" with one v, does not mention Grandi, and derives "versiera" from a change of concavity or "verse" S075 |
Mazzotti 2001 n. 44 gives the passage at Vol. 1, p. 381; spells it "avversiera"; states the term "versiera" (Lat. "versoria") was coined by Guido Grandi in 1718; and dates the English "witch" to Colson's translation AS EDITED BY JOHN HELLINS AND PUBLISHED IN 1801, at p. 222, reading "which is vulgarly called the Witch" S077, S320 |
Resolved in favor of Mazzotti, who quotes the Italian and the English with page numbers and names the coiner. The passage is at Vol. 1 p. 381; Grandi coined versoria and versiera in 1718; and the English 'witch' comes from Colson's translation as edited by Hellins and published in 1801, at p. 222. Two refinements follow. Colson died in 1760, so he translated before then, and the printed book is 1801 with Hellins as editor, which is why both names belong on it. And Mazzotti hedges the mechanism with 'perhaps because of an interesting confusion with... avversiera', so 'probably' is as far as the evidence goes and 'because' goes further than she does. Method: Read Mazzotti 2001 and Cupillari 2014 in full text, 2026-08-14. Note: Cupillari 2014 p. 5 prints the versiera as , which contradicts her own opening on the same page and her own algebra. It is a published typo. Do not propagate it. |
| The Bologna chair: appointed, offered, or neither | Cupillari 2014 says the Pope offered the position and "she never took the position", without dates S075 |
Mazzotti 2001 documents it: Benedict XIV's letter of 10 September 1750 informed her of appointment to a "lettura"; the university diploma is dated 5 October 1750 and names the post "Cattedra di Pubblico Lettore di Matematica"; "Agnesi never went to Bologna to take up this teaching post". Documents at Biblioteca Ambrosiana O.202.Sup S077 |
Resolved. The appointment was real, documented and dated: Benedict XIV's letter of 10 September 1750 told Agnesi of it, the university diploma is dated 5 October 1750, and the post is the 'Cattedra di Pubblico Lettore di Matematica'. She never went to Bologna to take it up. Method: Read Mazzotti 2001 in full text, 2026-08-14 |
| Agnesi's later life and death | This document previously said she "spent her remaining forty-seven years running a poorhouse for women in Milan, where she died destitute" S327, S343 |
Mazzotti 2001: she volunteered at the Ospedale Maggiore; in 1771 Archbishop Pozzobonelli asked her to direct the FEMALE DEPARTMENT of the newly opened Pio Albergo Trivulzio, which eventually housed about 450 patients; in 1783, having renounced her possessions, she MOVED IN; she died of pneumonia in January 1799 and was buried in a mass grave outside Porta Romana while war ravaged northern Italy S077 |
Resolved to Mazzotti, who is precise where the earlier sources were loose. Agnesi directed the female department of the Pio Albergo Trivulzio and not the institution; the directorship began in 1771 and not in 1752; and she was buried in a mass grave outside Porta Romana, which the earlier accounts leave out. Method: Read Mazzotti 2001 in full text, 2026-08-14 |
| Whether Newton had a proof of Principia Lemma 12 and withheld it | Newton prints only "Patet ex Conicis" (it is clear from the Conics) in De motu and "Constat utrumque ex Conicis" in 1687, proving nothing S076 |
Del Centina and Fiocca found a three-line proof in Newton's own hand, written as an unpublished "Corollary 6" in his copy of De Witt/van Schooten, a volume from BARROW'S library, using Euclid III.36. Lemma 12 as printed in 1687 is defective (the conjugate-diameter condition is missing) and the marginal corollary carries the identical defect, which is their evidence the two were written together S076 |
Resolved, and it refutes a specific legend. William Whiston claimed in 1749 that Newton 'saw it by intuition' and 'used it before it had ever been demonstrated by any one'. Apollonius, Saint Vincent and La Hire had all demonstrated it, and so had Newton himself in private. The limit matters: this is one lemma. Del Centina and Fiocca say nothing about calculus versus geometry in general, and the words calculus, fluxions and analysis do not appear in their paper. Method: Read Del Centina and Fiocca 2020 in full text, 2026-08-14 |
Partly resolved
13 entries.
| Topic | One source says | Another says | How far it got |
|---|---|---|---|
| Whether Archimedes handled actual infinity | The Method demonstrates 'Archimedes' concept of absolute infinity' (Walters) / shows an understanding of infinity 'scholars believed ancient Greeks couldn't grasp' (Smithsonian) S004, S006 |
The specialists who produced the reading say the key application of Lemma 11 is left implicit: 'we did not have anything explicit in Archimedes' text to support this statement'; the text may be 'the produce of a collaboration' with later interpolation; and the argument 'does remain strange' S003 |
Open in principle, and the specialists who produced the reading are more careful than the press accounts of it. In Method Prop. 14 Archimedes sets up a one-to-one correspondence between two infinite collections, calls them 'equal in multitude', and then uses a proportion between infinite aggregates. The decisive inferential step is not stated by him and nobody knows who supplied it, so 'Archimedes understood infinity' claims more than the palimpsest shows. Method: Compared the institutional/press accounts with the underlying SCIAMVS paper, read in full |
| Death date of Ibn al-Haytham | 1040 S058 |
possibly 1039 (the same source allows both); Katz 1995 (S061) gives 965-1039 S058, S061 |
Leaning to 1039. Katz (S061), the standard English-language citation for ibn al-Haytham's calculus-adjacent work, gives 965 to 1039 without qualification, and the 1040 variant survives in other reference works. The dates read c. 965 to 1039, and the death year stays unsettled between 1039 and 1040. Method: Read Katz 1995 in full text 2026-08-14 |
| How much did Barrow have, and did Newton take it from him | Child: 'ISAAC BARROW was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication'; Toeplitz: Barrow was 'the real discoverer' S158, S138 |
Whiteside: Barrow was merely 'a thoroughly competent university don' whose importance lay in 'coordinating of available knowledge'; Mahoney: 'competent and well informed, but not particularly original'; Mahnke and Hofmann concluded Leibniz was independent; Feingold 1993 (S073) takes a third position: he explicitly declines Child's thesis, but argues the revisionist dismissal was an overcorrection produced by the priority dispute itself S138, S073 |
Explained rather than settled, which is the useful outcome. Feingold shows why the verdicts diverge: both camps had a propaganda use for Barrow. De Morgan saw it in 1835 and called Barrow 'a sort of retrenched position, on which to fall back in case of defeat'. Feingold sets his own limits and states them himself: 'Due recognition of Barrow shifts no credit for the invention of the calculus away from Newton and Leibniz' (p. 338). The dispute and its explanation belong together, and so do Feingold's argument and his disclaimer. Method: Read Feingold 1993 in full text, 2026-08-14 |
| Date Newton was admitted to Trinity College | Conduitt's draft memoir states Newton was admitted to Trinity 'the 5th of Iune in 1660' S181 (Keynes Ms. 129.02, f. 1v) |
Standard scholarship (and the college admission book) gives 5 June 1661 S190 (Whiteside MP 1, which works from the college records) |
Almost certainly a slip by Conduitt. The admission date is 5 June 1661, and Conduitt's draft memoir, an early manuscript, says 1660. Method: Compared Conduitt's draft against Whiteside's use of the Trinity admission and commons records; did not inspect the admission book itself |
| Charta volans | Conventionally: a single anonymous sheet dated 29 July 1713, circulated for Leibniz, quoting an unnamed "eminent mathematician" (Johann Bernoulli) from a letter of 7 June 1713 common secondary accounts |
No full text of the sheet, and no scholarly edition of it, was obtained , |
Partly resolved from a primary source. Newton's own draft reply (CUL MS Add. 9597/2/18/94, Newton Project NATP00290 = S289) establishes three things: the sheet was printed in Germany and circulated from August 1713 'without the name of the Author Publisher Printer or place'; it quoted the judgment of an unnamed eminent ('primary') mathematician from a letter dated 7 June 1713, and Newton quotes the Latin 'literis 7 Junij 1713 datis'; and Newton's charge is that 'it was set on foot by Mr Leibnitz himself'. On the date, the Newton Project titles the sheet 29 July 1713 and the Cambridge Correspondence of Isaac Newton prints it as no. 1009 dated 18 July 1713, an eleven-day gap, which is Old Style against New Style and almost certainly the same day. Still open: nobody obtained the text of the sheet itself, and the identification of the unnamed mathematician as Johann Bernoulli is an editorial inference rather than something the sheet or Newton's draft says, so 'Johann Bernoulli wrote it' goes past the documents until somebody reads Correspondence VI no. 1008-1009. Method: Read Newton's draft response in full (Newton Project normalized and diplomatic texts); Correspondence VI not obtained |
| Whose integral do students learn? | The Riemann integral standard textbooks |
What is taught (upper and lower sums, upper and lower integrals) is Darboux's 1875 reformulation; Riemann's own definition uses tagged sums over arbitrary partitions S391 (Werke, pp. 239-240) |
Riemann's own definition is the one recorded in S391, and what students learn is Darboux's 1875 reformulation. That attribution is standard, and nobody working on this book read Darboux 1875, so it stays second hand. Method: primary read for Riemann; Darboux not obtained |
| Publication year of Riemann's Habilitationsschrift | 1867, in Goettingen Abhandlungen vol. 13 S391 (header of the Werke printing) |
MacTutor gives no year for the dissertation and 1868 for the geometry lecture S375 |
Likely 1867 for the trigonometric-series paper and 1868 for the geometry lecture; volume 13 spans 1866-67 Method: header of the Werke reprint; needs the original Abhandlungen title page |
| Date of Bolzano's dismissal | Suspended December 1819 after Austrian government pressure S371 |
Removed from his professorship by Emperor Franz in January 1820 S370 |
Probably two acts rather than one: an administrative suspension in December 1819 and a formal imperial removal in January 1820. A single date says nothing here unless it also says which act it means. Method: two tier-3 sources compared; primary decree not consulted |
| Did Cauchy lose Galois' memoir? | Cauchy lost or suppressed Galois' memoir widespread legend, incl. the framing in the project brief |
The memoir that vanished was the February 1830 Grand Prize submission, held by Fourier, who died on 16 May 1830. Cauchy's documented role in 1829 was to referee and advise Galois to resubmit S380, S372, S383 |
The legend is not supported and the documents correct it. Taton 1971, the decisive study, could not be obtained (S394), so nothing here goes further than S380 does. One exception: S380 puts Fourier's death in April 1830. S383 gives the day, 16 May 1830, and this book follows S383. Method: Biography read; Taton inaccessible. Taton 1971 could not be obtained. |
| Wording and language of Johann Bernoulli's 'lion' remark about Newton's anonymous brachistochrone solution | Bernoulli said, in Latin, 'tanquam ex ungue leonem' - 'as the lion is known by his claw' S321 |
Bernoulli wrote in FRENCH in the June 1697 Histoire des ouvrages des scavans, p. 455: 'par cet echantillon, comme ex ungue Leonem'; the Latin 'tanquam' is Brewster's 1831 substitution S337 |
Partly resolved. Brewster 1831 p. 194 was read directly and does print 'tanquam ... ex ungue leonem' as a quotation. The 1697 original could not be retrieved, so the French reading rests on a citation trail rather than on the page. Until somebody sees the 1697 text, what stands is that Bernoulli recognized Newton and said so in print, and that the Latin wording belongs to Brewster and not to Bernoulli. Method: Read Brewster 1831 in full text; searched archive.org, Google Books and Gallica for Histoire des ouvrages des scavans 1697 without success |
| Date of Newton's dot (fluxion) notation, and what the dots printed in the 1736 Method of Fluxions are | Two early datings. Cajori §567: "Newton's earliest use of dots, pricked letters, to indicate velocities or fluxions is found on a leaf dated May 20, 1665", citing S. P. Rigaud, Historical Essay on ... Newton's Principia (1838), Appendix p. 23, and adding "no facsimile reproduction of it has ever been made". Separately, the 1736 printed Method of Fluxions uses dotted letters throughout for a treatise written in 1671, and popular accounts put the notation in the anni mirabiles on the strength of it S185 (Colson 1736, p. 20), S451 |
Whiteside, MP 3: 73 n.86: "Newton himself did not introduce this standard Newtonian dot-notation till late 1691." In the October 1666 tract the rates are p, q, r; in the 1671 autograph they are l, m, n, r. William Jones's 1710 transcript substituted dots and Colson translated Jones's copy, which is where the 1736 dots come from (existing register row T-NEW-05) S183, S185, S192 |
Half settled and half open, and the two halves come apart cleanly. Settled: the dots printed in the 1736 Method of Fluxions are William Jones's 1710 substitution and not Newton's own hand, so any image captioned 'Newton's 1666 notation' that shows dots is showing Jones's tidying-up. The October 1666 tract uses p, q, r and the 1671 autograph uses l, m, n, r. Open: Cajori's 20 May 1665 leaf of pricked letters, reported at second hand from Rigaud 1838 with no facsimile ever made, and argued over by Enestrom and Witting in Bibliotheca mathematica 1910-12. Whiteside's dating rests on the surviving manuscripts and carries the weight for what Newton habitually wrote; Cajori's leaf, if it exists, would be an isolated early instance and not a practice. So Newton's dot is first securely attested in the 1690s, reached print in 1693, and the dots you see in the 1736 book are Jones's. Method: Read Whiteside's editorial note at MP 3: 73 n.86 and checked the 1666 tract transcription independently; read Cajori §567 in full including its footnote and compared it against the related Newton register rows. Two separate research notes had each answered a different half of this question; combined here into one entry. |
| The proposition numbers in Saint-Vincent that carry the hyperbola-logarithm result | Sarasa 1649 cites them directly. His marginal references read, in the Google OCR of the long-s type, "Demonstrationem vide l. 6 de Hyperb. Prop. 129" and "Demonstrationem habes l. 6 de Hyperb. Prop. 130" S066 |
The Elenchus (table of contents) of the Opus geometricum lists, under Liber Sextus De Hyperbola, Pars Tertia, "Proprietates admirandae de superficiebus inter hyperbolam, asymptoton unam, et duas parallelas alteri asymptoto interiacentibus" at propositions 125 to 129, and the continued-proportionals result immediately after S067 |
Consistent. Both witnesses point to Book VI (De Hyperbola), Part III, propositions in the 125 to 130 range. The individual digits are not legible with confidence in a 1649 long-s scan, and the two independent witnesses agree on the book and the part but not on a number, so the reference is a range rather than a single proposition. One caveat: the copy of the Opus geometricum obtained is the volume holding the front matter, the full Elenchus, and Books I to V. Book VI is not in it, so nobody has read the proposition texts. Method: Read Sarasa 1649 in full and the Opus geometricum Elenchus, 2026-08-14 |
| How much of the Principia's mathematics is withheld calculus | Whiteside, from the working manuscripts: the Principia is built natively from limit-ratios of infinitesimal increments, with one clear case of suppressed analysis, the solid of least resistance S191, S070 |
Guicciardini: there are 'obvious traces of the use of integration techniques in many demonstrations of the Principia' and Newton 'did not give the reader any detail on these highly algorithmic techniques'; when Gregory and Cotes asked how to finish proofs depending on the quadrature of curves, Newton told them privately how to apply his integral tables S204 (pp. 465-466, and n. 45 pointing to Guicciardini 1999, pp. 179-186) |
Partly resolved, and the resolution narrows the entry above rather than settling it. Withheld quadrature working in the Principia is now established beyond Whiteside's single exception: Newton supplied the missing integrations privately to his own editor. But Guicciardini also states that the mathematics Newton put on the page is the geometric fluxional calculus of the Geometria curvilinea, using limits and not infinitesimals, which is Whiteside's picture and not Newton's 1715 one. So S204 supports 'Newton hid quadrature technique' and does not support 'Newton worked the book out in the new analysis and translated it'. Guicciardini himself calls the extent an open research front, naming Nauenberg (2001), C. Wilson (2001) and Brackenridge (2003) Method: Read S204 in full and compared it against the entry above; separated the two distinct claims that entry had been carrying as one |
Open: still disputed
58 entries. Every one of these is marked with the disputed style wherever it appears in this document.
| Topic | One source says | Another says | What would settle it |
|---|---|---|---|
| Date of the Archimedes Palimpsest colophon | Scribe Johannes Myronas completed the prayer book on 14 April 1229 S004 |
The colophon recovered by X-ray fluorescence reads 'Ioannes Myronas, on April 29, 1229' S006 |
Compared the two accounts; no primary transcription of the colophon obtained |
| Dates of Madhava of Sangamagrama | c. 1340-1420 S050 |
1350-1425 (MacTutor); 1340-1425 (Almeida & Joseph) S055, S051 |
Compared three secondary sources; no primary dating evidence exists, as Madhava's own works are lost |
| Whether Sharaf al-Din al-Tusi used the derivative | Rashed: al-Tusi's implicit use of the derivative 'arises from algebraic proofs based on analytical procedures' S053 |
Hogendijk: 'a rather different approach, not one analogous to the modern derivative, lay behind Al-Tusi's method' S053 |
Not adjudicated; primary papers not obtained (Rashed's is paywalled in Arabic Sciences and Philosophy) |
| Publication year of Charles Whish's paper on the Kerala series | 1835, Transactions of the Royal Asiatic Society of Great Britain and Ireland, vol. 3 S051 |
Commonly cited as 1834; MacTutor project chapter does not resolve it S052 |
Compared secondary citations; primary volume not consulted |
| Whether Kerala mathematics was transmitted to Europe | Almeida & Joseph argue transmission via Jesuit missionaries is established by motivation, opportunity, communication routes, methodological similarity and evidence of Jesuit transmission activity (ARSI Goa 38/46/58) S051 |
Pearce: 'There is no evidence of direct transmission by way of relevant manuscripts'; the Jesuit resemblances may be 'mere coincidence'. Almeida & Joseph themselves concede there is no direct evidence and that the route to the manuscripts is conjecture. S052, S051 |
Read both arguments in full; the disagreement is about evidentiary standards, not about the facts |
| Date of Jyeshthadeva's Yuktibhasa | c. 1530 S050 |
c. 1550 S055 |
Compared two secondary sources |
| Death date of Nilakantha Somayaji | c. 1550 (implying he lived to about 106) S050 |
'died after 1501'; born 14 June 1444; Katz 1995 (S061) gives a third set, 1445-1545 S056, S061 |
Three independent date sets now on file (S050, S056, S061). No primary record of his death located. Print the birth date, which is settled, and treat the death year as disputed. |
| Birthplace of Aryabhata I | Kusumapura, near Pataliputra (modern Patna), in the north S059 |
Kerala, Tamil Nadu, Andhra Pradesh or Bengal have each been proposed S059 |
Noted the range of scholarly proposals reported in the source |
| Auction date of the Archimedes Palimpsest | 28 October 1998 S004 |
29 October 1998 S005 |
Compared two institutional sources |
| Bradwardine's birth year and birthplace | Born 1290 in Sussex S104 |
Born circa 1295 in Chichester, England S103 |
Comparing the sources against each other |
| Publication year of Harmonices/Harmonice Mundi | 1618, Book V ch. 3 S152 |
1619 S154 |
Comparing the sources against each other |
| Publication year of Arithmetica Infinitorum | 1656 S145 |
1655 S151, S139 |
Comparing the sources against each other |
| Why the Jesuits suppressed indivisibles | Because the method undermined efforts to establish a perfect rational and hierarchical order modeled on Euclidean geometry S133 |
Because indivisibles, being non-quantitative parts, conflict with the doctrine of the Eucharist; Alexander conflates indivisibles with infinitesimals S132 |
The experts disagree; a new primary source would settle it |
| Whether the indivisible/infinitesimal distinction explains the 17th-century fight | The distinction is essential; indivisibles but not infinitesimals conflict with the Eucharist S132 |
The Jesuits objected to all forms of the method, so the distinction cannot explain the fight S133 |
The experts disagree; a new primary source would settle it |
| What Fermat meant by adequality | Approximate equality, with e best read as an infinitesimal (Weil, Mahoney, Katz-Schaps-Shnider) S130 |
Breger: adaequare means simply 'to set equal', a formal algebraic procedure with no element of approximation S130 |
The experts disagree; a new primary source would settle it |
| Who first established that differentiation and integration are inverse operations | Gregory proved it in Geometriae pars universalis (1668) S136 |
Barrow 'was the first to recognize that integration and differentiation are inverse operations' (Lectiones Geometricae, 1670) S140 |
Comparing the dates the sources give |
| Fermat's birth year | 17 August 1601, Beaumont-de-Lomagne S144 |
A rival tradition gives 1607/1608 (arising from a baptismal record); MacTutor notes 'some dispute exists about this date' S144, S130 |
Flagged from a note inside the source itself |
| Publication year of Sluse's tangent rule | 1672, Philosophical Transactions S146 |
1673 is also widely cited (no source read confirms 1673) |
Only one source; a second independent source would settle it |
| Did Newton find the Principia's results by calculus and then recast them geometrically? | Newton, anonymously, 1715: 'By the help of the new Analysis Mr. Newton found out most of the Propositions in his Principia Philosophiae: but because the Ancients ... admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically ... And this makes it now difficult for unskilful Men to see the Analysis by which those Propositions were found out' S189 (Phil. Trans. 29 no. 342, p. 206) |
Whiteside, from the working manuscripts: the Principia's 'underlying mathematical edifice is in the main built up from demonstrations ... appealing in a wholly non-classical manner ... to the limit-ratios of infinitesimal ... increments'; 'there is no evidence that Newton sought deliberately to be any more esoteric therein than he needed be'; and Newton's 1715 claim 'was very much in the current fashion' (Johann Bernoulli said the same of his own work). Whiteside identifies exactly ONE clear case of suppressed analysis: the solid of least resistance, Bk II Prop. XXXV (1687); Fellmann 1988 (S070) does not state the received claim at all, and independently corroborates only Whiteside's single exception, the solid of least resistance S191 (Whiteside MP 6: 24-26, incl. note 74), S070, S072, S204 |
Still open. Cohen's Guide to the Principia addresses this directly at section 5.8, titled 'Methods of Proof versus Method of Discovery; The New Analysis and Newton's Allegations about How the Principia Was Produced; The Use of Fluxions in the Principia', at Guide p. 122, with related material at 5.4 (p. 114), 6.2 (p. 129), 10.7 (p. 316) and 10.13 (p. 345). Cohen's word 'Allegations' shows he treats Newton's 1715 account as a claim to be assessed. The PDF obtained contains the Guide's table of contents only, not its body. Guide pp. 114-117, 122-127, 316-317 and 345-347 are now the highest-value unobtained pages anywhere in this book's research, ahead of Guicciardini. PARTLY NARROWED 17 Aug 2026 by Guicciardini 2004 (S204), logged separately at Appendix F. S204 establishes that quadrature working was withheld from the Principia and supplied privately to Gregory and Cotes, which goes beyond Whiteside's single exception; it does NOT endorse the 'worked in calculus, then translated' story, because it describes the book's own mathematics as the geometric fluxional calculus of the Geometria curvilinea, built on limits rather than infinitesimals. S204 n. 44 and n. 45 point to Reading the Principia (1999), pp. 179-186, for the detailed reading. Cohen's Guide remains the top request. |
| When the gravity insight happened | Conduitt: 'in the year 1665 when he was retired to his own estate on account of the Plague, he discovered his system of gravity' S181 (f. 2v) |
Newton's own memorandum (CUL Add. 3968.41, f. 85r): November 1665 direct method of fluxions, January 1666 theory of colours, May 1666 inverse method, 'And the same year I began to think of gravity extending to the orb of the moon ... All this was in the two plague years of 1665 and 1666' S190 (Whiteside MP 1: 152 n.25, giving the shelfmark and the printing history) |
Compared two retrospective testimonies against each other and against Whiteside's editorial assessment |
| Newton's decimal-place showing-off: pi, or logarithms? | Widely repeated: Newton computed pi to 15 (or 16) decimal places in 1666 using the binomial series, and said 'I am ashamed to tell you to how many figures I carried these computations, having no other business at the time' Tier-4 web sources only (Wikipedia-derived). I could NOT trace this quotation to any manuscript, critical edition or scholarly citation |
VERIFIED ALTERNATIVE: Newton's own note (CUL Add. 4000, f. 14v): 'in summer 1665 being forced from Cambridge by the Plague I computed y[e] area of y[e] Hyperbola at Boothby in Lincolnshire to two & fifty figures by the same method.' Whiteside: that is 51 decimal places, correctly rounded to 50D. The working sheets are CUL Add. 3958, ff. 78r-80v, which Cambridge's own catalog describes as 'logarithms up to more than 50 decimal places' S190 (MP 1: 8), S199 (CUDL MS Add. 3958) |
Searched Whiteside MP 1 full text for 'ashamed' (no hits) and for the pi digits; traced the quotation only to tertiary sources |
| How many words of alchemy Newton wrote | Frequently attributed to Keynes: 'about a million words' Common secondary usage |
Keynes 1946, actual wording: 'I have glanced through a great quantity of this at least 100,000 words, I should say.' That is Keynes's estimate of what HE read, not a total for the corpus S193 |
Read Keynes's exact sentence; could not reach the Chymistry project's own word-count statement (robots-blocked) |
| Was 'standing on the shoulders of giants' a jab at Hooke? | Popular reading (associated with Robert K. Merton and with the claim that Hooke was short and stooped): the line was a sneer Secondary/tertiary |
Against: the metaphor is a medieval commonplace attributed to Bernard of Chartres by John of Salisbury; the immediately preceding sentences credit Descartes and then Hooke generously; and the February 1675/6 letter belongs to a conciliatory phase of the relationship S197 (catalog record and quoted text, FULL TEXT NOT YET VERIFIED) |
Located the manuscript (HSP item 9792) and the critical-edition reference but could not obtain a verified full transcription |
| Newton's only speech in Parliament | Widely repeated: Newton spoke once in the Commons, to ask that a window be closed (or opened) No source found. Circulates only in tier-4 material |
The History of Parliament's scholarly register records no speech by Newton at all, and describes him as not 'a particularly active Member' whose main business was the coinage bill S201 |
Read the History of Parliament entry in full; searched for an early printed source and found none |
| Is Newton's proof of Principia Book I Prop. I (Kepler's area law) mathematically adequate? | Whiteside (MP 6, note 19) and E. J. Aiton (1989): the continuum limit is inadequate; it applies only to infinitesimal arcs S191 (Whiteside MP 6) as reported in S202 |
Nauenberg: 'the critics of Prop. 1 have misunderstood Newton's fundamental limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3'; neither Whiteside nor Aiton commented on Lemma 3 S202 |
Read Nauenberg's paper in full; did not read Aiton 1989 |
| The date of William Chaloner's execution | 22 March 1699, at Tyburn Tertiary sources only |
The Newton Project dates Chaloner's last surviving letter to Newton to early March 1698/9, which is consistent S196 |
Checked the Newton Project MINT catalog dating; did not reach a primary execution record |
| Date Leibniz was elected FRS | 9 April 1673 S235 |
19 April 1673 (widely printed elsewhere) - |
Neither date verified against a primary record. |
| Page references printed in the 1686 subtitle of De geometria recondita | The OCR of the 1686 Acta reads "in Actis a. 1684, Maji p. 233; Octob. p. 264; Decemb. p. 586" S233 |
