The History of Mathematics: A Brief Course

3. ARCHIMEDES 301

References by Archimedes himself and other mathematicians tell of the exis-

tence of other works by Archimedes of which no manuscripts are now known to

exist. These include works on the theory of balances and levers, optics, the regular

polyhedra, the calendar, and the construction of mechanical representations of the

motion of heavenly bodies. In 1998 a palimpsest of Archimedes' work was sold at

auction for $2 million (see Plate 6).

From this list we can see the versatility of Archimedes. His treatises on the

equilibrium of planes and floating bodies contain principles that are now fundamen-

tal in mechanics and hydrostatics. The works on the quadrature of the parabola,

conoids and spheroids, the measurement of the circle, and the sphere and cylinder

extend the theory of proportion, area, and volume found in Euclid for polyhedra

and polygons to the more complicated figures bounded by curved lines and surfaces.

The work on spirals introduces a new class of curves, and develops the theory of

length, area, and proportion for them.

Since we do not have space to discuss all of Archimedes' geometry, we shall

confine our discussion to what may be his greatest achievement: finding the surface

area of a sphere. In addition, because of its impact on the issues involving proof

that we have been discussing, we shall discuss his Method.

3.1. The area of a sphere. Archimedes' two works on the sphere and cylinder

were sent to Dositheus. In the letter accompanying the first of these he gives some

of the history of the problem. Archimedes considered his results on the sphere to

be rigorously established, but he did have one regret:

It would have been beneficial to publish these results when Conon

was alive, for he is the one we regard as most capable of under-

standing and rendering a proper judgment on them. But, as we

think it well to communicate them to the initiates of mathemat-

ics, we send them to you, having rewritten the proofs, which those

versed in the sciences may scrutinize.

The fact that a pyramid is one-third of a prism on the same base and altitude

is Proposition 7 of Book 12 of Euclid's Elements. Thus Archimedes could say

confidently that this theorem was well established. Archimedes sought the surface

area of a sphere by finding the lateral surface area of a frustum of a cone and the

lateral area of a right cylinder. In our terms the area of a frustum of a cone with

upper radius r, lower radius R, and side of slant height h is nh(R + r). Archimedes

phrased this fact by saying that the area is that of a circle whose radius is the mean

proportional between the slant height and the sum of the two radii; that is, the

radius is \Jh{R + r). Likewise, our formula for the lateral surface area of a cylinder

of radius r and height h is 2irrh. Archimedes said it was the area of a circle whose

radius is the mean proportional between the diameter and height of the cylinder.

These results can be applied to the figures generated by revolving a circle about

a diameter with certain chords drawn. Archimedes showed (Proposition 22) that

(BB' + CC + -- + KK' + LM) : AM = A'Β : ΒΑ

in Fig. 18.

This result is easily derived by connecting B' to C, C to K, and K' to L and

considering the ratios of the legs of the resulting similar triangles. These ratios can

be added. All that then remains is to cross-multiply this proportion and use the

expressions already derived for the area of a frustum of a cone. One finds easily