Chapter 6
Newton
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 |
(Source 200, Westminster Abbey, 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. (Source 190, Whiteside, 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 (Source 190, Whiteside, MP 1: 8 n.21).
Newton's own famous recollection, written decades later, is at CUL Add. 3968.41, f. 85r: he was "in the prime of my age for invention" (Source 190, Whiteside, MP 1: 152 n.25). Whiteside, who spent his life on these manuscripts and had every reason to be skeptical, acquits him of lying: the weaknesses in Newton's chronology are "lapses in memory... rather than the product of deliberate distortion" (Source 190, Whiteside, 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 (Source 190, Whiteside).
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" (Source 180, Stukeley, f. 15r).
He saw it fall. It did not hit him. No eighteenth-century source has a head strike (Source 180, Stukeley). The setting of the telling mirrors the setting of the event, which is a gift for a textbook: an old man and a younger admirer, under apple trees, and 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 (Source 198, Cambridge University Library). Newton did his gravity calculations on the back of a legal document, because he was a poor student and parchment was expensive (Source 199, Cambridge University Library). On the same shelf: the calculus, the colors of light, the laws of motion, drawings of dissected eyes, and a scheme for dividing the octave. He was 23 (Source 199, Cambridge University Library).
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 (Source 198, Cambridge University Library).
The October 1666 tract (CUL Add. 3958.3) is the hardest dated evidence for the anni mirabiles, and 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 (Source 183, Newton).
Guess before you read on
De analysi is the paper that shows Newton had the method first, and 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 (Source 184, Newton). 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, and was not published until 1711, forty-two years later, and only because the priority war made publication a weapon (Source 184, Newton). 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, 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 (Source 185, Newton).
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 (Source 187, Newton).
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 (Source 204, Guicciardini, p. 460). It was a normal thing to do in Restoration England, and 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, Edmond Halley, David Gregory, and Nicolas Fatio de Duillier (Source 204, Guicciardini, 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", who got the results second hand from the acolytes, "sometimes in mutilated form" (Source 204, Guicciardini, 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" (Source 204, Guicciardini, p. 461). Handing out a damaged copy keeps the students dependent on the teacher and stops a pirate printer 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 (Source 204, Guicciardini, p. 460). Jones transcribed De methodis around 1710, Wilson copied Jones around 1720, and Samuel Horsley built the collected Opera omnia of 1779 to 1785 on those copies rather than on Newton's originals (Source 204, Guicciardini, p. 460). The first collected Newton is an edition of copies of copies. Colson's 1736 translation runs off the Jones transcript too (Source 204, Guicciardini, p. 460), which is the same chain that put Jones's tidied notation into the printed text (Source 192, Whiteside).
By December 1720 Newton had lost the thread completely, 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. (Source 204, Guicciardini, p. 461)
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 and found it "very much worn by having been lent out" (Source 204, Guicciardini, 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 (Source 204, Guicciardini, p. 455).
Early Newton built on three things: Wallis's habit of accepting a formula once it worked for enough particular cases, 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 (Source 204, Guicciardini, p. 463).
Then he turned on it. He wanted certainty rather than plausibility, and he knew the standing of infinitesimals was disputed (Source 204, Guicciardini, p. 463). At the same time he was deep in Pappus and trying to restore lost books of Apollonius, and 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" (Source 204, Guicciardini, p. 455). He had judged his own best work unprintable. Right after finishing De methodis he drafted an addendum in synthetic form, and reworked it about 1680 as the Geometria curvilinea: the calculus redone in geometry, with limits in place of infinitesimals (Source 204, Guicciardini, p. 463).
And the silence is not a gap in his mathematics. Whiteside's volumes show Newton working on mathematics without interruption until 1696 (Source 204, Guicciardini, 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, heard that Wallis was about to publish a summary of Newton's method for squaring curves, and was allowed to transcribe part of De methodis and of the Epistola posterior (Source 204, Guicciardini). The year is disputed: Guicciardini gives 1684 in one place and 1685 in another, in the same article. See Appendix F, CN-14.
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, and it 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 (Source 204, Guicciardini, p. 465).
