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 pre-eminence 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 simple rules, 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 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, incidentally, much admirable work. The problem is, of course, that of determining the cube root of 2.

The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the a's and the b's; each starts with 1. The next a, at each stage, is formed by adding the last a and b already obtained; the next b is formed by adding twice the previous a to the previous b. The first 6 pairs so obtained are (1, 1), (2, 3), (5, 7), (12, 17), (29, 41), (70, 99). In each pair, 2a2 – b2 is 1 or –1. Thus b/a is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisfy himself that the square of 99/70 is very nearly equal to 2.

Pythagoras—always a rather misty figure—is described by Proclus as the first who made geometry a liberal education. Many authorities, including Sir Thomas Heath,1 believe that he probably discovered the theorem that bears his name, to the effect that, in a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. In any case, this theorem was known to the Pythagoreans at a very early date. They knew also that the sum of the angles of a triangle is two right angles.

Irrationals other than the square root of two were studied, in particular cases by Theodorus, a contemporary of Socrates, and in a more general way by Theaetetus, who was roughly contemporary with Plato, but somewhat older. Democritus wrote a treatise on irrationals, but very little is known as to its contents. Plato was profoundly interested in the subject; he mentions the work of Theodorus and Theaetetus in the dialogue called after the latter. In the Laws (819–820), he says that the general ignorance on this subject is disgraceful, 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 . 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, 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 even a system of numeration based upon it. The Greeks were fond of attributing the wisdom of their pioneers to travels in Egypt, but what had really been achieved before the Greeks was very little. The prediction of an eclipse by Thales was, however, an example of foreign influence; there is no reason to suppose that he added anything to what he learnt from Egyptian or Babylonian sources, and it was a stroke of luck that his prediction was verified.

Let us begin with some of the earliest discoveries and correct hypotheses. Anaximander thought that the earth floats freely, and is not supported on anything. Aristotle,2 who often rejected the best hypotheses of his time, objected to the theory of Anaximander, that the earth, being at the centre, remained immovable because there was no reason for moving in one direction rather than another. If this were valid, he said, a man placed at the centre of a circle with food at various points of the circumference would starve to death for lack of reason to choose one portion of food rather than another. This argument reappears in scholastic philosophy, not in connection with astronomy, but with free will. It reappears in the form of 'Buridan's ass', which was unable to choose between two bundles of hay placed at equal distances to right and left, and therefore died of hunger.

Pythagoras, in all probability, was the first to think the earth spherical, but his reasons were (one must suppose) aesthetic rather than scientific. Scientific reasons, however, were soon found. Anaxagoras discovered that the moon shines by reflected light, and gave the right theory of eclipses. He himself still thought the earth flat, but the shape of the earth's shadow in lunar eclipses gave the Pythagoreans conclusive arguments in favour of its being spherical. They went further, and regarded the earth as one of the planets. They knew—from Pythagoras himself, it is said—that the morning star and the evening star are identical, and they thought that all the planets, including the earth, move in circles, not round the sun, but round the 'central fire'. They had discovered that the moon always turns the same face to the earth, and they thought that the earth always turns the same face to the 'central fire'. The Mediterranean regions were on the side turned away from the central fire, which was therefore always invisible. The central fire was called 'the house of Zeus', or 'the Mother of the gods'. The sun was supposed to shine by light reflected from the central fire. In addition to the earth, there was another body, the counter-earth, at the same distance from the central fire. For this, they had two reasons, one scientific, one derived from their arithmetical mysticism. The scientific reason was the correct observation that an eclipse of the moon sometimes occurs when both sun and moon are above the horizon. Refraction, which is the cause of this phenomenon, was unknown to them, and they thought that, in such cases, the eclipse must be due to the shadow of a body other than the earth. The other reason was that the sun and moon, the five planets, the earth and counter-earth, and the central fire, made ten heavenly bodies, and ten was the mystic number of the Pythagoreans.

This Pythagorean theory is attributed to Philolaus, a Theban, who lived at the end of the fifth century B.C. Although it is fanciful and in part quite unscientific, it is very important, since it involves the greater part of the

imaginative effort required for conceiving the Copernican hypothesis. To conceive of the earth, not as the centre of the universe, but as one among the planets, not as eternally fixed, but as wandering through space, showed an extraordinary emancipation from anthropocentric thinking. When once this jolt had been given to men's natural picture of the universe, it was not so very difficult to be led by scientific arguments to a more accurate theory.

To this various observations contributed. Oenopides, who was slightly later than Anaxagoras, discovered the obliquity of the ecliptic. It soon became clear that the sun must be much larger than the earth, which fact supported those who denied that the earth is the centre of the universe. The central fire and the counter-earth were dropped by the Pythagoreans soon after the time of Plato. Heraclides of Pontus (whose dates are about 388 to 315 B.C., contemporary with Aristotle) discovered that Venus and Mercury revolve about the sun, and adopted the view that the earth rotates on its own axis once every twenty-four hours. This last was a very important step, which no predecessor had taken. Heraclides was of Plato's school, and must have been a great man, but was not as much respected as one would expect; he is described as a fat dandy.

