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A History of Greek Mathematics
Volume I From Thales to Euclid
By Thomas Heath Dover Publications, Inc.
Copyright © 1981 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-16269-0
CHAPTER 1
INTRODUCTORY
The Greeks and mathematics.
IT is an encouraging sign of the times that more and more effort is being directed to promoting a due appreciation and a clear understanding of the gifts of the Greeks to mankind. What we owe to Greece, what the Greeks have done for civilization, aspects of the Greek genius: such are the themes of many careful studies which have made a wide appeal and will surely produce their effect. In truth all nations, in the West at all events, have been to school to the Greeks, in art, literature, philosophy, and science, the things which are essential to the rational use and enjoyment of human powers and activities, the things which make life worth living to a rational human being. 'Of all peoples the Greeks have dreamed the dream of life the best.' And the Greeks were not merely the pioneers in the branches of knowledge which they invented and to which they gave names. What they began they carried to a height of perfection which has not since been surpassed; if there are exceptions, it is only where a few crowded centuries were not enough to provide the accumulation of experience required, whether for the purpose of correcting hypotheses which at first could only be of the nature of guesswork, or of suggesting new methods and machinery.
Of all the manifestations of the Greek genius none is more impressive and even awe-inspiring than that which is revealed by the history of Greek mathematics. Not only are the range and the sum of what the Greek mathematicians actually accomplished wonderful in themselves; it is necessary to bear in mind that this mass of original work was done in an almost incredibly short space of time, and in spite of the comparative inadequacy (as it would seem to us) of the only methods at their disposal, namely those of pure geometry, supplemented, where necessary, by the ordinary arithmetical operations. Let us, confining ourselves to the main subject of pure geometry by way of example, anticipate so far as to mark certain definite stages in its development, with the intervals separating them. In Thales's time (about 600 B.C.) we find the first glimmerings of a theory of geometry, in the theorems that a circle is bisected by any diameter, that an isosceles triangle has the angles opposite to the equal sides equal, and (if Thales really discovered this) that the angle in a semicircle is a right angle. Rather more than half a century later Pythagoras was taking the first steps towards the theory of numbers and continuing the work of making geometry a theoretical science; he it was who first made geometry one of the subjects of a liberal education. The Pythagoreans, before the next century was out (i. e. before, say, 450 B.C.), had practically completed the subject-matter of Books I–II, IV, VI (and perhaps III) of Euclid's Elements, including all the essentials of the 'geometrical algebra' which remained fundamental in Greek geometry; the only drawback was that their theory of proportion was not applicable to incommensurable but only to commensurable magnitudes, so that it proved inadequate as soon as the incommensurable came to be discovered. In the same fifth century the difficult problems of doubling the cube and trisecting any angle, which are beyond the geometry of the straight line and circle, were not only mooted but solved theoretically, the former problem having been first reduced to that of finding two mean proportionals in continued proportion (Hippocrates of Chios) and then solved by a remarkable construction in three dimensions (Archytas), while the latter was solved by means of the curve of Hippias of Elis known as the quadratrix; the problem of squaring the circle was also attempted, and Hippocrates, as a contribution to it, discovered and squared three out of the five lunes which can be squared by means of the straight line and circle. In the fourth century Eudoxus discovered the great theory of proportion expounded in Euclid, Book V, and laid down the principles of the method of exhaustion for measuring areas and volumes; the conic sections and their fundamental properties were discovered by Menaechmus; the theory of irrationals (probably discovered, so far as [square root of 2] is concerned, by the early Pythagoreans) was generalized by Theaetetus; and the geometry of the sphere was worked out in systematic treatises. About the end of the century Euclid wrote his Elements in thirteen Books. The next century, the third, is that of Archimedes, who may be said to have anticipated the integral calculus, since, by performing what are practically integrations, he found the area of a parabolic segment and of a spiral, the surface and volume of a sphere and a segment of a sphere, the volume of any segment of the solids of revolution of the second degree, the centres of gravity of a semicircle, a parabolic segment, any segment of a paraboloid of revolution, and any segment of a sphere or spheroid. Apollonius of Perga, the 'great geometer', about 200 B.C., completed the theory of geometrical conics, with specialized investigations of normals as maxima and minima leading quite easily to the determination of the circle of curvature at any point of a conic and of the equation of the evolute of the conic, which with us is part of analytical conics. With Apollonius the main body of Greek geometry is complete, and we may therefore fairly say that four centuries sufficed to complete it.
But some one will say, how did all this come about? What special aptitude had the Greeks for mathematics? The answer to this question is that their genius for mathematics was simply one aspect of their genius for philosophy. Their mathematics indeed constituted a large part of their philosophy down to Plato. Both had the same origin.
