On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.
Measuring the area of curved figures is one of the oldest problems in mathematics. Around 250 BC Archimedes showed that the area of a parabolic segment is $\frac43$ of the area of a triangle inscribed in it. It was a stroke of genius, but made to measure: every new curve needed a new idea. By the 17th century, Fermat, Cavalieri, Wallis and others could already compute the area under $y=x^n$ for many values of $n$. What was missing was a general rule. And what was missing was the realization that Newton and Leibniz each reached in their own way: computing areas and computing rates of change are inverse operations.
Before calculus, tangents and areas were two unrelated problems. Finding the tangent to a curve was one art; finding the area beneath it, another. The great discovery of the seventeenth century, by Isaac Barrow, Newton and Leibniz, was that one is the inverse of the other. This is the fundamental theorem of calculus, and Leibniz’s notation writes it in a way that makes it look almost obvious. Leibniz: like sums and differences Leibniz thought of $\int$ as a sum and of $d$ as a difference. And adding up differences undoes the difference, as on a staircase: the sum of the steps is the total height. He wrote that $\int$ and $d$ are inverse to each other, “like powers and roots in ordinary calculation”. In symbols:
The expression $$\int_0^2 x^2\,dx$$has five pieces: the sign $\int$, the numbers $0$ and $2$, the function $x^2$ and the $dx$. Each does a different job, and each came from somewhere. Chapter 1 told the passage from Archimedes’ exhaustion to the integral. Here the integral is read slowly, the way one reads a sentence in a new language. A sentence in five pieces The $\int$ is an elongated S, for summa, sum. Leibniz first wrote it in a manuscript dated 29 October 1675. It is an iconic sign in a curious way: it does not look like an area, but like the word that describes how the area is made. The limits $0$ and $2$, below and above, say from where to where one sums. They are not Leibniz’s: it was the Frenchman Joseph Fourier who, around 1820, began writing them attached to the sign. Before that, they were written out in words, alongside. The function $x^2$ is the height of each strip. The $dx$ is the width of each strip, infinitely thin. And it is also the marker of which letter varies: if it said $dt$, the variable would be $t$. The letter $x$ is a dummy letter: $\int_0^2x^2\,dx$ and $\int_0^2t^2\,dt$ have the same value. The result, $\frac83$, does not depend on $x$. Read as a whole, the sentence says: add up the heights $x^2$ times the widths $dx$, for all $x$ from $0$ to $2$.
Working out the area of a rectangle is easy: base times height. A triangle’s is too: half of that. The trouble begins when the figure is curved. There is no way to tile a circle with little squares without leaving or overshooting a piece. For two thousand years mathematicians attacked this problem with the same idea: approximate the curve by straight figures that get smaller and smaller. What changed was the way of writing that idea.