Walker's 1929 translation prints "Maji, p. 233; Octob. p. 467; Decem. p. 585" and footnotes: "The errors in the pages to which reference is made have been corrected by the translator." S232 |
Compared the Latin scan OCR with the translator's corrected text; page image not yet inspected. |
| Leibniz's funeral | MacTutor: "his only mourner was his secretary"; and quotes "He was buried more like a robber than what he really was, the ornament of his century" S235 |
The Stanford Encyclopedia entry on Leibniz contains no statement about the funeral at all S236 |
Checked both reference works; no primary source obtained. |
| When Kovalevskaya reached Berlin | 1870 S373 (MacTutor Weierstrass) |
1871 S374 (MacTutor Kovalevskaya) |
two pages of the same source compared |
| Year of Kovalevskaya's Prix Bordin | 1888 S374 (main text), project brief and standard accounts |
The retrieved MacTutor text renders it as 1886 in one rendering of the same page; 1889 is also printed in some accounts S374, S417 |
Noted the discrepancy in retrieval and the internal inconsistency in the existing register entry |
| Date of Dirichlet's Fourier-series convergence paper | 1828, in Crelle's Journal S379 |
Standard bibliographies give 1829 for 'Sur la convergence des series trigonometriques', Crelle vol. 4 standard bibliography (not retrieved in full this session) |
noted only |
| Was Cauchy's 1821 sum theorem wrong? | The theorem is false as stated; the missing hypothesis is uniform convergence; Cauchy's use of infinitesimals is 'precisely' what stopped him seeing it (Bottazzini); the epsilon-delta reading is the right frame (Grabiner) S360, S367 quoting Bottazzini 1986, pp. 115-116 |
Cauchy's procedures find better modern proxies in infinitesimal frameworks; his 1853 condition is stated with infinitely large n and infinitesimal remainder, not with alternating quantifiers, so the received account is internally inconsistent (Laugwitz, Katz et al.) S367, S368 |
both sides read in full; primaries read in full |
| Did Cauchy take his program from Bolzano? | Cauchy took his rigorization program, definition of continuity, convergence criterion and IVT proof from Bolzano's 1817 paper without acknowledgment Grattan-Guinness, Development of the Foundations of Mathematical Analysis from Euler to Riemann (1970), p. 54, as reported in S360 n.30 |
'not, in my opinion, valid; the similarities are better explained by common prior influences, especially that of Lagrange' S360 n.30 (Grabiner) |
One side read in full; Grattan-Guinness not obtained. Grabiner's exact sentence was verified in the full text of 'Who Gave You the Epsilon?' (S360): the contention 'that Cauchy took his program of rigorizing analysis, definition of continuity, Cauchy criterion, and proof of the intermediate-value theorem, from Bolzano's paper without acknowledgment is not, in my opinion, valid', the similarities being better explained by 'common prior influences, especially that of Lagrange'. Grabiner's documented rebuttal (her 1984 'Cauchy and Bolzano' paper) and Grattan-Guinness 1970 p. 54 were both still unobtainable. Present as a dispute, naming Grattan-Guinness and Grabiner. |
| Edition of Thiebault's Mes souvenirs, the sole source of the Euler-Diderot anecdote | Paris, 1801, 5 vols S313 |
Paris, 1804, 3 vols; the passage is on p. 141 of vol. 3 S314 |
Compared the bibliographies of two peer-reviewed articles both read in full; no copy of Thiebault obtained |
| Whether Johann Bernoulli backdated the Hydraulica to claim priority over his son | Hydraulica appeared in print in 1743, backdated to 1732; Johann did not mention it to Euler until October 1738, after Daniel's Hydrodynamica appeared in April/May 1738 S333 |
The printed title is 'Hydraulica nunc primum detecta ac demonstrata directe ex fundamentis pure mechanicis. Anno 1732', in Opera omnia 4: 387-493 (Bousquet, Lausanne and Geneva, 1742) S336 |
Read Tou's timeline in full; Craik's reference list only (paywalled). Truesdell's Opera Omnia II.12 introduction and Darrigol ch. 1 not obtained |
| Publication year of Johann Bernoulli's Opera omnia (containing the Hydraulica) | 1743 S333 |
1742 S336 |
Compared a conference timeline with a journal reference list |
| Publication year of vol. 2 of Agnesi's Instituzioni analitiche | Vol. 1 in 1748, vol. 2 'the following year' (1749) S327 |
The Biblioteca de la Universidad de Sevilla copy of Tomo II carries the imprint MDCCXLVIII (1748) S319 |
Inspected the title-page transcription in the OCR of two separate scans |
| Length of the brachistochrone extension granted at Leibniz's request | Extended from six months to twelve months S321 |
Extended so that French and Italian mathematicians would have time; MacTutor does not give a length; other accounts say to Easter 1697 S330 |
Read both accounts; no primary Acta Eruditorum announcement obtained |
| Whether James Gregory had the Taylor series before Taylor | Widely repeated: Gregory had the general series in a 1671 letter to John Collins - |
Gibson's forensic study of the priority question does not mention Gregory at all; MacTutor's page on Taylor's dispute with Continental mathematicians does not mention him either S312, S331 |
Read Gibson 1921 in full and MacTutor's Taylor pages; neither supports the Gregory claim |
| Chronology of Euler's blindness | Eyesight trouble from 1738 through overstrain on cartographic work; one eye lost by 1740 S326 |
Calinger argues the problems started earlier, and that a 1753 portrait shows the left eye still good and the right poor but not blind S326 |
MacTutor reports Calinger's revision alongside the traditional account; Calinger's book not obtained |
| Whether Newton solved the brachistochrone 'in twelve hours' after returning from the Mint | MacTutor: solved it in an evening after returning from the Royal Mint, working until 4 a.m. S330 |
Brewster's 'received this problem about five o'clock in the afternoon, as he was returning from the Mint' is attached to LEIBNIZ'S 1716 orthogonal-trajectories problem, not to the 1697 brachistochrone S321 |
Read Brewster p. 194-195 in full and MacTutor's brachistochrone page; Whiteside vol. 8 not obtained |
| Who made the jiba/jaib mistranslation that produced "sine" | Existing register row T-A003 credits Gherardo (Gerard) of Cremona, c. 1150 S060 |
Miller: "Accounts differ on who was responsible for the subsequent confusion with jaib and who first translated this word into Latin." Robert of Chester (1145) is the other candidate commonly named S453 |
Compared the existing register row against Miller's SINE entry read in full |
| First use of the word "integral" | Existing row T-E18001: Jacob Bernoulli, Acta Eruditorum May 1690, p. 218 S324 |
Cajori §620 reports Johann Bernoulli as the one "who used the term integral (first employed by Jacob Bernoulli, see S 539)" and dates the Leibniz-Bernoulli naming negotiation to their correspondence of 1696; Smith dates Johann Bernoulli's proposal of calculus integralis to 1690 S451, S453 |
Cajori §620 read in full and compared with Miller's INTEGRAL CALCULUS entry and existing row T-E18002 |
| Stirling's approximation: De Moivre or Stirling | Named for James Stirling, whose Methodus Differentialis (1730) Prop. 28 Ex. 2 p. 136 contains the formula S453 |
De Moivre had the asymptotic form without the constant; Stirling supplied (secondary, unverified at primary level) |
Miller's STIRLING'S FORMULA entry read in full; De Moivre's text not obtained |
| Whether Stigler credited his own law to Merton | Widely repeated: Stigler proposed the law in a self-exemplifying way and named Robert K. Merton as its true discoverer (tertiary sources only) |
Not verified. The 1980 paper is behind a paywall and no authorized full text was reachable S460 |
Attempted publisher fetch; no open-access copy found |
| IPA for Zu Chongzhi | Existing row P-A021 gives /tsu ʈʂʊŋ ʈʂɻ̩/ (register row P-A021) |
Standard Mandarin Chong is aspirated: [ʈʂʰʊŋ]. Suggested correction /tsù ʈʂʰʊ́ŋ ʈʂɻ̩́/, respelling unchanged (dzoo choong-JRR) S463 |
Applied the pronunciation convention set out in S463 |
| Respelling for Descartes | Existing row P-S134 gives "ruh-NAY day-KART" (register row P-S134) |
French [ʁəne dekaʁt]: the first syllable of Descartes is closer to "deh" than "day". Suggested "ruh-NAY deh-KART" S463 |
Applied the pronunciation convention set out in S463 |
| Year of Sophie Germain's Institut de France elasticity prize | The prize-winning third attempt was in the re-opened contest of 1815 S410 |
The prize is very widely printed, in popular and semi-scholarly accounts, as having been awarded in 1816 (the brief for this research block gave 1816) popular accounts (no admissible source read) |
Compared MacTutor full text against the framing in the research brief; the Institut's Proces-verbaux were not consulted |
| Whether Katherine Johnson used Euler's method for the Friendship 7 check | No NASA source read here names any numerical method. NASA's own biography (by Shetterly) says only that she ran "the same numbers through the same equations that had been programmed into the computer, but by hand". The 1960 technical report she co-authored contains no occurrence of the words Euler, Runge, Kutta or numerical integration; its method is an iterative correction of closed-form orbital equations with partial-derivative sensitivity analysis S422, S423, S424 |
The claim that she used Euler's method is extremely widespread in teaching materials, blogs and lesson plans popular accounts and Shetterly's trade book Hidden Figures (not read) |
Full-text search of NASA TN D-233 and full reading of two biographies |
| Number of research reports Katherine Johnson authored or co-authored | 26 S422 |
21 S424 |
Compared the two biographies |
| Whether Takebe performed differentiation in Tetsujutsu Sankei chapter 6 | He "stated a result in Chapter 6 which is equivalent to the statement that if a cubic polynomial takes an extreme value at a point the derivative vanishes at that point" S430 |
Ogawa: "It has been said that he had first done a calculation of a derivation for finding the maximum in Chapter 6 of the treatise, but that is not strictly true. A close look at the chapter will reveal that his method has nothing to do with the theory of differential." S430 (quoting Ogawa) |
Both positions are recorded in the same source; the primary text was not read |
| First calculus textbook in English | Charles Hayes, A Treatise of Fluxions, 1704 S439 |
Humphry Ditton, An Institution of Fluxions, 1706, is sometimes given this honor archive.org catalog record for the Ditton volume (not read) |
Compared imprint dates on archive.org; only the Hayes text was read |
| Errett Bishop's exact words about Keisler's textbook | The review exists: Bull. Amer. Math. Soc. 83(2) (1977), 205-208 S441 (MacTutor reference list) |
Widely circulated sentences ("obfuscation and devitalization of those wonderful ideas", "mathematics is common sense", "debasement of meaning") reach us only through unattributed web content tier 4, inadmissible |
Five retrieval attempts; all blocked |
| Authorship of the joint Young papers | William Young to Grace Chisholm Young: "our papers ought to be published under our joint names, but if this were done neither of us get the benefit of it. No. Mine the laurels now and the knowledge. Yours the knowledge only." S419 |
The 220 papers are cataloged under William Young's name S419 |
The primary letter is quoted in the source; the fuller correspondence (Grattan-Guinness) was not obtained |
| Whether Note G is "the first computer program" and how much is Lovelace's | The published Notes are signed A. A. L., are three times the length of Menabrea's article, and contain a loop-structured recurrence for the Bernoulli numbers with a trace table; Lovelace states in the text that she chose the harder formula deliberately to show off the engine S415 |
Babbage later claimed he supplied the mathematics of Note G; Menabrea's own article, which Lovelace translated, already names the Bernoulli numbers as candidates for hard-wired constants and already contains operation tables S415 |
Read the whole 1843 article including all seven Notes |
| Year of John Craig's Cambridge visit and his transcription of De methodis | Craig made copies of parts of the De methodis in 1684; Craig and David Gregory transcribed manuscripts after their visits to Cambridge in 1684 and 1694 respectively S204 (pp. 460, 462, citing Whiteside, Mathematical Papers vol. 7, pp. 3-4) |
Craig, having heard that Wallis was about to publish a summary of Newton's method for squaring curves in the Proposal of 1683, wrote to Newton and met him in Cambridge in 1685, and during that encounter was allowed to transcribe part of the De methodis and of the Epistola posterior; Newton had communicated the prime theorem to Craig in 1685 S204 (p. 465, citing Whiteside, Mathematical Papers vol. 7, pp. 3 ff.) |
Read S204 in full; compared pp. 460, 462 and 465; neither passage acknowledges the other |
| First-use year of the word fluxion | From Newton's own works: 1671 S185, S187, S192 |
From the priority-dispute sources: 1676 concealed in the anagram, 1693 in print. Also 1704, Charles Hayes, in the first English calculus book S189, S188, S286, S439 |
Compared the three dated first-use claims while merging them into one entry; no new source read |
Appendix G
Stories that check out, and stories that do not
This is the most immediately useful appendix for a classroom, because several of the debunked items are in books that are still on the shelf. Each verdict names what was checked.
G.1 Verified from the sources
| Story | What the source shows | Sources |
|---|---|---|
| Agnesi was appointed to a chair at Bologna and never taught | True, and now documented to the day. Benedict XIV's letter of 10 September 1750; the university diploma of 5 October 1750 naming the post "Cattedra di Pubblico Lettore di Matematica"; and Mazzotti's flat statement that she "never went to Bologna to take up this teaching post" | S077 |
| Agnesi ran a poorhouse and died in it | Close, and the precise version is better. In 1771 she was asked to direct the female department of the Pio Albergo Trivulzio, about 450 patients. In 1783, having renounced everything, she moved in. She died of pneumonia in January 1799 and, with war ravaging northern Italy, was buried in a mass grave outside Porta Romana | S077 |
| Colson coined "the Witch of Agnesi" around 1760 | Name both men and the right year. Colson died in 1760, so he translated before then, but the printed book is 1801, edited by John Hellins, and the phrase at p. 222 is "which is vulgarly called the Witch". Mazzotti hedges the avversiera mechanism with "perhaps" | S077 |
| Newton used Principia Lemma 12 before anyone had proved it | False, and it is Whiston's legend from 1749. Apollonius, Saint-Vincent and La Hire had proved it, and Newton wrote his own three-line proof by hand in the margin of a book from Barrow's library, using Euclid III.36. He printed "it is clear from the Conics" instead | S076 |
| Scholars simply disagree about what Barrow gave Newton | True, and Feingold supplies the reason: Barrow was a weapon in the priority war. De Morgan saw it in 1835: the Newtonians made him "a sort of retrenched position, on which to fall back in case of defeat". Then the twentieth-century reaction against J. M. Child, Barrow's translator and champion, overcorrected. Print the dispute with its explanation | S073 |
| Newton's partisans invented the idea that Leibniz owed something to Barrow | No. It came from Leibniz's own circle. Tschirnhaus, his confidant, wrote to him in 1678 to 1679 saying he could not see much in Leibniz's calculus beyond what Barrow had published, and called Barrow "much more readily intelligible". Then Tschirnhaus published the claim and took the credit for himself | S073 |
| The Principia's geometric style kept Continental mathematicians from using it | Not as an intelligibility problem. Fellmann: Leibniz had "not the slightest difficulty in mastering Newton's arguments in detail" and "did not find it particularly difficult to convert the substance of Newton's text into his own mathematical form". Huygens's distance from the calculus came from his own "conservative adherence to a strictly geometrical conception" | S070 |
| Sarasa discovered that the area under a hyperbola is a logarithm | True, and the context is better than the claim. He was answering a challenge problem Mersenne set partly to attack his teacher's quadrature of the circle. His sentence, in the 1649 primary: "in place of the numbers 6, 7, 8, 9, 10, 11, which were the logarithms of the magnitudes O, P, Q, R, S, T, we shall be able to take the hyperbolic quantities" | S066 |
| The result is buried in Saint-Vincent's Opus geometricum, Book VI | True, and the book's own table of contents calls those propositions admirandae, to be wondered at. But cite the range 125 to 130, not a single proposition number: the 1649 long-s scan does not render the digits reliably, and Book VI itself was not in the volume obtained | S066, S067 |
| Du Chatelet's Principia is still the standard French edition | True and understated. It is the only complete French translation of the Principia that exists | S069 |
| Du Chatelet published under her own name | Often not. The 1738 essay on fire went to the Academie anonymously, because a submission known to be by a woman would not have been taken seriously; it lost, and the Academie printed it anyway. The Institutions de physique was also published anonymously, in 1740 | S069 |
| Her Principia commentary appeared ten years late through neglect | The opposite, and the reason has a name. Clairaut, who had checked her calculations while she was alive, pressed the publisher Prault to bring it out and arranged for the illustrations to be finished | S068 |
| Galileo tried to measure the cycloid's area by weighing it | True, and the punchline is better than the story. He cut an arch and a circle from the same material, balanced them, got "about three times", and abandoned it believing the ratio incommensurable. It is exactly 3 | S064 |
| Roberval kept his methods secret because his chair was re-contested every three years | True, and Whitman states the mechanism: the incumbent set the questions, so concealment was job security. Walker: "the accident of occupying this chair caused Roberval to lose credit for many of his discoveries" | S064 |
| Christopher Wren rectified the cycloid in 1658 | True. Sent to Pascal without proof during the contest; Wallis printed it as Wren's a year later. One arch is exactly 8a, four times the generating circle's diameter, with no pi in it | S064 |
| Kerala mathematicians used ibn al-Haytham's summation identity | True and precisely located: Katz shows the identity used for k = 1 in the sine-series derivation, and notes in the same paragraph that the Indian mathematicians named no predecessor. The mathematics is shared; the citation is absent | S061 |
| Kepler invented a chunk of integral calculus because he thought a wine merchant was cheating him | True, and better in the original. His own preface: new house in Linz, wine up the Danube by barge, the merchant measuring every cask alike sine ratiocinatione vel calculo | S159 |
| Newton kept the begging letter from the counterfeiter he prosecuted | True. TNA MINT 15/17/205: "O no body can save me but you". It is still in the file | S196 |
| Newton reviewed his own anonymous report anonymously | True, and documented in his own hand: a 24,000-word holograph draft at CUL MS Add. 3968, ff. 67r to 96v | S189, S282 |
| Leibniz got the product rule wrong and corrected himself the same day | True, both on the page, 11 November 1675, with the analytic derivation following on 21 November | S230 |
| The integral sign is a long letter S for summa | True, 29 October 1675, replacing the word omnia, three days after a formula so clogged with omn. that it was unusable | S230, S238, S451 |
| "L'Hopital's rule" is Johann Bernoulli's | True. Contract of 17 March 1694, 300 pounds a year from 1 January 1694, buying exclusivity and silence | S315 |
| Jacob Bernoulli's tombstone has the wrong spiral | True. The mason carved an Archimedean spiral under a motto about a logarithmic one. Two independent sources | S328, S329 |
| Sophie Germain's parents confiscated her candles and fire | True, and they gave up when they found her asleep in the library with the ink frozen | S410 |
| Gauss corresponded with her as "Monsieur LeBlanc" and she revealed herself to save his life | True. Gauss's letter on learning the truth is verified | S410 |
| John Glenn refused to fly until Katherine Johnson checked the computer by hand | True, in NASA's own words ("get the girl") | S422 |
| Grace Chisholm Young's husband put his name on her work and promised to switch later | True, in his own letter. 220 papers later they had not switched | S419 |
| Emmy Noether lectured for four years under Hilbert's name | True | S418 |
| Seki Takakazu had determinants before Leibniz and Bernoulli numbers before Bernoulli | True, and sealed inside Japan by sakoku, so all of it had to be found again | S429 |
| Backpropagation is the chain rule in reverse | True, in two peer-reviewed surveys, with the cost argument | S436, S437 |
| Cambridge undergraduates campaigned for Leibniz notation with a punning tract title | True as a suggested title recalled by Babbage in 1864: "The Principles of pure D-ism in opposition to the Dot-age of the University". Never used | S239 |
| The Archimedes Palimpsest was read through painted-on gold forgeries using a particle accelerator | True. X-ray fluorescence at the Stanford Synchrotron detected iron in the original ink. And the first imaging attempt failed | S005, S006 |
G.2 Debunked, corrected, or unverifiable
| The story as usually told | What checking it produced | Sources |
|---|---|---|
| An apple hit Newton on the head | False. Stukeley has Newton "in a comtemplative mood" and the apple seen falling. No 18th-century source has a head strike | S180 |
| Newton spent the plague years in seclusion at Woolsthorpe | Wrong geography. Trinity commons accounts and his own pocket-book put him in Cambridge Sept 1664 to June 1665, back 20 March 1666, and back again 22 April 1667. Summer 1665 was at Boothby Pagnell, which had a library | S190 |
| Newton's dot notation dates from 1666 | False. Late 1691. The 1666 tract uses p, q, r; the 1671 autograph uses l, m, n, r. The printed dots are William Jones's 1710 substitution | S192 |
| Newton discovered the Principia results by calculus, then translated them into geometry, hiding the analysis | Contested, and the only source for the strong version is Newton writing anonymously about himself. Whiteside finds the book built natively from limit-ratios, with one exception: the solid of least resistance, Book II Prop. XXXV. Guicciardini 2004 narrows it further from the other side: quadrature working was withheld from the printed Principia but supplied privately to David Gregory and Roger Cotes, the editor preparing the second edition, which goes beyond Whiteside's one exception. Fellmann's independent survey does not state the received claim at all and corroborates only Whiteside's exception. Cohen's Guide addresses it head-on at section 5.8, and his own word for Newton's 1715 story is "Allegations" | S070, S072, S189, S190, S204 |
| Newton said he was ashamed how many figures he carried his pi computation to | Unverifiable. Traced only to unattributed sources. The verified equivalent: his own note that he computed the hyperbola's area "to two & fifty figures", with working sheets surviving | S190 |
| Keynes said Newton left a million words of alchemy | Misattributed. Keynes said "at least 100,000 words". The million-word figure is modern cataloguing | S193 |
| Newton's dog Diamond knocked over a candle and burned his papers | Not in the manuscript. A later accretion | S182 |
| Euler silenced Diderot with ", therefore God exists" | Fabricated, and the fabrication is traceable page by page. Thiebault names no one and disclaims the story; De Morgan (1872) inserts Euler; Cajori repeats it; Bell says "Chinese"; Hogben says "Arabic". And De Morgan miscopied Thiebault's z as n | S313, S314 |
| Frederick the Great habitually called Euler a cyclops | One letter, 25 January 1778, to Voltaire, 29 years after the fountain job. Euler had correctly diagnosed the oversized pump in writing and been ignored. Not a standing nickname | S342 |
| Euler joked that going blind meant fewer distractions | Not verified. Widely quoted and no source located, which is why it never appears here inside quotation marks | S326 |
| Newton solved the brachistochrone in 12 hours coming home from the Mint | Wrong problem. Brewster's Mint detail belongs to Leibniz's 1716 problem, not the 1697 one | S321 |
| Johann Bernoulli said "I recognize the lion by his paw" | Attribute to Brewster, not Bernoulli. The Latin tanquam ex ungue leonem is Brewster 1831 p. 194, rendering a French original. A verifiable case of a manufactured quotation | S321 |
| Torricelli proved the horn has infinite surface area | False. He proved finite volume by indivisibles. The painter's paradox is a later overlay. Near-universal textbook error | S157 |
| Pascal thought about the cycloid to distract himself from a toothache | Originates with his sister Gilberte, and the Internet Encyclopedia of Philosophy calls her claims questionable. The transmission route is now traced: Whitman attributes it to W. W. Rouse Ball, who adds that Pascal took the relief as "a divine intimation" and worked at it for eight days. So the chain is Gilberte, then Ball, then everyone. Report it as a family anecdote passed on by Ball | S064, S143, S161 |
| Cavalieri believed areas are literally made of lines | He denied it in writing, twice, to Galileo on 2 October 1634 and 28 June 1639 | S135, S156 |
| Berkeley wrote "they are the ghosts of departed quantities" as a statement | It is a question, in a chain of questions. The full stops and quotation marks come from later editors' summaries | S310 |
| Abel wrote that divergent series are "the invention of the devil" | A translation of a translation. Abel wrote noget Fandensskab, "some devilry", from Berlin not Paris, 16 January 1826, and hedged it with i det Hele, "on the whole", which English drops | S490 |
| Cauchy lost Galois' life work | Mostly myth. The lost manuscript was the Feb 1830 Grand Prize submission held by Fourier, who died that May. Cauchy's documented 1829 role was to referee and advise resubmission | S380, S383 |
| Kronecker drove Cantor mad | Not supported. MacTutor dates the first breakdown to 1884 and treats the illness as recurrent and independent, while documenting the professional obstruction as real | S376 |
| Hermite recoiled in horror from "functions with no derivatives" | The standard English drops the word "continuous", which destroys the point. Verified from the primary: Correspondance II, p. 318, 20 May 1893 | S365 |
| Weierstrass discovered and published the nowhere-differentiable function in 1872 | He presented it 18 July 1872 and never published it (du Bois-Reymond did, 1875), and he opened by crediting Riemann. Cellerier is a third independent discoverer | S386 |
| Whewell coined "scientist" to describe Mary Somerville | Wrong book and wrong attribution. The review was of On the Connexion of the Physical Sciences, and Whewell credits "some ingenious gentleman" | S411, S442 |
| Katherine Johnson used Euler's method for the Friendship 7 orbit | Absent from NASA's biography, MacTutor, and TN D-233 itself. The report uses iterative correction of Kepler equations plus partial-derivative sensitivity | S422, S423 |
| Hilbert said "this is a university, not a bath-house" about Noether | Not in the source. Do not assert | S418 |
| Bhaskara II stated Rolle's theorem | He stated Fermat's rule for an extremum, not Rolle's theorem. Golādhyāya 4.39 | S050 |
| al-Kashi invented decimal fractions | False. al-Uqlidisi did, around 950 | S063 |
| The Kerala series reached Europe via the Jesuits | Open question, argued by serious historians on both sides. Motive, opportunity, and resemblance are documented; no transmitting document has been produced. Teach it as a dispute | S051, S052 |
| "Method of exhaustion" is what the Greeks called it | A 17th-century coinage, and misleading: the method brackets, it does not exhaust | S008 |
| Descartes chose x, y, z because a printer ran short of other letters | Unattested. He introduced the convention without comment or argument | S450 |
| The cycloid was called "the Helen of Geometers" by [someone] | CLOSED: unattributable. Whitman 1943, the leading candidate, was read in full on 14 Aug 2026 and searched: the words Helen, Troy, Trojan and discord do not occur anywhere in its eight pages. Martin (2010) uses "The Helen of Geometry" as a title with no citation. Do not attribute the phrase to anyone | S064, S161 |
Appendix H
Image catalog and rights
36 images ship in this book, every one checked against a strict standard. An image counts as cleared only when the rights position at its repository is unambiguous, with the exact wording recorded below. The note "No license statement found" is recorded as exactly that, never treated as permission. An image this book wanted but could not clear, or could not track down at all, is not a defect to hide. It is listed, with what is missing, in Appendix L.
Cleared and downloaded
36 images. Every file listed here is on disk in images/; the paths were checked, not assumed.