Conditions favouring the development of philosophy among the Greeks.
All men by nature desire to know, says Aristotle. The Greeks, beyond any other people of antiquity, possessed the love of knowledge for its own sake; with them it amounted to an instinct and a passion. We see this first of all in their love of adventure. It is characteristic that in the Odyssey Odysseus is extolled as the hero who had 'seen the cities of many men and learned their mind', often even taking his life in his hand, out of a pure passion for extending his horizon, as when he went to see the Cyclopes in order to ascertain 'what sort of people they were, whether violent and savage, with no sense of justice, or hospitable and godfearing'. Coming nearer to historical times, we find philosophers and statesmen travelling in order to benefit by all the wisdom that other nations with a longer history had gathered during the centuries. Thales travelled in Egypt and spent his time with the priests. Solon, according to Herodotus, travelled 'to see the world' ([TEXT NOT REPRODUCIBLE IN ASCII]), going to Egypt to the court of Amasis, and visiting Croesus at Sardis. At Sardis it was not till 'after he had seen and examined everything' that he had the famous conversation with Croesus; and Croesus addressed him as the Athenian of whose wisdom and peregrinations he had heard great accounts, proving that he had covered much ground in seeing the world and pursuing philosophy. (Herodotus, also a great traveller, is himself an instance of the capacity of the Greeks for assimilating anything that could be learnt from any other nations whatever; and, although in Herodotus's case the object in view was less the pursuit of philosophy than the collection of interesting information, yet he exhibits in no less degree the Greek passion for seeing things as they are and discerning their meaning and mutual relations; 'he compares his reports, he weighs the evidence, he is conscious of his own office as an inquirer after truth'.) But the same avidity for learning is best of all illustrated by the similar tradition with regard to Pythagoras's travels. Iamblichus, in his account of the life of Pythagoras, says that Thales, admiring his remarkable ability, communicated to him all that he knew, but, pleading his own age and failing strength, advised him for his better instruction to go and study with the Egyptian priests. Pythagoras, visiting Sidon on the way, both because it was his birthplace and because he properly thought that the passage to Egypt would be easier by that route, consorted there with the descendants of Mochus, the natural philosopher and prophet, and with the other Phoenician hierophants, and was initiated into all the rites practised in Biblus, Tyre, and in many parts of Syria, a regimen to which he submitted, not out of religious enthusiasm, 'as you might think' ([TEXT NOT REPRODUCIBLE IN ASCII]), but much more through love and desire for philosophic inquiry, and in order to secure that he should not overlook any fragment of knowledge worth acquiring that might lie hidden in the mysteries or ceremonies of divine worship; then, understanding that what he found in Phoenicia was in some sort an offshoot or descendant of the wisdom of the priests of Egypt, he concluded that he should acquire learning more pure and more sublime by going to the fountain-head in Egypt itself.
'There', continues the story, 'he studied with the priests and prophets and instructed himself on every possible topic, neglecting no item of the instruction favoured by the best judges, no individual man among those who were famous for their knowledge, no rite practised in the country wherever it was, and leaving no place unexplored where he thought he could discover something more.... And so he spent 22 years in the shrines throughout Egypt, pursuing astronomy and geometry and, of set purpose and not by fits and starts or casually, entering into all the rites of divine worship, until he was taken captive by Cambyses's force and carried off to Babylon, where again he consorted with the Magi, a willing pupil of willing masters. By them he was fully instructed in their solemn rites and religious worship, and in their midst he attained to the highest eminence in arithmetic, music, and the other branches of learning. After twelve years more thus spent he returned to Samos, being then about 56 years old.'
Whether these stories are true in their details or not is a matter of no consequence. They represent the traditional and universal view of the Greeks themselves regarding the beginnings of their philosophy, and they reflect throughout the Greek spirit and outlook.
From a scientific point of view a very important advantage possessed by the Greeks was their remarkable capacity for accurate observation. This is attested throughout all periods, by the similes in Homer, by vase-paintings, by the ethnographic data in Herodotus, by the 'Hippocratean' medical books, by the biological treatises of Aristotle, and by the history of Greek astronomy in all its stages. To take two commonplace examples. Any person who examines the underside of a horse's hoof, which we call a 'frog' and the Greeks called a 'swallow', will agree that the latter is the more accurate description. Or again, what exactness of perception must have been possessed by the architects and workmen to whom we owe the pillars which, seen from below, appear perfectly straight, but, when measured, are found to bulge out ([TEXT NOT REPRODUCIBLE IN ASCII]).
