Chapter 5
The seventeenth century: everybody gets close
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 the honest summary is that 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, in Kepler's own words sine discrimine, sine respectu figurae, sine ratiocinatione vel calculo: without distinction, without regard to shape, without reasoning or calculation (Source 159, Kepler, 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. (Source 151, [attributed to Jourdain, P. E. B. - UNVERIFIED])
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 (Source 152, Davis). 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 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, 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 (Source 154, O'Connor). 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. It is Proposition II.4, and the name "Cavalieri's principle" is not his (Source 135, O'Connor).
He sat on the manuscript for eight years, writing to Galileo more than once to say, in effect, please do not make me commit to what my own method seems to say (Source 156, Andersen). And he denied the atomistic reading in writing, twice, on 2 October 1634 and 28 June 1639 (Source 135, O'Connor, 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 (Source 132, Sherry; Source 133, Alexander). The two appeared in the same journal issue, both with evidence, and the dispute is unresolved. 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 and mostly refused to publish, sending results to friends in letters, sometimes as challenges (Source 144, O'Connor).
His method of maxima and minima works like this. To maximize a quantity, perturb the variable by a small amount e, 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:
Read this equation in words
f of x equals x times, a minus x. Then f of x plus e, minus f of x, equals a e minus two x e minus e squared. Dividing by e gives a minus two x minus e. Suppress e: zero equals a minus two x, so x equals a over two.
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 (Source 130, Katz). Fermat's own French phrase for the procedure was "une comparaison feinte ou une adégalité", a feigned comparison, or an adequality (Source 130, Katz, p. 6).
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 another (Source 130, Katz). The one thing the primary text settles: the last step is a suppression of the e-terms, not setting e = 0 (Source 130, Katz). 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, and he deliberately made it hard because he wanted readers to have to work (Source 148, O'Connor).
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 (Source 148, O'Connor). 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 (Source 450, Cajori). 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 (Source 450, Cajori).
1643 to 1670: the rest of the field
Evangelista Torricelli got about three months with Galileo before the old man died, and turned that into the barometer and the acute hyperbolic solid: an infinitely long solid with finite volume, which he had by 1641 and published in the 1644 Opera geometrica (Source 131, O'Connor). That same 1644 volume carried the first printed article on the cycloid, which brought a letter from Roberval charging plagiarism; Cajori attributed Torricelli's death in 1647 to the charge, which is a claim about a man's state of mind and not a medical record (Source 64, Whitman, p. 314). His own statement of the hyperbolic solid:
An acute hyperbolic solid, infinitely long ... is equal to that right cylinder. (Source 157, Wijeratne)
Guess before you read on
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, and the painter's paradox is a nineteenth and twentieth-century overlay on a seventeenth-century theorem (Source 157, Wijeratne). 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. Torricelli did not prove the infinite surface area at all; the painter's paradox every student meets is a nineteenth and twentieth-century embellishment on a seventeenth-century theorem (Source 157, Wijeratne). That is a near-universal textbook error and it is worth correcting in your book.
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, put them on a balance, and found the arch came out at about three times the circle. Then he abandoned the result, because he believed the ratio was incommensurable (Source 64, Whitman, 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 Cantor credits Galileo with making the curve well known and with naming it.
Gilles Personne de Roberval got the quadrature around 1634 and did not publish it until it appeared posthumously in the Traité des Indivisibles of 1693 (Source 64, Whitman). The institutional reason is clean, and Whitman states it exactly: the Chair of Ramus at the Collège Royal, which Roberval won in 1634, "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" (Source 64, Whitman, pp. 310 to 311). Secrecy was not vanity, it was tenure (Source 137, O'Connor). Evelyn Walker's verdict, which Whitman quotes: "the accident of occupying this chair caused Roberval to lose credit for many of his discoveries" (Source 64, Whitman, 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: cash prizes, a pen name (Amos Dettonville, an anagram of Louis de Montalte, the pen name from the Provincial Letters), judges, deadlines, angry losers, and a winner declared: nobody (Source 153, O'Connor). 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, whose characteristic triangle Leibniz would later say switched on a light in his head.
