than 3 10/71. The method could be carried to any required degree of approximation, and that is all
that any method can do in this problem. The use of inscribed and circumscribed polygons for
approximations to π goes back to Antiphon, who was a contemporary of Socrates.
Euclid, who was still, when I was young, the sole acknowledged text-book of geometry for boys,
lived at Alexandria, about 300 B.C., a few years after the death of Alexander and Aristotle. Most
of his Elements was not original, but the order of propositions, and the logical structure, were
largely his. The more one studies geometry, the more admirable these are seen to be. The
treatment of parallels by means of the famous postulate of parallels has the twofold merit of rigour
in deduction and of not concealing the dubiousness of the initial assumption. The theory of
proportion, which follows Eudoxus, avoids all the difficulties connected with irrationals, by
methods essentially similar to those introduced by Weierstrass into nineteenthcentury analysis.
Euclid then passes on to a kind of geometrical algebra, and deals, in Book X, with the subject of
irrationals. After this he proceeds to solid geometry, ending with the construction of the regular
solids, which had been perfected by Theaetetus and assumed in Plato's Timaeus.
Euclid Elements is certainly one of the greatest books ever written, and one of the most perfect
monuments of the Greek intellect. It has, of course, the typical Greek limitations: the method is
purely deductive, and there is no way, within it, of testing the initial assumptions. These
assumptions were supposed to be unquestionable, but in the nineteenth century non-Euclidean
geometry showed that they might be in part mistaken, and that only observation could decide
whether they were so.
There is in Euclid the contempt for practical utility which had been inculcated by Plato. It is said
that a pupil, after listening to a demonstration, asked what he would gain by learning geometry,
whereupon Euclid called a slave and said "Give the young man threepence, since he must needs
make a gain out of what he learns." The contempt for practice was, however, pragmatically
justified. No one, in Greek times, supposed that conic sections had any utility; at last, in the
seventeenth century, Galileo discovered that projectiles move in parabolas, and Kepler discovered
that planets move in ellipses. Sud-