to learn geometry from Theodorus, and then to the Pytha goreans in Italy. Is it likely, then, that Plato, who, as far as we know, never solved a geometrical question, should have invented this method of solving problems in geometry and taught it to Archytas, who was probably his teacher, and who certainly was the foremost geometer of that time, and that thereby Archytas was led to his celebrated solution of the Delian problem? The former of the two reasons advanced by Bretschneider, and given above, has reference to and is based upon the following well-known and remarkable passage of the Republic of Plato. The question under consideration is the order in which the sciences should be studied: having placed arithmetic first, and geometry-i. e. the geometry of plane surfaces-second, and having proposed to make astronomy the third, he stops and proceeds : "Then take a step backward, for we have gone wrong in the order of the sciences.' 'What was the mistake?' he said. 'After plane geometry,' I said, 'we took solids in revolution, instead of taking solids in themselves; whereas, after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.' 'That is true, Socrates; but these subjects seem to be as yet hardly explored. 'Why, yes,' I said, 'and for two reasons: in the first place, no government patronises them, which leads to a want of energy in the study of them, and they are difficult; in the second place, students cannot learn them unless they have a teacher. But then a teacher is hardly to be found; and even if one could be found, as matters now stand, the students of these subjects, who are very conceited, would not mind him. That, however, would be otherwise if the whole state patronised and honoured this science; then they would listen, and there would be continuous and earnest search, and discoveries would be made; since even now, disregarded as these studies are by the world, and maimed of their fair proportions, and although none of their votaries can tell the use of them, still they force their way by their natural charm, and very likely they may emerge into light.' Yes,' he said, 'there is a remarkable charm in them. But I do not clearly understand the change in the order. First you began with a geometry of plane surfaces?' 'Yes,' I said. 'And you placed astronomy next, and then you made a step backward?' 'Yes,' I said, 'the more haste the less speed; the ludicrous state of solid geometry made me pass over this branch and go on to astronomy, or motion of solids.' 'True,' he said. 'Then regarding the science now omitted as supplied, if only encouraged by the State, let us go on to astronomy.' 'That is the natural order,' he said." Cantor, too, says that 'stereometry proper, notwithstanding the knowledge of the regular solids, seems on the whole to have been yet [at the time of Plato] in a very backward state,'" and in confirmation of his opinion quotes part of a passage from the Laws.58 It will be seen, however, on reading it to the end, that the ignorance of the Hellenes referred to by Plato, and denounced by him in such strong language, is an ignorance-not, as Cantor thinks, of stereometry-but of incommensurables. We do not know the date of the Republic, nor that of the discovery of the cubature of the pyramid by Eudoxus, 56 Plato, Rep., VII. 528; Jowett, the Dialogues of Plato, vol. II., pp. 363, 364. 57 Cantor, Gesch. der Math., p. 193. 58 Plato, Leges, VII., 819, 820; Jowett, op. cit., vol. IV., pp, 333, 334. which founded stereometry," and which was an important advance in the direction indicated in the passage given above it is probable, however, that Plato had heard from his Pythagorean teachers of this desideratum; and I have in the last chapter (p. 86, sq.) pointed out a problem of high philosophical importance to the Pythagoreans at that time, which required for its solution a knowledge of stereometry. Further, the investigation given above shows, as Cantor remarks, that Archytas formed an honourable exception to the general ignorance of geometry of three dimensions complained of by Plato. It is noteworthy that this difficult problem-the cubature of the pyramid-was solved, not through the encouragement of any State, as suggested by Plato, but, and in Plato's own lifetime, by a solitary thinker-the great man whose important services to geometry we have now to consider. 59 It should be noticed, however, that with the Greeks Stereometry had the wider signification of geometry of three dimensions, as may be seen from the following passage in Proclus: ἡ μὲν γεωμετρία διαιρεῖται πάλιν εἴς τε τὴν ἐπίπεdov bewρlav Kal Tǹv σtepeoμetpíav.