Calculus

Descartes' coordinates turned curves into equations, and in the seventeenth century mathematicians such as Fermat, Wallis and Barrow found tricks for tangents and areas. Two people turned the tricks into a method. Isaac Newton, at home during the plague years of 1665–66, developed his "fluxions" and saw that finding tangents and finding areas are inverse problems. Gottfried Wilhelm Leibniz, a diplomat and philosopher, reached the same insight in the 1670s with a notation so good ($dx$, $\frac{dy}{dx}$, $\int$) that it has never been replaced.

Leibniz published in 1684, and Newton had published almost nothing. What followed was one of the ugliest disputes in science. Newton's supporters accused Leibniz of stealing from Newton's unpublished papers, and a Royal Society inquiry, secretly drafted by Newton, agreed. Historians now credit both as independent inventors. The feud is often blamed for cutting British mathematicians off from the far more productive Continental school for a century.

Calculus worked astonishingly well. Euler made the function its central object and poured out results: infinite series for $e^x$, $\sin x$ and $\cos x$, and the formula $e^{ix} = \cos x + i \sin x$, which gives $e^{i\pi} + 1 = 0$. Newton's mechanics, the physics of the entire eighteenth century, was calculus in action.

But no one could say what it was about. In 1734 George Berkeley, Bishop of Cloyne, published The Analyst, addressed to "an infidel mathematician". To find a derivative, he noted, you divide by a small increment, treating it as nonzero, and then set it to zero to get a clean answer. The vanishing increments of the calculus were "neither finite quantities, nor quantities infinitely small, nor yet nothing". They were "the ghosts of departed quantities". Mathematicians knew the objection was fair and had no good answer.

Find the slope of $y = x^2$ at a point $x$. Early calculus did it like this. Move a tiny distance $h$ along the curve. The height changes by

so the average slope over that step is $\frac{2xh + h^2}{h} = 2x + h$. Now let $h$ be "infinitely small" and throw it away: the slope is $2x$.

This is exactly what Berkeley objected to. To divide by $h$ it must be nonzero, and to throw it away it must be zero. Newton spoke of "ultimate ratios" and Leibniz of infinitesimals, but neither could say what $h$ was at the moment of discarding it.

The nineteenth-century answer, the limit, changes the question. Do not set $h$ to anything. Instead observe that $2x + h$ can be made as close to $2x$ as you like by taking $h$ small enough, and define the derivative as the number the averages approach:

The calculation is the same; only its justification changed.

The fundamental theorem of calculus then turns the problem around. The area under $y = x^2$ from 0 to 1 is found by asking which function has derivative $x^2$. The answer is $\frac{x^3}{3}$, so the area is $\frac{1}{3}$. Archimedes had found the same fact about the parabola, after pages of exhaustion arguments. Calculus gets it in one line, which is why it swept the world before anyone could justify it.

The answer took a century and changed mathematics. Questions about which functions could be written as sums of waves (Fourier analysis) and about calculus with complex numbers (complex analysis) pushed calculus into territory where intuition failed. The effort to put it on solid ground, with precise limits and precisely defined real numbers, became real analysis. Calculus itself became the tool every other science used, and differential geometry was one of the first fields built on it.