A lot of people have asked me to write something about “Archimedes Integration”, and I’m finally getting around to fulfilling that request.
As most of you already know, Archimedes was a philosopher in ancient Greece who, among other things, studied mathematics. He invented a technique for computing areas that’s the closest thing to calculus before Newton and Leibniz. Modern mathematicians call Archimedes technique “the method of exhaustion”.
The basic idea of the method of exhaustion is to take the figure whose area you want to compute, and to divide it into pieces whose area you already know how to compute; and to make the divisions smaller and smaller, *exhausting* the area not included.
For a simple example, we can start with a circle, and inscribe it with *equilateral* triangles that run from the center to the edge of the circle. For each of those triangles, the length of the base is the radius *r* of the circle; the height of the triangle is *r×(sqrt(3)/2)*. So the area of one triangle is *(1/2)(sqrt(3)/2)r2; and the approximate area of the circle with six triangles inscribed is *6(1/4)(sqrt(3)r2 = (3sqrt(3)/2)r2* or roughly 2.6r2. Not bad, but not great. Now, Archimedes wouldn’t have said it quite like that. You see, the Greeks didn’t use equational reasoning; they worked in terms of comparisons and ratios. So Archimedes wouldn’t say “1/4 r2“; he’d figure out a single triangle whose area was equivalent to the sums of the 6 triangles – a triangle whose height was the same as the radius of the circle, and whose base was *3sqrt(3)r*. The following diagram shows the circle with triangles inscribed, and the triangle with an area equal to the sum of the inscribed triangles area drawn over the triangle with area equal to the circle.
Now, moving on, suppose we made the triangles smaller. Cut the angle at the center of the circle in half, so that we had twice as many triangles, each with a central angle of 30 degrees rather than 60. Then our approximation would be (1/4)r2; with 12 triangles, that would give us 3r2; or an area equal to a triangle whose height was the radius, and whose width was 6, as in the following diagram:
Suppose we kept making the triangles smaller. Each time we reduce the size of the triangles, we remove (exhaust) more of the uncovered part of the circle; and length of the triangle whose area equals the sum of the areas of the inscribed triangles gets closer and closer to *2πr*. Or in the terms Archimedes would have used: the length of the base gets closer and closer to the radius of the circle. So the area of the circle gets closer and closer to the area of a triangle whose height is the radius of the circle, and whose width is circumference of the circle. So in equational terms, we end up with *(1/2)(2πr)(r) = πr2*.
The Greek mathematicians had no *explicit* concept of limits or how to work with them. But you can clearly see the origin of the concepts of limits here. And Archimedes didn’t stop there. He went on
to improve the method – to take a figure, and *enclose* the figure in triangles as well as inscribe triangles inside the figure. The area of the triangles *enclosing* the figure is an upper bound; the area *inscribed* in the figure is a lower bound; and the average of the two is a pretty good estimate.
The way that Archimedes did it – basing it on triangles and ratios – is obviously a very limited technique. But it’s very impressive for reasoning without the tools of algebra, and it *does* work on a good number of very regular figures.