Finally: The view that the heavenly bodies are eternal and incorruptible has had to be abandoned. The sun and stars have long lives, but do not live for ever. They are born from a nebula, and in the end they either explode or die of cold. Nothing in the visible world is exempt from change and decay; the Aristotelian belief to the contrary, though accepted by medieval Christians, is a product of the pagan worship of sun and moon and planets.
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CHAPTER XXIV Early Greek Mathematics and Astronomy
I AM concerned in this chapter with mathematics, not on its own account, but as it was related to Greek philosophy—a relation which, especially in Plato, was very close. The preeminence of the Greeks appears more clearly in mathematics and astronomy than in anything else. What they did in art, in literature, and in philosophy, may be judged better or worse according to taste, but what they accomplished in geometry is wholly beyond question. They derived something from Egypt, and rather less from Babylonia; but what they obtained from these sources was, in mathematics, mainly rules of thumb, and in astronomy records of observations extended over very long periods. The art of mathematical demonstration was, almost wholly, Greek in origin.
There are many pleasant stories, probably unhistorical, showing what practical problems stimulated mathematical investigations. The earliest and simplest relates to Thales, who, when in Egypt, was asked by the king to find out the height of a pyramid. He waited for the time of day when his shadow was as long as he was tall; he then measured the shadow of the pyramid, which was of course equal to its height. It is said that the laws of perspective were first studied by the geometer Agatharcus, in order to paint scenery for the plays of Aeschylus. The problem of finding the distance of a ship at sea, which was said to have been studied by Thales, was correctly solved at an early stage. One of the great problems that occupied Greek geometers, that of the duplication of the cube, originated, we are told, with the priests of a certain temple, who were informed by the oracle that the god wanted a statue twice as large as the one they had. At first
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they thought simply of doubling all the dimensions of the statue, but then they realized that the result would be eight times as large as the original, which would involve more expense than the god had demanded. So they sent a deputation to Plato to ask whether anybody in the Academy could solve their problem. The geometers took it up, and worked at it for centuries, producing, incidentall y, much admirable work. The problem is, of course, that of determinin g the cube
* G r e e k M a t h e m a ti c s, V ol . I, p. 1 4 5.
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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
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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-
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denly 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 Athelhard 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, certainly in Babylonia and probably in Egypt, which made the prediction of lunar eclipses fairly reliable, but not of solar eclipses, since those were not always visible at a given spot. We owe to the Babylonians the division of the right angle into ninety degrees, and of the degree into sixty minutes; they had a liking for the number sixty, and