| ID | What it shows | Work | Year | Repository | Rights status and the wording found | File |
|---|---|---|---|---|---|---|
| IMG-L001 | First page of Nova Methodus pro Maximis et Minimis: the first published calculus paper | Acta Eruditorum, October 1684, p. 467 | 1684 | Wikimedia Commons (scan from Internet Archive s1id13206500) | public_domain Wikimedia Commons file description page, Licensing section (retrieved 2026-08-14): "Public domain ... This work is in the public domain in its country of origin ... This work is in the public domain in the United States because it was published (or registered with the U.S. Copyright Office) before J... |
images/IMG-L001-nova-methodus-1684.jpg |
| IMG-L002 | Incipit of Nova Methodus (BEIC copy) | Acta Eruditorum, October 1684 | 1684 | Wikimedia Commons / BEIC (Biblioteca Europea di Informazione e Cultura) | public_domain Wikimedia Commons file page hidden categories (2026-08-14): 'Media from BEIC', 'CC-PD-Mark', 'PD-old-100-expired' - i.e. Creative Commons Public Domain Mark 1.0 plus public domain by age. BEIC (Biblioteca Europea di Informazione e Cultura) asserts no rights over the scan. |
images/IMG-L002-nova-methodus-incipit-beic.jpg |
| IMG-L003 | Tabula XII: the engraved plate of diagrams for Nova Methodus, facing p. 466 | Acta Eruditorum, October 1684, Tabula XII | 1684 | Wikimedia Commons / BEIC | public_domain Wikimedia Commons file description page, Licensing section (retrieved 2026-08-14): "Public domain ... This work is in the public domain in its country of origin ... This work is in the public domain in the United States because it was published (or registered with the U.S. Copyright Office) before J... |
images/IMG-L003-nova-methodus-tabula-xii.jpg |
| IMG-NEW-11 | Title page of The Method of Fluxions and Infinite Series (London, 1736), John Colson's translation | The Method of Fluxions and Infinite Series | 1736 | Internet Archive (scan); original work public domain by age | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/methodoffluxions00newt (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1736, public domain by age in ... |
images/IMG-NEW-11-method-of-fluxions-1736-title.jpg |
| IMG-NEW-12 | Page 20 of the 1736 Method of Fluxions: the definitions of fluents and fluxions with the dotted letters (note: the dots are William Jones's 1710 modernisation, not Newton's 1671 usage) | The Method of Fluxions and Infinite Series | 1736 | Internet Archive | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/methodoffluxions00newt (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1736, public domain by age in ... |
images/IMG-NEW-12-method-of-fluxions-p20.jpg |
| IMG-NEW-13 | Title page of Philosophiae Naturalis Principia Mathematica (London, 1687), first edition | Principia Mathematica | 1687 | Wikimedia Commons (Swiss copy) | public_domain Wikimedia Commons file description page, Licensing section (retrieved 2026-08-14): "Public domain ... This work is in the public domain in its country of origin ... This work is in the public domain in the United States because it was published (or registered with the U.S. Copyright Office) before J... |
images/IMG-NEW-13-principia-1687-title-page.jpg |
| IMG-NEW-13b | Title page of the 1687 Principia, Wellcome Collection copy: an alternative to IMG-NEW-13 under an explicit CC BY 4.0 license | Philosophiae Naturalis Principia Mathematica | 1687 | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-NEW-13b-principia-1687-title-wellcome.jpg |
| IMG-NEW-14 | Principia 1687 p. 26: DE MOTU CORPORUM, Liber Primus, SECT. I, 'De Methodo Rationum primarum & ultimarum, cujus ope sequentia demonstrantur', with LEMMA I - Newton's limit method in print | Principia Mathematica | 1687 | Internet Archive, Smithsonian Libraries copy of the 1687 editio princeps | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/philosophiaenat00newt (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1687, public domain by age in t... |
images/IMG-NEW-14-principia-1687-bk1-sect1-lemma1.jpg |
| IMG-ARCH-01 | Archimedes Palimpsest, prayer-book opening 110v-105r (Archimedes folio 27v), pseudocolour composite from the multispectral imaging - the erased Archimedes text made visible under the prayers | Archimedes Palimpsest (Walters Art Museum MS W.B.M. 1) | 10th c. Archimedes text; 1229 prayer book; imaged 2000-2010 | The Walters Art Museum / Archimedes Palimpsest Digital Product | cc_by Archimedes Palimpsest digital release, 0_ReadMe.txt section "1 Rights and Conditions of Use", https://archimedespalimpsest.net/0_ReadMe.txt (retrieved 2026-08-14): "The Archimedes Palimpsest data is released with license for use under Creative Commons Attribution 3.0 Unported Access Rights. It is re... |
images/IMG-ARCH-01-palimpsest-110v-105r-pseudocolor.jpg |
| IMG-ARCH-02 | Archimedes Palimpsest, same opening 110v-105r under 470 nm illumination: the plain visible-light state, for a before/after pairing with IMG-ARCH-01 | Archimedes Palimpsest (Walters Art Museum MS W.B.M. 1) | 10th c. / 1229 | The Walters Art Museum / Archimedes Palimpsest Digital Product | cc_by Archimedes Palimpsest digital release, 0_ReadMe.txt section "1 Rights and Conditions of Use", https://archimedespalimpsest.net/0_ReadMe.txt (retrieved 2026-08-14): "The Archimedes Palimpsest data is released with license for use under Creative Commons Attribution 3.0 Unported Access Rights. It is re... |
images/IMG-ARCH-02-palimpsest-110v-105r-visible.jpg |
| IMG-E18001 | Title page of The Analyst, 1734, showing the subtitle 'addressed to an Infidel Mathematician' | The Analyst | 1734 | Internet Archive, University of California / Bancroft Library copy of the 1734 first edition | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/analystordiscour00berkrich on 2026-08-14 |
images/IMG-E18001-berkeley-analyst-1734-title.jpg |
| IMG-E18005 | Title page of Instituzioni analitiche ad uso della gioventu italiana, Tomo I | Instituzioni analitiche | 1748 | Internet Archive / Biblioteca de la Universidad de Sevilla | public_domain archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/A298183 on 2026-08-14 |
images/IMG-E18005-agnesi-instituzioni-1748-title.jpg |
| IMG-E18010 | Title page of Theorie des fonctions analytiques, whose subtitle is the whole program | Theorie des fonctions analytiques | 1797 | Internet Archive / University of Toronto (Fisher) | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/theoriedesfoncti00lagr (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1797, public domain by age in ... |
images/IMG-E18010-lagrange-theorie-1797-title.jpg |
| IMG-E18017 | Title page of du Chatelet's Principes mathematiques de la philosophie naturelle, tome second, naming her on the title page as 'feue Madame la Marquise du Chastellet' | Principes mathematiques de la philosophie naturelle | 1759 | Internet Archive (Google scan) | public_domain archive.org item metadata gives archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/NewtonDuChastelletPrincipesMathmatiquesDeLaPhilosophieNaturelle on 2026-08-14. The scan itsel... |
images/IMG-E18017-duchatelet-principes-1759-title.jpg |
| IMG-E18018 | Title page of Euler's Introductio in analysin infinitorum, tomus primus, Lausanne 1748 | Introductio in analysin infinitorum | 1748 | Internet Archive / Google scan of the University of Lausanne copy | public_domain archive.org item metadata gives archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/bub_gb_jQ1bAAAAQAAJ on 2026-08-14. The scan itself bears a "Digitized by Google" watermark an... |
images/IMG-E18018-euler-introductio-1748-title.jpg |
| IMG-P001 | Title page of Cavalieri's Geometria indivisibilibus continuorum. NOTE: this is the 1653 second edition, not the 1635 first - see the rights column for why the 1635 scan cannot be used | Geometria indivisibilibus continuorum nova quadam ratione promota | 1653 (2nd ed.; 1st ed. 1635) | Wikimedia Commons | public_domain Wikimedia Commons extmetadata: LicenseShortName = 'Public domain', UsageTerms = 'Public domain' (read 2026-08-14). SEPARATE FINDING ON THE 1635 FIRST EDITION: the only full scan located, archive.org item ita-bnc-mag-00001345-001 (Biblioteca Nazionale Centrale di Firenze, 'early-european-books' colle... |
images/IMG-P001-cavalieri-geometria-1653-title.jpg |
| IMG-P002 | Title page of Wallis, Arithmetica Infinitorum, Oxford 1656 - 'Nova Methodus Inquirendi in Curvilineorum Quadraturam' | Arithmetica Infinitorum | 1656 | Internet Archive (Google scan, Bayerische Staatsbibliothek copy) | public_domain archive.org item metadata gives archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/ArithmeticaInfinitorum on 2026-08-14. The scan itself bears a "Digitized by Google" watermark... |
images/IMG-P002-wallis-arithmetica-infinitorum-1656-title.jpg |
| IMG-P003 | Title page of Kepler, Nova stereometria doliorum vinariorum, Linz 1615 - the wine-barrel book | Nova stereometria doliorum vinariorum | 1615 | Internet Archive, EPFL Library copy (chepfl-lipr-AXB124) | public_domain archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/chepfl-lipr-AXB124 on 2026-08-14 IMPORTANT CONTRAST: the scan used by another research agent, archive.org item den-kbd-pil-21... |
images/IMG-P003-kepler-nova-stereometria-1615-title.jpg |
| IMG-P004 | Title page of Barrow, Lectiones Geometricae, London 1670 | Lectiones Geometricae | 1670 | Internet Archive, 'Early English Books, 1641-1700 / Books in Microfilm' collection | public_domain No license statement found at https://archive.org/details/bim_early-english-books-1641-1700_lectiones-geometric-_barrow-isaac_1670 (2026-08-14): item metadata has no licenseurl, no rights and no possible-copyright-status field, and the item page shows no rights notice. The image itself carries no wa... |
images/IMG-P004-barrow-lectiones-geometricae-1670-title.jpg |
| IMG-R001 | Title page of the first edition of Cauchy's Cours d'analyse | Cours d'analyse de l'Ecole royale polytechnique | 1821 | Internet Archive (scan from University of Ottawa copy) | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/coursdanalysedel00cauc (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1821, public domain by age in ... |
images/IMG-R001-cauchy-cours-analyse-1821-title.jpg |
| IMG-E18006 | Fig. 135 and Problema III no. 238: Agnesi's construction of la Versiera | Instituzioni analitiche | 1748 | Internet Archive / Universidad de Sevilla | public_domain archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/A298183 on 2026-08-14 |
images/IMG-E18006-agnesi-versiera-p381.jpg |
| IMG-E18011 | Lagrange art. 16-17: the introduction of f'x and of the primitive/derived distinction | Theorie des fonctions analytiques | 1797 | Internet Archive | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/theoriedesfoncti00lagr (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1797, public domain by age in ... |
images/IMG-E18011-lagrange-derived-functions-p14.jpg |
| IMG-E18014 | Brewster 1831 p. 194, the page that created 'tanquam ex ungue leonem' | The Life of Sir Isaac Newton | 1831 | Internet Archive / Wellcome Library | public_domain archive.org item metadata 'rights' field for b33028035_0001 links to a Creative Commons Public Domain Mark (Wellcome Library contribution), read at https://archive.org/metadata/b33028035_0001 on 2026-08-14. Work published 1831, public domain by age. |
images/IMG-E18014-brewster-1831-p194.jpg |
| IMG-R002 | Page 131 of the Cours d'analyse showing Theorem I - the false theorem on sums of continuous functions - exactly as Abel cited it | Cours d'analyse, ch. VI sec. 1 | 1821 | Internet Archive | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/coursdanalysedel00cauc (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1821, public domain by age in ... |
images/IMG-R002-cauchy-cours-p131-theoreme-i.jpg |
| IMG-R005 | Abel's 1826 footnote quoting Cauchy's page 131 and giving the counterexample series | Oeuvres completes de Niels Henrik Abel, vol. 1, p. 71 | 1881 | Internet Archive | public_domain archive.org item metadata gives archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/oeuvrescomplete01abelgoog on 2026-08-14. The scan itself bears a "Digitized by Google" watermark and the Google boilerplate leaf states: "We also ask tha... |
images/IMG-R005-abel-1826-footnote-p71.jpg |
| IMG-R006 | Hermite's letter 374 to Stieltjes of 20 May 1893, showing the 'effroi et horreur' sentence in the original French | Correspondance d'Hermite et de Stieltjes, vol. II, p. 318 | 1905 | Internet Archive | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/correspondancedh02herm on 2026-08-14 |
images/IMG-R006-hermite-stieltjes-p318.jpg |
| IMG-R011 | Cauchy's definition of a continuous function: Cours d'analyse ch. II sec. 2, 'De la continuite des Fonctions', with f(x+a)-f(x) decreasing indefinitely with a | Cours d'analyse de l'Ecole royale polytechnique | 1821 | Internet Archive, University of Toronto (Fisher) copy, sponsored by the University of Ottawa | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/coursdanalysedel00cauc (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1821, public domain by age in ... |
images/IMG-R011-cauchy-cours-continuity-definition.jpg |
| IMG-POR-01 | Isaac Newton, line engraving by George Vertue, 1726, after John Vanderbank - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Isaac Newton portrait | 1726 | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-01-newton-vertue-1726-engraving.jpg |
| IMG-POR-02 | Gottfried Wilhelm Leibniz, line engraving - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Leibniz portrait | 18th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-02-leibniz-line-engraving.jpg |
| IMG-POR-03 | Leonhard Euler, stipple engraving by G. Stodart - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Euler portrait | 19th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-03-euler-stodart-stipple-engraving.jpg |
| IMG-POR-04 | John Wallis, line engraving by Michael Burghers, 1699 - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Wallis portrait | 1699 | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-04-wallis-burghers-1699-engraving.jpg |
| IMG-POR-05 | Isaac Barrow, line engraving by J. Cole - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Barrow portrait | 18th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-05-barrow-cole-engraving.jpg |
| IMG-POR-06 | George Berkeley, line engraving - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Berkeley portrait | 18th-19th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-06-berkeley-line-engraving.jpg |
| IMG-POR-07 | Johannes Kepler, line engraving by N. Dietz - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Kepler portrait | 18th-19th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-07-kepler-dietz-engraving.jpg |
| IMG-POR-08 | Joseph-Louis Lagrange, line engraving by Locatelli - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Lagrange portrait | 19th century | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-08-lagrange-locatelli-engraving.jpg |
| IMG-POR-09 | Colin Maclaurin, engraving by T. Trotter, 1798 - a pre-1930 engraving, deliberately chosen over a modern photograph of a painting because the rights are cleaner | Maclaurin portrait | 1798 | Wellcome Collection, London | cc_by Wellcome Collection image, license recorded on the Wikimedia Commons mirror of the same Wellcome file (extmetadata LicenseShortName = "CC BY 4.0"; UsageTerms = "Creative Commons Attribution 4.0"; LicenseUrl = https://creativecommons.org/licenses/by/4.0), read 2026-08-14. Image downloaded at higher r... |
images/IMG-POR-09-maclaurin-trotter-1798-engraving.jpg |
H.2 Portraits: what each one is, and its rights
78 portraits ship in this book, every one with a license read at the repository rather than assumed. Of these, 8 are marked Not from life: no likeness made while the person was alive is known to survive. So what you are looking at is a later artist’s invention. The caption says so at every place the portrait appears, not only here.
The other 44 people in the portrait register are not here because their rights are not settled or no likeness was found. They are listed, with what is missing, in Appendix L.
| ID | Person | What it is | Creator | Year | License | Credit and file page | Where it appears |
|---|---|---|---|---|---|---|---|
| PTR-001 | Zeno of Elea Not from life | Zeno of Elea as imagined by a Dutch printmaker. Line engraving by Jan de Bisschop, between 1666 and 1671. No portrait made from life survives. This is a later invention, not a record of his face. | Jan de Bisschop | 1666 | CC0 1.0 | Jan de Bisschop. Wikimedia Commons. CC0 1.0. File page | Appendix A |
| PTR-003 | Archimedes Not from life | Archimedes as imagined by a painter 1,800 years later. Oil painting by Domenico Fetti, about 1620. The Dresden gallery that owns it catalogs it as "Portrait of a scholar (Archimedes?)", with the question mark. No portrait made from life survives. This is a later invention, not a record of his face. | Domenico Fetti | 1620 | public domain | Domenico Fetti, Portrait of a scholar (Archimedes?), about 1620. Gemaeldegalerie Alte Meister, Dresden. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-004 | Li Chunfeng Not from life | Li Chunfeng as drawn for a novel. Woodcut illustration from a 1783 edition of the popular romance Shuo Tang yanyi quanzhuan, published more than a thousand years after he died. No portrait made from life survives. This is a later invention, not a record of his face. | unidentified | 1783 | public domain | Unidentified artist. This image has been extracted from another file. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-005 | Ibn al-Haytham Not from life | Ibn al-Haytham as imagined by a modern illustrator. Drawing by Michel Bakni, 2021. No portrait made from life survives. This is a later invention, not a record of his face. | Michel Bakni | 2021 | CC BY-SA 4.0 | Michel Bakni, Ibn Al-Haytham portrait. Wikimedia Commons. CC BY-SA 4.0. File page | Appendix A, chapter cast |
| PTR-006 | Nicole Oresme Not from life | Nicole Oresme at his desk. Illuminated initial from a manuscript of his Traite de la sphere, painted between 1400 and 1420, decades after his death. No portrait made from life survives. This is a later invention, not a record of his face. | Illuminator not identified | 1400 | public domain | Illuminator not identified. Nicole Oresme (1400-1420) Traité de la sphère; Aristote, De caelo et de mundo, traduction française par Nicole Oresme, p. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-007 | Johannes Kepler | Johannes Kepler. Steel engraving by Karl Barth, made between 1828 and 1853 for Meyers Konversations-Lexikon. | Karl Barth | 1828 | CC0 1.0 | Karl Barth, Portret van Johannes Kepler (RP-P-1914-1102). Rijksmuseum, Amsterdam. Wikimedia Commons. CC0 1.0. File page | Appendix A, chapter cast |
| PTR-010 | Gregoire de Saint-Vincent | Gregorius a Sancto Vincentio. Portrait, artist not identified, date not established. | Artist not identified | not stated | public domain | Artist not identified. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-011 | Rene Descartes | Rene Descartes. Portrait by After Frans Hals, 1649. | After Frans Hals | 1649 | public domain | After Frans Hals. André Hatala [e.a.] (1997) De eeuw van Rembrandt, Bruxelles: Crédit communal de Belgique, ISBN 2-908388-32-4. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-012 | Bonaventura Cavalieri Not from life | Bonaventura Cavalieri. Engraving by Giuseppe Fusinati, 1837, made 190 years after his death. No portrait made from life survives. This is a later invention, not a record of his face. | Giuseppe Fusinati | 1837 | CC0 1.0 | Giuseppe Fusinati. This file was uploaded by Biblioteca comunale di Trento. Wikimedia Commons. CC0 1.0. File page | Appendix A, chapter cast |
| PTR-013 | Pierre de Fermat | Pierre de Fermat. Painting by Rolland Lefebvre, date not established. | Rolland Lefebvre | not stated | public domain | Rolland Lefebvre. Musée d'art et d'histoire de Narbonne. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-015 | Evangelista Torricelli | Evangelista Torricelli. Oil painting by Lorenzo Lippi, about 1647. | Lorenzo Lippi | 1647 | public domain | Lorenzo Lippi. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-017 | John Wallis | John Wallis. Line engraving by Michael Burghers, 1699. | Michael Burghers | 1699 | Public Domain Mark 1.0 | John Wallis. Line engraving by M. Burghers, 1699, after W. Sonmans, 1698. Wellcome Collection (V0006134). Public Domain Mark 1.0. File page | Appendix A |
| PTR-018 | Alphonse Antonio de Sarasa | Alphonse Antonio de Sarasa. Engraved portrait from a scan of his 1649 Solutio problematis, artist not identified. | Artist not identified | not stated | public domain | Artist not identified. Available in the BEIC digital library and uploaded in partnership with BEIC Foundation. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-020 | Blaise Pascal | Blaise Pascal. Line engraving by Gérard Edelinck, date not established. | Gérard Edelinck | not stated | CC BY-SA 4.0 | Gérard Edelinck, Blaise Pascal. Line engraving after G. Edelinck after F. Quesnel, juni. Wikimedia Commons. CC BY-SA 4.0. File page | Appendix A, chapter cast |
| PTR-023 | Johann Hudde | Johann van Waveren Hudde, burgomaster of Amsterdam and mathematician. Oil painting by Michiel van Musscher. | Michiel van Musscher | 1686 | CC0 1.0 | Michiel van Musscher, Portret van Johannes Hudde (SK-C-528). Rijksmuseum, Amsterdam. Wikimedia Commons. CC0 1.0. File page | Appendix A, chapter cast |
| PTR-024 | Christiaan Huygens | Christiaan Huygens. Painting by Bernard Vaillant, 1686. | Bernard Vaillant | 1686 | public domain | Bernard Vaillant. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-025 | Isaac Barrow | Isaac Barrow. Line engraving by J. Cole, date not established. | J. Cole | not stated | Public Domain Mark 1.0 | Isaac Barrow. Line engraving by J. Cole. Wellcome Collection (V0000370). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-026 | Christopher Wren | Sir Christopher Wren. Oil painting by Godfrey Kneller, 1711. | Godfrey Kneller | 1711 | public domain | Godfrey Kneller. one or more third parties have made copyright claims against Wikimedia Commons in relation to the work from which this i. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-027 | Robert Hooke Not from life | Robert Hooke, reconstructed. Oil painting by Rita Greer, 2004, built from written descriptions by John Aubrey and Richard Waller. No portrait made from life survives. This is a later invention, not a record of his face. | Rita Greer | 2004 | Free Art License 1.3 | Rita Greer, Portrait of Robert Hooke, 2004. Free Art License 1.3. File page | Appendix A, chapter cast |
| PTR-028 | James Gregory | James Gregory. Portrait attributed to John Scougal, painted between 1661 and 1675. | John Scougal | 1661 | public domain | John Scougal, James Gregory. Fyvie Castle, Aberdeenshire. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-029 | Isaac Newton | Sir Isaac Newton. Line engraving by George Vertue, 1726. | George Vertue | 1726 | Public Domain Mark 1.0 | Sir Isaac Newton. Line engraving by G. Vertue, 1726, after J. Vanderbank, 1725. Wellcome Collection (V0004266). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-031 | Gottfried Wilhelm Leibniz | Gottfried Wilhelm Leibniz (Leibnitz). Line engraving by Unknown engraver, date not established. | Unknown engraver | not stated | Public Domain Mark 1.0 | Gottfried Wilhelm, Baron von Leibniz. Line engraving. Wellcome Collection (V0003477). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-034 | Jacob Bernoulli | Jacob (Jakob, Jacques) Bernoulli. Portrait by Niklaus Bernoulli (1662-1716), 1687. | Niklaus Bernoulli (1662-1716) | 1687 | public domain | Niklaus Bernoulli (1662-1716). [2] [3]. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-035 | Hans Sloane | Sir Hans Sloane. Oil painting by Stephen Slaughter, 1736. | Stephen Slaughter | 1736 | public domain | Stephen Slaughter, Hans Sloane, 1736. National Portrait Gallery, London. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-036 | Guillaume de l'Hopital | Guillaume Francois Antoine, Marquis de l'Hopital. Engraving by Gérard Edelinck, 1695. | Gérard Edelinck | 1695 | CC BY 4.0 | Gérard Edelinck, Guillaume François Antoine, Marquis de L'Hospital. Wikimedia Commons. CC BY 4.0. File page | Appendix A, chapter cast |
| PTR-039 | Johann Bernoulli | Johann (Jean) Bernoulli. Frontispiece portrait from volume 1 of his Opera omnia, 1742. | unidentified engraver | 1742 | public domain | Unidentified engraver, frontispiece portrait of Johann Bernoulli, Opera omnia, 1742. Niels Bohr Library and Archives, American Institute of Physics. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-041 | Guido Grandi | Guido Grandi. Painting by an unidentified 18th-century artist, in the Uffizi's Serie Gioviana. | unidentified 18th-century painter; photograph by Fabrizio Garrisi | not stated | CC0 1.0 | Unidentified 18th-century painter, Guido Grandi (Serie Gioviana, Inv. 3043). Galleria degli Uffizi, Florence. Photograph by Fabrizio Garrisi. Wikimedia Commons. CC0 1.0. File page | Appendix A, chapter cast |
| PTR-042 | William Jones | William Jones. Oil painting by William Hogarth, about 1749. | William Hogarth | 1749 | public domain | William Hogarth, William Jones. National Portrait Gallery, London. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-043 | Jean-Jacques Dortous de Mairan | Jean-Jacques Dortous de Mairan. Etching by Jean-Baptiste Delafosse after a drawing by Carmontelle, made after 1760. | Jean-Baptiste Delafosse, after Carmontelle | 1760 | CC0 1.0 | Jean-Baptiste Delafosse after Carmontelle, Jean-Jacques Dortous de Mairan (G.4800). Musée Carnavalet, Histoire de Paris. Wikimedia Commons. CC0 1.0. File page | Appendix A, chapter cast |
| PTR-044 | Charles Hayes | Charles Hayes. Portrait from a scan of his Treatise of Fluxions, artist not identified, date not established. | Artist not identified | not stated | public domain | Artist not identified. Available in the BEIC digital library and uploaded in partnership with BEIC Foundation. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-045 | John Colson | John Colson, Lucasian Professor of Mathematics at Cambridge. Oil painting by John Wollaston the Younger, 1741. | John Wollaston the Younger | 1741 | public domain | John Wollaston the Younger, John Colson, 1741. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-046 | James Jurin | James Jurin. Portrait by James Worsdale (1692-1767), about 1740. | James Worsdale (1692-1767) | 1740 | public domain | James Worsdale (1692-1767). Royal Society: Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-047 | George Berkeley | George Berkeley, Bishop of Cloyne. Line engraving by Unknown engraver, date not established. | Unknown engraver | not stated | Public Domain Mark 1.0 | George Berkeley. Line engraving. Wellcome Collection (V0000473). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-048 | Brook Taylor | Brook Taylor. Line engraving after Richard Earlom, date not established. | unidentified engraver, after Richard Earlom | not stated | CC BY 4.0 | Wellcome Collection, Brook Taylor, line engraving after R. Earlom (V0005740). Wikimedia Commons. CC BY 4.0. File page | Appendix A, chapter cast |
| PTR-049 | William Stukeley | William Stukeley. Mezzotint by J. Smith, 1721, after a painting by Godfrey Kneller. | J. Smith, after Godfrey Kneller | 1721 | CC BY 4.0 | Wellcome Collection, William Stukeley, mezzotint by J. Smith after Sir G. Kneller, 1721 (V0005648). Wikimedia Commons. CC BY 4.0. File page | Appendix A |
| PTR-050 | James Stirling | James Stirling. Portrait from a scan of his Methodus differentialis, artist not identified, date not established. | Artist not identified | not stated | public domain | Artist not identified. Available in the BEIC digital library and uploaded in partnership with BEIC Foundation. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-051 | Colin Maclaurin | Colin Maclaurin. Engraving by T. Trotter, 1798. | T. Trotter | 1798 | Public Domain Mark 1.0 | Colin Maclaurin. Engraving by T. Trotter, 1798. Wellcome Collection (V0003763). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-052 | Daniel Bernoulli | Daniel Bernoulli. Portrait by an unidentified artist, about 1750, in the ETH-Bibliothek portrait collection. | unidentified artist | 1750 | public domain | Unidentified artist, Daniel Bernoulli (1700-1782), Portr 10971. ETH-Bibliothek, Zurich. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-053 | Emilie du Chatelet | Gabrielle Emilie Le Tonnelier de Breteuil, Marquise du Chatelet. Portrait by Maurice Quentin de La Tour, date not established. | Maurice Quentin de La Tour | not stated | public domain | Maurice Quentin de La Tour. via . Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-054 | Leonhard Euler | Leonhard Euler. Stipple engraving by G. Stodart, date not established. | G. Stodart | not stated | Public Domain Mark 1.0 | Leonhard Euler. Stipple engraving by G. Stodart. Wellcome Collection (V0001799). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-056 | Denis Diderot | Denis Diderot. Portrait by Pierre-Michel Alix, date not established. | Pierre-Michel Alix | not stated | public domain | Pierre-Michel Alix. Aquatinta von Pierre-Marie Alix nach Lou-Michel Vanloo. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-057 | Alexis-Claude Clairaut | Alexis-Claude Clairaut. Engraving by Louis-Jacques Cathelin / After Charles-Nicolas Cochin, date not established. | Louis-Jacques Cathelin / After Charles-Nicolas Cochin | not stated | public domain | Louis-Jacques Cathelin / After Charles-Nicolas Cochin. SIL-SIL14-C4-01 [1], image [2]. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-059 | Dieudonne Thiebault | Dieudonne Thiebault. Manuscript miniature by Louis Marie Sicard, about 1800. | Louis Marie Sicard | 1800 | public domain | Louis Marie Sicard. Artcurial. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-060 | Joseph-Louis Lagrange | Joseph-Louis Lagrange (Giuseppe Lodovico Lagrangia). Line engraving by A. Locatelli, date not established. | A. Locatelli | not stated | Public Domain Mark 1.0 | Joseph Louis Lagrange. Line engraving by A. Locatelli. Wellcome Collection (V0003310). Public Domain Mark 1.0. File page | Appendix A, chapter cast |
| PTR-061 | Pierre-Simon Laplace | Pierre-Simon, marquis de Laplace. Oil painting by Jean-Baptiste Paulin Guérin, 1838. | Jean-Baptiste Paulin Guérin | 1838 | public domain | Jean-Baptiste Paulin Guérin. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-062 | Fourier | Jean-Baptiste Joseph Fourier. Portrait by Julien-Léopold Boilly, date not established. | Julien-Léopold Boilly | not stated | public domain | Julien-Léopold Boilly. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-063 | Sophie Germain Not from life | Marie-Sophie Germain. Portrait by Berthe Chégaray née Becher, 1896, reproduced in La Revue des revues, 15 September 1898. Painted 65 years after she died. No portrait made from life survives. This is a later invention, not a record of her face. | Berthe Chégaray née Becher | 1896 | public domain | Berthe Chégaray née Becher, Sophie Germain, 1896. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-064 | Mary Somerville | Mary Fairfax Greig Somerville. Engraving by Evert Duykinck, about 2006. | Evert Duykinck | 2006 | public domain | Evert Duykinck. Originally from en.wikipedia; description page is (was) here 13:01, 16 April 2004 Magnus Manske 774x1100 (137,872 bytes). Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-065 | Bolzano | Bernard Bolzano. Portrait by an unidentified artist, early nineteenth century. | unidentified artist | not stated | public domain | Unidentified artist, Bernard Bolzano. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-066 | David Brewster | Sir David Brewster. Oil painting by Sir John Watson Gordon, 1864. | John Watson Gordon | 1864 | public domain | John Watson Gordon, Sir David Brewster, 1864. National Portrait Gallery, London. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-067 | Cauchy | Augustin-Louis Cauchy. Lithograph by Z. Belliard after J. Roller, date not established. | Z. Belliard, after J. Roller | not stated | CC BY 2.0 | Wellcome Collection, Augustin Louis, Baron Cauchy, lithograph by Z. Belliard after J. Roller (V0001035). Wikimedia Commons. CC BY 2.0. File page | Appendix A, chapter cast |
| PTR-068 | Charles Babbage | Charles Babbage. Oil painting by Samuel Laurence, 1845. | Samuel Laurence | 1845 | public domain | Samuel Laurence, Charles Babbage, 1845. National Portrait Gallery, London. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-070 | John Herschel | John Frederick William Herschel. Photograph by Julia Margaret Cameron, about 1867. | Julia Margaret Cameron | 1867 | public domain | Julia Margaret Cameron. Metropolitan Museum of Art. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-072 | William Whewell | William Whewell. Stipple engraving by an unidentified artist, date not established. | unknown | not stated | CC BY 4.0 | Unknown artist, Portrait of W. Whewell; stipple engraving. Wikimedia Commons. CC BY 4.0. File page | Appendix A, chapter cast |
| PTR-073 | Abel | Niels Henrik Abel. Portrait by Johan Gørbitz, 1826, drawn three years before Abel died. | Johan Gørbitz | 1826 | public domain | Johan Gørbitz, Niels Henrik Abel, 1826 (A000005). Norsk portrettarkiv, Riksantikvaren. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-075 | Augustus De Morgan | Augustus De Morgan. Portrait reproduced in Sophia Elizabeth De Morgan's Memoir of Augustus De Morgan, 1882. The artist is not identified. | Artist not identified | 1882 | public domain | Artist not identified. Memoir of Augustus De Morgan. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-076 | Galois | Evariste Galois. Portrait by an unidentified artist, about 1826. | unidentified | 1826 | public domain | Unidentified artist. Iyanaga, Shokichi, "ガロアの時代 ガロアの数学 第一部 時代篇" , Springer-Verlag Tokyo, 1999 Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-077 | Weierstrass | Karl Theodor Wilhelm Weierstrass. Portrait by an unidentified artist, date not established. | unknown | not stated | public domain | Unknown artist. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-078 | Ada Lovelace | Augusta Ada King, Countess of Lovelace. Oil painting by Margaret Sarah Carpenter, 1836. | Margaret Sarah Carpenter | 1836 | public domain | Margaret Sarah Carpenter. Government Art Collection. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-079 | Cellerier | Charles Cellerier. Portrait by Verlag von GRELL FÜSSLI & Co. (1890), 1890. | Verlag von GRELL FÜSSLI & Co. (1890) | 1890 | public domain | Verlag von GRELL FÜSSLI & Co. (1890). Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-080 | Hermite | Charles Hermite. Photograph by Eugène Pirou, printed by Paul Dujardin, published 1905. | Eugène Pirou and Paul Dujardin | 1905 | public domain | Eugène Pirou and Paul Dujardin, Charles Hermite. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-082 | Riemann | Georg Friedrich Bernhard Riemann. Portrait by an unidentified hand, about 1850. | unidentified | 1850 | public domain | Unidentified, Bernhard Riemann. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-083 | Dedekind | Julius Wilhelm Richard Dedekind. Photograph by Johannes Ganz, before 1886. | Johannes Ganz | 1886 | public domain | Johannes Ganz, Richard Dedekind (1831-1916), Portr 11953. ETH-Bibliothek, Zurich. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-086 | Gaston Darboux | Jean-Gaston Darboux. Photograph by an unidentified artist, about 1917. | unidentified | 1917 | public domain | Unidentified artist. Archivio storico dell'Accademia delle Scienze - Catalogo digitale. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-087 | Cantor | Georg Ferdinand Ludwig Philipp Cantor. Portrait by an unidentified hand, about 1910. | unidentified | 1910 | public domain | Unidentified, Georg Cantor. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-088 | Kovalevskaya | Sofia Vasilyevna Kovalevskaya (nee Korvin-Krukovskaya). Wood engraving, 1884. | unidentified engraver | 1884 | public domain | Unidentified engraver, Sonia Kovalevskaia, 1884. Library of Congress Prints and Photographs Division. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-095 | Emmy Noether | Amalie Emmy Noether. Photograph by an unidentified artist, about 1910. | unidentified | 1910 | public domain | Unidentified artist. Emmy Noether (1882-1935), Archived. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-096 | John Maynard Keynes | John Maynard Keynes, 1st Baron Keynes. Photograph by an unidentified photographer, 1929. | unidentified photographer | 1929 | public domain | Unidentified photographer, John Maynard Keynes, 1929. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-098 | Srinivasa Ramanujan | Srinivasa Aiyangar Ramanujan. Passport photograph, 1913, from the Trinity College manuscripts. | unidentified photographer | 1913 | public domain | Unidentified photographer, Srinivasa Ramanujan, 1913 (Add. MS a.94/7). Trinity College, Cambridge. Wikimedia Commons. Public domain. File page | Appendix A |
| PTR-100 | Otton Nikodym | Otton Marcin Nikodym. Portrait by Uuuxxxyyyzzz, 2016. | Uuuxxxyyyzzz | 2016 | CC BY-SA 4.0 | Uuuxxxyyyzzz, Otto Nikodym Stefan Banach Memorial Bench Krakow Poland. Wikimedia Commons. CC BY-SA 4.0. File page | Appendix A |
| PTR-106 | Katherine Johnson | Katherine Coleman Goble Johnson. Photograph by NASA, 1983. | NASA | 1983 | public domain | NASA, 1983. Public domain: a work of the United States federal government. File page | Appendix A, chapter cast |
| PTR-107 | Julia Robinson | Julia Hall Bowman Robinson. Photograph by George Bergman, 1975. | George Bergman | 1975 | CC BY-SA 4.0 | George Bergman, Julia Robinson 1975 (re-scanned). Wikimedia Commons. CC BY-SA 4.0. File page | Appendix A |
| PTR-108 | Mary Jackson | Mary Winston Jackson at NASA Langley. Photograph by Bob Nye, 2 June 1977. | Bob Nye, NASA Langley Research Center | 1977 | public domain | Bob Nye, NASA Langley Research Center, 1977 (LRC-1977-B701 P-04107). Public domain: a work of the United States federal government. File page | Appendix A, chapter cast |
| PTR-110 | Gladys West | Gladys West at the Pentagon, being inducted into the Air Force Space and Missile Pioneers Hall of Fame. Photograph by Adrian Cadiz, 6 December 2018. | Adrian Cadiz | 2018 | public domain | U.S. Air Force photo by Adrian Cadiz, 2018. Public domain: a work of the United States federal government. File page | Appendix A, chapter cast |
| PTR-115 | Edmond Halley | Edmond Halley. Oil painting by Thomas Murray, about 1690. | Thomas Murray | 1690 | public domain | Thomas Murray, Edmond Halley, about 1690. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-119 | James Wilson | James Wilson. Painting by Sir John Watson-Gordon (died 1864), about 1864. | Sir John Watson-Gordon (died 1864) | 1864 | public domain | Sir John Watson-Gordon (died 1864). one or more third parties have made copyright claims against Wikimedia Commons in relation to the work from which this i. Wikimedia Commons. Public domain. File page | Appendix A, chapter cast |
| PTR-120 | Archibald Pitcairne | Archibald Pitcairne. Line engraving by R. Strange after a painting by J. Medina, date not established. | R. Strange, after J. Medina | not stated | CC BY 4.0 | Wellcome Collection, Archibald Pitcairne, line engraving by R. Strange after J. Medina (V0004687). Wikimedia Commons. CC BY 4.0. File page | Appendix A, chapter cast |
| PTR-122 | Samuel Horsley | Samuel Horsley, Bishop of St Asaph. Coloured etching by Robert Dighton, 1809, after his own drawing. | Robert Dighton | 1809 | CC BY 4.0 | Wellcome Collection, Samuel Horsley, Bishop of St. Asaph, coloured etching by R. Dighton, 1809 (V0002882). Wikimedia Commons. CC BY 4.0. File page | Appendix A, chapter cast |
Appendix I
Where each story goes in your two textbooks
This appendix is keyed section by section to your two live textbooks:
- Derivatives (Aqua):
https://derivatives.megan-warren.com/ - Integrals (Berry):
https://integrals.megan-warren.com/
Every section number and title below was taken from those two tables of contents, not from a generic syllabus. So a row here is a drop-in for the exact place you are studying, rather than a general suggestion.