A still more essential fact is that the Greeks were a race of thinkers. It was not enough for them to know the fact (the [TEXT NOT REPRODUCIBLE IN ASCII]); they wanted to know the why and wherefore (the [TEXT NOT REPRODUCIBLE IN ASCII]), and they never rested until they were able to give a rational explanation, or what appeared to them to be such, of every fact or phenomenon. The history of Greek astronomy furnishes a good example of this, as well as of the fact that no visible phenomenon escaped their observation. We read in Cleomedes that there were stories of extraordinary lunar eclipses having been observed which 'the more ancient of the mathematicians' had vainly tried to explain; the supposed 'paradoxical' case was that in which, while the sun appears to be still above the western horizon, the eclipsed moon is seen to rise in the east. The phenomenon was seemingly inconsistent with the recognized explanation of lunar eclipses as caused by the entrance of the moon into the earth's shadow; how could this be if both bodies were above the horizon at the same time? The 'more ancient' mathematicians tried to argue that it was possible that a spectator standing on an eminence of the spherical earth might see along the generators of a cone, i.e. a little downwards on all sides instead of merely in the plane of the horizon, and so might see both the sun and the moon although the latter was in the earth's shadow. Cleomedes denies this, and prefers to regard the whole story of such cases as a fiction designed merely for the purpose of plaguing astronomers and philosophers; but it is evident that the cases had actually been observed, and that astronomers did not cease to work at the problem until they had found the real explanation, namely that the phenomenon is due to atmospheric refraction, which makes the sun visible to us though it is actually beneath the horizon. Cleomedes himself gives this explanation, observing that such cases of atmospheric refraction were especially noticeable in the neighbourhood of the Black Sea, and comparing the well-known experiment of the ring at the bottom of a jug, where the ring, just out of sight when the jug is empty, is brought into view when water is poured in. We do not know who the 'more ancient' mathematicians were who were first exercised by the 'paradoxical' case; but it seems not impossible that it was the observation of this phenomenon, and the difficulty of explaining it otherwise, which made Anaxagoras and others adhere to the theory that there are other bodies besides the earth which sometimes, by their interposition, cause lunar eclipses. The story is also a good illustration of the fact that, with the Greeks, pure theory went hand in hand with observation. Observation gave data upon which it was possible to found a theory; but the theory had to be modified from time to time to suit observed new facts; they had continually in mind the necessity of 'saving the phenomena' (to use the stereotyped phrase of Greek astronomy). Experiment played the same part in Greek medicine and biology.
Among the different Greek stocks the Ionians who settled on the coast of Asia Minor were the most favourably situated in respect both of natural gifts and of environment for initiating philosophy and theoretical science. When the colonizing spirit first arises in a nation and fresh fields for activity and development are sought, it is naturally the younger, more enterprising and more courageous spirits who volunteer to leave their homes and try their fortune in new countries; similarly, on the intellectual side, the colonists will be at least the equals of those who stay at home, and, being the least wedded to traditional and antiquated ideas, they will be the most capable of striking out new lines. So it was with the Greeks who founded settlements in Asia Minor. The geographical position of these settlements, connected with the mother country by intervening islands, forming stepping-stones as it were from the one to the other, kept them in continual touch with the mother country; and at the same time their geographical horizon was enormously extended by the development of commerce over the whole of the Mediterranean. The most adventurous seafarers among the Greeks of Asia Minor, the Phocaeans, plied their trade successfully as far as the Pillars of Hercules, after they had explored the Adriatic sea, the west coast of Italy, and the coasts of the Ligurians and Iberians. They are said to have founded Massalia, the most important Greek colony in the western countries, as early as 600 B.C. Cyrene, on the Libyan coast, was founded in the last third of the seventh century. The Milesians had, soon after 800 B.C., made settlements on the east coast of the Black Sea (Sinope was founded in 785); the first Greek settlements in Sicily were made from Euboea and Corinth soon after the middle of the eighth century (Syracuse 734). The ancient acquaintance of the Greeks with the south coast of Asia Minor and with Cyprus, and the establishment of close relations with Egypt, in which the Milesians had a large share, belongs to the time of the reign of Psammetichus I (664–610 B.C.), and many Greeks had settled in that country.
The free communications thus existing with the whole of the known world enabled complete information to be collected with regard to the different conditions, customs and beliefs prevailing in the various countries and races; and, in particular, the Ionian Greeks had the inestimable advantage of being in contact, directly and indirectly, with two ancient civilizations, the Babylonian and the Egyptian.
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Excerpted from A History of Greek Mathematics by Thomas Heath. Copyright © 1981 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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