Chapter 7
Leibniz
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" (Source 451, Cajori).
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 (Source 238, Miller).
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. (Source 230, Child, 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. (Source 230, Child, Child p. 82)
He knew.
Guess before you read on
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 (Source 230, Child). 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 (Source 230, Child, 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. (Source 230, Child, Child p. 100)
He tested it on one example, got the right answer by accident, believed it, and then 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. (Source 230, Child, Child p. 102)
Child's marginal exasperation, written in 1920 about a page from 1675: "Leibniz, as a logician, should have known better than to trust a single example" (Source 230, Child).
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):
Read this equation in words
x plus d x, times y plus d y, minus x y, equals x d y, plus y d x, plus d x times d y. With x equal to three, y equal to five, and d x and d y each one one-thousandth: the actual change is zero point zero zero eight zero zero one. d x times d y is zero point zero zero zero zero zero one, which was Leibniz's first guess. x d y plus y d x is zero point zero zero eight zero zero zero, the correct rule. The actual change minus the correct rule is d x times d y, zero point zero zero zero zero zero one.
The guess is smaller than the actual change by a factor of 8001, 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 (Source 230, Child).
Where the ideas came from
Huygens set him a test. In Paris in 1672, testing a 26-year-old diplomat who claimed an interest in mathematics, Huygens asked Leibniz 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:
Read this equation in words
The sum from n equals one to infinity of two over, n times n plus one, equals two times the sum of, one over n minus one over n plus one. The terms cancel in pairs, leaving two times, one minus the limit of one over capital N plus one, which equals two.
Huygens was impressed (Source 237, Arthur). 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", and it becomes "the basis for the fundamental theorem of the calculus: the sum (integral) of the differentials equals the difference of the sums" (Source 237, Arthur). Pascal's triangle adds downwards; Leibniz's harmonic triangle subtracts upwards. 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" (Source 232, Leibniz (trans. Walker)). 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, with plate Tabula XII facing p. 466, signed "per G.G.L." (Source 231, Leibniz, 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 (Source 231, Leibniz).
- 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 (Source 231, Leibniz). 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 (Source 231, Leibniz). The 1929 translator quietly fixed the printer's errors and left a footnote about it (Source 232, Leibniz (trans. Walker)).
- It shares its issue with an anatomical review of the structure of the ear (Source 231, Leibniz).
The ∫ sign reached print two years later, in De geometria recondita (1686), almost in passing, inside a squabble about whether the circle can be squared (Source 233, Leibniz). Its own subtitle misprints the page number of Nova methodus (Source 233, Leibniz). 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" (Source 232, Leibniz (trans. Walker), p. 625, Evelyn Walker's translation).
And then the joke that turned into a field. In a 1695 letter to Wallis, Leibniz noticed that if ∫ and d behave like roots and powers, and powers can be one-half, then so, "usefully", can d. He wrote d^(1/2)y in his own hand (Source 241, Leibniz (ed. Gerhardt)). 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 (Source 241, Leibniz (ed. Gerhardt)).
The man
Law degree at 20. Allegedly refused a doctorate at Leipzig over his age, and the story that the dean's wife was responsible is unverified (Source 235, O’Connor & Robertson). 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. He spent decades trying 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 (Source 235, O’Connor & Robertson).
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 (Source 236, Look). 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, and the standard account is that his secretary was effectively the only mourner (Source 235, O’Connor & Robertson; Source 243, Westminster Abbey). The Royal Society, of which he had been a Fellow for forty-three years, marked his death with nothing. The Académie des Sciences in Paris, where he was a foreign member, had Fontenelle deliver a formal éloge (that document is real but was not read in full here; do not attribute wording to it).
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, the pall borne by two dukes, three earls, and the Lord Chancellor, under a monument by the King's own architect (Source 243, Westminster Abbey; Source 200, Westminster Abbey).
Eleven years, eight hundred miles, and two funerals.
One question before you go
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. Richard Arthur states the principle Leibniz took out of that afternoon: "the sum of the differences is the difference between the first term and the last" (Source 237, Arthur).
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.