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.

Archimedes and the circle hemmed in

Around 250 BC, in Syracuse, Archimedes drew a hexagon inside a circle and another hexagon outside it. The length of the circumference lies between the perimeters of the two. Then he doubled the number of sides: $12$, $24$, $48$, $96$. With the 96-sided polygons he arrived at

$$3\tfrac{10}{71}<\pi<3\tfrac17,$$

that is, $3.1408\ldots<\pi<3.1428\ldots$ The method, which seventeenth-century mathematicians called exhaustion, “exhausts” the difference between the curved figure and the straight ones. It is an idea of a limit, two thousand years before the word. But Archimedes had no script for it: each problem required a new argument, written in sentences and figures, and the passage to infinity was avoided by a double proof by contradiction.

A name for the number

For centuries the ratio of circumference to diameter had no symbol: it was described in words, “the quantity which, multiplied by the diameter, gives the circumference”. In 1706 the Welsh teacher William Jones used the Greek letter $\pi$, the initial of periphéreia, the periphery, the way round. Euler adopted the letter in his books and it became standard. Getting a name changed the number’s status: it stopped being a relation between two lengths and became an object that can be added, squared and put inside formulas, such as $A=\pi r^2$.

Kepler and the wine barrel

In 1613 Johannes Kepler married for the second time and bought wine for the wedding. The merchant measured the barrels’ volume by pushing a rod through the bung-hole to the opposite corner and reading the wet length. Kepler, who had already discovered the laws of planetary motion, found the method suspicious: barrels of different shapes could have the same diagonal and different volumes. In 1615 he published the Nova stereometria doliorum vinariorum, the “new measurement of the solids of wine barrels”, in which he cut barrels into extremely thin slices and added up their volumes.

Twenty years later the Italian Bonaventura Cavalieri systematised this idea with indivisibles: a plane figure is made of infinitely many lines, a solid of infinitely many surfaces. It was powerful and dangerous. Adding infinitely many things of zero thickness gives what?

The sum that becomes an integral

The answer came with Leibniz, and it was above all an answer of writing. Cut the region under a curve $y=f(x)$, between $a$ and $b$, into $n$ rectangles of width $\Delta x$. The approximate area is

$$A\approx f(x_1)\,\Delta x+f(x_2)\,\Delta x+\cdots+f(x_n)\,\Delta x=\sum_{k=1}^{n} f(x_k)\,\Delta x.$$

The thinner the rectangles, the better the approximation. In the limit, as the width tends to zero, the sum becomes

$$A=\int_a^b f(x)\,dx.$$

Look at the two formulas side by side. The structure is the same: a summation sign, a height $f$, a width. What changes is the letters. The Greek $\Sigma$, the letter S for “sum”, becomes the Latin $\int$, also an S, stretched out. The Greek $\Delta$, the letter D for “difference”, becomes the Latin $d$. Greek letters mark the finite, Latin letters mark the infinitely small, and the passage from one world to the other is drawn in the very change of alphabet. The old texts on this site called this mirabilis abbreviatio: the integral’s script remembers where it came from. Whoever looks at $\int f(x)\,dx$ still sees the rectangles of height $f(x)$ and width $dx$.

The rigour of this passage only arrived in the nineteenth century. Augustin Cauchy, in 1823, and Bernhard Riemann, in 1854, defined the integral as the limit of these sums, now called Riemann sums. Leibniz’s notation of 1675 did not have to change: it already had the right shape before the theory knew how to justify it.

Exercise: the area of a triangle by integration

The site’s old texts asked: why is the area of a triangle $\frac{B\cdot H}2$? Place the triangle with one vertex at the origin, base $B$ on the $x$-axis and the opposite vertex at $(B,H)$. The sloping side is the line $y=\frac HB x$. Then

$$A=\int_0^B \frac HB\,x\,dx=\frac HB\cdot\frac{B^2}2=\frac{B\cdot H}2.$$

The memorised formula becomes the result of a calculation.

Exercise: a Riemann sum by hand

Approximate $\int_0^1 x^2\,dx$ with $4$ rectangles of width $\frac14$, using the height at the right-hand side of each:

$$\frac14\left[\left(\tfrac14\right)^2+\left(\tfrac24\right)^2+\left(\tfrac34\right)^2+\left(\tfrac44\right)^2\right]=\frac14\cdot\frac{1+4+9+16}{16}=\frac{30}{64}\approx0.469.$$

The exact value is $\frac13\approx0.333$. With thinner rectangles the sum approaches $\frac13$: with $100$ rectangles it is about $0.338$.

Sources: Archimedes, Measurement of a Circle · Florian Cajori, A History of Mathematical Notations · Johannes Kepler, MacTutor · Bonaventura Cavalieri, MacTutor.

Chapter index · ← Leibniz against the dots: active and sterile notation · Euler and functions: giving a rule a name →