That is the pattern for everything after. Guicciardini names three forces that pushed Newton to the press: priority, over Gregory first and Leibniz above all; his new standing as the undisputed leader of British mathematics after the Principia; and a change in what counted as publishing at all, once scientific journals existed and the Continental calculus was being built in the Acta Eruditorum and the Paris Mémoires (Source 204, Guicciardini, pp. 465, 468).
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 (Source 204, Guicciardini, 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" (Source 204, Guicciardini, p. 466). A man in his sixties went back into manuscripts he had written in his twenties and took the infinitesimals out, 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 (Source 192, Whiteside, MP 3: 73 n.86). Reproductions of "Newton's 1666 notation" that you will meet in textbooks are reproducing Jones (Source 192, Whiteside).
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:
Read this equation in words
One plus x, all to the one-half power, equals one, plus one half x, minus one eighth x squared, plus one sixteenth x cubed, minus five over one twenty-eight x to the fourth, plus seven over two fifty-six x to the fifth, and so on. Twelve terms give the square root of one point two as one point zero nine five four four five one one five zero three four seven two seven; the exact value begins one point zero nine five four four five one one five zero one zero three three two.
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 (Source 189, [Newton]). Whiteside contextualizes it: Johann Bernoulli said the same thing about his own work, and Whiteside finds the Principia built natively from limit-ratios, with one genuine exception, the solid of least resistance (Source 190, Whiteside, MP 6: 24 n.74 and MP 6: 26).
Two 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, and independently corroborates only Whiteside's single exception, using Whiteside's own phrase "Newton's suppressed analysis" for the solid of least resistance (Source 70, Fellmann, 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 (Source 72, Newton). 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", which 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" (Source 204, Guicciardini, p. 465). When 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 (Source 204, Guicciardini, 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, built on limits and not on infinitesimals (Source 204, Guicciardini, p. 465). 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 (Source 204, Guicciardini). Guicciardini calls the size of the hidden part an open research question and points at his own Reading the Principia, pp. 179 to 186, for the detail (Source 204, Guicciardini, p. 466). See Appendix F, CN-15.
The question is still open here, because the copy of Cohen obtained contains the Guide's contents and not its body. But it is no longer a vague gap: Guide pp. 114 to 117, 122 to 127, and 345 to 347 would settle it, and they are still the highest-value request. See Appendix K.
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 (Source 186, Newton); students enjoy discovering that the famous heckle was pre-refuted. And its reputation for impossibility is, in Whiteside's word, "ill-deserved": he quotes Todhunter's estimate that "probably a dozen pages would supply the necessities of a student who wished to master even the Principia" (Source 191, Whiteside, MP 6: 25 n.78). A manuscript at King's College records Cambridge students muttering, as Newton walked past, that there goes "the man who has writt a book that neither he nor any one else understands" (Source 191, Whiteside).
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, and Halley had to talk him down (Source 203, Chin). Newton's view was that Hooke had done nothing and left "the drudgery of calculations" to others (Source 203, Chin).
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, who 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, and it is 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 (Source 76, Del Centina).
William Whiston later turned that into a legend: Newton "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: in Newton's own copy of De Witt and van Schooten, a volume that came from Barrow's library, Newton wrote a three-line proof in his own hand as an unpublished "Corollary 6", using Euclid III.36 (Source 76, Del Centina).
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 catalogue, drafted on Newton's advice, held two copies of Borelli's Apollonius of 1661 alongside Saint-Vincent's 1647 Opus geometricum (Source 76, Del Centina).
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 (Source 76, Del Centina).
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" (Source 193, Keynes). Keynes wrote "at least 100,000 words"; the widely repeated "million words" figure is modern cataloguing wrongly attributed to him (Source 193, Keynes). Keynes, in the middle of a world war and while designing the post-war financial system, spent his spare hours buying back and reading a dead alchemist's notebooks, and 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 (Source 195, Newton). In 1675 he obtained a royal dispensation exempting the Lucasian Professor from taking holy orders (Source 194, Snobelen). 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 K.2 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, petitioned Parliament twice claiming he could fix the Mint's security while accusing the Mint of incompetence, and 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. (Source 196, Chaloner, 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 (Source 196, Chaloner).
One question before you go
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 (Source 192, Whiteside).
You meet the fluxion itself in Derivatives 1.4, where a derivative stops being something you do at a point and becomes a function. The symbol on that page belongs to a later editor, not to the man who did the mathematics.