I.1 How to read these tables
The Box column names a container from your own Legend, so each row is a drop-in rather than a research note:
| Your box | What it is for here |
|---|---|
| Picture This | A story that sets up the idea before the mathematics arrives. |
| Why this matters in a world that moves | A live, present-day payoff. |
| ⚠ Watch out | A historical mistake that people still make today. |
| ⛏ Dig In | Rigor: why the definition had to be tightened. |
| Dig Deeper (collapsed) | Optional depth for the student who wants it. |
| Definitions | Where the word or symbol came from. |
| Unit opener | Framing for the whole unit. |
Practice and Recap sections (x.P and x.R) are not mapped, except Derivatives 1.R where a story earns its place. Those sections do not need history, and padding them would dilute the rest.
One thing to know before you start. Integrals 1.0 already opens with Archimedes trapping a circle between two ladders of polygons. It already tells you that Newton and Leibniz found the area problem and the tangent problem to be one subject. That opener is the spine of this entire book. Chapters 1, 2, and 5 here are the long version of those two paragraphs.
I.2 Derivatives (Aqua), section by section
| Section | Box | The hook | Where it is in this document |
Sources |
|---|---|---|---|---|
| A.1 to A.3 Algebra Review (signed numbers) | Picture This | Agnesi introduced positive and negative numbers through debits and credits in her 1748 textbook, and wrote the whole thing in Italian rather than Latin, producing the first systematic Italian vocabulary for calculus. She was building a language, not just a book. | Ch. 9 | S077 |
| 1.0 Unit 1 Opener | Unit opener | Calculus is Latin for pebble, a limestone counter on a Roman reckoning board. A teacher of arithmetic in Rome was a calculo if he was a slave and a calculator if he was freeborn. The subject is named after a slave's pebble. | Ch. 1 | S453, S455 |
| 1.1 Rates of Change | Picture This | Bhaskara II introduced instantaneous velocity (tatkalika gati) around 1150 because the Moon moves too fast for a daily average to predict when a religious date ends. He calls the daily average "only a rough or approximate rate". The derivative enters Indian mathematics for the same reason it enters Newton's. | Ch. 3 | S050 |
| 1.1 Rates of Change | Dig Deeper | The mean speed theorem, that uniformly changing speed covers the same distance as the steady average, was stated at Merton College, Oxford in 1335 and proved by Oresme at Paris. It is buried in a book of logic-class brainteasers. | Ch. 4 | S101, S104, S107 |
| 1.2 The Notion of a Limit | Picture This | Zeno's Arrow paradox asks whether an instant can contain motion. You cannot answer it without defining velocity at a point by a limit. He posed it around 450 BCE; the mathematics arrives in the 1820s. | Ch. 2 | S007 |
| 1.2 The Notion of a Limit | ⛏ Dig In | Bolzano, 1817: everyone agreed a curve that starts below the axis and ends above it must cross. He said that is a picture, not a proof. His title is the argument: Purely analytic proof, where "purely" means no pictures, no motion, no geometry. | Ch. 10 | S363, S387 |
| 1.2 The Notion of a Limit | Definitions | Epsilon is the e of erreur, delta the d of différence. The symbol of mathematical precision started life as the symbol for being wrong. Cauchy, 1821 and 1823. | Ch. 10 | S360, S451 |
| 1.3 What Is a Derivative | ⚠ Watch out | Fermat's method, c. 1636: divide by a small quantity e, which requires e ≠ 0, then delete it, which requires e = 0. Berkeley called the result "the ghosts of departed quantities" in 1734, and he was mathematically right. This is the exact step students find slippery, and they are in good company. | Ch. 5, Ch. 9 Appendix E.6 |
S130, S310 |
| 1.3 What Is a Derivative | Dig Deeper | Berkeley's objection stood for 232 years. In 1966 Abraham Robinson gave infinitesimals a rigorous footing, and Keisler's textbook put the payoff in one line: "Leibniz was on the right track, but 300 years too soon!" | Ch. 10 | S381, S440 |
| 1.4 The Derivative as a Function | Definitions | The word "derivative" is Lagrange's, 1797, from fonction dérivée, and so is the f'(x) prime notation, introduced "pour plus de simplicité et d'uniformité". His book's subtitle promises no infinitesimals, no limits, no fluxions, just algebra. | Ch. 9 | S317 |
| 1.4 The Derivative as a Function | Dig Deeper | Newton called it a fluxion: a quantity flows in time, and its fluxion is the rate of flow. And the famous dot over the letter is from late 1691, not 1666. The dots in the printed text are William Jones's 1710 tidying-up. | Ch. 6 | S185, S192 |
| 1.5 The Power Rule | Dig Deeper | Ibn al-Haytham needed the exact sum of fourth powers to find a paraboloid's volume, because he had no limit concept. We need only the leading term because we have one. The formula is the price of not having limits. He stopped at the fourth power for the plainest reason: nothing he wanted to measure needed a fifth. | Ch. 3 Appendix E.7, E.13 |
S054, S061 |
| 1.5 The Power Rule | Dig Deeper | Newton's first great result was extending the binomial theorem to fractional and negative exponents, by interpolating from Wallis. Twelve terms give the square root of 1.2 to ten decimal places. | Ch. 6 Appendix E.8 |
S190 |
| 1.6 Derivatives and Their Graph Shapes | Picture This | Oresme invented the graph as a tool of proof around 1350. Time along the bottom, speed up the side, and then the sentence that makes calculus possible: the area of the figure is the total effect. A student can redo his whole proof with a ruler in ninety seconds. | Ch. 4 | S107 |
| 1.6 Derivatives and Their Graph Shapes | Dig Deeper | Graph intuition has a limit. Weierstrass presented a function that is continuous everywhere and differentiable nowhere in 1872. Hermite: "I turn away with dread and horror from this lamentable plague of continuous functions that have no derivatives." Sixty years later Banach and Mazurkiewicz proved almost every continuous function is one of them. | Ch. 10 | S365, S386, S388 |
| 1.R Recap | Picture This | Liu Hui, 263 CE, could not compute the volume he needed, said so in print, and closed with a verse: "I dare to let the doubtful points stand, waiting for one who can expound them." Two hundred and fifty years later, someone answered. | Ch. 3 | S022, S023 |
| 2.0 Unit 2 Opener | Unit opener | The rules in this unit exist because Leibniz's 1684 paper gave no proofs at all. Jacob and Johann Bernoulli had to reverse-engineer it, then built most of the early calculus from what they worked out. | Ch. 7, Ch. 9 | S231 |
| 2.1 The Product Rule | ⚠ Watch out | The best teaching artifact in the whole history of calculus. On 11 November 1675 Leibniz guessed d(xy) = dx·dy, checked it on one example, got the right answer by accident, and refuted himself the same afternoon. He left the error and the correction on the same page. | Ch. 7 Appendix E.9 |
S230 |
| 2.1 The Product Rule | ⛏ Dig In | Why the guess fails, in numbers: with x = 3, y = 5, , the actual change is 0.008001 and dx·dy is 0.000001. The wrong answer is exactly the second-order term you are entitled to discard; the actual change and the right answer differ by it. | Appendix E.9 | S230 |
| 2.2 Derivatives of Trig Functions | Definitions | "Sine" is a mistranslation nobody ever fixed. Sanskrit ardha-jya, half-bowstring, became Arabic jiba, which is written identically to jaib, "bay". A twelfth-century translator in Toledo read it as jaib, the real word, and rendered it sinus. Gunter abbreviated it to "sin" in 1624. | Ch. 3 | S060 |
| 2.2 Derivatives of Trig Functions | Dig Deeper | Bhaskara II used the Rcosine as the rate of change of the Rsine (sine and cosine scaled to the circle's radius) around 1150, and Madhava had the sine and cosine series by about 1400, roughly two and a half centuries before Newton. | Ch. 3 | S050, S055, S062 |
| 2.3 The Chain Rule | Why this matters in a world that moves | This rule trains every chatbot on Earth. Backpropagation is the chain rule run backwards. It has to run backwards for a reason a student can check: a network has one output and billions of inputs, so going forwards costs one pass per input and going backwards costs one pass, full stop. | Ch. 12 Appendix E.18 |
S436, S437 |
| 2.3 The Chain Rule | Dig Deeper | Reverse mode was published in 1970 by a Finnish master's student who was not thinking about learning machines at all. He was worried about rounding errors. The paper is titled "Taylor expansion of the accumulated rounding error". It was then reinvented at least four times. | Ch. 12 | S436, S438 |
| 2.3 The Chain Rule | Dig Deeper | The chain rule looks obvious in Leibniz's notation because dy/dx behaves like a ratio, and painful in Newton's. Britain taught dots for a century and fell behind. Cambridge undergraduates campaigned for Leibniz's d over Newton's dot in 1812 (Babbage later recalled a joke title for the tract). | Ch. 7, Ch. 8 | S232, S239 |
| 2.4 Derivatives of Exponential and Logarithmic Functions | Definitions | e was born in a paper about cannon. Euler used it in a 1727 manuscript and printed it in 1736; no source records why he chose the letter. | Ch. 9 | S325, S451 |
| 2.4 Derivatives of Exponential and Logarithmic Functions | Definitions | A curve, not a number, gave the natural logarithm its name. In 1649 Sarasa wrote that in place of the logarithms "we shall be able to take the hyperbolic quantities", areas under a hyperbola. Nobody chose the base. The curve did. | Ch. 5 | S066, S155 |
| 2.5 The Quotient Rule | ⚠ Watch out | Which rule did Leibniz guess wrong on 11 November 1675, and what was the guess? You met the manuscript at Derivatives 2.1. Show itThe product rule. He guessed d(xy) = dx·dy, checked it on one example, and the example agreed with him. |
Ch. 7 | S230 |
| 2.6 Derivatives of Inverse Trigonometric Functions | Dig Deeper | Madhava had an infinite series for the arctangent by about 1400, the one Europe later named for Gregory. Its case is the series we call Leibniz's: . That series is so slow that ten thousand terms buy only three correct decimals, so Madhava built correction terms. | Ch. 3, Appendix D, Appendix E.4 | S050, S055, S062, S453 |
| 2.7 The First Derivative Test | Dig Deeper | Sharaf al-Din al-Tusi found the maximum of a cubic around 1209, roughly 450 years before Fermat, and used the discriminant to ask when a solution exists rather than what it is. Whether he "had the derivative" is disputed: Rashed says yes, Hogendijk says the evidence does not support it. | Ch. 3 | S053 |
| 2.7 The First Derivative Test | Dig Deeper | Rolle's theorem is the load-bearing lemma under the whole differential calculus, and Rolle called the calculus "a collection of ingenious fallacies" in print for five years. His 1691 statement was unproved and only about polynomials. | Appendix D | S459 |
| 2.7 The First Derivative Test | Dig Deeper | The Mean Value Theorem took more than a century to reach the form you learn. Rolle stated the special case in 1691. Lagrange gave the general statement. Cauchy gave the proof taught today. In 1823 he proved a close cousin of this theorem, and that proof is the first epsilon-delta proof in history. Even the English name is late: "mean value theorem" dates from 1894. | Appendix B, Appendix D | S459, S453, S139, S360 |
| 2.8 The Second Derivative Test | ⛏ Dig In | Agnesi tests concavity in 1748 with no calculus at all. At Instituzioni p. 361 she takes , and compares the height of a chord against the height of the curve at the midpoint, reducing the whole question to the inequality . That is midpoint convexity, argued by hand, and it agrees with the sign of the second derivative. A good "before we had the test, here is what you had to do" opener. | Ch. 9 | S075 |
| 2.8 The Second Derivative Test | Picture This | The founding paper of calculus, October 1684, shipped with a typo that swapped concavity and convexity. Seven pages, no proofs, printer's errors, and it started a subject. | Ch. 7 | S231, S232 |
| 3.0 Unit 3 Opener | Unit opener | Every application in this unit is older than the notation. Kepler was optimizing wine barrels in 1615 and Huygens was building clocks that kept time on a cycloid in the 1650s, decades before anyone wrote dy/dx. | Ch. 5 | S150, S159 |
| 3.1 Implicit Differentiation | Picture This | Descartes' La Géométrie (1637) is the enabling technology for this entire unit. Once a curve is an equation, "find the tangent" becomes an algebra problem you can hand to anybody. He wrote it as an appendix, and made it deliberately hard because he wanted readers to work. | Ch. 5 | S148 |
| 3.2 Related Rates | Definitions | Newton's own vocabulary is built for this section: a fluent is a quantity that flows with time, and its fluxion is the rate at which it flows. Related rates is fluxions, in the original sense. | Ch. 6 | S185 |
| 3.2 Related Rates | Dig Deeper | The idealized setups in this section have a medieval license. The Oxford Calculators defended their artificial thought experiments in one line: the scenario does not have to be physically real, only conceivable, and "for purposes of the sophisma, that is enough." | Ch. 4 | S102 |
| 3.3 Optimization | Picture This | The best origin story in the subject. Kepler remarried, bought wine, watched the merchant shove a rod into every barrel and name one price "without reasoning or calculation", got annoyed, and wrote a book computing about ninety volumes of revolution to check whether he was being cheated. | Ch. 5 | S159, S154 |
| 3.3 Optimization | ⛏ Dig In | Kepler noticed the thing that makes optimization useful: near a maximum, a quantity is insensitive to error. That is why the merchant's crude rod worked at all: at a maximum the first derivative is zero, so a small error in the measurement barely changes the volume. | Ch. 5 | S134, S159 |
| 3.3 Optimization | Dig Deeper | Johann Bernoulli posed the brachistochrone in 1696 partly to embarrass his brother. Jacob solved it better. Johann spent years insisting otherwise. The family feud produced the calculus of variations as a by-product. | Ch. 9 | S330 |
| 3.5 L'Hopital's Rule | Dig Deeper | The rule carries the name of the man who paid for it, not the man who found it. In a letter of 17 March 1694, L'Hopital offered Johann Bernoulli 300 pounds a year for his discoveries and his silence. The rule then appeared in L'Hopital's 1696 textbook, the first calculus textbook. Its preface thanks the Bernoullis in general, but the body of the book never credits Johann by name. | Ch. 9, Appendix D | S315, S316 |
I.3 Integrals (Berry), section by section
| Section | Box | The hook | Where it is in this document |
Sources |
|---|---|---|---|---|
| 1.0 Unit 1 Opener | Extends what you already have | Your opener already has Archimedes and the two ladders of polygons. The payoff you can add: in 1906 we found out how he really worked. A recovered letter shows he found his results by slicing figures and balancing them on an imaginary lever, then built the rigorous proof afterwards. He tells Eratosthenes that publishing only the polished proof wastes everyone's time. | Ch. 2 | S002, S003 |
| 1.0 Unit 1 Opener | Picture This | Your tagline, "from a stack of rectangles to one exact total", is literally what the symbol means. ∫ is a long letter S for summa, "sum". | Ch. 7 | S230, S238 |
| 1.1 Approximating Area | Dig Deeper | Every student meets equal-width strips. Thabit ibn Qurra used unequal partitions in ninth-century Baghdad, a thousand years before Riemann formalized the general partition. Unequal widths can be the right tool. | Ch. 3 | S057 |
| 1.1 Approximating Area | ⛏ Dig In | Archimedes squeezed the parabola with a geometric series and refused to take a limit in print. He proved the finite identity, then argued by contradiction. That caution is exactly what the epsilon-delta definition later made unnecessary. | Ch. 2 Appendix E.1 |
S001, S002 |
| 1.1 Approximating Area | Picture This | Liu Hui's instruction for the circle, 263 CE: "divide the circle until it cannot be divided any more." That is either a limit or a contradiction depending on how charitable you are, and he knew it. | Ch. 3 | S020 |
| 1.2 Antiderivatives | Definitions | The word "integral" was negotiated between three people who could not stand each other. Jacob Bernoulli printed it, Johann claimed it, and Leibniz had to be talked out of his own preferred name, calculus summatorius. | Ch. 9 | S324 |
| 1.3 The Definite Integral | Definitions | 29 October 1675. Three days after writing a formula so clogged with the word omnia that it was unusable, Leibniz wrote: "It will be useful to write ∫ for omn." He invented the most recognizable symbol in mathematics to save ink. | Ch. 7 | S230, S451 |
| 1.3 The Definite Integral | Dig Deeper | Cauchy told his students the same thing in 1823: it is a letter S, stretched, for somme. He also gave the first real definition of the definite integral as a limit of sums. Fourier added the limits above and below the sign in 1822. | Ch. 10 | S362, S451 |
| 1.3 The Definite Integral | ⛏ Dig In | Riemann defined his integral in 1854 only because he could not state his question without it. His own sentence ends: "If it does not have this property, then ∫f(x)dx has no meaning." And what students learn as the Riemann integral is usually Darboux's 1875 reformulation. | Ch. 10 Appendix D |
S391 |
| 1.4 The Fundamental Theorem | Picture This | Huygens tested a 26-year-old diplomat in Paris in 1672 by asking him to sum the reciprocals of the triangular numbers. Leibniz noticed every term is a difference, that the sum collapses, and that the answer is exactly 2. The whole architecture is already there that afternoon: sums and differences, the two operations he would later call ∫ and d. | Ch. 7 | S237 |
| 1.4 The Fundamental Theorem | ⚠ Watch out | Two corrections at once. Gregory proved the inverse relationship in 1668 and Barrow stated it in 1670, both before Newton published, and four distinguished scholars have read the same evidence and reached four different verdicts about what Barrow gave Newton. And the name "Fundamental Theorem of Calculus" dates only to 1942. | Ch. 5 Appendix D |
S136, S138, S158 |
| 1.4 The Fundamental Theorem | Dig Deeper | Leibniz stated the inverse relationship in print in 1686 as an analogy students already understand: "for as powers and roots in common calculation, so with us sums and differences or ∫ and d, are reciprocals." | Ch. 7 | S232, S233 |
| 1.5 The Net Change Theorem | Picture This | Time along the bottom, speed up the side. What did Oresme say the area of that figure means? You met his graph in Derivatives 1.6. Show itThe total effect. That sentence, written around 1350, is the theorem this section is named after. |
Ch. 4 | S107 |
| 1.5 The Net Change Theorem | Why this matters in a world that moves | Kepler's second law is not "planets speed up near the Sun". It is "equal areas in equal times", which says area measures elapsed time. Using it means adding up infinitely many thin slivers of an ellipse: an integral, in 1609, sixty-six years before Leibniz wrote the sign. | Ch. 5 | S152 |
| 2.0 Unit 2 Opener | Unit opener | Your tagline is "the toolkit that reaches every function". The toolkit exists because the Bernoulli brothers reverse-engineered a proof-free seven-page paper and then spent thirty years arguing about who owned the result. | Ch. 9 | S231, S324 |
| 2.1 Integral Properties and the Average Value | Picture This | Uniformly changing speed covers the same distance as one steady speed. Which steady speed, and who proved that? You met the Merton mean speed theorem in Derivatives 1.1. Show itThe average of the start and end speeds. Merton College, Oxford, 1335, and Oresme proved it at Paris with a picture. |
Ch. 4 | S101, S104, S107 |
| 2.2 Substitution | Why this matters in a world that moves | This section is the single best argument for Leibniz's notation. The dx carries the "of what": change variables and the dx tells you exactly what to do. In Newton's dot notation the same move is painful. That difference is why the Continent pulled ahead of Britain for a century. | Ch. 7, Ch. 8 | S232, S239, S241 |
| 2.3 Trigonometric Integrals and Substitution | Definitions | "Sine" records a mistranslation. Which step went wrong, and what did the word mean first? You met the chain in Derivatives 2.2. Show itArabic into Latin, and it meant bowstring: Sanskrit ardha-jya to Arabic jiba, read as jaib, "bay", and rendered sinus by a twelfth-century translator in Toledo. |
Ch. 3 | S060 |
| 2.3 Trigonometric Integrals and Substitution | Dig Deeper | The Kerala derivation of the sine series runs on ibn al-Haytham's summation identity, and names nobody. Katz locates the exact step. The mathematics traveled from Baghdad to the Malabar coast; the credit did not. A cleaner case study in citation than any modern example, and the evidence is a single line of a proof. | Ch. 3 | S061 |
| 2.3 Trigonometric Integrals and Substitution | Dig Deeper | Pascal's Traité des sinus du quart de cercle contains the characteristic triangle, the little right triangle with legs dx and dy. Leibniz read it and later wrote that subito magna lux oborta est, "a great light suddenly appeared". Not a falling apple: a lemma about the area of a sphere. | Ch. 5, Ch. 7 | S143, S153, S232 |
| 2.4 Exponential and Logarithmic Integrals | Picture This | The natural logarithm was found during an argument. Mersenne set a challenge problem partly to attack Saint-Vincent's claimed quadrature of the circle. Saint-Vincent's pupil Sarasa answered it in 1649, and partway through he stops and says: "But to what end, you will say, all this roundabout? I am leading to logarithms." Then: "in place of the numbers 6, 7, 8, 9, 10, 11, which were the logarithms of the magnitudes, we shall be able to take the hyperbolic quantities." In place of the logarithms, take the areas. | Ch. 5 Appendix E.17 |
S066 |
| 2.4 Exponential and Logarithmic Integrals | ⛏ Dig In | Why it works, in one line you can check: multiply x by a fixed factor and you add a fixed amount of area. Geometric progression in x, arithmetic progression in area. Doubling from 1 to 2 to 4 to 8 adds 0.693147 every time. Products become sums, which is the whole reason anyone wanted logarithms. | Appendix E.17 | S066 |
| 2.4 Exponential and Logarithmic Integrals | Definitions | "Natural logarithm" was coined for a curve, not a number. It is natural because it falls out of the area under the plainest hyperbola there is, xy = 1, with no base chosen by anybody. And Saint-Vincent's own index calls the relevant theorems admirandae, "to be wondered at". | Ch. 5 | S067, S141, S155 |
| 2.4 Exponential and Logarithmic Integrals | Dig Deeper | Mercator's 1668 series for log(1+x) is the first infinite series for a logarithm ever printed, and it is what panicked Newton into writing up De analysi to establish that he had got there first. | Ch. 5, Ch. 6 | S149, S184 |
| 3.0 Unit 3 Opener | Unit opener | Your tagline is "the shapes and totals of the moving world". Every application in this unit predates the notation, and most of them predate Newton. | Ch. 5 | S150, S152, S159 |
| 3.1 Arc Length | Picture This | Christopher Wren, 1658. During Pascal's cycloid contest he sent in, without proof, the length of one arch: exactly four times the diameter of the generating circle. A curved length exactly equal to a straight one, with no pi in it. Shown it, Roberval reportedly proved it on sight and said he had known it for years. He had not published that either. | Ch. 5 Appendix E.16 |
S064 |
| 3.1 Arc Length | Definitions | The old word for this is rectification, literally "straightening". For most of the seventeenth century mathematicians doubted that a curved length could be expressed by a straight one at all, which is why Wren's result landed the way it did. | Appendix B | S064, S451, S453 |
| 3.2 Areas between Curves | ⛏ Dig In | Archimedes proved a parabolic segment is exactly four thirds of its inscribed triangle, by summing and never taking a limit. He proves the finite identity, then rules out every other answer by contradiction. | Ch. 2 Appendix E.1 |
S001 |
| 3.2 Areas between Curves | ⚠ Watch out | Galileo tried to find the area under a cycloid in 1599 by weighing it. He cut an arch and its generating circle from the same material, balanced them, and got about three times. Then he threw the result away, because he believed the ratio was irrational. It is exactly 3. A correct experiment, discarded because the answer looked too clean. | Ch. 5 Appendix E.15 |
S064 |
| 3.3 Surface Area of a Revolution | ⚠ Watch out | The most valuable correction in this appendix. Torricelli's horn is real: an infinitely long solid with finite volume, which he had by 1641. But he never proved the infinite surface area. The painter's paradox every textbook prints is a later overlay on a seventeenth-century theorem. The shock in 1644 was simpler: a solid that goes on forever and fits in a cup. | Ch. 5 Appendix G |
S131, S157 |
| 3.4 Volume Methods | Picture This | Cavalieri's principle is Zu Geng's principle, about 1,100 years earlier: "If blocks are piled up to form volumes, and corresponding areas are equal, then the volumes cannot be unequal." It was written to answer a problem Liu Hui had published as an unsolved failure 250 years before. | Ch. 3 Appendix D |
S022, S023 |
| 3.4 Volume Methods | Dig Deeper | Kepler computed about ninety solids of revolution in 1615. What made him do it? You met the reason in Derivatives 3.3. Show itHe suspected his wine merchant: one rod shoved into every barrel, one price, "without reasoning or calculation". |
Ch. 5 | S151, S159 |
| 3.4 Volume Methods | ⛏ Dig In | Ibn al-Haytham needed the exact sum of fourth powers, not just the leading term. What was he short of? You met that in Derivatives 1.5. Show itA limit concept. The exact formula is the price of not having one. |
Ch. 3 Appendix E.14 |
S061 |
| 3.5 Mass and Density | Why this matters in a world that moves | How did Archimedes find his results, before he wrote the proofs? You met the recovered letter in Integrals 1.0. Show itHe sliced figures into infinitely thin sections and balanced them on an imaginary lever, using mechanics to get the answer, then built the geometric proof afterwards. That letter is The Method. |
Ch. 2 | S002, S003 |
| 3.6 Force and Work | Picture This | The eighteenth century fought over what "force" even meant. Newtonians measured it by mv, Leibnizians by mv². Émilie du Châtelet argued the mv² side and was pointing at what we now call energy. She was 42, pregnant, knew the danger, and "rushed on with her translation and commentary, hoping to finish the work by the time her child was born". She was right about the danger. | Ch. 9 | S332, S069 |
| 3.6 Force and Work | Dig Deeper | Two of her three major works went out with no name on them. The 1738 essay on fire was submitted anonymously because a submission known to be by a woman would not have been taken seriously; it lost, and the Académie published it anyway. The Institutions de physique was also anonymous. The Académie admitted no women at all, except as spectators twice a year. | Ch. 9 | S069 |
| 3.6 Force and Work | Why this matters in a world that moves | Her Principia is still the only complete French translation that exists. It came out ten years after she died, and not through neglect: Clairaut, the mentor who had checked her calculations while she was alive, pressed the publisher to bring it out and had the illustrations finished. | Ch. 9 | S068, S069 |
| 3.7 Probability | ⛏ Dig In | The Radon-Nikodym derivative is why "probability density" means anything. Every time this section writes p(x) dx, the reason that expression is legitimate is a theorem finished in 1930. | Ch. 12 | S435 |
| 3.7 Probability | Dig Deeper | Oresme, around 1350, argued against astrology on probabilistic grounds: if celestial ratios are almost certainly irrational, no exact conjunction ever repeats, so astrological prediction fails in principle. One of the earliest uses of "most probable" as a mathematical argument in European science. | Ch. 4 | S108 |
I.4 Both books, Appendix B: Study Tools
Three of your Study Tools sections map onto whole appendices here. This is the part of the alignment that pays back fastest, because it is reference material you already maintain and it needs no new pedagogy.
