142 P Y T P Y T the distinction between commensurable and incommensurable quantities. A reference to Euclid X. 2 will show that the method above is the one used to prove that two magnitudes are incommen- surable ; and in Euclid X. 3 it will be seen that the greatest common measure of two commensurable magnitudes is found by this process of continued subtraction. It seems probable that Pythagoras, to whom is attributed one of the rules for representing the sides of right-angled triangles in numbers, tried to find the sides of an isosceles right-angled triangle numerically, and that, failing in the attempt, he suspected that the hypotenuse and a side had no common measure. He may have demonstrated the incommensurability of the side of a square and its diagonal. The nature of the old proof which consisted of a reductio ad absurd- urn, showing that, if the diagonal be commensurable with the side, it would follow that the same number would be odd and even 1 makes it more probable, however, that this was accom- plished by his successors. The existence of the irrational as well as that of the regular dodecahedron appears to have been regarded by the school as one of their chief discoveries, and to have been preserved as a secret ; it is remarkable, too, that a story similar to that told by lamblichus of Hippasus is narrated of the person who first published the idea of the irrational, namely, that he suffered shipwreck, &c. 2 Eudemus ascribes the problems concerning the application of figures to the Pythagoreans. The simplest cases of the problems, Euclid VI. 28, 29 those, namely, in which the given parallelogram is a square correspond to the problem : To cut a given straight line internally or externally so that the rectangle under the segments shall be equal to a given rectilineal figure. The solution of this problem in which the solution of a quadratic equation is implicitly contained depends on the problem, Euclid II. 14, and the theorems, Euclid II. 5 and 6, together with the theorem of Pythagoras. It is probable that the finding of a mean proportional between two given lines, or the construction of a square which shall be equal to a given rectangle, is due to Pythagoras himself. The solution of the more general problem, Euclid VI. 25, is also attributed to him by Plutarch (7). The solution of this problem depends on that of the particular case and on the application of areas ; it requires, moreover, a knowledge of the theorems : Similar rectilineal figures are to each other as the squares on their homo- logous sides (Euclid VI. 20) ; and, If three lines are in geometrical proportion, the first is to the third as the square on the first is to the square on the second. Now Hippocrates of Chios, about 440 B.C., who was instructed in geometry by the Pythagoreans, possessed this knowledge. We are justified, therefore, in ascrib- ing the solution of the general problem, if not (with Plutarch) to Pythagoras, at least to his early successors. The theorem that similar polygons are to each other in the duplicate ratio of their homologous sides involves a first sketch, at least, of the doctrine of proportion and the similarity of figures. 3 That we owe the foundation and development of the doctrine of proportion to Pythagoras and his school is confirmed by the testi- mony of Nicomachus (14) and lamblichus (15 and 16). From these passages it appears that the early Pythagoreans were acquainted, not only with the arithmetical and geometrical means between two magnitudes, but also with their harmonical mean, which was then called " subcontrary." The Pythagoreans were much occupied with the representation of numbers by geometrical figures. These speculations originated with Pythagoras, who was acquainted with the summation of the natural numbers, the odd numbers, and the even numbers, all of which are capable of geometrical representa- tion. See the passage in Lucian (17) and the rule for finding Pythagorean triangles (12) and the observations thereon supra. On the other hand, there is no evidence to support the statement of Montucla that Pythagoras laid the foundation of the doctrine of isoperimetry, by proving that of all figures having the same peri- meter the circle is the greatest, and that of all solids having the 1 For this proof, see Euclid X. 117 ; see also Aristot., Analyt. Pr., i. c. 23 and c. 44. 2 Knoche, Untersuchungen her die neu aufgefundenen Scholien des Prokliis Diadochus zu Euclid's Elementen, pp. 20 and 23, Herford, 1865. 