Adding Up Infinitely Many Nothings

There is no formula for the area under a curve, because the curve has no straight edges and every area formula you were taught assumes straight edges.

So cheat. Fill the space with rectangles. Each rectangle has a flat top and the curve does not, so you miss thin slivers along the top of every one, and your answer is wrong by the total of the slivers. Use narrower rectangles. There are now more slivers and each is much smaller, and the total error has gone down rather than up.

Keep going and the errors shrink towards nothing. The area is what the sums approach.

That is the integral, and it is the same construction as the previous chapter pointed at a different problem: a sequence of honest approximations, and a limit they converge on. The slope needed intervals to shrink; the area needs slices to.

Archimedes did this in the third century BC and the method deserves its name.

To find the area of a circle he drew a polygon inside it and a polygon outside it. The inner one is definitely smaller than the circle and the outer one is definitely bigger, so the circle's area is trapped between two numbers he could compute. Then he doubled the number of sides. The gap narrowed. He got to ninety-six sides and squeezed π between 3 + 10/71 and 3 + 1/7, which is correct to two decimal places and arrived at without decimals, without algebra, and without the concept of a limit.

The method of exhaustion is what this is called, and it is the integral with no notation and no general theory. Each problem had to be solved from scratch by someone willing to do an enormous amount of arithmetic.

His result on the parabola is the better one and the less famous.

He found the exact area of a parabolic segment, which is harder than the circle, and the way he found it is the interesting part. He imagined the region cut into slices and balanced on a lever against slices of a shape he already understood, and read the answer off the balance point. It is a physical argument, and it gave him the answer.

He then threw it away and wrote an exhaustion proof instead, because he did not consider the lever to be a proof.

That distinction is worth more than the theorem. Archimedes had a method for finding things and a method for establishing them, he knew they were different, and he published the second while privately relying on the first. The lever argument survives because a copy of the text was rediscovered in 1906 on a palimpsest, written over by a prayer book. Without that accident we would have the proofs and no idea how he thought.

The general theory he lacked is the subject of the next chapter, and it is the reason nobody has to repeat his labour. He computed one area with a lifetime's ingenuity. The fundamental theorem makes most such areas a two-line calculation, and that gap between genius and procedure is what a good piece of mathematics buys.