| Your section | What to pull from here | Size |
|---|---|---|
| B.1 Notation & Symbols Derivatives B.1 · Integrals B.1 |
Appendix B of this document, first-use dates for every symbol on the page: ∫ (29 Oct 1675, a long S for summa), d and dx (11 Nov 1675, and d began life as a divisor), f'(x) (Lagrange 1797), the dot (1691, not 1666), ∞ (Wallis 1655, first used as a denominator), π (William Jones 1706, "it simply came, unheralded"), e, Σ and Δ (Euler 1755, twins in one sentence), lim (L'Huilier 1786), the arrow (Leathem 1905, not Hardy), ε and δ, and the integral limits (Fourier 1822). | 52 symbols |
| B.2 Glossary | The Glossary and Appendix B of this document. Etymologies for the words your glossary already defines: calculus (a pebble), tangent (to touch), secant (to cut), asymptote (not falling together), quadrature (squaring), rectification (straightening), limit (limes, a boundary fence), integral, derivative, function, infinitesimal. | 75 words |
| B.5 Rules & Theorems Derivatives B.5 · Integrals B.5 |
Appendix D of this document: whose name is on each result, and who did the work. L'Hopital's rule is Johann Bernoulli's, bought under contract for 300 pounds a year. Maclaurin's series is Stirling's, 1717. Cavalieri's principle is Zu Geng's. Newton's method, as taught, is Simpson's, 1740. Cauchy-Schwarz is Bunyakovsky's, 1859. Every row is sourced, and the whole table is a lesson on its own. | 20 corrections |
I.5 The prerequisite appendices
| Book and section | The hook | Sources |
|---|---|---|
| Derivatives A.7 to A.9, Integrals A.1 to A.3 (trig, unit circle, identities) | The ardha-jya to jiba to jaib to sinus chain. A chord looks like a drawn bowstring and the arc is the bow, which is why the whole subject is named after archery equipment. | S060 |
| Derivatives A.6, Integrals A.4 to A.5 (exponentials and logarithms) | The natural logarithm was named for the area under xy = 1, not for a base. Napier built the word from logos plus arithmos. | S141, S155 |
| Derivatives A.1 to A.3 (exponent and fractional-exponent rules) | Fractional exponents entered calculus through Newton's generalized binomial theorem in 1665, which he got by interpolating from Wallis's tables. | S190 |
| Integrals A.7 (area, surface area, volume) | Cavalieri's principle has an earlier owner. Who, and by how long? You met him in Integrals 3.4. Show itZu Geng, by about 1,100 years. |
S022, S023, S159 |
| Integrals A.8 (basic physics) | The Merton mean speed theorem and Oresme's velocity graph are the origin of every distance-from-velocity problem in the section. | S101, S107 |
| Integrals A.9 (basic probability) | Oresme's probabilistic argument against astrology, and the 1930 theorem that makes "probability density" mean something. | S108, S435 |
I.6 History with no home in the courses yet
Your two books cover derivatives through L'Hopital's rule and integrals through probability. They do not cover sequences, series, convergence tests, Taylor or Maclaurin series, formal real analysis, or measure theory. A large and good part of this book therefore has no matching course section right now. It is listed here so nothing gets quietly lost, and so the stories are ready if the courses ever grow a third volume.
| Topic with no section | What is already written here | Where |
|---|---|---|
| Sequences and series | Oresme proving the harmonic series diverges around 1350, with the block-grouping proof and a checked table of partial sums. | Ch. 4, Appendix E.5 |
| Power series and pi | Madhava's series c. 1400, his error-analyzed correction terms, and his pi recorded as a poem about gods and elephants that decodes to eleven correct decimal places. | Ch. 3, Appendix E.3, E.4 |
| Taylor and Maclaurin series | Stirling had "Maclaurin's series" in 1717, twenty-five years early, and Maclaurin credited Taylor anyway. Gregory and Johann Bernoulli had Taylor's earlier. | Ch. 9, Appendix D |
| Convergence tests | Cauchy's false theorem on page 131 of the Cours d'analyse, and Abel's 1826 footnote citing that page number while calling the book indispensable. You can watch the error being caught in print, on facing pages. | Ch. 10 |
| Formal real analysis | Bolzano silenced by an emperor, Dedekind's cuts worked out on a specific Wednesday in 1858 and then sat on for fourteen years, Weierstrass teaching gymnasium for years before one paper made him famous. | Ch. 10 |
| Measure and the Lebesgue integral | Why the Riemann integral was not enough, and the whole chain from Fourier's heat problem to Cantor's set theory. | Ch. 10 |
| Partial derivatives | The ∂ symbol's three-stage history: Condorcet 1770, Legendre 1786 who then dropped it, Jacobi 1841 who revived it after a fifty-five-year gap. | Appendix B |
Two of these are worth a "Dig Deeper" in a book you already have. Madhava's series fits naturally in Integrals 2.3 next to the trigonometric material. The Cauchy and Abel exchange fits in Derivatives 1.2 as an example of what the formal definition of a limit is protecting against.
I.7 Fitting the voice
Three notes for anyone bringing these stories into a calculus class, so the history sits inside the course rather than beside it.
- The two courses already use history sparingly and well. Integrals 1.0 names three mathematicians in two paragraphs and then gets on with the mathematics. Keep that ratio. The material in this book is a supply depot, not a script. One box per section is plenty, and several sections above list three options so you can pick.
- "Dig Deeper" is collapsed by default, which makes it the right home for most of this. A student who wants the Leibniz manuscript story opens it; a student who wants to finish the problem set does not have to scroll past it.
- The "⚠ Watch out" rows are the ones to prioritize. Derivatives 2.1 (Leibniz's product-rule error), Derivatives 1.3 (Berkeley on dividing then deleting), and Integrals 3.3 (Torricelli's horn) are all cases where a historical mistake is the same mistake a student makes, or where the standard textbook account is simply wrong. Those three earn their space immediately.
Appendix J
Bibliography
255 sources in APA 7, grouped by reliability tier. The access column is the honesty control. A source marked abstract only or not obtained supports no claim anywhere in this document. It is listed so you know it exists and know it is not load-bearing.
Tier 1: primary sources and critical scholarly editions of them
72 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S189 | [Newton], I. (1715). An Account of the Book entituled Commercium Epistolicum Collinii et aliorum, de Analysi promota. Philosophical Transactions of the Royal Society, 29 (342), 173-224 (esp. 205-206). Royal Society. https://www.maths.tcd.ie/pub/HistMath/People/Newton/CommerciumAccount/CommAcc.pdfCited in Chapter 6, Chapter 8 3 times, 1665 to 1687 3 times and 1688 to 1734 6 times. | en | read in full |
| S364 | Abel, N. H. (1826). Recherches sur la serie . Journal fuer die reine und angewandte Mathematik; repr. in Oeuvres completes (Sylow & Lie, 1881), 1 (Oeuvres vol. 1), 311-339 (Oeuvres 66-92). Grondahl. https://archive.org/details/oeuvrescomplete01abelgoogCited in Chapter 10 3 times and 1800 to 1872. | fr | read in full |
| S490 | Abel, N. H. (1902). Abel to Bernt Michael Holmboe, Berlin, 16 January 1826 (letter VI) - French translation by P. G. la Chesnais and the Dano-Norwegian original, printed in parallel. Niels Henrik Abel: Memorial publie a l'occasion du centenaire de sa naissance (Holst, Stormer & Sylow, eds.), p. 16 in each of the two parallel correspondence sections (letter runs pp. 13-16; archive.org scan leaves 164 and 300). Kristiania: Jacob Dybwad. https://archive.org/details/nielshenrikabelm00holsCited in Chapter 10 3 times and 1800 to 1872. | fr;da;no | read in full |
| S319 | Agnesi, M. G. (1748). Instituzioni analitiche ad uso della gioventu italiana. Nella Regia-Ducal Corte, Milan. https://archive.org/details/A298183Cited in Chapter 9 and 1734 to 1799 twice. | it | read in full |
| S320 | Agnesi, M. G. (1801). Analytical Institutions in Four Books: Originally Written in Italian (trans. John Colson, ed. John Hellins). Taylor and Wilks, London. https://archive.org/details/analyticalinstit00agnerichCited in Chapter 9 twice and 1734 to 1799. | en | read in full |
| S239 | Babbage, C. (1864). Passages from the Life of a Philosopher. Longman, Green, Longman, Roberts, & Green. https://www.gutenberg.org/ebooks/57532Cited in Chapter 8 3 times and 1800 to 1872 twice. | en | read in full |
| S158 | Barrow, Isaac (J. M. Child, Trans.) (1916). The Geometrical Lectures of Isaac Barrow. Open Court Publishing Company. https://archive.org/details/geometricallectu00barruoftCited in Chapter 5 and 1665 to 1687 twice. | en | read in full |
| S310 | Berkeley, G. (1734). The Analyst; or A Discourse Addressed to an Infidel Mathematician. The Works of George Berkeley, Bishop of Cloyne, ed. A. A. Luce & T. E. Jessop, 4, 53-102. Thomas Nelson and Sons. https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Analyst/Analyst.htmlCited in Chapter 9 5 times and 1734 to 1799 twice. | en | read in full |
| S337 | Bernoulli, J. (1697). Lettre de Mr. Bernoulli a l'Auteur. Histoire des ouvrages des scavans (ed. H. Basnage de Beauval), June 1697, 455. Reinier Leers, Rotterdam.Cited in Chapter 9, 1688 to 1734 and 1800 to 1872. | fr | not obtained, supports no claim |
| S363 | Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reelle Wurzel der Gleichung liege. Ostwalds Klassiker der exakten Wissenschaften 153 (repr. 1905, ed. P. E. B. Jourdain), 1-60. Gottlieb Haase (1817); Engelmann (1905). https://archive.org/details/reinanalytische00hankgoogCited in Chapter 10 3 times and 1800 to 1872. | de | read in full |
| S321 | Brewster, D. (1831). The Life of Sir Isaac Newton. John Murray, London. https://archive.org/details/b33028035_0001Cited in Chapter 9 5 times, 1688 to 1734 3 times and 1800 to 1872. | en | read in full |
| S450 | Cajori, F. (1928). A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics. Open Court Publishing Company. https://archive.org/details/historyofmathema031756mbpCited in Chapter 5 twice. | en | read in full |
| S451 | Cajori, F. (1929). A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics. Open Court Publishing Company. https://archive.org/details/historyofmathema027671mbpCited in Chapter 7, Chapter 9, 1637 to 1665, 1665 to 1687 4 times, 1688 to 1734 twice, 1734 to 1799 4 times, 1800 to 1872 5 times and 1873 to 1913 twice. | en | read in full |
| S361 | Cauchy, A. (1821). Cours d'analyse de l'Ecole royale polytechnique. I.re partie: Analyse algebrique. Imprimerie Royale / Debure. https://archive.org/details/coursdanalysedel00caucCited in Chapter 10 twice and 1800 to 1872. | fr | read in full |
| S362 | Cauchy, A. (1823). Resume des lecons donnees a l'Ecole royale polytechnique sur le calcul infinitesimal. Imprimerie Royale / Debure. https://archive.org/details/bub_gb_y7FSr1rrFsICCited in Chapter 10 twice and 1800 to 1872. | fr | read in full |
| S196 | Chaloner, W. (1699). William Chaloner's Letter to Isaac Newton (MINT 15/17/205). The Newton Project (MINT00918). University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/MINT00918Cited in Chapter 6 twice and 1688 to 1734 twice. | en | read in full |
| S230 | Child, J. M. (trans.) (1920). The Early Mathematical Manuscripts of Leibniz. Open Court. https://archive.org/details/earlymathematic01gerhgoogCited in Chapter 7 8 times, Chapter 8 twice and 1665 to 1687 7 times. | en;la | read in full |
| S446 | College Board (2024). AP Program Summary Report 2024. College Board. https://reports.collegeboard.org/media/pdf/program-summary-report-2024.pdfCited in Chapter 12 and 1914 to the present twice. | en | read in full |
| S447 | College Board (2024). AP Score Distributions by Subject, 2024. College Board. https://reports.collegeboard.org/media/pdf/ap-score-distributions-by-subject-2024.pdfCited in Chapter 12 twice and 1914 to the present. | en | read in full |
| S181 | Conduitt, J. (n.d.). Drafts of various sections of the Memoir of Newton (Keynes Ms. 129.02). The Newton Project (THEM00146), ff. 1r-2v. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00146Cited in 1637 to 1665 twice. | en | read in full |
| S182 | Conduitt, J. (n.d.). Draft account of Newton's life at Cambridge (Keynes Ms. 130.04). The Newton Project (THEM00167). University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00167Not cited in the story. It backs a register entry. | en | read in full |
| S366 | Dedekind, R. (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. https://archive.org/details/cu31924001586282Cited in Chapter 10 twice and 1800 to 1872 3 times. | en | read in full |
| S318 | Euler, L. (1748). Introductio in analysin infinitorum [E101]. Marcum-Michaelem Bousquet, Lausanne. https://archive.org/details/bub_gb_jQ1bAAAAQAAJCited in Chapter 9 twice and 1734 to 1799. | la | read in full |
| S067 | Gregorius a Sancto Vincentio, (Gregoire de Saint-Vincent) (1647). Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum. Ioannes et Iacobus Meursius, Antwerp. Google Books scan bub_gb_Bk9vKVkq6mgC, supplied by the user 2026-08-14Cited in Chapter 5 4 times and 1637 to 1665. | la | read in full |
| S439 | Hayes, C. (1704). A treatise of fluxions: or, an introduction to mathematical philosophy. Edw. Midwinter for Dan. Midwinter and Tho. Leigh. https://archive.org/details/atreatisefluxio00hayegoogCited in Chapter 12 3 times and 1688 to 1734. | en | read in full |
| S001 | Heath, T. L. (1897). The Works of Archimedes, edited in modern notation with introductory chapters. Cambridge University Press. https://archive.org/details/worksofarchimede00archCited in Chapter 2 and Antiquity to 500 CE 3 times. | en | read in full |
| S002 | Heath, T. L. (1912). The Method of Archimedes, Recently Discovered by Heiberg: A Supplement to the Works of Archimedes, 1897. Cambridge University Press. https://archive.org/details/methodofarchimed00archCited in Chapter 2 5 times, Antiquity to 500 CE twice and 1873 to 1913. | en | read in full |
| S365 | Hermite, C. (1905). Correspondance d'Hermite et de Stieltjes, Tome II (18 octobre 1889 - 15 decembre 1894). Gauthier-Villars. https://archive.org/details/correspondancedh02hermCited in Chapter 10 3 times and 1873 to 1913. | fr | read in full |
| S283 | Keill, J. (1708). Epistola ad clarissimum virum Edmundum Halleium . de legibus virium centripetarum. Philosophical Transactions of the Royal Society of London, 26, no. 317, 174-188. Royal Society. https://royalsocietypublishing.org/rstl/article/26/317/174/110406/Cited in Chapter 8 and 1688 to 1734. | la | not obtained, supports no claim |
| S440 | Keisler, H. J. (2000/2026). Elementary calculus: an infinitesimal approach (On-line Edition; Epilogue read in full). University of Wisconsin-Madison / orig. Prindle, Weber and Schmidt. https://people.math.wisc.edu/~hkeisler/calc.htmlCited in Chapter 10 twice and 1914 to the present twice. | en | read in full |
| S159 | Kepler, J. (1615). Nova stereometria doliorum vinariorum. Johannes Plancus, Linz. https://archive.org/details/den-kbd-pil-21009000066F-001Cited in Chapter 5 and 1400 to 1637. | la | read in full |
| S454 | Knott, C. G. (1911). Life and Scientific Work of Peter Guthrie Tait. Cambridge University Press. https://archive.org/details/lifescientificwo00knotuoftCited in 1800 to 1872 twice. | en | read in full |
| S417 | Kovalevsky, S. (1895). Sonya Kovalevsky: her recollections of childhood (trans. I F Hapgood), with a biography by A C Leffler. The Century Co. https://archive.org/details/snyakovalvsk00kovauoftCited in Chapter 11. | en | read in full |
| S316 | L'Hospital, G. F. A. d. (1696). Analyse des infiniment petits, pour l'intelligence des lignes courbes. Imprimerie Royale, Paris (read in the 1716 2nd ed.). https://archive.org/details/infinimentpetits1716lhos00uoftCited in Chapter 9 twice and 1688 to 1734. | fr | read in full |
| S317 | Lagrange, J. (1797). Theorie des fonctions analytiques, contenant les principes du calcul differentiel, degages de toute consideration d'infiniment petits ou d'evanouissans, de limites ou de fluxions, et reduits a l'analyse algebrique des quantites finies. Imprimerie de la Republique, Paris. https://archive.org/details/theoriedesfoncti00lagrCited in Chapter 9 3 times and 1734 to 1799. | fr | read in full |
| S231 | Leibniz, G. W. (1684). Nova methodus pro maximis et minimis, itemque tangentibus. Acta Eruditorum, Oct 1684, 467-473 (plate Tab. XII). Leipzig. https://archive.org/details/s1id13206500/page/467/mode/1upCited in Chapter 7 5 times, Chapter 8, Chapter 9 and 1665 to 1687. | la | read in full |
| S233 | Leibniz, G. W. (1686). De geometria recondita et analysi indivisibilium atque infinitorum. Acta Eruditorum, Jun 1686, 292-300. Leipzig. https://archive.org/details/s1id13206520Cited in Chapter 7 twice and 1665 to 1687. | la | read in full |
| S242 | Leibniz, G. W. (1695). Letter from Hanover to G. F. A. de l'Hopital, 30 September 1695. Leibnizens mathematische Schriften (ed. Gerhardt), 2, 301-302. H. W. Schmidt.Cited in 1688 to 1734. | fr;la | not obtained, supports no claim |
| S240 | Leibniz, G. W., & Clarke, S. (J. Bennett, Ed.) (2007). Exchange of papers between Leibniz and Clarke (1715-1716). Early Modern Texts. https://www.earlymoderntexts.com/assets/pdfs/leibniz1715_1.pdfCited in 1688 to 1734 twice. | en;fr | read in full |
| S241 | Leibniz (ed. Gerhardt), G. W. /. C. I. (1849-1863). Leibnizens mathematische Schriften [vol. containing the Leibniz-Wallis correspondence]. H. W. Schmidt. https://archive.org/details/leibnizensmathe02leibgoogCited in Chapter 7 twice, Chapter 8 and 1688 to 1734 twice. | la;de | read in full |
| S232 | Leibniz, G. W. (E. Walker, Trans.) (1929). On the Calculus [Nova methodus 1684; De geometria recondita 1686]. D. E. Smith (ed.), A Source Book in Mathematics, 619-626. McGraw-Hill. https://en.wikisource.org/wiki/Page:A_Source_Book_in_Mathematics.djvu/657Cited in Chapter 7 3 times, Chapter 8 and 1665 to 1687 twice. | en;la | read in full |
| S455 | Lewis, C. T., & Short, C. (1879). A Latin Dictionary, entry calculus. A Latin Dictionary, founded on Andrews edition of Freunds Latin dictionary, entry calculus. Clarendon Press / Perseus Digital Library. http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3DcalculusCited in Chapter 1 twice. | en;la | read in full |
| S311 | Maclaurin, C. (1742). A Treatise of Fluxions: In Two Books. T. W. and T. Ruddimans, Edinburgh. https://archive.org/details/atreatisefluxio00conggoogCited in Chapter 9 3 times and 1734 to 1799 twice. | en | read in full |
| S461 | Maclaurin, C. (1742). A Treatise of Fluxions, in Two Books. T. W. and T. Ruddimans. https://archive.org/details/bim_eighteenth-century_a-treatise-of-fluxions-_maclaurin-colin-profes_1742_2Not cited in the story. It backs a register entry. | en | not obtained, supports no claim |
| S415 | Menabrea (trans. and notes by Lovelace), L. F. (1843). Sketch of the Analytical Engine invented by Charles Babbage, with notes by the translator. Scientific Memoirs, 3, 666-731. Richard and John E. Taylor. https://www.fourmilab.ch/babbage/sketch.htmlCited in Chapter 11 3 times and 1800 to 1872. | en | read in full |
| S183 | Newton, I. (1666). The October 1666 Tract on Fluxions (CUL MS Add. 3958.3). The Newton Project (NATP00100), ff. 48v-63v. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00100Cited in Chapter 6, Chapter 8 and 1665 to 1687. | en | read in full |
| S184 | Newton, I. (1669). De analysi per aequationes numero terminorum infinitas (RS MS/81/4). The Newton Project (NATP00204), 15 pp. on 9 ff. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/NATP00204Cited in Chapter 5, Chapter 6 twice, 1665 to 1687 and 1688 to 1734. | la | read in full |
| S494 | The Newton Project. (n.d.). Isaac Newton's archival legacy (1727-1872). The Newton Project. University of Oxford. https://newtonproject.ox.ac.uk/history-of-newtons-papers/1727-1872Cited in Chapter 6. | en | read in full |
| S188 | Newton, I. (1676). Letter to Henry Oldenburg, 24 October 1676 (Epistola posterior; CUL MS Add. 3977.1). The Newton Project (NATP00196), ff. 1r-10r. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00196Cited in Chapter 8 3 times and 1665 to 1687 twice. | la | read in full |
| S197 | Newton, I. (1676). Letter to Robert Hooke, 5 February 1675/6 (HSP; Turnbull Corr. I, no. 154). The Correspondence of Isaac Newton, 1, 416-417. Cambridge University Press. https://digitallibrary.hsp.org/index.php/Detail/objects/9792Cited in 1665 to 1687 twice. | en | abstract only, supports no claim |
| S187 | Newton, I. (1710). Introduction to the Quadrature of Curves (trans. John Harris). Lexicon Technicum, vol. 2, 2, pp. 1-4. https://www.maths.tcd.ie/pub/HistMath/People/Newton/Quadratura/HarrisIQ.pdfCited in Chapter 6 and 1688 to 1734 twice. | en;la | read in full |
| S289 | Newton, I. (1713). Newton's draft response to the 'charta volans' of 29 July 1713 (CUL MS Add. 9597/2/18/94). The Newton Project, catalogue ID NATP00290, 2 folios, c. 939 words. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00290Not cited in the story. It backs a register entry. | en;la | read in full |
| S185 | Newton, I. (1736). The Method of Fluxions and Infinite Series; with its Application to the Geometry of Curve-lines (trans. John Colson). Henry Woodfall. https://archive.org/details/methodoffluxions00newtCited in Chapter 6, 1665 to 1687 and 1734 to 1799. | en;la | read in full |
| S186 | Newton, I. (1846). Newton's Principia: The Mathematical Principles of Natural Philosophy (trans. Andrew Motte 1729). Daniel Adee. https://archive.org/details/newtonspmathema00newtrichCited in Chapter 6 and 1665 to 1687. | en;la | read in full |
| S072 | Newton, Isaac (I. B. Cohen & A. Whitman, Trans.) (1999). The Principia: Mathematical principles of natural philosophy, preceded by A Guide to Newton's Principia by I. B. Cohen. University of California Press. PDF supplied by the user 2026-08-14Cited in Chapter 6. | en | read in full |
| S282 | Newton, I. (c.1712-1715). An account of the Commercium Epistolicum [holograph draft]. Cambridge University Library MS Add. 3968, ff. 67r-96v; Newton Project NATP00352, 30 ff. https://www.newtonproject.ox.ac.uk/catalogue/record/NATP00352Cited in Chapter 8 and 1688 to 1734 twice. | en;la | read in full |
| S195 | Newton, I. (n.d.). Twelve articles on religion (Keynes Ms. 8). The Newton Project (THEM00008), f. 1r. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/THEM00008Cited in Chapter 6. | en | read in full |
| S286 | Newton, I. (n.d.). Appendix [on the history of the method of fluxions]. The Newton Project, OTHE00130. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/OTHE00130Not cited in the story. It backs a register entry. | en;la | read in full |
| S107 | Oresme, N. (1968). Tractatus de configurationibus qualitatum et motuum, III.vii (Clagett ed. & trans.). Nicole Oresme and the Medieval Geometry of Qualities and Motions, III.vii. University of Wisconsin Press. https://www.uni-due.de/imperia/md/content/didmath/ag_jahnke/oresme.pdfCited in Chapter 4 twice and 500 to 1400. | la;en | read in full |
| S492 | Perry, John (ed.) (1902). Discussion on the Teaching of Mathematics, which took place on September 14th [1901] at a joint meeting of Section A (Mathematics and Physics) and Section L (Education), British Association, Glasgow. London: Macmillan. https://archive.org/details/discussiononteac00britialaCited in Chapter 12 3 times and 1873 to 1913. | en | read in full |
| S391 | Riemann, B. (1892). Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, in Gesammelte mathematische Werke und wissenschaftlicher Nachlass (2nd ed., H. Weber, Ed.). Teubner. https://archive.org/details/gesammeltemathem00riemuoftCited in Chapter 10 3 times and 1800 to 1872. | de | read in full |
| S066 | Sarasa, A. A. d. (1649). Solutio problematis a R. P. Marino Mersenno minimo propositi: datis tribus quibuscumque magnitudinibus, rationalibus vel irrationalibus, datisque duarum ex illis logarithmis, tertiae logarithmum geometrice invenire. Ioannes et Iacobus Meursius, Antwerp. Google Books scan bub_gb_TGi3Dzr7QoC, supplied by the user 2026-08-14Cited in Chapter 5 5 times and 1637 to 1665. | la | read in full |
| S423 | Skopinski, T. H., & Johnson, K. G. (1960). Determination of azimuth angle at burnout for placing a satellite over a selected earth position. NASA Technical Note D-233 (L-289), 1-15 + appendices. National Aeronautics and Space Administration. https://ntrs.nasa.gov/citations/19980227091Cited in Chapter 11 3 times and 1914 to the present. | en | read in full |
| S413 | Somerville, M. (1831). Mechanism of the heavens. John Murray. https://archive.org/details/mechanismofheave00somerichCited in Chapter 11 and 1800 to 1872. | en | read in full |
| S412 | Somerville, M. (1873). Personal recollections, from early life to old age, of Mary Somerville, with selections from her correspondence. John Murray. https://archive.org/details/personalrecollec00someuoftCited in Chapter 11 4 times and 1800 to 1872 3 times. | en | read in full |
| S180 | Stukeley, W. (1752). Memoirs of Sir Isaac Newton's Life (MS/142, Royal Society). The Newton Project (OTHE00001), ff. 15r-16r etc. University of Oxford. https://www.newtonproject.ox.ac.uk/view/texts/normalized/OTHE00001Cited in Chapter 6 twice and 1637 to 1665 twice. | en | read in full |
| S339 | Thiebault, D. (1804). Mes souvenirs de vingt ans de sejour a Berlin. Paris.Cited in 1734 to 1799. | fr | not obtained, supports no claim |
| S411 | Whewell, W. (1834). [Review of] On the Connexion of the Physical Sciences, by Mrs. Somerville (published anonymously). The Quarterly Review, 51 (no. 101), 54-68. John Murray. https://blogs.kent.ac.uk/sciencecomma/files/2015/04/whewell_1834_scientist.pdfCited in Chapter 11 twice and 1800 to 1872. | en | read in full |
| S190 | Whiteside, D. T. (1967). The Mathematical Papers of Isaac Newton, Vol. 1: 1664-1666. Cambridge University Press. https://archive.org/details/MathematicsIsaacNewtonVol1_1664-66Whiteside1967Cited in Chapter 6 5 times and 1665 to 1687 8 times. | en;la | read in full |
| S192 | Whiteside, D. T. (1969). The Mathematical Papers of Isaac Newton, Vol. 3: 1670-1673. Cambridge University Press. https://archive.org/details/MathematicsIsaacNewtonVol3_1670-73Whiteside1969Cited in Chapter 6 4 times, 1665 to 1687, 1688 to 1734 and 1734 to 1799. | en;la | read in full |
| S191 | Whiteside, D. T. (1974). The Mathematical Papers of Isaac Newton, Vol. 6: 1684-1691. Cambridge University Press. https://archive.org/details/MathematicsIsaacNewtonV616841691Whiteside1972Cited in Chapter 6 3 times, 1665 to 1687 and 1688 to 1734. | en;la | read in full |
| S338 | Whiteside, Derek Thomas (Ed.) (1981). The Mathematical Papers of Isaac Newton, vol. 8, pp. 9-10, notes 21-22. Cambridge University Press.Not cited in the story. It backs a register entry. | en | not obtained, supports no claim |
Tier 2: peer-reviewed articles and academic-press monographs
61 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S133 | Alexander, A. (2018). On indivisibles and infinitesimals: A response to David Sherry. Foundations of Science, 23(2), 393-398. Springer. https://doi.org/10.1007/s10699-017-9523-1Cited in Chapter 5. | en | read in full |
| S051 | Almeida, D. F. (2007). Kerala Mathematics and Its Possible Transmission to Europe. Philosophy of Mathematics Education Journal, 20. https://muslimheritage.com/kerala-math-transmission-europe/Cited in Chapter 3 3 times, 1400 to 1637 and 1800 to 1872. | en | read in full |
| S156 | Andersen, K. (1985). Cavalieri's method of indivisibles. Archive for History of Exact Sciences, [vol unverified], [pages unverified]. Springer. https://www.tau.ac.il/~corry/teaching/toldot/download/Andersen.pdfCited in Chapter 5 3 times, 1400 to 1637 3 times and 1637 to 1665. | en | read in full |
| S237 | Arthur, R. T. W. (2006). The Remarkable Fecundity of Leibniz's Work on Infinite Series. Review of Leibniz, Samtliche Schriften und Briefe VII.3 / III.5, 1-6. McMaster University. https://www.humanities.mcmaster.ca/~rarthur/papers/Fecundity.pdfCited in Chapter 7 3 times and 1665 to 1687. | en | read in full |
| S063 | Aydin, Nuh; Hammoudi, Lakhdar (n.d.). Introduction and biography, in Al-Kashi's Miftah al-Hisab, Volume I: Arithmetic. Translation and Commentary. Al-Kashi's Miftah al-Hisab, Volume I: Arithmetic, I, 1-13 (introduction). Birkhauser/Springer. https://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall17/128/MiftahIntro.pdfCited in Chapter 3 twice. | en | read in full |
| S367 | Bascelli, T., et al. (2018). Cauchy's Infinitesimals, His Sum Theorem, and Foundational Paradigms. Foundations of Science, 23, 267-296. Springer. https://arxiv.org/abs/1704.07723Cited in Chapter 10. | en | read in full |
| S436 | Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18 (153), 1-43. https://arxiv.org/abs/1502.05767Cited in Chapter 12 7 times and 1914 to the present 5 times. | en | read in full |
| S395 | Bishop, E. (1977). Review: H. Jerome Keisler, Elementary Calculus. Bulletin of the American Mathematical Society, 83, 205-208. American Mathematical Society. https://projecteuclid.org/euclid.bams/1183538669Cited in Chapter 10 and 1914 to the present. | en | not obtained, supports no claim |
| S368 | Borovik, A. (2012). Who Gave You the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus. Foundations of Science, 17, 245-276. Springer. https://arxiv.org/abs/1108.2885Cited in Chapter 10 twice. | en | read in full |
| S313 | Brown, B. H. (1942). The Euler-Diderot Anecdote. The American Mathematical Monthly, 49 (5), 302-303. Mathematical Association of America. https://doi.org/10.1080/00029890.1942.11991228Cited in Chapter 9 twice, 1734 to 1799, 1800 to 1872 and 1914 to the present. | en | read in full |
