2.1 Avoiding Infinite Summations

Figure 2.1 Approximating triangles below

The triangle with vertices at (±1,0) and (0,1) has area 1. The two triangle that lie above this and have vertices at (±1/2,3/4) have a combined area of 1/4. If we put four triangles above these two, adding vertices at (±1/4,15/16) and (±3/4,7/16), then these four triangles will add a combined area of 1/16. In general, what Archimedes showed is that no matter how many triangles we have placed inside this region, we can put in two new triangles for each one we just inserted and increase the total area by one-quarter of the amount by which we last increased it.

As we take more triangles, we get successive approximations to the total area:

Archimedes then makes the observation that each of these sums brings us closer to 4/3:

(2.1.1)
A modern reader is inclined to make the jump to an infinite summation at this point and say that the actual area is

This is precisely what Archimedes did not do. He proceeded very carefully. Imitating him, we let K denote the area to be calculated. Archimedes showed that K could not be larger than 4/3 nor less than 4/3.

His argument, shown on the next webpage, is important because it points to what eventually would become the definition of the infinite series though this definition would not be firmly established until the early nineteenth century.

In the seventeenth and eighteenth centuries, there was a free-wheeling style in which it appeared that sceintists treated infinite series as finite summations with a very large number of terms. In fact, scientists of this time were very aware of the distinction between series with a large number of summands and infinite series. They knew you could get into serious trouble if you did not make this distinction. But they also knew that treating infinite series as if they really were summations led to useful insights such as the fact that the integral of a power series could be found by integrating each term, just as in a finite summation. They developed a sense for what was and was not legitimate. But by the early 1800s, the sense for what should and should not work was proving insufficient, as exemplified by the strange behavior of Fourier's trigonometric series. Cauchy and others returned to Archimedes' example of how to handle such series.