The Two Problems Were One Problem

Slopes and areas have nothing obviously to do with each other. One is about steepness at a point, the other about accumulation across a stretch. They were studied separately for two thousand years.

They are inverse operations, and the statement of that is the fundamental theorem of calculus.

Strip the symbols off and it is almost a triviality. Water runs into a tank at a rate that varies. If you know the rate at every moment, you can work out how much is in the tank, which is accumulation. If you know how much is in the tank at every moment, you can work out the rate it was going in, which is a rate of change. Each of those recovers the other. The total and the rate are two descriptions of one situation.

What that buys is enormous, and it is computational.

The previous chapter found areas by adding up slices, which is laborious and has to be redone for every curve. The theorem says you need not add anything. Find a function whose rate of change is your curve, evaluate it at the two ends of the interval, subtract. An infinite sum collapses into one subtraction, and the only skill required is recognising what differentiates to what.

Almost everything a calculus course spends its time on is that recognition, which is why the subject feels like a bag of tricks. The tricks are a search for antiderivatives, and the theorem is the reason the search is worth making.

Newton and Leibniz both arrived here in the 1670s, independently, and what followed is the ugliest episode in the history of the subject.

Newton had the results first, in the mid-1660s, and did not publish. Leibniz developed his version later and published first, in 1684. From there it became a question of national credit rather than of mathematics: accusations of plagiarism, anonymous pamphlets, and in 1712 a Royal Society committee appointed to settle the matter which found for Newton. Newton was president of the Royal Society. He also drafted the report.

The modern view is that both invented it, which is unsurprising for a result that was ready to be found.

The damage is the part worth taking from this, because it fell on the winning side.

British mathematicians kept Newton's notation out of loyalty. His dots over letters are serviceable for simple cases and cramped for anything harder. Leibniz's symbols, the integral sign and the dy/dx form, carry the structure of the operation in the way they are written, so they do some of the thinking for you. The continent used them and British analysis fell behind for most of a century.

That is the clearest case in mathematics of notation having consequences. The theorems were the same on both sides of the Channel. One side had symbols that made the next step easy to see and the other had symbols that made it hard, and a hundred years of relative decline followed from a choice made for reasons of honour.