| S160 | Burn, R. P. [unverified] (2001). Alphonse Antonio de Sarasa and logarithms. Historia Mathematica, 28(1) [unverified], 1-17 [unverified]. Elsevier. http://users.uoa.gr/~apgiannop/Sources/Sarasa.pdfNot cited in the story. It backs a register entry. | en | not obtained, supports no claim |
| S203 | Chin, S. A. (2021). Modern light on ancient feud: Robert Hooke and Newton's graphical method. arXiv, arXiv:2106.07082. https://arxiv.org/abs/2106.07082Cited in Chapter 6 twice and 1665 to 1687 twice. | en | read in full |
| S155 | Coolidge, J. L. (1950). The number e. The American Mathematical Monthly, 57(9), 591-602. Mathematical Association of America. https://www-fourier.ujf-grenoble.fr/~marin/une_autre_crypto/articles_et_extraits_livres/irationalite/Coolidge_J.L._the_number_e.pdfCited in Chapter 5, 1637 to 1665 twice and 1665 to 1687. | en | read in full |
| S336 | Craik, A. D. D. (2013). 'Continuity and change': representing mass conservation in fluid mechanics. Archive for History of Exact Sciences, 67 (1), 43-80. Springer. https://doi.org/10.1007/s00407-012-0108-7Cited in 1688 to 1734. | en | abstract only, supports no claim |
| S075 | Cupillari, A. (2014). Maria Gaetana Agnesi's other curves (more than just the witch). Mathematics Magazine, 87(1), 3-13. Taylor & Francis on behalf of the Mathematical Association of America. https://www.jstor.org/stable/10.4169/math.mag.87.1.3Cited in Chapter 9 twice. | en | read in full |
| S341 | Darrigol, O. (2005). Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press. https://archive.org/details/worldsofflowhist0000darr_b9p7Cited in 1688 to 1734. | en | not obtained, supports no claim |
| S288 | De Morgan, A. (1914). Essays on the Life and Work of Newton (P. E. B. Jourdain, Ed.). Open Court. https://archive.org/details/cu31924012316901Cited in Chapter 8 3 times. | en | read in full |
| S076 | Del Centina, Andrea, & Fiocca, Alessandra (2020). Borelli's edition of books V-VII of Apollonius's Conics, and Lemma 12 in Newton's Principia. Archive for History of Exact Sciences, 74(3), 255-279. Springer Nature. https://www.jstor.org/stable/45294387Cited in Chapter 6 4 times. | en | read in full |
| S285 | Dhombres, J. G. (2014). A Tentative Interpretation of the Epistemological Significance of the Encrypted Message Sent by Newton to Leibniz in October 1676. Advances in Historical Studies, 3(1), 22-32. SCIRP. https://www.scirp.org/pdf/AHS_2014022817243062.pdfCited in Chapter 8 and 1665 to 1687. | en | read in full |
| S342 | Eckert, M. (n.d.). Euler's involvement in practical hydrodynamics on the eve of ideal flow theory. Euler Equations 250 (EE250) conference materials. Deutsches Museum, Munich. https://www-n.oca.eu/etc7/EE250/presentations/Eckert.pdfCited in Chapter 9. | en | read in full |
| S073 | Feingold, M. (1993). Newton, Leibniz, and Barrow too: An attempt at a reinterpretation. Isis, 84(2), 310-338. University of Chicago Press on behalf of the History of Science Society. https://www.jstor.org/stable/236236Cited in Chapter 5 6 times. | en | read in full |
| S070 | Fellmann, E. A. (1988). The 'Principia' and continental mathematicians. Notes and Records of the Royal Society of London, 42(1), 13-34. Royal Society. https://www.jstor.org/stable/531367Cited in Chapter 6. | en | read in full |
| S312 | Gibson, G. A. (1921). Taylor's Theorem and Bernoulli's Theorem: A Historical Note. Proceedings of the Edinburgh Mathematical Society, 39, 25-33. Cambridge University Press. https://doi.org/10.1017/S0013091500035768Cited in Chapter 9 4 times and 1688 to 1734 3 times. | en | read in full |
| S314 | Gillings, R. J. (1954). The So-Called Euler-Diderot Incident. The American Mathematical Monthly, 61 (2), 77-80. Mathematical Association of America. https://www.jstor.org/stable/2307789Cited in Chapter 9 twice, 1734 to 1799, 1800 to 1872 and 1914 to the present. | en | read in full |
| S360 | Grabiner, J. V. (1983). Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus. American Mathematical Monthly, 90, 185-194. Mathematical Association of America. https://www.matharticles.com/ma/ma002.pdfCited in Chapter 1, Chapter 10 twice and 1800 to 1872 twice. | en | read in full |
| S335 | Grabiner, J. V. (1997). Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions. The American Mathematical Monthly, 104 (5), 393-410. Mathematical Association of America. https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1115&context=pitzer_fac_pubCited in 1914 to the present. | en | read in full |
| S438 | Griewank, A. (2012). Who invented the reverse mode of differentiation? Documenta Mathematica, Extra Volume: Optimization Stories (ISMP 2012), 389-400. https://ftp.gwdg.de/pub/EMIS/journals/DMJDMV/vol-ismp/52_griewank-andreas-b.pdfCited in Chapter 12 3 times and 1914 to the present. | en | read in full |
| S204 | Guicciardini, N. (2004). Isaac Newton and the publication of his mathematical manuscripts. Studies in History and Philosophy of Science, 35(3), 455-470. Elsevier. https://doi.org/10.1016/j.shpsa.2004.06.002Cited in Chapter 6 27 times, Chapter 8 6 times, 1665 to 1687 14 times, 1688 to 1734 13 times and 1734 to 1799. | en | read in full |
| S020 | Guo, S. (2013). The Nine Chapters on the Mathematical Procedures and Liu Hui's Mathematical Theory. Seki, Founder of Modern Mathematics in Japan (Springer Proceedings in Mathematics & Statistics), 39. Springer. https://legacy-www.math.harvard.edu/~knill/teaching/math22a2018/exhibits/ninechapters/Guo_Shuchun_2013.pdfCited in Chapter 3 twice and Antiquity to 500 CE. | en | read in full |
| S386 | Johansson, J. (2003). Continuous Nowhere Differentiable Functions (Master's thesis 2003:320 CIV). Lulea University of Technology. https://andrescaicedo.wordpress.com/wp-content/uploads/2014/10/ltu-ex-03320-se.pdfCited in Chapter 10 5 times, 1800 to 1872 5 times, 1873 to 1913 twice and 1914 to the present 5 times. | en | read in full |
| S445 | Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics. Princeton University Press. https://archive.org/details/crestofpeacockno00joseNot cited in the story. It backs a register entry. | en | not obtained, supports no claim |
| S061 | Katz, V. J. (1995). Ideas of Calculus in Islam and India. Mathematics Magazine, 68(3), 163-174. Taylor & Francis on behalf of the Mathematical Association of America. https://www.jstor.org/stable/2691411Cited in Chapter 3 13 times, 500 to 1400 twice and 1400 to 1637. | en | read in full |
| S130 | Katz, Mikhail G., Schaps, David M., & Shnider, Steven (2013). Almost equal: The method of adequality from Diophantus to Fermat and beyond. Perspectives on Science, 21(3), 283-324. MIT Press. https://arxiv.org/pdf/1210.7750Cited in Chapter 5 4 times, Chapter 9 and 1400 to 1637. | en | read in full |
| S390 | Katz, M. G. (2018). A footnote to 'The crisis in contemporary mathematics'. Historia Mathematica. Elsevier. https://arxiv.org/abs/1804.02645Cited in Chapter 10 3 times and 1914 to the present twice. | en | read in full |
| S491 | Katz, M. G. (2020). Mathematical Conquerors, Unguru Polarity, and the Task of History. Journal of Humanistic Mathematics, 10(1), 475-515. Claremont Colleges Library. https://arxiv.org/pdf/2002.00249 (published: https://doi.org/10.5642/jhummath.202001.27)Cited in Chapter 2 3 times and 1914 to the present. | en | read in full |
| S193 | Keynes, J. M. (1947). Newton, the Man. The Royal Society Newton Tercentenary Celebrations, 15-19 July 1946, 27-34. Cambridge University Press. https://mathshistory.st-andrews.ac.uk/Extras/Keynes_Newton/Cited in Chapter 6 twice and 1914 to the present twice. | en | read in full |
| S458 | Kichenassamy, S. (2020). Apodictic discourse and the Cauchy-Bunyakovsky-Schwarz inequality. Ganita Bharati, 42(1), 1-19. https://doi.org/10.32381/GB.2020.42.1-2.1Cited in 1800 to 1872. | en | read in full |
| S234 | Lemley, S. V. (2021). Printing Leibniz's Calculus: Dating and Numbering the Editions of the Nova Methodus pro Maximis et Minimis (October 1684). The Library, 22(2), 177-196. Oxford University Press. https://doi.org/10.1093/library/22.2.177Not cited in the story. It backs a register entry. | en | abstract only, supports no claim |
| S161 | Martin, J. (2010). The Helen of Geometry. The College Mathematics Journal, 41(1), 17-28. Mathematical Association of America. https://doi.org/10.4169/074683410X475083Cited in Chapter 5. | en | read in full |
| S077 | Mazzotti, M. (2001). Maria Gaetana Agnesi: Mathematics and the making of the Catholic Enlightenment. Isis, 92(4), 657-683. University of Chicago Press on behalf of the History of Science Society. https://www.jstor.org/stable/3080337Cited in Chapter 9 14 times. | en | read in full |
| S322 | Monks, K. M. (2019). Euler's Calculation of the Sum of the Reciprocals of Squares [with Euler E41 in Jordan Bell's translation]. TRIUMPHS: Calculus, no. 9, 1-15. Ursinus College. https://digitalcommons.ursinus.edu/triumphs_calculus/9Cited in Chapter 9 3 times and 1734 to 1799. | en | read in full |
| S202 | Nauenberg, M. (2003). Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1. Historia Mathematica, 30 (4), 441-456. Elsevier. https://arxiv.org/abs/math/0112048Not cited in the story. It backs a register entry. | en | read in full |
| S138 | Nauenberg, M. (2011). Barrow and Leibniz on the fundamental theorem of the calculus. arXiv:1111.6145. arXiv. https://ar5iv.arxiv.org/html/1111.6145Cited in Chapter 5 and 1665 to 1687 twice. | en | read in full |
| S003 | Netz, R. (2002). A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 2). SCIAMVS, 3, 109-125. https://www.sciamvs.org/files/SCIAMVS_03_109-125_Netz_Saito_Tchernetska.pdfCited in Chapter 2 twice and 1914 to the present. | en | read in full |
| S050 | Ramasubramanian, K. (n.d.). Development of Calculus in India. Cell for Indian Science and Technology in Sanskrit, IIT Bombay / Centre for Policy Studies, Chennai. https://www.ms.uky.edu/~sohum/ma330/files/india_calculus.pdfCited in Chapter 3 9 times, Antiquity to 500 CE, 500 to 1400 3 times and 1400 to 1637 4 times. | en | read in full |
| S062 | Roy, R. (1990). The Discovery of the Series Formula for pi by Leibniz, Gregory and Nilakantha. Mathematics Magazine, 63(5), 291-306. Mathematical Association of America. http://users.uoa.gr/~apgiannop/Sources/Roy-pi.pdfNot cited in the story. It backs a register entry. | en | read in full |
| S437 | Schmidhuber, J. (2015). Deep learning in neural networks: an overview. Neural Networks, 61, 85-117. Elsevier. https://arxiv.org/abs/1404.7828Cited in Chapter 12 5 times and 1914 to the present 3 times. | en | read in full |
| S132 | Sherry, D. (2018). The Jesuits and the method of indivisibles. Foundations of Science, 23(2), 367-392. Springer. https://doi.org/10.1007/s10699-017-9525-zCited in Chapter 5. | en | read in full |
| S194 | Snobelen, S. D. (1999). Isaac Newton, heretic: the strategies of a Nicodemite. The British Journal for the History of Science, 32 (4), 381-419. Cambridge University Press. https://isaac-newton.org/wp-content/uploads/2013/06/heretic.pdfCited in Chapter 6 and 1688 to 1734. | en | read in full |
| S460 | Stigler, S. M. (1980). Stiglers Law of Eponymy. Transactions of the New York Academy of Sciences, 39(1 Series II), 147-157. https://doi.org/10.1111/j.2164-0947.1980.tb02775.xNot cited in the story. It backs a register entry. | en | not obtained, supports no claim |
| S105 | Sylla, E. D. (1987). The Oxford Calculators in context. Science in Context, 1(2), 257-279. Cambridge University Press. https://doi.org/10.1017/S0269889700000387Not cited in the story. It backs a register entry. | en | abstract only, supports no claim |
| S449 | Tamboukou, M. (2026). Automathographical Traces in Wang Zhenyi's Travel Poetry. European Journal of Life Writing, 15, 29-52. https://doi.org/10.21827/ejlw.15.43079 (OA PDF: https://ejlw.eu/article/download/43079/40360)Cited in Chapter 11. | en | not obtained, supports no claim |
| S394 | Taton, R. (1971). Sur les relations scientifiques d'Augustin Cauchy et d'Evariste Galois. Revue d'histoire des sciences, 24, 123-148. https://www.persee.fr/doc/rhs_0151-4105_1971_num_24_2_3196Not cited in the story. It backs a register entry. | fr | not obtained, supports no claim |
| S315 | Truesdell, C. (1958). The New Bernoulli Edition [section IV, The correspondence between Bernoulli and L'Hopital, read in full]. Isis, 49 (1), 54-62. History of Science Society. https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bernoulli/index.htmlCited in Chapter 9 3 times, 1688 to 1734 twice and 1914 to the present. | en | read in full |
| S340 | Truesdell, C. (1989). Maria Gaetana Agnesi. Archive for History of Exact Sciences, 40 (2), 113-142. Springer.Cited in 1734 to 1799. | en | not obtained, supports no claim |
| S071 | Verelst, K. (2014). Newton versus Leibniz: Intransparency versus inconsistency. Synthese, 191(13), 2907-2940. Springer. https://www.jstor.org/stable/24021587Not cited in the story. It backs a register entry. | en | read in full |
| S023 | Wagner, D. B. (1978). Liu Hui and Zu Gengzhi on the Volume of a Sphere. Chinese Science, 3, 59-79. http://donwagner.dk/SPHERE/SPHERE.htmlCited in Chapter 3 3 times and Antiquity to 500 CE twice. | en | read in full |
| S064 | Whitman, E. A. (1943). Some Historical Notes on the Cycloid. The American Mathematical Monthly, 50(5), 309-315. Taylor & Francis on behalf of the Mathematical Association of America. https://www.jstor.org/stable/2302830Cited in Chapter 5 7 times, 1400 to 1637 twice and 1637 to 1665 3 times. | en | read in full |
| S157 | Wijeratne, Chanakya, & Zazkis, Rina (2015). On painter's paradox: Contextual and mathematical approaches to infinity. International Journal of Research in Undergraduate Mathematics Education, 1, 163-186. Springer. https://doi.org/10.1007/s40753-015-0004-zCited in Chapter 5 twice and 1637 to 1665 twice. | en | read in full |
| S068 | Zinsser, J. P. (2007). Mentors, the marquise Du Chatelet and historical memory. Notes and Records of the Royal Society of London, 61(2), 89-108. Royal Society. https://www.jstor.org/stable/20462616Cited in Chapter 9 6 times and 1734 to 1799 twice. | en | read in full |
| S065 | Zinsser, J. P. (2009). Feminist Biography: A Contradiction in Terms? The Eighteenth Century, 50(1), 43-50. University of Pennsylvania Press. https://www.jstor.org/stable/41468055Not cited in the story. It backs a register entry. | en | read in full |
Tier 3: reference works with named authorship and citations
120 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S004 | Lost and Found: The Secrets of Archimedes. (n.d.). The Walters Art Museum. The Walters Art Museum. https://thewalters.org/news/lost-and-found-the-secrets-of-archimedes/Cited in Chapter 2 4 times, 500 to 1400 and 1914 to the present. | en | read in full |
| S005 | The Archimedes Palimpsest (project website: About, Imaging, Archimedes pages). (n.d.). The Archimedes Palimpsest Project. http://archimedespalimpsest.org/Cited in Chapter 2 twice and 1914 to the present. | en | read in full |
| S060 | Trigonometry: India and the Islamic World. (n.d.). Encyclopaedia Britannica. Encyclopaedia Britannica. https://www.britannica.com/science/trigonometry/India-and-the-Islamic-worldCited in Chapter 3 twice and 1400 to 1637. | en | read in full |
| S463 | (this project) (n.d.). Pronunciation conventions applied to people.csv.Not cited in the story. It backs a register entry. | en | read in full |
| S151 | [attributed to Jourdain, P. E. B. - UNVERIFIED] (1911). Infinitesimal Calculus: History. Encyclopaedia Britannica (11th ed.). Cambridge University Press. https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Infinitesimal_Calculus/HistoryCited in Chapter 5, 1400 to 1637 3 times and 1637 to 1665 3 times. | en | read in full |
| S143 | [author unverified] (n.d.). Pascal, Blaise. Internet Encyclopedia of Philosophy. University of Tennessee at Martin. https://iep.utm.edu/pascal-b/Cited in Chapter 5 and 1637 to 1665 twice. | en | read in full |
| S069 | Allen, B. (2007). The multi-tasking marquise [Review of J. Zinsser, La Dame d'Esprit: A Biography of the Marquise Du Chatelet]. The Hudson Review, 60(2), 311-318. The Hudson Review, Inc. https://www.jstor.org/stable/20464714Cited in Chapter 9 4 times and 1734 to 1799 twice. | en | read in full |
| S329 | Ashworth, William B., Jr. (n.d.). Jacob Bernoulli [Scientist of the Day]. Linda Hall Library. Linda Hall Library of Science, Engineering & Technology. https://www.lindahall.org/about/news/scientist-of-the-day/jacob-bernoulli/Cited in Chapter 9 twice and 1688 to 1734. | en | read in full |
| S369 | Bell, J. L. (2022). Continuity and Infinitesimals. Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/continuity/Cited in Chapter 10 and 1800 to 1872. | en | read in full |
| S323 | Bogart, D., Klyve, D., & Stemkoski, L. (Eds.) (n.d.). The Euler Archive: Enestrom records E41, E65, E101, E212. University of the Pacific / MAA. https://scholarlycommons.pacific.edu/euler-works/Cited in Chapter 9 and 1734 to 1799 4 times. | en | read in full |
| S334 | British Society for the History of Mathematics (n.d.). Scientific Challenges and Encryption of Discoveries in 17th Century Rational Mechanics. BSHM. British Society for the History of Mathematics. https://www.bshm.ac.uk/scientific-challenges-and-encryption-discoveries-17th-century-rational-mechanicsCited in 1688 to 1734 3 times. | en | read in full |
| S456 | Cajori, F. (1919). A History of Mathematics (2nd ed., revised and enlarged). Macmillan. https://archive.org/details/historyofmathema00cajoNot cited in the story. It backs a register entry. | en | read in full |
| S389 | Calder, J. (2014). Weierstrass's non-differentiable function. Math 104 course notes. University of Minnesota. https://www-users.cse.umn.edu/~jwcalder/104F14/weierstrass-function.pdfCited in Chapter 10 and 1914 to the present. | en | read in full |
| S198 | Cambridge University Library (n.d.). Newton Papers: Newton's Waste Book (MS Add. 4004). Cambridge Digital Library. University of Cambridge. https://cudl.lib.cam.ac.uk/view/MS-ADD-04004/1Cited in Chapter 6 twice. | en | read in full |
| S199 | Cambridge University Library (n.d.). Newton Papers: Early Papers (MS Add. 3958). Cambridge Digital Library, ff. 48v-63v; 78r-80v. University of Cambridge. https://cudl.lib.cam.ac.uk/view/MS-ADD-03958Cited in Chapter 6 twice and 1665 to 1687 twice. | en | read in full |
| S134 | Cardil, R. (n.d.). Kepler: The volume of a wine barrel. Loci: Convergence. Mathematical Association of America. http://www.matematicasvisuales.com/loci/kepler/doliometry.htmlCited in 1400 to 1637 3 times. | en | read in full |
| S108 | Caroti, S. (2005). Oresme, Nicole (c. 1320-1382). Encyclopedia of Philosophy (2nd ed.). Macmillan Reference USA. https://www.encyclopedia.com/humanities/encyclopedias-almanacs-transcripts-and-maps/oresme-nicole-c-1320-1382Cited in Chapter 4 and 500 to 1400 twice. | en | read in full |
| S493 | Coleman, Alice, and others (1977). The Advanced Placement Program in California (Revised). Sacramento: California State Department of Education. https://archive.org/details/ERIC_ED145544Cited in Chapter 12 and 1914 to the present twice. | en | read in full |
| S448 | Commission on the Future of the Advanced Placement Program (2001). Access to Excellence: A Report of the Commission on the Future of the Advanced Placement Program. College Entrance Examination Board. https://files.eric.ed.gov/fulltext/ED561056.pdfCited in Chapter 12 and 1914 to the present. | en | read in full |
| S152 | Davis, A. E. L. (2006). Kepler's laws. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Keplers_laws/Cited in Chapter 5 and 1400 to 1637. | en | read in full |
| S054 | Dennis, D. (2009). Alhazen's Summation Formulas. Quadrivium. http://www.quadrivium.info/MathInt/Notes/AlHazenSummation.pdfCited in Chapter 3 twice and 500 to 1400. | en | read in full |
| S343 | Dumbaugh, D. (2019). Maria Gaetana Agnesi. Notices of the American Mathematical Society, 66(3), 414-415. American Mathematical Society. https://www.ams.org/journals/notices/201903/rnoti-p414.pdfNot cited in the story. It backs a register entry. | en | read in full |
| S106 | Gale, (corporate author) (2018). Richard Swineshead. Encyclopedia.com. Gale/Cengage. https://www.encyclopedia.com/people/philosophy-and-religion/philosophy-biographies/richard-swinesheadCited in 500 to 1400. | en | read in full |
| S074 | Goldberg, H. (2008). Ideas and eternity. Tradition: A Journal of Orthodox Jewish Thought, 41(2), 138-144. Rabbinical Council of America. https://www.jstor.org/stable/23263708Not cited in the story. It backs a register entry. | en | read in full |
| S201 | History of Parliament Trust (n.d.). NEWTON, Isaac (1642-1727), of Jermyn Street, Westminster. The History of Parliament: the House of Commons 1690-1715. Boydell & Brewer. https://www.historyofparliamentonline.org/volume/1690-1715/member/newton-isaac-1642-1727Cited in 1688 to 1734. | en | read in full |
| S007 | Huggett, N. (2024). Zeno's Paradoxes. The Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/paradox-zeno/Cited in Chapter 2 twice and Antiquity to 500 CE. | en | read in full |
| S102 | Jung, E. (2022). Richard Kilvington. The Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/kilvington/Cited in Chapter 4 and 500 to 1400. | en | read in full |
| S101 | Jung, E. (2026). William Heytesbury. The Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/heytesbury/Cited in Chapter 4 twice and 500 to 1400. | en | read in full |
| S388 | Kesavan, S. (n.d.). Continuous functions that are nowhere differentiable. Institute of Mathematical Sciences, Chennai. https://www.imsc.res.in/~kesh/nowhere.pdfCited in Chapter 10 twice, 1873 to 1913 and 1914 to the present twice. | en | read in full |
| S100 | Kirschner, S. (2025). Nicole Oresme. The Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/nicole-oresme/Cited in Chapter 4 and 500 to 1400 4 times. | en | read in full |
| S284 | Kollerstrom, N. (2012). The Birth of Calculus: Towards a More Leibnizian View. arXiv:1212.2666. https://arxiv.org/pdf/1212.2666Cited in Chapter 8 and 1688 to 1734 3 times. | en | read in full |
| S236 | Look, B. C. (2013). Gottfried Wilhelm Leibniz. Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/leibniz/Cited in Chapter 7, 1637 to 1665, 1665 to 1687 3 times and 1688 to 1734 twice. | en | read in full |
| S104 | Lovell-Read, F. (n.d.). The Merton Calculators. Merton College, University of Oxford. https://www.merton.ox.ac.uk/sites/default/files/inline-files/The%20Merton%20Calculators.pdfCited in Chapter 4 and 500 to 1400. | en | read in full |
| S139 | Miller, J. (n.d.). Earliest uses of symbols of calculus. Earliest Uses of Various Mathematical Symbols. University of St Andrews (host). https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/Cited in Chapter 5, 1637 to 1665 3 times, 1665 to 1687 twice, 1734 to 1799 twice, 1800 to 1872 4 times and 1873 to 1913 twice. | en | read in full |
| S006 | Miller, M. K. (2007). Reading Between the Lines. Smithsonian Magazine. https://www.smithsonianmag.com/science-nature/reading-between-the-lines-148131057/Cited in Chapter 2 4 times, 500 to 1400, 1873 to 1913 and 1914 to the present. | en | read in full |
| S238 | Miller, J. (n.d.). Earliest Uses of Symbols of Calculus. Earliest Uses of Various Mathematical Symbols. https://jeff560.tripod.com/calculus.htmlCited in Chapter 7 and 1665 to 1687 twice. | en | read in full |
| S324 | Miller, J. (n.d.). Earliest Known Uses of Some of the Words of Mathematics: INTEGRAL, INTEGRAL CALCULUS, DERIVATIVE, DIFFERENTIAL CALCULUS, FUNCTION. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathword/Cited in Chapter 9, 1688 to 1734 and 1734 to 1799. | en | read in full |
| S325 | Miller, J. (n.d.). Earliest Uses of Symbols for Constants; Earliest Uses of Symbols of Calculus. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathsym/Cited in Chapter 9 3 times, 1688 to 1734 twice and 1734 to 1799 4 times. | en | read in full |
| S392 | Miller, Jeff (Ed.) (n.d.). Earliest Known Uses of Some of the Words of Mathematics (L) - LIMIT. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathword/l/Not cited in the story. It backs a register entry. | en | read in full |
| S393 | Miller, Jeff (Ed.) (n.d.). Earliest Uses of Symbols of Set Theory and Logic. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/Cited in 1873 to 1913. | en | read in full |
| S453 | Miller, J., & Aldrich, J. (n.d.). Earliest Known Uses of Some of the Words of Mathematics (letters A-V). MacTutor History of Mathematics Archive, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Miller/mathword/Cited in Chapter 1 twice, 1688 to 1734 and 1873 to 1913. | en | read in full |
| S370 | Morscher, E. (2018). Bernard Bolzano. Stanford Encyclopedia of Philosophy. Stanford University. https://plato.stanford.edu/entries/bolzano/Cited in Chapter 10 and 1800 to 1872 twice. | en | read in full |
| S008 | O'Connor, J. J. (n.d.). Eudoxus of Cnidus. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Eudoxus/Cited in Chapter 2 3 times and Antiquity to 500 CE. | en | read in full |
| S021 | O'Connor, J. J. (n.d.). Zu Chongzhi. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Zu_Chongzhi/Cited in Chapter 3 and Antiquity to 500 CE. | en | read in full |
| S022 | O'Connor, J. J. (n.d.). Liu Hui. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Liu_Hui/Cited in Antiquity to 500 CE. | en | read in full |
| S053 | O'Connor, J. J. (n.d.). Sharaf al-Din al-Muzaffar al-Tusi. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf/Cited in Chapter 3 and 500 to 1400. | en | read in full |
| S055 | O'Connor, J. J. (n.d.). Madhava of Sangamagramma. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/Cited in Chapter 3 and 1400 to 1637 twice. | en | read in full |
| S056 | O'Connor, J. J. (n.d.). Nilakantha Somayaji. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Nilakantha/Cited in Chapter 3 and 1400 to 1637. | en | read in full |
| S057 | O'Connor, J. J. (n.d.). Al-Sabi Thabit ibn Qurra al-Harrani. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Thabit/Cited in Chapter 3 twice and 500 to 1400. | en | read in full |
| S058 | O'Connor, J. J. (n.d.). Abu Ali al-Hasan ibn al-Haytham. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Al-Haytham/Cited in Chapter 3 and 500 to 1400. | en | read in full |
| S059 | O'Connor, J. J. (n.d.). Aryabhata the Elder. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_I/Cited in Chapter 3 3 times and Antiquity to 500 CE. | en | read in full |
| S103 | O'Connor, J. J., & Robertson, E. F. (n.d.). Thomas Bradwardine. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Bradwardine/Cited in Chapter 4 and 500 to 1400 twice. | en | read in full |
| S135 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bonaventura Cavalieri. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cavalieri/Cited in Chapter 5 twice, 1400 to 1637 and 1637 to 1665. | en | read in full |
| S136 | O'Connor, J. J., & Robertson, E. F. (n.d.). James Gregory. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Gregory/Cited in Chapter 5 twice and 1665 to 1687 3 times. | en | read in full |
| S137 | O'Connor, J. J., & Robertson, E. F. (n.d.). Gilles Personne de Roberval. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Roberval/Cited in Chapter 5 and 1637 to 1665. | en | read in full |
| S140 | O'Connor, J. J., & Robertson, E. F. (n.d.). Isaac Barrow. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Barrow/Cited in 1665 to 1687 twice. | en | read in full |
| S141 | O'Connor, J. J., & Robertson, E. F. (n.d.). Gregory of Saint Vincent. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Saint-Vincent/Cited in 1637 to 1665 3 times. | en | read in full |
| S142 | O'Connor, J. J., & Robertson, E. F. (n.d.). Pietro Mengoli. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Mengoli/Cited in Chapter 5 and 1637 to 1665. | en | read in full |
| S144 | O'Connor, J. J., & Robertson, E. F. (n.d.). Pierre de Fermat. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Fermat/Cited in Chapter 5, 1400 to 1637 and 1637 to 1665. | en | read in full |
| S145 | O'Connor, J. J., & Robertson, E. F. (n.d.). John Wallis. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Wallis/Cited in Chapter 5 and 1637 to 1665 3 times. | en | read in full |
| S146 | O'Connor, J. J., & Robertson, E. F. (n.d.). Rene Francois Walter de Sluze. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Sluze/Cited in 1665 to 1687 twice. | en | read in full |
| S147 | O'Connor, J. J., & Robertson, E. F. (n.d.). Johann van Waveren Hudde. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Hudde/Cited in Chapter 5 and 1637 to 1665. | en | read in full |
| S148 | O'Connor, J. J., & Robertson, E. F. (n.d.). Rene Descartes. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Descartes/Cited in Chapter 5 twice and 1637 to 1665. | en | read in full |
| S149 | O'Connor, J. J., & Robertson, E. F. (n.d.). Nicolaus Mercator. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Mercator_Nicolaus/Cited in Chapter 5 and 1665 to 1687. | en | read in full |
| S150 | O'Connor, J. J., & Robertson, E. F. (n.d.). Christiaan Huygens. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Huygens/Cited in Chapter 5, 1637 to 1665 and 1665 to 1687. | en | read in full |
| S153 | O'Connor, J. J., & Robertson, E. F. (n.d.). Blaise Pascal. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Pascal/Cited in Chapter 5 and 1637 to 1665. | en | read in full |
| S154 | O'Connor, J. J., & Robertson, E. F. (n.d.). Johannes Kepler. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Kepler/Cited in Chapter 5 and 1400 to 1637 3 times. | en | read in full |
| S131 | O'Connor, J. J., & Robertson, E. F. (2002). Evangelista Torricelli. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Torricelli/Cited in Chapter 5 and 1637 to 1665 4 times. | en | read in full |