3 It is agreed on all hands that these two theories were treated at length by Pythagoras and his school. It is almost certain, however, that the theorems arrived at were^ proved for commensurable magni- tudes only, and were assumed to hold good for all. The Pythagoreans themselves seem to have been aware that their proofs were not rigor- ous, and were open to serious objection ; in this we may have the explanation of the secrecy which was attached by them to the idea of the incommensurable and to the pentagram which involved, and indeed represented, that idea. Now it is remarkable that the doctrine of proportion is twice treated in the Elements of Euclid first, in a general manner, so as to include incommensurables, in Book V., which tradition ascribes to Eudoxus, and then arithmetically in Book VII., which, as Hankel has supposed, contains the treatment of the subject by the older Pythagoreans. same surface the sphere js the greatest. We must also deny to Pythagoras and his school a knowledge of the conic sections, "ami in particular of the quadrature of the parabola, attributed to him by some authors ; and we have noticed the misconception which gave rise to this erroneous inference. Let us now see what conclusions can be drawn from the foregoing examination of the mathematical work of Pythagoras and his school, and thus form an estimate of the state of geometry about 480 B.C. First, as to matter. It forms the bulk of the first two books of Euclid, and includes a sketch of the doctrine of proportion which was probably limited to commensurable magnitudes together with some of the contents of the sixth book. It contains too the discovery of the irrational (dAoyov) and the construction of the regular solids, the latter requiring the description of certain regular polygons the founda- tion, in fact, of the fourth book of Euclid. Secondly, as to form. The Pythagoreans first severed geometry from the needs of practical life, and treated it as a liberal science, giving definitions and introducing the manner of proof which has ever since been in use. Further, they distinguished between discrete and continuous quantities, and regarded geometry as a branch of mathematics, of which they made the fourfold division that lasted to the Middle Ages the quadrivium (fourfold way to knowledge) of Boetius and the scholastic philosophy. And it may be observed that the name of "mathematics," as well as that of "philosophy," is ascribed to them. Thirdly, as to method. One chief characteristic of the mathematical work of Pythagoras was the combination of arithmetic with geometry. The notions of an equation and a propor- tion which are common to both, and contain the first germ of algebra were introduced among the Greeks by Thales. These notions, especially the latter, were elabo- rated by Pythagoras and his school, so that they reached the rank of a true scientific method in their theory of proportion. 4 To Pythagoras, then, is due the honour of having supplied a method which is common to all branches of mathematics, and in this respect he is fully comparable to Descartes, to whom we owe the decisive combination of algebra with geometry. See C. A. Bretschneider, Die Geometric u. die Geometer vor Eu- klides (Leipsic, 1870) ; H. Hankel, Zur Gcschichte dcr Mathcmatik (Leipsic, 1874) ; F. Hoefer, Histoire des Mathematiqucs (Paris, 1874); G. J. Allman, "Greek Geometry from Thales to Euclid," in Hermathena, Nos. v. , vii. , and x. (Dublin, 1877, 1881, and 1884) ; M. Cantor, Vorlesungen iibcr Gcschichte der Mathematik (Leipsic, 1880). The recently published Short History of Greek Mathematics by James Gow (Cambridge, 1884) will be found a convenient compilation. (G. J. A. ) PYTHEAS of Massilia was a celebrated Greek navi- gator and geographer, to whom the Greeks appear to have been indebted for the earliest information they possessed, of at all a definite character, concerning the western regions of Europe, and especially the British Islands. The period at which he lived cannot be accurately determined ; but it is certain that he wrote, not only before Eratosthenes, who relied much upon his authority, but before Dicaearchus, who was a pupil of Aristotle, and died about 285 B.C. Hence he may probably be regarded as about contem- porary with Alexander the Great. His work is now wholly lost, and appears to have been consulted in the original by comparatively few ancient writers, most of the 4 Proportion was not regarded by the ancients merely as a branch of arithmetic. We learn from Proclus that "Eratosthenes looked on proportion as the bond of mathematics" (op. cit., p. 43). We are also told in an anonymous scholium on the Elements of Eucli<l, which Knoche attributes to Proclus, that the fifth book, which treats of pro- portion, is common to geometry, arithmetic, music, and, in a wonl, to all mathematical science. And Kepler, who lived near enough to the ancients to reflect the spirit of their methods, says that one part of geometry is concerned with the comparison of figures and quantities, whence proportion arises. He also adds that arithmetic and geometry afford mutual aid to each other, and that they cannot be separated.
