Some problems can not be solved using elegant, intuitive methods. Some problems can only be solved with ugly, brute force. Yet, even in the solving of problems through the use of force rather than finesse, there is a glimmer of beauty. The method of exhaustions, credited to Eudoxus, is one such kernel, which Leibnitz and Newton nurtured into a rose.

The question is simple. What is the formula for the volume of a pyramid? The answer is just as simple. 1/3(height of pyramid)(Area of base). Unfortunately, the path between is more difficult than passage up the Congo.

One's first instinct may be to cut up the pyramid, and reassemble the pieces to construct a figure for which the volume is already known. It works well with triangles, after all. Unfortunately, it is a little more difficult to do with pyramids.

Then one might try to work it out using proportions and side lengths, but this would meet with no more success. Once again, the Greeks had stuck upon a problem beyond the bounds of the straight edge and compass.

What then? The answer lies in the work of Eudoxus, who is generally credited with formalizing the method of exhaustions. Hippocrates and some other mathematicians are given credit for observing the result, but without proof, the observations were worthless.

The basic premise of the method is as follows:

Given a big thing and a small thing, take away more than half of the big thing. Then take away more than half of what is left. Do it again, and again, as long as you can. Then eventually, what is left of the big one will be smaller than the small thing.

Thus, Eudoxus introduced the concept of infinite series and limits. By combining this method with Bryson's "squeeze," the volume of a pyramid can be determined. Begin by inscribing a triangular prism inside the pyramid. The height of the first should be 1/2 the height of the pyramid(H). Then the area of the base of the prism is needed. In this first case, it can the base is a triangle with a base 1/2 as long as the original base(b), and a "height" 1/2 the original triangle's height(h). Thus the volume of the first prism is 1/2H[1/2(1/2h*1/2b)], or 1/16 Hbh. Then, new prisms are constructed in the gaps left by the old prisms in a similar fashion, endlessly. By adding all the volumes up, we end up with an "infinite series."

The same method can be performed using prisms that are bigger than the tetrahedron(3-sided pyramid) and the end result is the same. And, since a pyramid with any number of sides can be divided up into a tetrahedrons, the volume of all pyramids is equal to 1/3 height times base. A cone can be viewed as a pyramid with an infinite number of sides, so it also fits the formula.

And Eudoxus did this almost 2000 years ago. That is awesome.

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