two great inquiries which seem to share between them the domain of geometry. Apollonius is concerned with the Geometry of Forms and Situations, while in Archimedes we find the Geometry of Measurements, dealing with the quadrature of curvilinear plane figures and with the quadrature and cubature of curved surfaces, investigations which gave birth to the calculus of the infinite conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and Newton.

In geometry Archimedes stands, as it were, on the shoulders of Eudoxus in that he applied the method of exhaustion to new and more difficult cases of quadrature and cubature. Further, in his use of the method he introduced an interesting variation of the procedure as we know it from Euclid. Euclid (and presumably Eudoxus also) only used inscribed figures, “exhausting” the figure to be measured, and had to invert the second half of the reductio ad absurdum to enable approximation from below (so to speak) to be applied in that case also.

Archimedes, on the other hand, approximates from above as well as from below; he approaches the area or volume to be measured by taking closer and closer circumscribed figures, as well as inscribed, and thereby compressing, as it were, the inscribed and circumscribed figure into one, so that they ultimately coincide with one another and with the figure to be measured. But he follows the cautious method to which the Greeks always adhered; he never says that a given curve or surface is the limiting form of the inscribed or circumscribed figure; all that he asserts is that we can approach the curve or surface as nearly as we please.

The deductive form of proof by the method of exhaustion is apt to obscure not only the way in which the results were arrived at but also the real character of the procedure followed. What Archimedes actually does in certain cases is to perform what are seen, when the analytical equivalents