tians than was Thales. This is seen by the attention he gave to the geometry of areas and volumes, and to arithmetic. The school of Pythagoras was undoubtedly familiar with many of the propositions in the first two books of Euclid and with parts of the fifth and sixth books; that is, it was familiar with the ordinary theorems in plane geometry concerning equality of lines and of angles and with many of the theorems on equivalent and congruent areas. The geometry of the circle was not developed by the Pythagoreans, but was studied later at the Athenean school. It is said that the Pythagoreans knew that the angle sum of a triangle equals two right angles. Hence they held a conception of parallel lines if we are to suppose that any proof of the above theorem was accomplished. The theorem of the three squares (Euclid, I, 47), that in any right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, is known as the Pythagorean theorem. The Egyptians knew the truth of this theorem where the sides were in the ratio of 3:45, but Pythagoras was the first to see the truth of this relation in any right triangle.

Pythagoras is also credited with the discovery of the geometric irrational and of the three kinds of proportion, arithmetical geometric, and harmonic. The Pythagoreans were much interested in the study of the regular solids and are credited with their constructions. This being true, they were certainly familiar with the construction of the regular plane polygons of 3, 4, 5, sides. The construction of the regular polygon of five sides depends upon the division of a line in extreme and mean ratio. Allman' contends that Pythagoras was familiar with this, but Gow3 quotes from the Eudemian Summary, which attributes the discovery of this problem, known as the Golden Section, to the school of Plato. Gow's conclusion admits of less speculation and perhaps is nearer the truth.

We must note that in all that is known regarding the contributions of Thales and Pythagoras to the development of geometry there is more or less speculation as to the nature of the material, but in particular there is no definite statement re1 Gow, p. 153.

'Allman, Greek Geometry from Thales to Euclid, p. 40. Hereafter referred to as Allman

garding the methods of proof. We are, perhaps, safe in judging that a large part of the work of these early schools was in finding out geometric truths, and that the sequence of proved theorems was not then held in any hard and fast line.

We have no record that the school of Pythagoras was concerned with the practical, and we may conclude with Dr. Allman1 that the Pythagoreans were the first to sever geometry from the needs of practical life and to treat it as a liberal science.

Thus far we have the growth of geometry-first, the practical stage as with the Egyptians; second, the beginning of the logical stage in the school of Thales, where practical applications were employed and the foundations of deductive geometry laid; and thirdly, under the Pythagoreans, we have the subject treated as a liberal science. But we have no proof that the subjectmatter was yet organized into any fixed sequence.

When the Pythagorean school at Croton in Southern Italy was disbanded for political reasons, its influence had already grown and other schools had been founded on the shores of the Mediterranean. About this time, fresh from her glories of the Persian wars, Athens, exceedingly wealthy, attracted people of all nations. Among these were teachers who were willing to work for hire. Such were the Sophists. To them and the school of Plato, we are principally indebted for the great mass of subjectmatter which was finally organized into a text by Euclid of Alexandria. We recall that the Pythagoreans developed the geometry of areas but neglected the geometry of the circle. This study was taken up by the Athenian Greeks and many theorems were discovered in their futile attempts to solve the so-called Three Problems of Antiquity: the trisecting of any angle, the duplication of the cube, and the quadrature of the circle.2

Something of the nature of the contributions to the subjectmatter of geometry during this period can be seen from the titles of some of the works. Euclid is universally credited with being the first to write a complete text on geometry, but he was not the first to write on particular portions of it. Although the school of Thales is not generally credited with adding a great 1Op. cit., p. 47.

2 For a scientific treatment of these famous problems, see Klein, Famous Problems of Elementary Geometry, trans. Beman and Smith,

