narrow strips and have a saw-like edge; all the elements are trapezia except two, which are triangles, one in each figure. Each trapezium (or triangle) is weighed where it is against another area hung at a fixed point of an assumed lever; thus the whole of the inscribed and circumscribed figures respectively are weighed against the sum of an indefinite number of areas all suspended from one point on the lever. The result is obtained by a real integration, confirmed as usual by a proof by the method of exhaustion.
The geometrical proof proceeds thus. Drawing in the segment the inscribed triangle with the same base and height as the segment, Archimedes next inscribes triangles in precisely the same way in each of the segments left over, and proves that the sum of the two new triangles is ¼ of the original inscribed triangle. Again, drawing triangles inscribed in the same way in the four segments left over, he proves that their sum is ¼ of the sum of the