in the same field of investigation, recognizing as their leader the one whose wisdom was the greatest. It was the same in the school of Pythagoras, in Southern Italy. We learn1 that the master lectured there on philosophy and mathematics, and that his listeners were of two kinds. In the first class the lectures were of a more general nature, and women were allowed to attend. In the second class women were debarred. We are to judge that only the select who had some skill as mathematicians and philosophers were allowed within the inner circle. Here the various contributions to the new science were made and the members took upon themselves a vow not to reveal these discoveries to the world. The Sophists are credited with having introduced higher education into Athens. There was a demand for the study of philosophy and mathematics, and so the Sophists came. Their schools were on the street corners or within the gymnasia. Their method was the Socratic, that of question and answer. So Socrates, who was by no means a Sophist, furnished a method of study that endures even to-day. Under such a developmental method the elements of geometry stood more ready to be appreciated by minds not yet matured. One has good reason then to believe that in the time of Plato and the Sophists, the study of geometry was becoming more common. That the Greek mind was more interested in the chain of reasoning than in the subject-matter itself is illustrated in some of the dialogues of Plato.2 Under the old Greek education, the youth from sixteen to eighteen continued his physical and social training. Under the new education more attention was given to the training of the intellect. We learn from the 'Republic" something of the place of geometry in the curriculum proposed by Plato. The pupil from the age of seven to about seventeen was to study music and gymnastics. Under music was included reading, writing, arithmetic, and geometry. In this early period there was to 1 Ball, A Short Account of the History of Mathematics, pp. 19-22. 2 See the dialogue on incommensurable lines between Socrates and one of Meno's slaves in Plato's Meno, 81-85, trans. Jowett. 3 Monroe, A Text-book in the History of Education, p. 115 * Republic, II-IV; VII, 537-540, trans. Jowett. See Bosanquet, The Education of the Young in the Republic of Plato, pp. 1-14. be no compulsion, it being sufficient that the student enter into the work as dictated by his interest and his aptitude. From seventeen to twenty the youth received the military gymnastic training of the ephebe. From twenty to thirty, the study of arithmetic, geometry, music, and astronomy was undertaken in a thoroughly systematic manner for those having special fitness and inclination. At the age of thirty-five, after five years' study of dialectics, the trained man returned to social and political life and was to be there a directing force. In the "Laws," written in the old age of Plato, there are the same general recommendations, only the later years of the man are to be devoted to mathematical and astrological studies. For The four-fold division of the mathematical sciences was instituted by the early Greeks. We have observed in the recommendations of Plato the study of arithmetic, geometry, music, and astronomy. Later in Rome, Cassiodorus embodied them in the quadrivium, which constituted the advanced course in the medieval monastic schools. But the Pythagoreans were the first to assign this four-fold division. From Proclus1 we learn that "the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the how many, and the other to the how much; and they assigned to each of these parts a two-fold division. they said that discrete quantity, or the how many, either subsists by itself, or must be considered with relation to some other; but that continued quantity, or the how much, is either stable or in motion. Hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable; but astronomy contemplates quantity so far as it is of a self-motive nature." Thus we see the origin of the plan of keeping motion out of the domain of geometry. It was essentially a study of forms in fixed positions. The idea of rotation, which is employed to-day in the study of elementary geometry, was therefore not permissible. The Greeks throughout were consistent with these ideas. The Pythagoreans founded the theory of proportion, giving a method applicable in the fields of both arithmetic and geometry, but, notwithstanding this close relationship, geometry 1 Allman, p. 23 (ref. Proclus, ed. Friedlein, p. 45). with the Greeks was developed within itself. Algebra was not yet invented, and a great deal of mathematical work, since simplified by the methods of algebra, was laboriously carried on by geometry alone. The idea of a duality, of a one-to-one correspondence between algebra and geometry, could not be utilized, and not until the time of Descartes (1637) was this principle fully recognized. Aristotle was more practical than some of his predecessors, but he showed this same tendency to confine each branch of mathematics to its own domain. He says, "We cannot prove anything by starting from a different genus, e.g., nothing geometrical by means of arithmetic. . . Where the subjects are so different as they are in arithmetic and geometry we cannot apply the arithmetical sort of proof to that which belongs to quantities in general, unless these quantities are numbers, which can only happen in certain cases." The theory of geometry was thus isolated by the Greeks from other branches of mathematics, and mathematical development was thereby retarded many centuries. ... The restriction that problems of construction in elementary geometry should be limited to the use of the