graceful, and implies that he himself began to know about it rather late in life. It had of course an
important bearing on the Pythagorean philosophy.
One of the most important consequences of the discovery of irrationals was the invention of the
geometrical theory of proportion by Eudoxus (ca. 408 -- ca. 355 B.C.). Before him, there was only
the arithmetical theory of proportion. According to this theory, the ratio of a to b is equal to the
ratio of c to d if a times d is equal to b times c. This definition, in the absence of an arithmetical
theory of irrationals, is only applicable to rationals. Eudoxus, however, gave a new definition not
subject to this restriction, framed in a manner which suggests the methods of modern analysis.
The theory is developed in Euclid, and has great logical beauty.
Eudoxus also either invented or perfected the "method of exhaustion," which was subsequently
used with great success by Archimedes. This method is an anticipation of the integral calculus.
Take, for example, the question of the area of a circle. You can inscribe in a circle a regular
hexagon, or a regular dodecagon, or a regular polygon of a thousand or a million sides. The area
of such a polygon, however many sides it has, is proportional to the square on the diameter of the
circle. The more sides the polygon has, the more nearly it becomes equal to the circle. You can
prove that, if you give the polygon enough sides, its area can be got to differ from that of the circle
by less than any previously assigned area, however small. For this purpose, the "axiom of
Archimedes" is used. This states (when somewhat simplified) that if the greater of two quantities
is halved, and then the half is halved, and so on, a quantity will be reached, at last, which is less
than the smaller of the original two quantities. In other words, if a is greater than b, there is some
whole number n such that 2n times b is greater than a.
The method of exhaustion sometimes leads to an exact result, as in squaring the parabola, which
was done by Archimedes; sometimes, as in the attempt to square the circle, it can only lead to
successive approximations. The problem of squaring the circle is the problem of determining the
ratio of the circumference of a circle to the diameter, which is called π. Archimedes used the
approximation 22/7 in calculations; by inscribing and circumscribing a regular polygon of 96
sides, he proved that π is less than 3 1/7 and greater