Aristarchus of Samos, who lived approximately from 310 to 230 B.C., and was thus about twenty-five years older than Archimedes, is the most interesting of all ancient astronomers, because he advanced the complete Copernican hypothesis, that all the planets, including the earth, revolve in circles round the sun, and that the earth, rotates on its axis once in twenty-four hours. It is a little disappointing to find that the only extant work of Aristarchus, On the Sizes and Distances of the Sun and the Moon, adheres to the geocentric view. It is true that, for the problems with which this book deals, it makes no difference which theory is adopted, and he may therefore have thought it unwise to burden his calculations with an unnecessary opposition to the general opinion of astronomers; or he may have only arrived at the Copernican hypothesis after writing this book. Sir Thomas Heath, in his work on Aristarchus,3 which contains the text of this book with a translation, inclines to the latter view. The evidence that Aristarchus suggested the Copernican view is, in any case, quite conclusive.

The first and best evidence is that of Archimedes, who, as we have seen, was a younger contemporary of Aristarchus. Writing to Gelon, King of Syracuse, he says that Aristarchus brought out 'a book consisting of certain hypotheses', and continues: 'His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit.' There is a passage

in Plutarch saying that Cleanthes 'thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe (i.e. the earth), this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis'. Cleanthes was a contemporary of Aristarchus, and died about 232 B.C. In another passage, Plutarch says that Aristarchus advanced this view only as a hypothesis, but that his successor Seleucus maintained it as a definite opinion. (Seleucus flourished about 150 B.C.) Aëtius and Sextus Empiricus also assert that Aristarchus advanced the heliocentric hypothesis, but do not say that it was set forth by him only as a hypothesis. Even if he did so, it seems not unlikely that he, like Galileo two thousand years later, was influenced by the fear of offending religious prejudices, a fear which the attitude of Cleanthes (mentioned above) shows to have been well grounded.

The Copernican hypothesis, after being advanced, whether positively or tentatively, by Aristarchus, was definitely adopted by Seleucus, but by no other ancient astronomer. This general rejection was mainly due to Hipparchus, who flourished from 161 to 126 B.C. He is described by Heath as 'the greatest astronomer of antiquity'.4 He was the first to write systematically on trigonometry; he discovered the precession of the equinoxes; he estimated the length of the lunar month with an error of less than one second; he improved Aristarchus's estimates of the sizes and distances of the sun and moon; he made a catalogue of eight hundred and fifty fixed stars, giving their latitude and longitude. As against the heliocentric hypothesis of Aristarchus, he adopted and improved the theory of epicycles which had been invented by Apollonius, who flourished about 220 B.C.; it was a development of this theory that came to be known, later, as the Ptolemaic system, after the astronomer Ptolemy, who flourished in the middle of the second century A.D.

Copernicus perhaps came to know something, though not much, of the almost forgotten hypothesis of Aristarchus, and was encouraged by finding ancient authority for his innovation. Otherwise, the effect of this hypothesis on subsequent astronomy was practically nil.

Ancient astronomers, in estimating the sizes of the earth, moon, and sun, and the distances of the moon and sun, used methods which were theoretically valid, but they were hampered by the lack of instruments of precision. Many of their results, in view of this lack, were surprisingly good. Eratosthenes estimated the earth's diameter at 7,850 miles, which is only about fifty miles short of the truth. Ptolemy estimated the mean distance of

the moon at 29 1/2 times the earth's diameter; the correct figure is about 30.2. None of them got anywhere near the size and distance of the sun, which all under-estimated. Their estimates, in terms of the earth's diameter, were:

Aristarchus, 180;

Hipparchus, 1,245;

Posidonius, 6,545.

The correct figure is 11,726. It will be seen that these estimates continually improved (that of Ptolemy, however, showed a retrogression); that of Posidonius5 is about half the correct figure. On the whole, their picture of the solar system was not so very far from the truth.

Greek astronomy was geometrical, not dynamic. The ancients thought of the motions of the heavenly bodies as uniform and circular, or compounded of circular motions. They had not the conception of force. There were spheres which moved as a whole, and on which the various heavenly bodies were fixed. With Newton and gravitation a new point of view, less geometrical, was introduced. It is curious to observe that there is a reversion to the geometrical point of view in Einstein's General Theory of Relativity, from which the conception of force, in the Newtonian sense, has been banished.

The problem for the astronomer is this: given the apparent motions of the heavenly bodies on the celestial sphere, to introduce, by hypothesis, a third co-ordinate, depth, in such a way as to make the description of the phenomena as simple as possible. The merit of the Copernican hypothesis is not truth, but simplicity; in view of the relativity of motion, no question of truth is involved. The Greeks, in their search for hypotheses which would 'save the phenomena', were in effect, though not altogether in intention, tackling the problem in the scientifically correct way. A comparison with their predecessors, and with their successors until Copernicus, must convince every student of their truly astonishing genius.

Two very great men, Archimedes and Apollonius, in the third century B.C., complete the list of first-class Greek mathematicians. Archimedes was a friend, probably a cousin, of the king of Syracuse, and was killed when that city was captured by the Romans in 212 B.C. Apollonius, from his youth, lived at Alexandria. Archimedes was not only a mathematician, but also a physicist and student of hydrostatics. Apollonius is chiefly noted for his work on conic sections. I shall say no more about them, as they came too late to influence philosophy.

After these two men, though respectable work continued to be done in Alexandria, the great age was ended. Under the Roman domination, the Greeks lost the self-confidence that belongs to political liberty, and in losing it acquired a paralysing respect for their predecessors. The Roman soldier who killed Archimedes was a symbol of the death of original thought that Rome caused throughout the Hellenic world.

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