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deal to the subject-matter of geometry, to two of its members have been attributed geometric writings. According to Suidas,1 Anaximander wrote a work with the title "A Collection of Figures Illustrative of Geometry," and Plutarch2 (De exilio c. 17) states that Anaxagoras of Clazomena wrote a treatise on the quadrature of the circle. According to Vitruvius2 (vii. Praef.), Anaxagoras also wrote a work on perspective. The Pythagoreans published nothing of their work on geometry, although later Philolaus, who lived in the time of Plato, published an account of their philosophy. What we know of the writings on geometry during its development at Athens is to be found in the Eudemian Summary to which reference has already been made 4 Hippocrates of Chios, next, who discovered the quadrature of the lune, and Theodorus of Cyrene became distinguished geometers; indeed Hippocrates was the first who is recorded to have written 'Elements.' Plato, who followed him, caused mathematics in general, and geometry in particular, to make great advances, by reason of his well-known zeal for the study, for he filled his writings with mathematical discourses, and on every occasion exhibited the remarkable connexion between mathematics and philosophy. To this time belong also Leodamas the Thasian and Archytas of Tarentum and Theatetus of Athens, by whom mathematical inquiries were greatly extended, and improved into a more scientific system. Younger than Leodamas were Neocleides and his pupil Leon, who added much to the work of their predecessors: for Leon wrote an ‘Elements' more carefully designed, both in the number and the utility of its proofs, and he invented, also, a diorismus (or test for determining) when the proposed problem is possible and when impossible. Eudoxus of Cnidus, a little later than Leon, and a student of the Platonic school, first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the (golden) section, to which he applied the analytical method. Amyclas of Heraclea, one of Plato's

Ibid., pp. 135-137 (ref. Proclus, ed. Friedlein)

5 The cutting of a line in extreme and mean ratio.

companions, and Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, and also Deinostratus, the brother of Menæchmus, made the whole of geometry yet more perfect. Theudius of Magnesia made himself distinguished as well in other branches of philosophy as also in mathematics; composed a very good book of 'Elements,' and made more general propositions which were confined to particular cases. Cyzicenus of Athens also about the same time became famous in other branches of mathematics, but especially in geometry. All these consorted together in the Academy and conducted their investigations in common. Hermotimus of Colophon pursued further the lines opened up by Eudoxus and Theætetus, and discovered many propositions of the 'Elements' and composed some on Loci. Philippus of Mende, a pupil of Plato and incited by him to mathematics, carried on his inquiries according to Plato's suggestions and proposed to himself such problems as, he thought, bore upon the Platonic philosophy."

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Besides these, others have been credited with writings on geometry. Diogenes Laertius relates that Democritus wrote on geometry, on numbers and perspective, and also two books on incommensurable lines and (?) solids. Another work ascribed to him bears the incomprehensible title, "The Difference of the Gnomon or the Contact of the Circle and the Sphere." Suidas2 credits Theætetus with having written on the five regular solids. Theophrastus, a pupil of Aristotle, according to Diogenes Laeritus, wrote a history of geometry in four books, together with six books of astronomy and one of arithmetic. According to Hypsicles, a book on "The Comparison of the Five Regular Solids," was written by Aristæus. This contained the theorem, "The same circle circumscribes the pentagon of the dodecahedron and the triangle of the icosahedron, these solids being inscribed in the same sphere."

As has already been mentioned, during the pre-euclidean period the subject-matter of elementary plane geometry was practically completed. In the time of the Pythagorean school, the five regular solids were studied, but stereometry as a science was not

yet established. The school of Plato also studied the regular solids.1 A decided advance was made in this study when Eudoxus' proved the theorem, "A triangular pyramid is onethird of a prism on the same base and between the same parallels." This established a relation between magnitudes of apparently unlike properties and hence tended to systematize the logic of solids. And so Eudoxus may be credited with having founded the science of solid geometry.

From our account thus far, we can see that the development of elementary geometry was not parallel with the sequence exhibited in the "Elements" of Euclid. In other words, this sequence was not according to the historic development of the subject. This will be considered later, when treating the sequence of Euclid; but may be illustrated briefly at this point. The subject-matter of the third book of Euclid, which treats of circles, was developed in the Athenian period. The theory of proportion which follows later in the "Elements" was systematized earlier by the Pythagorean school. This school also studied the properties of the five regular solids before the great amount of material of plane geometry had yet been worked over, much less systematized. This lack of agreement between the historic development of geometry and the common logical sequence will be referred to in the last chapter of this essay.

Educational Features of the Greek Geometry

It is not to be expected in the early development of any science that any but mature minds should engage in its study. So it was in the study of geometry by the Greeks. We know nothing of the conduct of the school of Thales at Miletus, but undoubtedly the word "school" is to be used only in the sense of a select body of men, probably few in number, working together

1 They have since been called the Platonic Bodies. 2 Allman, p. 88.

3 Archimedes states that Democritus was the first to state the above as a formula without proof. See Heiberg's German translation of the Greek text of a newly discovered MS. by Archimedes, published in Bibliotheca Mathematica, vol. 7. For a brief account of the same in English see an article by Professor Charles S. Slichter in Bulletin of the American Mathematical Society, vol. XIV, No. 8.