The toothache story, that Pascal thought about the cycloid because a toothache kept him awake and the pain stopped when he concentrated, rests on his sister Gilberte, and the Internet Encyclopedia of Philosophy explicitly calls her claims questionable (Source 161, Martin; Source 143, [author unverified]). Whitman shows the transmission route: he attributes the story to W. W. Rouse Ball, adding that Pascal took the relief as "a divine intimation to proceed with the problem" and then "worked incessantly at it for eight days" (Source 64, Whitman, 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 (Source 64, Whitman, p. 314).
The result is startling and a student can check it:
Read this equation in words
The arc length of one cycloid arch equals eight a, which is four times the diameter of the generating circle.
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. (Source 139, Miller, Wallis's Latin, translated)
Wallis is worth knowing outside the mathematics. He got his career from cryptography, breaking a Royalist cipher in 1642 until Parliament noticed, and he spent twenty years exchanging insulting pamphlets with Thomas Hobbes about whether the circle can be squared, with titles like School Discipline for not saying his Lessons Aright (Source 145, O'Connor).
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, and MacTutor calls it "the first attempt to write a systematic text-book on what we should call the calculus" (Source 136, O'Connor). He had the Taylor series forty years early in a private letter. He designed a reflecting telescope he could not build. He had a stroke at 36 while pointing a telescope at Jupiter for his students (Source 136, O'Connor).
Isaac Barrow stated the inverse relationship geometrically in Lectiones Geometricae (1670), resigned the Lucasian chair to his 26-year-old student in 1669, and posted that student's manuscript to London in the same year (Source 184, Newton). 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. (Source 158, Barrow, 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 (Source 138, Nauenberg).
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 (Source 73, Feingold, 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 admitting he could not see much in Leibniz's calculus that was new beyond what Barrow had published, and calling 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, which was used against Leibniz decades later (Source 73, Feingold, 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" (Source 73, Feingold, p. 325).
Feingold sets his own limits, and they are as sharp as his argument. He explicitly declines Child's thesis: it is not his aim "to argue (in the manner of Child) that... he is, in fact, the true inventor of the calculus" (Source 73, Feingold, p. 312). And his conclusion: "Due recognition of Barrow shifts no credit for the invention of the calculus away from Newton and Leibniz" (Source 73, Feingold, p. 338).
The question of what exactly Barrow gave Newton stays open. 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, and it contains a false claim to have squared the circle. A 22-year-old Dutchman 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 (Source 67, Gregorius a Sancto Vincentio).
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:
Read this equation in words
The integral from a to b of d x over x equals the natural log of b over a. The integral from a to a r, the integral from a r to a r squared, and the integral from a r squared to a r cubed of d x over x are all equal, each one the natural log of r. And the integral from one to u v of d x over x equals the integral from one to u, plus the integral from one to v, of d x over x.
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: that Saint-Vincent's quadrature of the circle "abeat in illud necdum solutum Problema", collapses into this still-unsolved problem (Source 66, Sarasa). 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." (Source 66, Sarasa, printed p. 8)
He takes magnitudes in continued proportion whose logarithms are 6, 7, 8, 9, 10, numbers which, 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."
In place of the logarithms, take the areas. That sentence is the birth of the logarithm as an area, and it appears inside a defence 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, with no base chosen by anybody (Source 155, Coolidge).
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 (Source 66, Sarasa; Source 67, Gregorius a Sancto Vincentio). 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 (Source 67, Gregorius a Sancto Vincentio).
Pietro Mengoli posed the Basel problem in Bologna in 1650 (Source 142, O'Connor). Johann Hudde was differentiating polynomials algebraically in 1657 and later ran the city of Amsterdam for thirty years (Source 147, O'Connor). Nicholas Mercator's 1668 series for log(1+x) is the first infinite series for a logarithm ever printed, and it is what pushed Newton to write up De analysi to establish priority (Source 149, O'Connor). Christiaan Huygens demolished Saint-Vincent's false proof, built the first pendulum clock that kept time on a cycloid rather than a circle, and then personally taught the young Leibniz the mathematics he needed (Source 150, O'Connor).
The scoreboard in 1669
By the time Newton was 26, Europe had: coordinate geometry, a general method for maxima and minima, a general method for tangents, quadrature of x^n, the area under a hyperbola as a logarithm, infinite series for logarithms and for π, 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.
One question before you go
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" (Source 66, Sarasa). 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.