Chapter 9
The eighteenth century: attacked, defended, and put to work
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 and then built most of the early subject from it (Source 231, Leibniz).
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 (Source 324, Miller). 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 (Source 330, O'Connor).
The famous line about Newton's anonymous solution needs care. The Latin "tanquam ex ungue leonem", "as the lion is known by his claw", is verified in David Brewster's 1831 Life of Newton, p. 194 (Source 321, Brewster):
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. (Source 321, Brewster, 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, adds "says he", and the Latin version outlives the original by two centuries (Source 321, Brewster). The Latin belongs to Brewster, writing in 1831, so the line is his and not Bernoulli's (Source 321, Brewster). 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 (Source 337, Bernoulli).
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 (Source 321, Brewster).
Jacob's tombstone. He loved the logarithmic spiral because it reproduces itself under scaling, and 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 (Source 328, O'Connor; Source 329, Ashworth, confirmed by two independent sources). The difference is a good two-minute exercise:
Read this equation in words
The Archimedean spiral: r equals a theta, its turns equally spaced. The logarithmic spiral: r equals a times e to the b theta, each turn a fixed multiple further out.
William Ashworth's dry moral: if you want your mathematics on your tombstone, supervise the mason (Source 329, Ashworth).
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 (Source 333, Tou). 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 (Source 333, Tou).
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 (Source 316, L'Hospital).
The contract is documented: 300 pounds a year from 1 January 1694, offered in a letter of 17 March 1694, buying both exclusivity and silence (Source 315, Truesdell). 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 (Source 315, Truesdell). L'Hôpital did not always pay in full (Source 315, Truesdell).
The complication that makes it a better story than "he stole it": l'Hôpital's 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 (Source 316, L'Hospital). 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:
Read this equation in words
Sine x over x equals one, minus x squared over three factorial, plus x to the fourth over five factorial, and so on. It also equals the infinite product over n of, one minus x squared over n squared pi squared. Matching the coefficient of x squared: negative one sixth equals negative the sum of one over n squared pi squared. So the sum from n equals one to infinity of one over n squared equals pi squared over six, which is one point six four four nine three four zero six six eight, and so on.
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. The teaching module's own footnote puts it well: 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" (Source 322, Monks). 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 (Source 322, Monks).
Two corrections your textbook probably needs:
- Euler's 1735 Basel paper (E41) uses
p, not π. He adopts π at Introductio §126 in 1748 (Source 318, Euler; Source 325, Miller). - Euler's identity is not in the Introductio. §138 gives only . Euler's
idates from 1777 and was published in 1794 (Source 318, Euler; Source 325, Miller).
Notation he standardized or introduced: f(x), e, Σ, Δ, i, and π by adoption. Δ and Σ arrive in one book, in one sentence, as explicit twins, in 1755 (Source 451, Cajori). e was born in a paper about cannon. i waited seventeen years between being written and being printed. π was a Welsh mathematician's abbreviation that a Swiss genius made compulsory (Source 325, Miller).
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 (Source 326, O'Connor). His papers were still being published from the backlog for 43 years after his death, which is why he needed his own catalog, the E-numbers (Source 323, Bogart).
The Diderot anecdote is fabricated, and the fabrication is fully traceable. The story that Euler silenced Diderot at the Russian court with "(a+bⁿ)/n = x, therefore God exists" collapses under Ronald Gillings' 1954 analysis. The chain: Thiébault reports an unnamed mathematician and 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 (a+bⁿ)/z = x, and De Morgan miscopied the z as an n (Source 313, Brown; Source 314, Gillings).
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 (a+bn)/z = x. De Morgan miscopied that z as an n in 1872, and every later teller copied De Morgan (Source 313, Brown; Source 314, Gillings).
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, correctly diagnosed the oversized pump, warned Frederick in writing, and was ignored (Source 342, Eckert). 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, and 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, so that modern analysis would "enrich, not undermine" the traditional framework of religious and metaphysical knowledge (Source 77, Mazzotti, p. 675). That is a deliberate intellectual strategy, not timidity. She held that the differential was equivalent to the Newtonian fluxion, insisted on the priority of geometrical evidence, and 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" (Source 77, Mazzotti, p. 675). She was building a vocabulary, not just a book.
- She introduced signed numbers through debits and credits (Source 77, Mazzotti, p. 675). In 1748.
- She took "unusual care... in explaining every single step of mathematical reasoning" (Source 77, Mazzotti, 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, "who had never before worked with the symbols of differential and integral calculus" (Source 77, Mazzotti, 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 (Source 77, Mazzotti, 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, and "neither more conservative traditional Catholicism nor the radical culture of the Enlightenment seems to have offered women comparable opportunities" (Source 77, Mazzotti, p. 658). After that pontificate the space closed, and "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 (Source 77, Mazzotti, 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" (Source 77, Mazzotti; Source 319, Agnesi).
