CHAPTER III. THE SCHOOLS OF ATHENS AND CYZICUS.1 CIRC. 420 B.C.-300 B.C. It was towards the close of the fifth century before Christ that Athens first became the chief centre of mathematical studies. Several causes conspired to bring this about. During that century she had become, partly by commerce, partly by appropriating for her own purposes the contributions of her allies, the most wealthy city in Greece; and the genius of her statesmen had made her the centre on which the politics of the peninsula turned. Moreover, whatever states disputed her claim to political supremacy her intellectual pre-eminence was admitted by all. There was no school of thought which had not at some time in that century been represented at Athens by one or more of its leading thinkers; and the ideas of the new science, which was being so eagerly studied in Asia Minor and Graecia Magna, had been brought before the Athenians on various occasions. 1 The history of these schools is discussed at length in G. Loria's Le Scienze Esatte nell' Antica Grecia, Modena, 1893-1900; in G. J. Allman's Greek Geometry from Thales to Euclid, Dublin, 1889; and in J. Gow's Greek Mathematics, Cambridge, 1884; it is also treated by Cantor, chaps. ix, x, and xi; by Hankel, pp. 111-156; and by C. A. Bretschneider in his Die Geometrie und die Geometer vor Eukleides, Leipzig, 1870; a critical account of the original authorities is given by S. P. Tannery in his Géométrie Grecque, Paris, 1887, and other papers. D Anaxagoras. Amongst the most important of the philosophers who resided at Athens and prepared the way for the Athenian school I may mention Anaxagoras of Clazomenae, who was almost the last philosopher of the Ionian school. He was born in 500 B.C., and died in 428 B.C. He seems to have settled at Athens about 440 B.C., and there taught the results of the Ionian philosophy. Like all members of that school he was much interested in astronomy. He asserted that the sun was larger than the Peloponnesus: this opinion, together with some attempts he had made to explain various physical phenomena which had been previously supposed to be due to the direct action of the gods, led to a prosecution for impiety, and he was convicted. While in prison he is said to have written a treatise on the quadrature of the circle. The Sophists. The sophists can hardly be considered as belonging to the Athenian school, any more than Anaxagoras can; but like him they immediately preceded and prepared the way for it, so that it is desirable to devote a few words to them. One condition for success in public life at Athens was the power of speaking well, and as the wealth and power of the city increased a considerable number of "sophists" settled there who undertook amongst other things to teach the art of oratory. Many of them also directed the general education of their pupils, of which geometry usually formed a part. We are told that two of those who are usually termed sophists made a special study of geometry-these were Hippias of Elis and Antipho, and one made a special study of astronomy-this was Meton, after whom the metonic cycle is named. Hippias. The first of these geometricians, Hippias of Elis (circ. 420 B.C.), is described as an expert arithmetician, but he is best known to us through his invention of a curve called the quadratrix, by means of which an angle can be trisected, or indeed divided in any given ratio. If the radius of a circle rotate uniformly round the centre O from the position OA through a right angle to OB, and in the same time a straight line drawn perpendicular to OB move uniformly parallel to itself from the position OA to BC, the locus of their intersection will be the quadratrix. Let OR and MQ be the position of these lines at any time; and let them cut in P, a point on the curve. Then angle AOP: angle 40B=OM: OB. Similarly, if OR' be another position of the radius, angle AOP: angle AOBOM': OB. Hence, if the angle AOP be given, and it be required to divide it in any given ratio, it is sufficient to divide OM in that ratio at M', and draw the line MP'; then OP will divide AOP in the required ratio. If OA be taken as the initial line, OP=r, the angle AOP = 0, and OA = a, we have 0 := r sin : a, and the equation of the curve is π= 2a0 cosec 0. Hippias devised an instrument to construct the curve mechanically; but constructions which involved the use of any mathematical instruments except a ruler and a pair of compasses were objected to by Plato, and rejected by most geometricians of a subsequent date. Antipho. The second sophist whom I mentioned was Antipho (circ. 420 B.C.). He is one of the very few writers among the ancients who attempted to find the area of a circle by considering it as the limit of an inscribed regular polygon with an infinite number of sides. He began by inscribing an equilateral triangle (or, according to some accounts, a square); on each side he inscribed in the smaller segment an isosceles triangle, and so on ad infinitum. This method of attacking the quadrature problem is similar to that described above as used by Bryso of Heraclea. No doubt there were other cities in Greece besides Athens where similar and equally meritorious work was being done, though the record of it has now been lost; I have mentioned here the investigations of these three writers, chiefly because they were the immediate predecessors of those who created the Athenian school. The Schools of Athens and Cyzicus. The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C.; the school was established on a permanent basis by the labours of Plato and Eudoxus; and, together with the neighbouring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 B.C.) of the university at Alexandria drew thither most of the talent of Greece. Eudoxus, who was amongst the most distinguished of the Athenian mathematicians, is also reckoned as the founder of the school at Cyzicus. The connection between this school and that of Athens was very close, and it is now impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, Menaechmus, and Aristaeus belonged to that of Cyzicus. There was always a constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil Theodorus of Cyrene, and there was no difference in their treatment of the subject, so that they may be conveniently treated together. Before discussing the work of the geometricians of these schools in detail I may note that they were especially interested in three problems: 1 namely (i), the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle, that is, the determination of a square whose area is equal to that of a given circle. Now the first two of these problems (considered analytically) require the solution of a cubic equation; and, since a construction by means of circles (whose equations are of the form x2 + y2+ax+by+c=0) and straight lines (whose equations are of the form ax+By+y=0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of the conic sections be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have been long known to be incommensurable, but it is only recently that it has been shewn by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence this problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes. Besides attacking these problems the later Platonic school collected all the geometrical theorems then known and arranged them systematically. These collections comprised the bulk of the propositions in Euclid's Elements, books I-IX, XI, and XII, together with some of the more elementary theorems in conic sections. Hippocrates. Hippocrates of Chios (who must be carefully 1 On these problems, solutions of them, and the authorities for their history, see my Mathematical Recreations and Problems, London, fourth edition, 1905, chap. viii. |