VI.3 Archimedes

b. Siracusa, Magna Graecia (now Syracuse, Italy), ca. 287 B.C.E.; d. Siracusa, 212 B.C.E.

Area of the circle; centers of gravity, method of exhaustion, volume of the sphere

Archimedes’ life was as spectacular as his scientific achievements: various sources attest that he built a ship, a cosmological model, and magnificent catapults with which he defended his native Syracuse during the Second Punic War. The Roman besiegers eventually took the city by deceit, and Archimedes was killed in the ensuing pillage. According to legend, his tomb was engraved with a sphere inscribed in a cylinder, to mark one of his most famous discoveries. Indeed, the first part of his Sphere and Cylinder reaches a climax with a proof that the volume of every sphere is two thirds of that of the cylinder circumscribing it. Archimedes’ interest in establishing the volume or area of curved figures is also attested by his discovery of the area of the circle and of a sphere, and by treatises on spiral curves, conoids, and paraboloids, and on the Quadrature of the Parabola.

While following an axiomatic-deductive framework, Archimedes’ style is distinctive. Many of his theorems about curved figures use the so-called METHOD OF EXHAUSTION [II.6 §2].

Take the problem of determining the area of a circle. Archimedes accomplished this by showing that it had the same area as that of a certain right-angled triangle. Since it was known how to calculate the area of a triangle, he was “reducing” a problem whose solution was unknown to one whose solution was known. Rather than establishing this directly, he proves that the area of the circle can be neither larger than nor smaller than the area of the triangle, so that only one possibility remains: that they are equal. This is achieved, here and in general, by inscribing and circumscribing rectilinear figures to the curvilinear figure under investigation, thus getting closer and closer to it. The leap from closer and closer approximation to equivalence of a rectilinear and curvilinear figure, however, can be accomplished only indirectly, by excluding the other possibilities. Such arguments usually employ a lemma, already found in Euclid, to the effect that if we start with a quantity and replace it by a quantity at most half as large, and then repeat this, then what remains can be made as small as we please.

Archimedes’ output also includes The Sand-Reck-oner, about astronomy and arithmetic, and works on the centers of gravity of plane figures and on bodies immersed in a fluid.

Above all, Archimedes provides unique insights into the processes of ancient Greek mathematics. The second part of Sphere and Cylinder contains problems about constructing given solid bodies. Several of the proofs are in two parts: analysis and synthesis. In the analysis, the result one wants to establish is taken as proved, and consequences are drawn from it, until one hits upon a result that is already proved elsewhere, and the process is then reconstituted in reverse (the “synthesis”). The recently rediscovered Method (addressed to Eratosthenes) reveals that Archimedes arrived at some of his most famous results, e.g., the area of a segment of a parabola, by imagining the two objects involved (say, a segment of a parabola and a triangle) as divided up into an infinite number of slices and lines, then placed at the two ends of a balance and set in equilibrium with each other. Archimedes underlined that this heuristic procedure was not a strict proof, but that only makes the Method all the more valuable a glimpse into the mind of a great mathematician.

Further Reading

Archimedes. 2004. The Works of Archimedes: Translation and Commentary. Volume 1: The Two Books On the Sphere and the Cylinder, edited and translated by R. Netz. Cambridge: Cambridge University Press.

Dijksterhuis, E. J. 1987. Archimedes, with a bibliographical essay by W. R. Knorr. Princeton, NJ: Princeton University Press.

Serafina Cuomo