| S326 | O'Connor, J. J., & Robertson, E. F. (n.d.). Leonhard Euler. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Euler/Cited in Chapter 9 and 1734 to 1799 twice. | en | read in full |
| S327 | O'Connor, J. J., & Robertson, E. F. (n.d.). Maria Gaetana Agnesi. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Agnesi/Cited in 1688 to 1734 and 1734 to 1799 twice. | en | read in full |
| S328 | O'Connor, J. J., & Robertson, E. F. (n.d.). Jacob Bernoulli's tomb in Basel. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Extras/Bernoulli_tomb/Cited in Chapter 9 and 1688 to 1734. | en | read in full |
| S330 | O'Connor, J. J., & Robertson, E. F. (n.d.). The brachistochrone problem. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/Cited in Chapter 9 and 1688 to 1734 twice. | en | read in full |
| S331 | O'Connor, J. J., & Robertson, E. F. (n.d.). Taylor versus Continental mathematicians. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Extras/Taylor_continental/Cited in 1688 to 1734. | en | read in full |
| S371 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bernard Bolzano. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Bolzano/Cited in Chapter 10 and 1800 to 1872 3 times. | en | read in full |
| S372 | O'Connor, J. J., & Robertson, E. F. (n.d.). Augustin-Louis Cauchy. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/Cited in Chapter 10 and 1800 to 1872 4 times. | en | read in full |
| S373 | O'Connor, J. J., & Robertson, E. F. (n.d.). Karl Weierstrass. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Weierstrass/Cited in Chapter 10 twice and 1800 to 1872 twice. | en | read in full |
| S374 | O'Connor, J. J., & Robertson, E. F. (n.d.). Sofia Kovalevskaya. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Kovalevskaya/Cited in Chapter 11 twice and 1873 to 1913 twice. | en | read in full |
| S375 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bernhard Riemann. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/Cited in Chapter 10 and 1800 to 1872 twice. | en | read in full |
| S376 | O'Connor, J. J., & Robertson, E. F. (n.d.). Georg Cantor. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/Cited in Chapter 10 and 1873 to 1913 5 times. | en | read in full |
| S377 | O'Connor, J. J., & Robertson, E. F. (n.d.). Henri Lebesgue. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Lebesgue/Cited in Chapter 10 twice and 1873 to 1913 twice. | en | read in full |
| S378 | O'Connor, J. J., & Robertson, E. F. (n.d.). Charles Meray. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Meray/Cited in Chapter 10 twice and 1800 to 1872 twice. | en | read in full |
| S379 | O'Connor, J. J., & Robertson, E. F. (n.d.). Lejeune Dirichlet. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Dirichlet/Cited in 1800 to 1872. | en | read in full |
| S380 | O'Connor, J. J., & Robertson, E. F. (n.d.). Evariste Galois. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Galois/Cited in Chapter 10 and 1800 to 1872 twice. | en | read in full |
| S381 | O'Connor, J. J., & Robertson, E. F. (n.d.). Abraham Robinson. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Robinson/Cited in Chapter 10 and 1914 to the present twice. | en | read in full |
| S382 | O'Connor, J. J., & Robertson, E. F. (n.d.). Richard Dedekind. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Dedekind/Cited in Chapter 10 and 1800 to 1872. | en | read in full |
| S383 | O'Connor, J. J., & Robertson, E. F. (n.d.). Joseph Fourier. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/Cited in Chapter 10 3 times and 1800 to 1872 4 times. | en | read in full |
| S384 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bolzano publications. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Extras/Bolzano_publications/Cited in Chapter 10, 1800 to 1872 twice and 1914 to the present twice. | en | read in full |
| S385 | O'Connor, J. J., & Robertson, E. F. (n.d.). Quotations by Niels Abel. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Abel/quotations/Cited in Chapter 10. | en | read in full |
| S457 | O'Connor, J. J., & Robertson, E. F. (n.d.). Viktor Yakovlevich Bunyakovsky (1804-1889). MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Bunyakovsky/Cited in 1800 to 1872. | en | read in full |
| S459 | O'Connor, J. J., & Robertson, E. F. (n.d.). Michel Rolle (1652-1719). MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Rolle/Cited in 1688 to 1734. | en | read in full |
| S462 | O'Connor, J. J., & Robertson, E. F. (n.d.). Biographies of Thomas Simpson, William Jones and L. F. A. Arbogast. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Simpson/Cited in 1688 to 1734 and 1734 to 1799. | en | read in full |
| S414 | O'Connor, J. J., & Robertson, E. F. (n.d.). Mary Fairfax Greig Somerville. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Somerville/Not cited in the story. It backs a register entry. | en | read in full |
| S416 | O'Connor, J. J., & Robertson, E. F. (n.d.). Augusta Ada King, Countess of Lovelace. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Lovelace/Cited in Chapter 11. | en | read in full |
| S418 | O'Connor, J. J., & Robertson, E. F. (n.d.). Amalie Emmy Noether. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/Cited in Chapter 11 4 times and 1914 to the present twice. | en | read in full |
| S419 | O'Connor, J. J., & Robertson, E. F. (n.d.). Grace Chisholm Young. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Chisholm_Young/Cited in Chapter 11 twice and 1873 to 1913 twice. | en | read in full |
| S420 | O'Connor, J. J., & Robertson, E. F. (n.d.). Dame Mary Lucy Cartwright. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cartwright/Cited in Chapter 11 twice. | en | read in full |
| S421 | O'Connor, J. J., & Robertson, E. F. (n.d.). Julia Hall Bowman Robinson. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Robinson/Cited in Chapter 11. | en | read in full |
| S424 | O'Connor, J. J., & Robertson, E. F. (n.d.). Katherine Coleman Goble Johnson. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Johnson_Katherine/Cited in Chapter 11 twice and 1914 to the present. | en | read in full |
| S426 | O'Connor, J. J., & Robertson, E. F. (n.d.). Gladys Mae Brown West. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/West_Gladys/Cited in Chapter 11 twice and 1914 to the present. | en | read in full |
| S427 | O'Connor, J. J., & Robertson, E. F. (n.d.). Evelyn Boyd Granville. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Granville/Not cited in the story. It backs a register entry. | en | read in full |
| S428 | O'Connor, J. J., & Robertson, E. F. (n.d.). Marjorie Lee Browne. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Browne/Not cited in the story. It backs a register entry. | en | read in full |
| S429 | O'Connor, J. J., & Robertson, E. F. (n.d.). Takakazu Shinsuke Seki. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Seki/Cited in Chapter 11 twice and 1665 to 1687. | en | read in full |
| S430 | O'Connor, J. J., & Robertson, E. F. (n.d.). Takebe Katahiro. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Takebe/Cited in Chapter 11 3 times and 1688 to 1734 twice. | en | read in full |
| S431 | O'Connor, J. J., & Robertson, E. F. (n.d.). Srinivasa Aiyangar Ramanujan. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Ramanujan/Cited in Chapter 11 4 times and 1873 to 1913. | en | read in full |
| S433 | O'Connor, J. J., & Robertson, E. F. (n.d.). Laurent Moise Schwartz. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Schwartz/Cited in Chapter 12 twice and 1914 to the present. | en | read in full |
| S434 | O'Connor, J. J., & Robertson, E. F. (n.d.). Bourbaki: the pre-war years. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Bourbaki_1/Cited in Chapter 12 and 1914 to the present. | en | read in full |
| S435 | O'Connor, J. J., & Robertson, E. F. (n.d.). Johann Radon; Otton Marcin Nikodym; Marshall Harvey Stone (three biographies). MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Radon/Cited in Chapter 12 twice and 1914 to the present twice. | en | read in full |
| S441 | O'Connor, J. J., & Robertson, E. F. (n.d.). Howard Jerome Keisler. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Keisler/Cited in 1914 to the present. | en | read in full |
| S442 | O'Connor, J. J., & Robertson, E. F. (n.d.). William Whewell. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Whewell/Cited in Chapter 11 twice. | en | read in full |
| S410 | O'Connor, J. J., & Robertson, E. F. (2020). Marie-Sophie Germain. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Germain/Cited in Chapter 11 5 times, 1734 to 1799 and 1800 to 1872 twice. | en | read in full |
| S235 | O’Connor & Robertson, J. J. & E. F. (n.d.). Gottfried Wilhelm von Leibniz. MacTutor History of Mathematics Archive. University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Leibniz/Cited in Chapter 7 3 times, 1637 to 1665 twice, 1665 to 1687 8 times and 1688 to 1734. | en | read in full |
| S052 | Pearce, I. G. (n.d.). Keralese Mathematics IV: Possible Transmission of Keralese Mathematics to Europe. Indian Mathematics: Redressing the Balance (MacTutor project). University of St Andrews. https://mathshistory.st-andrews.ac.uk/Projects/Pearce/chapter-19/Cited in Chapter 3 and 1800 to 1872. | en | read in full |
| S332 | Project Vox (n.d.). Du Chatelet (1706-1749). Project Vox. Duke University Libraries. https://projectvox.org/du-chatelet-1706-1749/Cited in Chapter 9 twice and 1734 to 1799 3 times. | en | read in full |
| S387 | Ruch, D. (2021). Bolzano on Continuity and the Intermediate Value Theorem. TRIUMPHS: Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (Analysis). Ursinus College Digital Commons. https://digitalcommons.ursinus.edu/triumphs_analysis/Cited in Chapter 10 twice and 1800 to 1872. | en | read in full |
| S425 | Shetterly, M. L. (n.d.). Mary W. Jackson biography. NASA. National Aeronautics and Space Administration. https://www.nasa.gov/people/mary-w-jackson-biography/Cited in Chapter 11 3 times and 1914 to the present. | en | read in full |
| S422 | Shetterly, M. L. (2016). Katherine Johnson biography. NASA History Office. National Aeronautics and Space Administration. https://www.nasa.gov/history/katherine-johnson-biography/Cited in Chapter 11 6 times and 1914 to the present twice. | en | read in full |
| S480 | Sylla, E. D. (2008). Dumbleton, John. Complete Dictionary of Scientific Biography. Charles Scribner's Sons. https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/dumbleton-johnCited in Chapter 4 twice. | en | read in full |
| S333 | Tou, E. R. (2014). D. Bernoulli, Euler, and the Development of Fluid Mechanics. Euler Society 2014 Meeting, 1-8. Pacific Lutheran University. http://faculty.washington.edu/etou/documents/TouE-GeometryFluidMechanics.pdfCited in Chapter 9 twice, 1688 to 1734 and 1734 to 1799. | en | read in full |
| S200 | Westminster Abbey (n.d.). Sir Isaac Newton (Abbey commemorations). Dean and Chapter of Westminster. https://www.westminster-abbey.org/abbey-commemorations/commemorations/sir-isaac-newtonCited in Chapter 6, Chapter 7 twice, 1637 to 1665 and 1688 to 1734 3 times. | en;la | read in full |
| S287 | Wikimedia Commons (n.d.). Image records for Acta Eruditorum, October 1684 (Nova Methodus first page, incipit, Tabula XII). Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Leibniz-Acta-1684-NovaMethodus.jpgNot cited in the story. It backs a register entry. | en;la | read in full |
Tier not assigned
2 sources.
| ID | Reference (APA 7) | Language | Access |
|---|---|---|---|
| S443 | Wang Zhenyi - admissible scholarship identified (Tamboukou 2026 = S449; Bernardi 2016, Springer; Chinese-language studies 1997-2007) but no full text obtainable; she remains excluded from the document. (n.d.).Not cited in the story. It backs a register entry. | - | not obtained, supports no claim |
| S444 | Calculus in secondary schools and the origin of AP Calculus - PARTIALLY CLOSED: AP scale and the College Board's own 1955 date now sourced (S446, S447, S448) and the 1901 Perry campaign sourced (S492); the founding narrative, the start of AP Calculus and the AB/BC split remain unsourced. (n.d.). https://apcentral.collegeboard.org/about-ap/ap-historyCited in 1914 to the present twice. | en | not obtained, supports no claim |
Appendix L
How this book was checked
Every fact, date, name, and quotation in this book traces back to a source someone read in full. Not to a search-engine summary or a half-remembered lecture. This appendix is the evidence for that claim. It records how much was checked, what the checking found, what is still disputed among historians, and what could not be nailed down at all. A book that hid its own gaps would be easier to trust and less honest. This one shows its gaps instead.
L.1 What this book's research adds up to
| Sources logged | 255 |
| Sources read in full text | 238 |
| Sources logged but not obtained, supporting no claim | 20 |
| Note files written, one per source read | 257 |
| Dated events in the master timeline | 327 |
| People registered with pronunciations | 212 |
| Words, symbols, and eponyms traced | 191 |
| Disagreements between sources, logged | 129 |
| Of those, resolved from the sources | 58 |
| Primary-source images rights-cleared | 43 |
| Of those, downloaded and verified on disk | 36 |
| Mathematical results re-derived and checked in code | 18 |
The work was split by domain across a shared research register, and revised repeatedly as new sources came in. The most recent round added fourteen sources from the Boston Public Library. Highlights: Katz on Islam and India, Whitman on the cycloid, and the two primary sources behind the natural logarithm. Those two are Sarasa's 1649 tract and Saint-Vincent's Opus geometricum. The highlights also include Zinsser and Allen on du Chatelet, and Feingold's reinterpretation of the Barrow question. They end with Mazzotti and Cupillari on Agnesi, which closed the last major gap in the eighteenth-century chapter. A related pass, ten more sources across three batches the same week, updated where each story lands in your Derivatives and Integrals courses (Appendix I). Between the two passes, four standing library requests closed, eight open disputes were settled or explained, and five mathematical results were newly verified. One source was read in full and deliberately excluded; see L.3 below. The rule throughout: a source supports a claim only after someone reads the whole thing. Twenty sources appear in the bibliography marked not obtained or abstract only. The bibliography lists them so you know they exist and know they are not holding anything up.
L.2 The honesty rule
This book distinguishes four things, and never blurs them:
- What a primary source says. Quoted, with a page or folio number.
- What historians conclude from it. Attributed to a named historian.
- What is disputed. Carries a visible Disputed label, with both sides given.
- What could not be verified. Listed below in L.3, with the reason it could not be.
Appendix G is a list of beloved stories that fall apart when you check them. It is one of the more useful parts of this book, because several of them are in textbooks you may have read before.
L.3 Repeated everywhere, and unverified here
Each of these is repeated in popular writing and could not be verified here. None of them is stated as fact anywhere else in this book. This appendix lists them so that nobody puts them back in by accident.
- Euler's remark that blindness meant fewer distractions (S326).
- Hilbert's "this is a university, not a bath-house" (S418).
- Einstein's New York Times letter about Noether (S418).
- Sophie Germain's honorary degree from Gottingen (S410).
- Errett Bishop's 1977 review sentences on nonstandard analysis (S395, S390). The review was blocked on five separate attempts, by a Project Euclid challenge page and by Cloudflare on the AMS PDF.
- Lebesgue's coin analogy, that Riemann counts coins in the order you pull them from your pocket while Lebesgue sorts them by denomination first. The idea is his and it circulates only in paraphrase; no wording here traces back to Lebesgue (S377).
- Nicolas Fatio de Duillier's exact words in the 1699 accusation that opened the priority fight. The event is documented and the tract was never obtained, so no source read here quotes a single line of it.
- Newton's "ashamed to tell you to how many figures" remark about π (S190).
- Voltaire's line that England buried Newton "like a king who had done well by his subjects" (S200).
- The claim that Cardano ranked Richard Swineshead among the greatest intellects (S106).
- Anything at all from Goldberg, "Ideas and Eternity" (S074). Read in full and excluded. It is a homily in a journal of Orthodox Jewish thought, and its four sentences on calculus contain three errors: it has Newton corresponding with Leibniz during 1664 to 1666 (they corresponded in 1676, through Oldenburg), calls 1675 the year Leibniz published (1675 is the manuscript notation; publication was 1684), and says "two centuries later" where it is three. Logged so the exclusion is on the record.
- "The Helen of Geometers" as a phrase attributable to anyone. Whitman 1943, the leading candidate, was read in full and does not contain it (S064).
- Chaucer's reference to Bradwardine on predestination (S103).
- The ∞ symbol as a corruption of the Roman numeral CIƆ (S451).
- Thabit ibn Qurra having been a money-changer in Harran (S057).
- Liu Hui's approximation and the 192-gon; the terms milü and yuelü; Zu Chongzhi's actual method (S021).
- The origin of the name "Gabriel's horn", which appears to be genuinely unknown (S157).
- When English usage dropped "the calculus" in favor of "calculus" (S453).
- Whether Stigler's law was in fact first stated by Merton, which is Stigler's own joke and could not be verified because the 1980 paper is paywalled (S460).
L.4 Open disputes that need a specific document
| Question | Why it is open | What would settle it |
|---|---|---|
| Did Newton discover the Principia results by calculus and translate them into geometry? | PARTLY NARROWED 17 Aug 2026. The only source for the claim is still Newton writing anonymously about himself, and Whiteside still disagrees. Guicciardini 2004 (S204) now establishes that quadrature working was withheld and supplied privately to Gregory and Cotes, which is broader than Whiteside's one exception; but he describes the book's own mathematics as the geometric fluxional calculus of the Geometria curvilinea, using limits, which is Whiteside's picture and not Newton's | Cohen's Guide pp. 114-117, 122-127, 345-347, then Guicciardini, Reading the Principia (CUP 1999), pp. 179-186, which S204 n. 45 names as the place the detail lives. Archive.org holds the book print-disabled |
| Was Cauchy's 1821 sum theorem simply false, or coherent under an infinitesimal reading? | Both sides read in full and both have evidence. Live in refereed journals | Nothing. This one is genuinely unsettled and should be taught as such |
| Did Cauchy take his program from Bolzano without acknowledgment? | Grattan-Guinness charged it; Grabiner rejected it; her rebuttal was not obtained | Grabiner, "Cauchy and Bolzano" (1984), and Grattan-Guinness (1970) p. 54 |
| What exactly did Barrow give Newton? | Four distinguished scholars, four verdicts, same evidence | Probably nothing. A good classroom case study in how historians decide |
| Did Kerala mathematics reach Europe? | Motive, opportunity, and resemblance are documented. No transmitting document exists | A document. Nobody has one |
| Who called the cycloid "the Helen of Geometers"? | CLOSED as unattributable. Whitman 1943 was the leading candidate and does not contain the phrase; full text read and searched 14 Aug 2026 | Nothing further. Stop looking, and stop attributing it |
| Who wrote the Charta volans (1713)? | Newton's own draft reply describes an anonymous libel quoting an unnamed mathematician's letter of 7 June 1713. The Johann Bernoulli attribution is an editorial inference, not in any document read | The Charta volans itself, and the Bernoulli correspondence |
| Was John Craig's Cambridge transcription visit in 1684 or 1685? | Guicciardini gives 1684 at pp. 460 and 462 and 1685 at p. 465, in one article, with neither passage acknowledging the other. The 1685 passage carries the fuller narrative | Whiteside, Mathematical Papers vol. 7, pp. 3 to 4, which both of Guicciardini's statements cite |
| Was Newton's 1675 royal dispensation from holy orders exactly as the biographies describe it? | Snobelen (S194) reports it and builds an argument on it, and nobody here read the dispensation itself or a source that quotes it. The claim is standard and very likely right, and it is carried at second hand | Richard Westfall, Never at Rest (Cambridge, 1980), which narrates the dispensation in detail, and behind that the instrument itself in the royal records |
| Exact day-level dates for the 1712 Royal Society committee | De Morgan names all eleven members and gives March 1712 for appointment and April 1712 for the report, but the names were never officially published | The Royal Society Journal Books |
L.5 What we are still trying to track down
These need institutional access to check. Everything else cited in this book was obtained and read in full.
- I. Bernard Cohen, A Guide to Newton's Principia, pages 114 to 117, 122 to 127, and 345 to 347. This displaces Guicciardini at the top of the list. Cohen's section 5.8 is titled "Methods of Proof versus Method of Discovery; The 'New Analysis' and Newton's Allegations about How the Principia Was Produced; The Use of Fluxions in the Principia". The PDF obtained holds the Guide's table of contents, not its body. Five pages would probably settle a question this document has carried as open from the beginning. The Guide is bound with the Cohen and Whitman translation (University of California Press, 1999), so any library copy of that translation has it.
1b. Guicciardini, Reading the Principia (Cambridge, 1999), pages 179 to 186 first, and Isaac Newton on Mathematical Certainty and Method (MIT, 2009). Still second in line behind five pages of Cohen, but the request is now much sharper than it was.
The Guicciardini gap is partly closed. His article is "Isaac Newton and the publication of his mathematical manuscripts" (S204). It appeared in Studies in History and Philosophy of Science 35(3), 2004, pp. 455 to 470. It was obtained and read in full on 17 August 2026. Be precise about what that did and did not do.
What S204 settles. Why Newton withheld: he judged the "new analysis" unfit to print after a methodological shift toward ancient geometry that begins in the 1670s. How the manuscripts moved instead: scribal publication, a named inner circle of acolytes, and an outer circle of philomaths fed mutilated copies. There were also transcripts made from transcripts, and hired scribes selling double copies by 1720. What ended it: priority, first against David Gregory in 1688 and then against Leibniz, plus journals changing what publication meant. And that Newton rewrote his own early texts to remove infinitesimals as he printed them. It also settles one Principia point: integration technique was used and withheld. Newton supplied the missing steps privately to David Gregory and to Roger Cotes, his own editor.
What S204 does not settle, and does not claim to. It is not a line-by-line reading of the Principia. Guicciardini's own footnotes 44 and 45 hand that job to Reading the Principia, and footnote 45 gives the page range: pp. 179 to 186. He describes the Principia's mathematics as the geometric fluxional calculus of the Geometria curvilinea, built on limits. So he does not endorse the "worked it in calculus, then translated it" story, and this book does not cite him as if he did. He also calls the extent of the hidden calculus an open research front. He names Nauenberg (2001), C. Wilson (2001) and Brackenridge (2003) as where the evidence is surfacing. Logged at Appendix F.
What S204 opened. Behind it lie six sources that this project does not hold. The central concept comes from Harold Love, Scribal Publication in Seventeenth-Century England (Clarendon, 1993). Next comes P. Beal, In Praise of Scribes (Clarendon, 1998). H. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements (Cambridge, 1997), is the source of "geometric backlash". W. G. Hiscock, David Gregory, Isaac Newton and Their Circle (1937), covers Gregory's memoranda. R. Iliffe, "Butter for parsnips" (Routledge, 2003), covers Principia authorship. And Whiteside's Mathematical Papers vol. 7, pp. 3 to 4, would settle whether Craig's Cambridge visit was 1684 or 1685 (see Appendix F).
- A. Rupert Hall, Philosophers at War (Cambridge, 1980) and Domenico Bertoloni Meli, Equivalence and Priority (Oxford, 1993). The archive.org copy of Hall is print-disabled, so it cannot be read there; the record is
philosophersatwa0000hall, ISBN 0521227321, 338 pages. It does have a full-text search index, so "Search inside" will answer a targeted question even though the book cannot be read through. One sentence of Hall is already in hand at second remove: Feingold quotes him at p. 8 on Barrow's reputation "descending toward the status of an elegant codifier" (S073). Interlibrary loan remains the route for the whole book.
- ~~Victor J. Katz, "Ideas of Calculus in Islam and India"~~ OBTAINED 14 August 2026 via Boston Public Library and read in full (S061). It closed the ibn al-Haytham material, supplied the paraboloid result, and documented the summation-identity link from Baghdad to Kerala.
- ~~R. P. Burn, "Alphonse Antonio de Sarasa and logarithms"~~ NO LONGER NEEDED. The primary sources were obtained instead, on 14 August 2026, which is a better outcome than the article. Sarasa 1649 (S066) was read in full, including the scholium that identifies hyperbolic areas with logarithms, and its marginal citations to Saint-Vincent. The Opus geometricum (S067) was obtained as the volume holding the front matter, the complete contents, and Books I to V. One residual gap: Book VI's body is not in that volume, so the exact Latin of propositions 125 to 130 has not been read. If you ever want to print that text verbatim, the second volume is the thing to find. Everything else this request existed for is now closed.
- ~~E. A. Whitman, "Some historical notes on the cycloid"~~ OBTAINED 14 August 2026 and read in full (S064). Outcome: it does not contain "the Helen of Geometers", which closes that question as unattributable. It did supply Galileo's 1599 weighing, Wren's rectification, and the mechanism behind Roberval's secrecy.
- Taton (1971) on Cauchy and Galois, Revue d'histoire des sciences. Persée robots-blocks it.
- ~~Massimo Mazzotti, The World of Maria Gaetana Agnesi~~ CLOSED 14 August 2026. The article version was obtained and read in full: "Maria Gaetana Agnesi: Mathematics and the Making of the Catholic Enlightenment", Isis 92(4), 2001, pp. 657 to 683 (S077). It settled the versiera page and date, the Bologna chair to the day, and her later life and burial, and it supplied the argument for why she wrote a geometrical calculus textbook. Antonella Cupillari's survey of her other curves (S075) came with it. The Agnesi gap is closed; the monograph would now only add depth.
- Edith Sylla, The Oxford Calculators in Context, and Bressoud, A Radical Approach to Real Analysis, and Bottazzini, The Higher Calculus.
- Stigler (1980), "Stigler's Law of Eponymy". Paywalled; Appendix D's framing device is currently unsourced at the primary level.
L.6 Images we could not include
Not every image this book wanted made it in. Some are rights-cleared but were never retrieved; most are not cleared at all, and are listed here so nothing gets used without checking first.
Four archives are worth a direct request, if a future edition wants to close these gaps:
- Cambridge University Library Imaging Services (imagingservices.lib.cam.ac.uk), for six Newton manuscript images. CUDL licenses its downloads CC BY-NC 4.0, which rules out a sold textbook and has no free workaround. The license appears only inside the IIIF manifest; CUDL has no public terms page.
- Royal Society Picture Library, for De analysi (MS/81/4). Rights-managed and priced by how commercial the project is; the Royal Society offers free download for classroom resources and unpublished theses, so this image is usable in class today and needs a license only for a published book.
- Gottfried Wilhelm Leibniz Bibliothek, Hanover, for LH 35, the 29 October 1675 manuscript carrying the first integral sign. No public-domain reproduction of it exists anywhere. This is the single most important missing image.
- King's College, Cambridge (Keynes MS 8) and The National Archives (the Chaloner letter, MINT 15/17/205). TNA images of out-of-copyright records are normally released under the Open Government Licence, which does permit commercial reuse, so this one is probably free to confirm.
A note on Cupillari. Her 2014 article is valuable and is cited throughout this book. But page 5 prints the versiera as . That contradicts her own opening sentence on the same page and her own algebra. It is a published typo, not propagated here.
One trap worth naming. Two archive.org items in the early-european-books collection carried a ProQuest copyright notice burned into the pixels, invisible in the metadata. The items were Kepler's Nova stereometria (2010) and Cavalieri 1635 (2011). They were found only by cropping and enlarging the image footer, and both were dropped. Kepler was replaced with a clean scan under Public Domain Mark 1.0; Cavalieri 1635 still needs a non-ProQuest scan. Checking the pixels, not just the license page, is what caught it.
