History of Western Philosophy - Bertrand Russell [125]
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 . Archimedes used the approximation in calculations; by inscribing and circumscribing a regular polygon of 96 sides, he proved that is less than and greater than . 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 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 in 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 nineteenth-century 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's 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. Suddenly the work that the Greeks had done from pure love of theory became the key to warfare and astronomy.
The Romans were too practical-minded to appreciate Euclid; the first of them to mention him is Cicero, in whose time there was probably no Latin translation; indeed there is no record of any Latin translation before Boethius (ca. A.D. 480). The Arabs were more appreciative: a copy was given to the caliph by the Byzantine emperor about A.D. 760, and a translation into Arabic was made under Harun al Rashid, about A.D. 800. The first still extant Latin translation was made from the Arabic by Adelard of Bath in A.D. 1120. From that time on, the study of geometry gradually revived in the West; but it was not until the late Renaissance that important advances were made.
I come now to astronomy, where Greek achievements were as remarkable as in geometry. Before their time, among the Babylonians and Egyptians, many centuries of observation had laid a foundation. The apparent motions of the planets had been recorded, but it was not known that the morning and evening star were the same. A cycle of eclipses had been discovered,