- 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" (Source 77, Mazzotti, n. 44).
- Colson's synopsis went further and made it hers: "which she calls the Witch" (Source 320, Agnesi). 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 (Source 77, Mazzotti, 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 (Source 77, Mazzotti).
Why the curve was 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" (Source 77, Mazzotti).
The irony is exact: the same John Colson annotated Newton's Method of Fluxions and was thanked by Maclaurin for defending Newton against Berkeley (Source 311, Maclaurin; Source 320, Agnesi).
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 (Source 75, Cupillari). One example, ready to teach: at page 361 Agnesi takes , finds both asymptotes, plots points, and then establishes concavity with no calculus at all, by comparing the height of a chord against the height of the curve at the midpoint and reducing the comparison to the inequality 9a²/3 > a². 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, give up balls and the theater, and 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 (Source 77, Mazzotti).
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 and is to this day the only complete French translation of that work (Source 69, Allen). Her addition beyond translation is substantive: the Exposition abrégée du système du monde commentary (Source 332, Project Vox).
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, the same text he had worked through under Nicole and Bernoulli. She found it dry and told him so (Source 68, Zinsser). Then Alexis-Claude Clairaut took over and pushed her through equations of the third and fourth degree, multiple unknowns, and operations on radicals. Zinsser's 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 (Source 68, Zinsser). 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" (Source 68, Zinsser).
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 (Source 332, Project Vox; Source 68, Zinsser). Her Institutions de Physique was written as a physics textbook for her thirteen-year-old son, and its last chapter is an open attack on Dortous de Mairan's 1728 Académie memoir on forces vives, published just as Mairan was about to become the Académie's perpetual secretary (Source 68, Zinsser).
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, because a submission known to be by a woman would not have been taken seriously by the judges. 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 (Source 69, Allen). The Académie royale des sciences admitted no women at all, except as spectators at its two public meetings a year (Source 69, Allen).
She was 42, pregnant, and knew what that meant. 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" (Source 69, Allen). 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 and arranged for the illustrations to be finished (Source 68, Zinsser). 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" (Source 310, Berkeley, 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? (Source 310, Berkeley, 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 (Source 310, Berkeley). Berkeley closes the book with 66 Queries, questions rather than assertions, which is a rhetorical trick worth showing students directly.
Guess before you read on
Fermat's method, and Newton's after it, does two things in a row: divide by a small quantity e, then delete 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, which is why Berkeley could call the results "the ghosts of departed quantities" and be mathematically right (Source 130, Katz; Source 310, Berkeley).
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, which 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, and then you discard it, which requires it to be zero. Berkeley is not wrong.
The responses: James Jurin's Geometry No Friend to Infidelity, Benjamin Robins, and Colin Maclaurin's Treatise of Fluxions (1742), which 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. (Source 311, Maclaurin, Preface p. i)
Maclaurin also concedes the substantive point: "the Supposition of an infinitely little Magnitude being too bold a Postulate" (Source 311, Maclaurin).
Both Berkeley and Jurin published anonymously or pseudonymously and then attacked each other for hiding (Source 310, Berkeley).
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: the principles of differential calculus, freed from any consideration of infinitesimals, vanishing quantities, limits, or fluxions, and reduced to the algebraic analysis of finite quantities (Source 317, Lagrange). 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:
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 the function fx the primitive function, and we shall call these the derived functions." (Source 317, Lagrange, arts. 16 and 17. English translated here, not quoted from a published translation.)
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:
Read this equation in words
f of x equals e to the negative one over x squared when x is not zero, and zero when x equals zero. Every derivative of f at zero is zero, so its Taylor series at zero is zero plus zero x plus zero x squared, and so on: identically zero. But f of x is not zero for any x other than zero.
The function is smooth everywhere, its Taylor series at the origin is identically zero, and the function is not (Source 317, Lagrange).
Taylor and Maclaurin did not invent their series
- Brook Taylor, Methodus Incrementorum (1715). Gregory, Leibniz, and Johann Bernoulli had it earlier (Source 312, Gibson). 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" (Source 312, Gibson).
- 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 (Source 312, Gibson, and the notation file).
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" (Source 312, Gibson).
One question before you go
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 a³ - zy² = 0, finds both asymptotes, plots points, and reduces the whole question to the inequality 9a²/3 > a² (Source 75, Cupillari). That is midpoint convexity, argued by hand, and it agrees with the sign of the second derivative.
Derivatives 2.7 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.