rule and compasses as instruments of construction dates from the Grecian period. Such a restriction barred out all so-called mechanical constructions, and as a result it was impossible to get a solution of the three famous problems, trisecting any angle, finding a square equal in area to a given circle, and duplicating a given cube. These were capable of solution by means of conic sections and certain special curves, but not by using circles and straight lines. is perhaps right in attributing this restriction to the influence of Plato. Eutocius (Torelli ed., p. 135) relates that Plato invented a mechanical device for inserting two mean proportionals between two given straight lines, Hippocrates having reduced the duplication problem to this one. But Plutarch in two of his writings relates "that Plato blamed Archytas and Eudoxus and Menæchmus for using such instruments for the purpose of solving the duplication-problem, and said that the good of geometry was spoilt and destroyed thereby," and that owing to this remonstrance of Plato, "mechanics were separated from geom1 Allman, p. 146 (ref. Anal. post. I, vii, p. 75a, ed. Bek.). 2 Gow, p. 181. 3 Ibid. (ref. Quaest. Conv. viii, 9, 2, c. 1). Here are etry and became a branch of the military art."1 two opposing statements, but Plato's general position as to the aim of education would lead one to the conclusion that the statements attributed to him are the correct ones, and that at least as early as his time the above restriction was imposed in the use of instruments in elementary geometry. Thus we see the expulsion of mechanics from geometry. But mechanics was studied nevertheless, and the science was developed under Aristotle and later made great progress at Alexandria. Archytas is mentioned as the first to place mechanics on a scientific basis and "to apply mechanical motion to the solution of a geometric problem, while trying to find by means of the section of a semi-cylinder two mean proportionals with a view to the duplication of the cube.""" During this period there appears to have been a neglect of solid geometry. It has already been mentioned that Hippocrates reduced the problem of the duplication of the cube to that of finding two mean proportionals between two given straight lines. He thus changed the problem from one of solid geometry to one of plane. This indicates a tendency of the times, or else Plato3 would not have complained that stereometry went entirely out of fashion. Furthermore, we know that by the time of Euclid solid geometry was not developed in the sense that characterized plane geometry. Although they made no use of them in theoretical geometry, we may mention that the Greeks were familiar with the square, the level, and the gnomen or carpenter's square. The inventions of the square and level are attributed by Pliny (Nat. Hist. VII, 57) to Theodorus of Samos, who lived contemporaneous with Thales.* According to Allman, they were known long before to the Egyptians: "So that to Theodorus is due at most the honour of having introduced them into Greece." Anaximander, of Miletus, was the first to introduce the gnomen and the sun-dial into Greece.5 These came, according to Herodotus, from the Babylonians в 1 Plutarch Marcellus, trans Langhorne, p. 106. 2 Allman, p. 110, quoting from Diog. Laert.; Plutarch, Marcellus, trans. Langhorne, p. 106. 3 Republic, VII, 528, trans. Jowett. 4 Allman, p. 15. 'Gow, p. 145. "Herodotus, II, 109. The lettering of geometric figures began with the Greeks. Cantor points out that the letters spelling the word "health" were placed at the vertices of the pentagram, the Pythagorean emblem.1 Hippocrates, later, in attempting the quadrature of the lune, used quaint descriptions in describing the lettering of his figures. Thus, he wrote, "the line on which AB is marked" and "the point on which K stands." Aristotle still later used a letter symbolism in his Physics (VII, 5, pp. 249-250 of the Berlin ed.). He says, "If A be the mover, B the moved thing, I the distance, and ▲ the time of the mo B tion, then A will move twice the distance П in the times or the whole distance T in half the time ▲." The value of this symbolism was fully appreciated by Aristotle when he said that much time and trouble is saved by a general symbolism.3 Recognizing the Greek love of oratory and in particular the propensity of the Sophists for verbal disputations, we can judge that geometry was taught orally, and only that was committed to writing which was no longer a subject for argument. It is probable that a board strewn with sand was in common use for the drawing of figures, for this was a common method among the orientals in doing their calculating, and the Greeks certainly made use of this convenient mode.1 It will be pointed out later that the tendency to hold to the special characterized the practical geometries of Latin Europe even up to the middle of the seventeenth century. The period was one of retrogression in this respect when compared with the time of the early Greeks. But we shall also see that Euclid was not entirely free from this tendency. This being the case, we should expect to find in the pre-Euclidean geometry like tendencies. "Eutocius, at the beginning of his commentary on the Conics of Apollonius (p. 9. Hallev's edn.), quotes from Geminus. 1 Cantor, I, p. 195. 2 Gow, p. 169. 3 Ibid., p. 105n. There is a tradition, that as Archimedes was contemplating some geometric figures drawn in the sand on the floor, a soldier was admitted and ordered Archimedes to follow him to Marcellus. Archimedes, refusing to do it until he had finished his problem, the soldier in a passion drew his sword and killed him. Plutarch, Marcellus, trans, Langhorne, p. 114, |