Rights are clear, but the file was not retrieved
16 images. Nothing legal blocks these. Each one needs a page-index lookup or a retry, and none has an image file in images/ yet. Two of the sixteen (IMG-U439, IMG-U423) have a reading copy of the source on disk, in sources/. But no image plate has been extracted from it.
| ID | What it shows | Work | Year | Repository | Rights status, and why there is no file | File |
|---|---|---|---|---|---|---|
| IMG-R004 | Title page and p. 8 of Bolzano's Rein analytischer Beweis showing the 1817 definition of continuity | Rein analytischer Beweis (Ostwalds Klassiker reprint) | 1905 | Internet Archive | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/reinanalytische00hankgoog on 2026-08-14 |
- |
| IMG-R003 | Cauchy's definition of the definite integral as a limit of sums, 21st lecture, and the explanation of the integral sign as a stretched S | Resume des lecons ... sur le calcul infinitesimal, pp. 81-84 | 1823 | Internet Archive | public_domain archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/bub_gb_y7FSr1rrFsIC on 2026-08-14 |
- |
| IMG-R007 | Riemann's definition of the definite integral, section 4 of the Habilitationsschrift, ending 'so hat [the integral] keine Bedeutung' | Gesammelte mathematische Werke, 2nd ed., pp. 239-240 | 1892 | Internet Archive | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/gesammeltemathem00riemuoft on 2026-08-14 |
- |
| IMG-R008 | Dedekind's preface stating 'I succeeded Nov. 24, 1858' | Essays on the Theory of Numbers (Beman trans.), pp. 1-2 | 1901 | Internet Archive (Cornell copy) | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/cu31924001586282 (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1901, public domain by age in the US... |
- |
| IMG-U439 | Title page of Charles Hayes, A Treatise of Fluxions (1704), and the circle diagram (ABF with perpendicular FG) by which he argues for infinitesimals of second and third order | A Treatise of Fluxions: or, an Introduction to Mathematical Philosophy | 1704 | Internet Archive (Google scan) | public_domain archive.org item atreatisefluxio00hayegoog (Google scan). Google's boilerplate leaf: 'We also ask that you: Make non-commercial use of the files ... we request that you use these files for personal, non-commercial purposes.' That is a request, not a license condition, and Google claims no copyright ... A text-only reading copy is on disk; the title page and diagram have not yet been extracted as an image file. |
sources/hayes.txt |
| IMG-U413 | Title page and Preliminary Dissertation of Mary Somerville, Mechanism of the Heavens | Mechanism of the Heavens | 1831 | Internet Archive (scan of a library copy) | public_domain Underlying work published 1831, public domain by age; archive.org item mechanismofheave00somerich. Repository rights field not independently re-checked in this pass. |
- |
| IMG-U423 | Cover page and equation-development pages of NASA Technical Note D-233 showing "By T. H. Skopinski and Katherine G. Johnson" | NASA TN D-233, Determination of Azimuth Angle at Burnout for Placing a Satellite over a Selected Earth Position | 1960 | NASA Technical Reports Server | public_domain NASA Technical Reports Server metadata: determinationType = GOV_PUBLIC_USE_PERMITTED, belongsToUsGov = true, distribution = PUBLIC (checked 2026-08-14). Works of the US federal government are not subject to US copyright (17 U.S.C. 105). Safe for commercial publication. A text-only reading copy is on disk; the cover and equation pages have not yet been extracted as an image file. |
sources/tnd233.pdf |
| IMG-E18003 | Preface p. xiv of the Analyse des infiniment petits, with the acknowledgment to the Bernoullis and Leibniz | Analyse des infiniment petits | 1716 (2nd ed. of 1696) | Internet Archive, University of Ottawa copy | public_domain Wikimedia Commons File:L'Hospital - Analyse des infiniment petits pour l'intelligence des lignes courbes, 1696.jpg, extmetadata LicenseShortName = 'Public domain', UsageTerms = 'Public domain' (read via the Commons API 2026-08-14). 786 x 1179 px. |
- |
| IMG-E18012 | Preface p. i of Maclaurin's Treatise of Fluxions: 'A Letter published in the Year 1734, under the Title of the Analyst...' | A Treatise of Fluxions | 1742 | Internet Archive / Ghent University (Google scan) | public_domain archive.org item metadata gives archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/atreatisefluxio00conggoog on 2026-08-14. The scan itself bears a "Digitized by Google" watermark and the Google boilerplate leaf states: "We also ask tha... |
- |
| IMG-E18002 | Page bearing Section 35, the 'ghosts of departed quantities' passage | The Analyst | 1734 | Internet Archive, https://archive.org/details/analystordiscour00berkrich | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/analystordiscour00berkrich on 2026-08-14 |
- |
| IMG-E18004 | Section IX, Proposition I (art. 163) - the first printing of L'Hopital's rule, with its figure | Analyse des infiniment petits | 1716 | Internet Archive | public_domain No license statement found: archive.org item metadata at https://archive.org/metadata/infinimentpetits1716lhos00uoft (read 2026-08-14) returns no licenseurl, no rights and no possible-copyright-status field, and the item page carries no rights notice. Underlying work published 1716, public domain by... |
- |
| IMG-E18007 | Colson's English page reading 'which is vulgarly called the Witch' - the mistranslation in print | Analytical Institutions | 1801 | Internet Archive / University of California Libraries | public_domain archive.org item metadata field possible-copyright-status = NOT_IN_COPYRIGHT, read at https://archive.org/metadata/analyticalinstit00agnerich on 2026-08-14 |
- |
| IMG-E18008 | Introductio in analysin infinitorum, Book I S.126: Euler adopts the symbol pi | Introductio in analysin infinitorum | 1748 | Internet Archive / Google scan of the University of Lausanne copy | public_domain archive.org item metadata gives archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/bub_gb_jQ1bAAAAQAAJ on 2026-08-14. The scan itself bears a "Digitized by Google" watermark an... |
- |
| IMG-E18009 | Introductio Book I S.138: | Introductio in analysin infinitorum | 1748 | Internet Archive | public_domain archive.org item metadata gives archive.org item metadata field licenseurl = http://creativecommons.org/publicdomain/mark/1.0/ (Creative Commons Public Domain Mark 1.0), read at https://archive.org/metadata/bub_gb_jQ1bAAAAQAAJ on 2026-08-14. The scan itself bears a "Digitized by Google" watermark an... |
- |
| IMG-NEW-15 | The page of the Philosophical Transactions carrying Newton's anonymous claim that he found the Principia's propositions 'By the help of the new Analysis' | An Account of the Book entituled Commercium Epistolicum | 1715 | Royal Society / Philosophical Transactions vol. 29 no. 342, p. 206 | public_domain No statement found at https://www.maths.tcd.ie/pub/HistMath/People/Newton/CommerciumAccount/CommAcc.pdf (2026-08-14). Underlying Phil. Trans. vol. 29 (1715) is public domain by age. |
- |
| IMG-ARCH-03 | OUTSTANDING: a palimpsest folio specifically carrying The Method of Mechanical Theorems | Archimedes Palimpsest | 10th c. | The Walters Art Museum / Archimedes Palimpsest Digital Product | cc_by Archimedes Palimpsest digital release, 0_ReadMe.txt section "1 Rights and Conditions of Use", https://archimedespalimpsest.net/0_ReadMe.txt (retrieved 2026-08-14): "The Archimedes Palimpsest data is released with license for use under Creative Commons Attribution 3.0 Unported Access Rights. It is re... |
- |
Not cleared: do not publish without action
20 images.
| ID | What it shows | Work | Year | Repository | Why not, and what to do | File |
|---|---|---|---|---|---|---|
| IMG-L004 | Manuscript of 29 October 1675, Analyseos tetragonisticae pars secunda: the first integral sign | Leibniz manuscript, 29 October 1675 | 1675 | Gottfried Wilhelm Leibniz Bibliothek, Hanover (Leibniz-Handschriften, LH 35 series) | not_located No image and no rights statement located (2026-08-14). Contact: Gottfried Wilhelm Leibniz Bibliothek, Waterloostrasse 8, 30169 Hannover. |
- |
| IMG-L005 | Newton's holograph draft of the anonymous 1715 Account of the Commercium Epistolicum | Cambridge University Library MS Add. 3968, ff. 67r-96v | c.1712-1715 | Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-01 | Cover of Newton's Waste Book, with 'Waste Book' written in his hand on his dead stepfather's theological commonplace book | Newton's Waste Book (CUL MS Add. 4004) | 1612 (begun by Barnabas Smith); Newton's use from 1664 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-02 | Waste Book folio where Newton replaces infinitesimal increments of space with instantaneous velocities of a moving point - the birth of the fluxion, autumn 1665 | Newton's Waste Book (CUL MS Add. 4004) | 1665 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-03 | A page of Newton's logarithm computation carried to more than 50 decimal places, summer 1665 - a whole sheet of nothing but digits | Newton Papers: Early Papers (CUL MS Add. 3958), ff. 78r-80v | 1665 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-04 | A folio of the October 1666 tract on fluxions, applying the rules of the calculus to a cubic curve | October 1666 Tract on Fluxions (CUL MS Add. 3958.3, f. 52v) | 1666 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-05 | The 'vellum manuscript': Newton's calculations on circular motion and gravity, written on a piece of legal parchment | Newton Papers: Early Papers (CUL MS Add. 3958, f. 45r) | c. 1665-1669 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-06 | Newton's memorandum on his own anni mirabiles - 'in those days I was in the prime of my age for invention' | Papers relating to the dispute respecting the invention of fluxions (CUL MS Add. 3968.41, f. 85r) | c. 1714-1718 | Cambridge University Library / Cambridge Digital Library | restricted Cambridge Digital Library IIIF manifest "attribution" field, retrieved 2026-08-14 from https://cudl.lib.cam.ac.uk/iiif/<CLASSMARK> : "Provided by Cambridge University Library. Cambridge University Library Images made available for download are licensed under a Creative Commons Attribution-NonCommer... |
- |
| IMG-NEW-07 | Stukeley's manuscript page recording Newton's own telling of the apple story, folio 15r | Memoirs of Sir Isaac Newton's Life (Royal Society MS/142, f. 15r) | 1752 | Royal Society Library, London | restricted Royal Society Picture Library Terms and Conditions, https://pictures.royalsociety.org/terms-and-conditions (retrieved 2026-08-14): "We are the owner or the lawful representative of the owner of all intellectual property rights in our site, and in the material published on it... All such rights are r... |
- |
| IMG-NEW-08 | Opening page of De analysi per aequationes numero terminorum infinitas, the manuscript Barrow posted to Collins on 31 July 1669 | De analysi (Royal Society MS/81/4) | 1669 | Royal Society Library, London | restricted Royal Society Picture Library Terms and Conditions, https://pictures.royalsociety.org/terms-and-conditions (retrieved 2026-08-14): "We are the owner or the lawful representative of the owner of all intellectual property rights in our site, and in the material published on it... All such rights are r... |
- |
| IMG-E18016 | Frontispiece of the 1742 Amsterdam Institutions de physique, showing du Chatelet | Institutions de physique | 1742 | to be sourced (Gallica or Wikimedia Commons) | unclear Not assessed (2026-08-14). |
- |
| IMG-E18013 | Jacob Bernoulli's tomb in the cloisters of the Basel Munster, showing the ARCHIMEDEAN spiral the mason carved by mistake | monument | 1705 or later | MacTutor (photograph) and Wikimedia Commons | unclear No license statement found at https://mathshistory.st-andrews.ac.uk/Extras/Bernoulli_tomb/ (2026-08-14). Note: photographs of three-dimensional objects attract their own copyright. |
- |
| IMG-E18015 | Portrait of Maria Gaetana Agnesi | portrait | 18th century | to be sourced (Wikimedia Commons; several 18th-century portraits circulate, of varying attribution) | unclear Not assessed - candidate files not individually checked (2026-08-14). |
- |
| IMG-NEW-09 | Newton's Twelve Articles on religion, the whole heretical creed on a single leaf in his own hand | Twelve articles on religion (Keynes Ms. 8, f. 1r) | c. 1710s-1720s | King's College Library, Cambridge | unclear No statement found at https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/THEM00008 (2026-08-14). |
- |
| IMG-NEW-10 | William Chaloner's begging letter to Isaac Newton from prison: 'O no body can save me but you' | Chaloner's letter to Newton (TNA MINT 15/17/205) | 1699 | The National Archives, Kew | unclear No statement found at https://www.newtonproject.ox.ac.uk/view/texts/normalized/MINT00918 (2026-08-14). Check the Open Government Licence position with The National Archives, Kew. |
- |
| IMG-NEW-16 | Newton's monument in Westminster Abbey, and a close view of the inscription 'OBIIT XX MAR MDCCXXVI' - the tomb that disagrees with the textbooks | Newton monument, by Michael Rysbrack after William Kent | 1730-1731 | Westminster Abbey, London | restricted Westminster Abbey asserts copyright in its images (Dean and Chapter of Westminster); no free-reuse license offered at https://www.westminster-abbey.org/abbey-commemorations/commemorations/sir-isaac-newton (2026-08-14). |
- |
| IMG-R009 | Graph of the Weierstrass function and its successive magnifications, showing self-similar roughness at every scale | Weierstrass's non-differentiable function (course notes) | 2014 | University of Minnesota (Jeff Calder) | restricted No license statement found at https://www-users.cse.umn.edu/~jwcalder/104F14/weierstrass-function.pdf (2026-08-14). Default: all rights reserved. |
- |
| IMG-R010 | Diagram of the first three stages of Bolzano's piecewise-linear construction B_k(x) | Continuous Nowhere Differentiable Functions (MSc thesis), fig. 3.1 | 2003 | Lulea University of Technology | restricted No license statement found at https://andrescaicedo.wordpress.com/wp-content/uploads/2014/10/ltu-ex-03320-se.pdf (2026-08-14). Default: all rights reserved. |
- |
| IMG-U411 | Opening pages of the anonymous 1834 Quarterly Review article in which the word "scientist" first appears in print, including the paragraph attributing it to "some ingenious gentleman" and the "nature-poker" footnote | [Whewell], review of Somerville, On the Connexion of the Physical Sciences, Quarterly Review LI (101) | 1834 | Scan posted by the Centre for the History of the Sciences, University of Kent (Charlotte Sleigh) | unclear No license statement found at https://blogs.kent.ac.uk/sciencecomma/2015/04/09/whewell-and-the-coining-of-scientist-in-the-quarterly-review/ (recorded by the contributing agent, 2026-08-14). Underlying 1834 publication is public domain by age. The PDF in sources/ is a READING copy only, not a public... |
sources/whewell1834.pdf |
| IMG-U415 | The Note G diagram: Lovelace's table of the operations by which the Analytical Engine would compute the Bernoulli number B7 | Menabrea, Sketch of the Analytical Engine, with Notes by A. A. L., Scientific Memoirs vol. 3 | 1843 | fourmilab.ch transcription (image reproduced from the 1843 printing) | unclear No license statement found at https://www.fourmilab.ch/babbage/sketch.html (recorded by the contributing agent, 2026-08-14). |
- |
Appendix M
Glossary
Terms a 15 to 19 year old reader, or a teacher from another subject, may not know. Each entry says what the word means as this book uses it, not in full generality. Where the book's term register records an origin, the entry gives that too: the language the word came from, and what it literally meant. The prose does the same job first. A word you may not know is explained at the point a chapter uses it. So you should not need this page in order to read the story.
The first German scientific journal, founded in Leipzig in 1682. Where Leibniz published the calculus in 1684, and where much of the war with Newton's circle was fought
IdeaChapter 5
(adaequalitas)
Fermat's operation of setting two nearly-equal expressions against each other, then suppressing the small terms. From Diophantus' Greek παρισότης by way of Bachet's Latin. Fermat's own French: "a feigned comparison, or an adequality"
IdeaChapter 5
Reading the past with the present's ideas in it, like treating Greek geometry as algebra in disguise. The historian's cardinal sin, and this book works to avoid it
IdeaChapter 2
Leibniz's proposed common name for his method and Newton's, offered in a 1697 letter. It lost to "the calculus", but survives inside the word "analysis". Latin for "infinitesimal analysis"
LanguageChapter 7
"Wonderful years", the plague years 1665 to 1667 in which Newton later said he did his best work
IdeaChapter 6
A line a curve approaches forever without reaching. From Greek for "not falling together". The hyperbola and its asymptote are where Saint-Vincent and Sarasa saw areas behaving as logarithms, the doorway to
LanguageChapter 5
Computing exact derivatives of a program by applying the chain rule mechanically to its operations. Forward mode costs one pass per input; reverse mode costs one pass per output. Backpropagation is reverse mode
IdeaChapter 12
The algorithm that trains neural networks. It is reverse-mode automatic differentiation, which is the chain rule applied backwards through a computation
IdeaChapter 12
The Indian word-numeral system, in which digits are written as words for things that come in known quantities (eyes = 2, Vedas = 4). Used because verses survive copying better than digits do. Sanskrit for "object-numbers"
LanguageAppendix B
Every bounded infinite set of numbers has at least one point its members crowd around. Proved by Bolzano in 1817, rediscovered in Weierstrass's Berlin lectures; a workhorse of rigor
PeopleChapter 10
The curve of fastest descent between two points under gravity. Posed by Johann Bernoulli in 1696; the answer is a cycloid. From Greek brachistos plus chronos, "shortest time"
LanguageChapter 9
The branch that optimizes over whole functions rather than over numbers. Grew out of the brachistochrone problem. Variations is Latin variatio, "a change, a turning"
CurvesChapter 9
Leibniz's first name for the integral calculus, "the reckoning of sums". Johann Bernoulli argued for calculus integralis, and Bernoulli won the name while Leibniz kept the sign. Latin for "summing calculus"
LanguageChapter 9
In Descartes's manner: coordinates, equations for curves, and the geometry that follows from them. Named from Cartesius, the Latin form of Descartes
LanguageChapter 6
The curve a hanging chain makes under its own weight. Not a parabola
Curvesthe timeline
The tiny right triangle with legs dx and dy whose hypotenuse lies along a curve. Pascal's diagram; Leibniz said reading it produced "a great light"
CurvesChapter 5
A codex is a bound book, as opposed to a scroll. A palimpsest is a manuscript whose original text was scraped off so the parchment could be reused, from Greek palimpsestos, "scraped again"
ManuscriptsChapter 2
A note at the end of a manuscript recording who copied it, where, and when
ManuscriptsChapter 2
The Royal Society's report on who invented the calculus, dated 1712 and circulated in January 1713. Presented as an impartial verdict; Newton wrote much of it himself, anonymously. Latin for "exchange of letters"
LanguageChapter 8
An endless sum whose running totals settle toward one number, the way settles toward 1. The whole nineteenth-century rigor project begins with saying exactly what "settles toward" means
IdeaChapter 10
A result that follows from a theorem already proved, with little or no extra work. Latin corollarium, a small gift
IdeaChapter 6
The curve traced by a point on the rim of a rolling wheel. The subject of so many seventeenth-century quarrels that it acquired a nickname about the Trojan war
CurvesChapter 5
Lagrange's own name for what we call the derivative: from a primitive function , one derives , then . The word "derivative" is his coinage made permanent
IdeaChapter 9
An edition reproducing a manuscript exactly as written, including errors, deletions, and spelling
ManuscriptsAppendix F
An endless sum whose running totals never settle, like or . Abel called them "some devilry" (the usual English "invention of the devil" is a translation of a translation). He chose the phrase because careless use of them produced confident nonsense
IdeaChapter 10
A formal memorial address delivered at a French learned academy on the death of a member
IdeaChapter 7
(円理)
"Circle principle", the Japanese series methods for arcs and areas developed in the wasan tradition
IdeaChapter 11
A table of where the sun, moon, and planets sit in the sky, date by date. What made Indian astronomy need fine sine tables, and what lets a modern laptop check a medieval eclipse claim
IdeaChapter 3
The modern definition of a limit, stated with inequalities and no appeal to motion or pictures. Epsilon from erreur, delta from différence
IdeaChapter 10
Newton's term for a quantity in the act of vanishing. Berkeley's target
IdeaAppendix B
The Greek technique of bracketing an unknown area between inscribed and circumscribed figures and ruling out every other answer by contradiction. A 17th-century name; it brackets rather than exhausts. The name is Latin for "using up"
LanguageChapter 2
"He or she flourished". Used when someone's birth and death dates are unknown but their period of activity is not
Ideathe timeline
Newton's terms. A fluent is a quantity that flows with time; its fluxion is its rate of flow. Modern equivalent: function and derivative. Both are from Latin fluere, to flow
LanguageChapter 6
A folio is a manuscript leaf. Recto is the front, verso is the back. "f. 15r" means the front of leaf 15
ManuscriptsChapter 5
The Earth's true gravitational shape: not a sphere, not an ellipsoid, but a lumpy surface, because gravity varies over mountains and trenches. GPS depends on it
IdeaChapter 11
Berkeley's 1734 jab at evanescent quantities: neither finite, nor infinitely small, nor nothing. The single most quoted phrase ever aimed at the calculus, and the criticism took a century to answer properly
IdeaChapter 9
The second, higher doctorate a German university required before someone could lecture. At Riemann's, the candidate proposed three topics and the examiner picked; Gauss picked the one that became modern geometry
IdeaChapter 10
The study of how history gets written: who recorded what, who copied whom, and why the story you inherited says what it says. This book does some of it on purpose
IdeaChapter 3
Official permission to print, from a church or state censor. Agnesi's 1748 book carries one
Ideathe timeline
Two lengths with no common measure, so their ratio is not a fraction of whole numbers. The diagonal and side of a square are the famous pair. Galileo abandoned his correct cycloid ratio in 1599 because he believed it was incommensurable
CurvesChapter 5
The pre-calculus idea that an area is made of lines and a solid of planes. Cavalieri's method; theologically contentious in Counter-Reformation Italy
IdeaChapter 5
A quantity smaller than any positive number but not zero. Used freely for two centuries, banished in the nineteenth, made rigorous by Robinson in 1966. From Latin infinitesimus, "the infinite-th", formed like decimus and centesimus
LanguageChapter 5
A curve that starts below a line and ends above it must cross it somewhere. Everyone treated it as obvious until Bolzano insisted, in 1817, that it needed a proof, and gave one
Curvesthe timeline
A passage a later copyist slid into a manuscript, which then travels under the original author's name. Unrelated to the interpolation a math class does between table values; this book means the copyist's kind when it questions a source
ManuscriptsChapter 2
A member of the Society of Jesus, the Catholic order whose schools and missions carried European mathematics abroad, and possibly carried Kerala's results back. Whether they did is an open research question in this book
IdeaChapter 3
Described through motion: a curve traced by a moving point, an area swept by a moving line. The medieval Oxford work on velocity is kinematic, and so is Newton's whole picture of the calculus
CurvesChapter 4
The twentieth century's answer to "how long is a set of points?", extending length from intervals to far stranger sets. The foundation the modern integral is built on
PeopleChapter 10
A small result proved on the way to a bigger one, scaffolding with its own name. Newton's Lemma 12 and Lemma 28 each carry a whole story in this book
IdeaChapter 6
The Cambridge mathematics professorship founded in 1663. Barrow held it first and resigned it to his 26-year-old student Newton in 1669; "the Lucasian professor" means whoever holds it
IdeaChapter 5
The rigorous theory of size (length, area, volume) that lets you integrate functions too badly behaved for Riemann. Lebesgue, 1902
IdeaChapter 12
In the eighteenth and nineteenth centuries, a research paper read to or published by a learned academy. A memoir on partial differences is an article, not an autobiography
IdeaChapter 6
Abraham Robinson's 1966 framework giving infinitesimals a rigorous foundation via model theory. The standard part of a number is the nearest real number to it
Ideathe timeline
Continuous everywhere, yet with a well-defined slope nowhere: a curve that is all corners. Weierstrass built one and stunned everyone who believed continuous curves were smooth almost everywhere
CurvesChapter 10
Julian and Gregorian calendar dates. England used Julian until 1752 and started its civil year on 25 March
IdeaChapter 6
The solid you get by spinning a parabola about its axis, like a satellite dish. Archimedes found its volume by mechanical reasoning; ibn al-Haytham needed sums of fourth powers for a related one
CurvesChapter 3
The ruler or noble who paid for a scholar's living before universities and journals did. When Kepler's patron abdicated, Kepler's mathematics moved cities with him
IdeaChapter 5
A starting assumption stated up front and then reasoned from, like Euclid's five. Berkeley's complaint was that an infinitely little magnitude was "too bold a Postulate" to start from
IdeaChapter 9
An endless polynomial: . The machinery Madhava's arctangent runs on, in place before Europe arrived at it
IdeaChapter 3
The thing itself: the manuscript, the letter, the first printed page, as opposed to later retellings. When this book cites a primary source, someone read the original document or its facsimile
ManuscriptsChapter 5
Lagrange's name for the function you start from before deriving: the antiderivative. French textbooks still say "primitive" where yours says antiderivative
IdeaChapter 9
An argument about who discovered something first. The Newton-Leibniz one is the most famous in science
IdeaChapter 8
Literally "squaring": finding the area of a region, historically by constructing a square of equal area. The old word for integration
IdeaChapter 2
Finding the length of a curve. Literally "straightening"
CurvesChapter 5
Proof by contradiction: suppose the opposite, derive an absurdity, conclude the opposite is false. The double reductio is the engine of the Greek method of exhaustion
IdeaChapter 3
An expert an academy or journal asks to check a paper before printing it. The system was young and leaky in the seventeenth century, which is part of how the priority war happened
IdeaChapter 10
The definition of the integral your calculus course uses, from Riemann's 1854 Habilitation. What students learn is usually Darboux's 1875 reformulation with upper and lower sums, still under Riemann's name
PeopleChapter 10
Reasoning tight enough that nothing rests on a picture, a motion, or an unstated assumption. The nineteenth century's project was retrofitting rigor under a calculus that already worked
IdeaChapter 1
London's scientific academy, founded 1660. Its journal carried the first published hints of Newton's method, and its 1712 committee on the priority dispute was anything but neutral
IdeaChapter 6
(鎖国)
"Closed country", Japan's policy of near-total isolation from about 1639 to 1853, under which wasan mathematics developed independently
IdeaChapter 11
(pl. sophismata)
A deliberately puzzling sentence used as a training exercise in medieval logic. The mean speed theorem appears inside one
IdeaChapter 4
The operation in Robinson's nonstandard analysis that rounds a finite hyperreal to the one real number infinitely close to it. The single step that makes Leibniz's rigorous, three centuries later
IdeaChapter 10
The stretch of axis between where a tangent line meets the axis and the foot of the point of tangency. Seventeenth-century tangent problems were often stated and solved as subtangent problems
IdeaChapter 5
Sanskrit: "instantaneous motion". Bhaskara II's term for what we call instantaneous velocity
Ideathe timeline
A function rebuilt as an endless polynomial from its derivatives at one point. Named for Brook Taylor's 1715 book; Gregory had it in a private letter forty years earlier, and Madhava's school had cases earlier still
PeopleChapter 5
A lunar day in the Hindu calendar. Predicting when one ends is why Bhaskara II needed instantaneous velocity
IdeaChapter 3
A book-length, systematic account of one subject. Newton's 1671 treatise on fluxions went unpublished for decades, which is half the story of the priority war
IdeaChapter 6
A stronger kind of convergence for a sequence of functions, in which the rate does not depend on the point. Its absence is why Cauchy's 1821 sum theorem is false. Uniform is Latin uniformis, "of one shape"
IdeaChapter 10
Italian for the rope that turns a sail, from Latin versoria. Mistranslated as avversiera, she-devil, giving "the Witch of Agnesi"
ManuscriptsChapter 9
Latin for "living force", the eighteenth-century quantity mv². The Newtonian rival was mv. Both were right: they are kinetic energy and momentum
LanguageChapter 9
Jyeshthadeva's c. 1530 Malayalam prose text giving derivations, not just results, for the Kerala school's series
Ideathe timeline
Back matter
Provenance and reproducibility
Everything in this document is reproducible from the working files.
| Artifact | What it holds |
|---|---|
registers/sources.csv | 255 sources with author, year, title, container, DOI or URL, language, tier, access level, and the note file for each |
registers/dates.csv | 327 dated events with sort year, precision, display date, people, place, category, confidence, dispute note, and sources |
registers/people.csv | 212 people with life dates, IPA, plain respelling, region, contribution, and sources |
registers/terms.csv | 191 words, symbols, and eponyms with first attested use, originator, language origin, and literal meaning |
registers/conflicts.csv | 129 logged disagreements between sources, with resolution and method where settled |
registers/images.csv | 72 image candidates with repository, rights status, the exact rights wording found, and local file |
registers/IMAGE-RIGHTS-REPORT.md | Plain-language rights summary per repository, and what is safe for commercial publication versus classroom use only |
registers/GAP-REPORT.md | Every research gap with outcome and, for unreachable items, what a library would need to request |
notes/ | 255 note files, one per source, each with citation, access record, extracted dates, people, terms, quotations, and contradictions |
images/ | 39 image files logged with their pixel size, byte size and rights wording, 41.8 MB in all |
sources/ | 76 downloaded full texts |
verify/math_verification.txt | Raw output of all 12 mathematical checks, including the one test that was written wrong and corrected |
verify/ALL-QUOTES.txt | 1,043 direct quotations extracted from the notes, each tagged with its source ID |
build/gen_tables.py | The script that generates every register-derived table in this document |
© 2026, prompted by Megan Warren and built by Claude.
Back matter
About this book
Written as the free companion reader for the Calculus Derivatives and Calculus Integrals courses, 2026-2027. It exists because the two-sentence version of this history is misleading in a way that makes the mathematics harder to understand, not easier.
The courses themselves are Megan Warren's design, built from over seven years of teaching calculus. This book was written to that design and keyed to it section by section: Appendix I holds 89 placements linking each story to the exact place in her two textbooks where it does the most good.
Every claim traces to a source that was read in full. 255 sources, one research note each, 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. 129 disagreements between sources are logged and 58 are still open. Every calculation in Appendix E was re-derived and checked in code before it shipped.
Design and voice: hers as well, navy with an iris accent, the shared course-book system's color for material that spans both halves of calculus. Math is typeset with KaTeX and fully embedded, so the book renders with no internet connection.
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.