THE STRUCTURE OF ARCHIMEDES’ ON SPIRALS

One may be forgiven for considering this, On Spirals, to be Archimedes’ finest. The figures bend and balance as the argument reaches – effortlessly, quickly, and yet, how, one cannot quite grasp – towards several magnificent results. These suggest no less than the squaring of the circle: first, a certain line (defined by a tangent to the circle) is equal to the circumference of the circle; second, a certain area is equal to the circle's third.

We are witnesses to Archimedes in action, as he engaged in a campaign of publications. At some early date, we are told in this treatise, he sent out via his mathematician friend Conon a complex geometrical challenge containing many claims. He had gradually discharged this challenge. Previously, he had sent to Dositheus the two books On the Sphere and the Cylinder (following on the Quadrature of the Parabola, which contained results independent from the original challenge sent via Conon). Now, he sends out On Spirals. This, once again, is sent to Dositheus. Archimedes once again proves some of the claims contained in that letter to Conon; he also reflects, briefly, on that geometrical challenge as a whole.

In this treatise, Archimedes promises to find not two, but four results. One of them is the result on the tangent mentioned above (being equal to the circumference of the circle). The result on the area of the spiral (being one-third the circle enclosing it) is proved and then further expanded to two extra, inherently interesting results, showing the ratios between the entire shells of spirals enclosing each other as well as the ratios of fragments of shells enclosing each other.