--Proclus, ed. Friedlein, p. 39: see also ibid., PP. 73, 116. CHAPTER V. EUDOXUS. Eudoxus of Cnidus.--His life.-Founded the School of Cyzicus.-Notices of his Geometrical work.-Examination of and inferences from these Notices.— Eudoxus discovered the Cubature of the Pyramid, invented the Method of Exhaustions, and was the Founder of the Doctrine of Proportion as given in the Fifth Book of Euclid.-κaμmúλai ypaμμal and Hippopede.—Retrospective view of the progress of Geometry.—Effect of the Dialectic Method in general, and, in particular, in Geometry.-Necessity of recasting the methods of investigation and proof.-Estimate of the services of Eudoxus.— Though his fame was very great in antiquity, yet he was for centuries unduly depreciated.-Justice is now done to him.-His place in the History of Science. EUDOXUS of Cnidus1-astronomer, geometer, physician, lawgiver-was born about 407 B. C., and was a pupil of Archytas in geometry, and of Philistion, the Sicilian [or Italian Locrian], in medicine, as Callimachus relates in his Tablets. Sotion in his Successions, moreover, says that he also heard Plato; for when he was twenty-three years of age and in narrow circumstances, he was attracted by the reputation of the Socratic school, and, in company with Theomedon the physician, by whom he was supported, he went to Athens, where-or rather at Piræus-he remained two months, going each day to the city to hear the lectures of the Sophists, Plato being one of them, by whom, however, he was coldly received. He then returned home, and, being again aided by the contributions of his friends, he set sail for Egypt with Chrysippus-also a physician, and who, as well as Eudoxus, learnt medicine from Philistion-bearing with him letters of recommendation from Agesilaus to Nectanabis, by whom he was commended to 1 Diog. Laert., VIII., c. viii; A. Boeckh, ueber die vierjährigen Sonnenkreise der Alten, vorzüglich den Eudoxischen, Berlin, 1863. the priests. When he was in Egypt with Chonuphis of Heliopolis, Apis licked his garment, whereupon the priests said that he would be illustrious (evdokov), but short-lived.' He remained in Egypt one year and four months, and composed the Octaëteris-an octennial period. Eudoxus then-his years of study and travel now over-took up his abode at Cyzicus, where he founded a school (which became famous in geometry and astronomy), teaching there and in the neighbouring cities of the Propontis; he also went to Mausolus. Subsequently, at the height of his reputation, he returned to Athens, accompanied by a great many pupils, for the sake, as some say, of annoying Plato, because formerly he had not held him worthy of attention. Some say that, on one occasion, when Plato gave an entertainment, Eudoxus, as there were many guests, introduced the fashion of sitting in a semicircle. Aristotle tells us that Eudoxus thought that pleasure was the summum bonum; and, though dissenting from his theory, he praises Eudoxus in a manner which with him is quite unusual :'And his words were believed, more from the excellence of his character than for themselves; for he had the repu- . tation of being singularly virtuous, owppwv: it therefore seemed that he did not hold this language as being a 2 Boeckh thinks, and advances weighty reasons for his opinion, that the voyage of Eudoxus to Egypt took place when he was still young-that is, about 378 B. C.; and not in 362 B. C., in which year it is placed by Letronne and others. Boeckh shows that it is probable that the letters of recommendation from Agesilaus to Nectanabis, which Eudoxus took with him, were of a much earlier date than the military expedition of Agesilaus to Egypt. In this view Grote agrees. See Boeckh, Sonnenkreise, pp. 140-148; Grote, Plato, vol. 1., pp. 120– 124. 3 The Octaëteris was an intercalary cycle of eight years, which was formed with the object of establishing a correspondence between the revolutions of the sun and moon; eight lunar years of 354 days, together with three months of 30 days each, make up 2922 days: this is precisely the number of days in eight years of 365 days each. This period, therefore, presupposes a knowledge of the true length of the solar year; its invention, however, is attributed by Censorinus to Cleostratus. 4 Is this the foundation of the statement in Grote's Plato, vol. I., p. 124the two then became friends'? K |