solids three-the tetrahedron, the cube and the octahedron—were
known to the Egyptians and are to be found in their architecture.
Let us now examine what is required for the construction of the
other two solids-the icosahedron and the dodecahedron. In the
formation of the tetrahedron three, and in that of the octahedron
four, equal equilateral triangles had been placed with a common
vertex and adjacent sides coincident; and it was known that if
six such triangles were placed round a common vertex with their
adjacent sides coincident, they would lie in a plane, and that,
therefore, no solid could be formed in that manner from them. It
remained, then, to try whether five such equilateral triangles could
be placed at a common vertex in like manner; on trial it would
be found that they could be so placed, and that their bases wou'ld
form a regular pentagon. The existence of a regular pentagon
would thus become known. It was also known from the formation
of the cube that three squares could be placed in a similar way
with a common vertex; and that, further, if three equal and
regular hexagons were placed round a point as common vertex
with adjacent sides coincident, they would form a plane. It remained
in this case, too, only to try whether three equal regular
pentagons could be placed with a common vertex and in a similar
way; this on trial would be found possible and would lead to the
construction of the regular dodecahedron, which was the regular
solid last arrived at.

We see that the construction of the regular pentagon is required for the formation of each of these two regular solids, and that, therefore, it must have been a discovery of Pythagoras. If we examine now what knowledge of geometry was required for the solution of this problem, we shall see that it depends on Euclid IV. 10, which is reduced to Euclid II. 11, which roblem is reduced to the following: To produce a given straight iine so that the rectangle under the whole line thus produced and the produced part shall be equal to the square on the given line, or, in the language of the ancients, To app y to a given straight line a rectangle which shall be equal to a given area-in this case the square on the given line-and which shall be excessive by a square. Now it is to be observed that the problem is solved in this manner by Euclid (VI. 30, 1st method), and that we know on the authority of Eudemus that the problems concerning the application of areas and their excess and defect are old, and inventions of the Pythagoreans (5). Hence the statements of Iamblichus concerning Hippasus (9)that he divulged the sphere with the twelve pentagons-and of Lucian and the scholiast on Aristophanes (ro)-that the pentagram was used as a symbol of recognition amongst the Pythagoreansbecome of greater importance.

Further, the discovery of irrational magnitudes is ascribed to Pythagoras by Eudemus (13), and this discove has been ever regarded as one of the greatest of antiquity. rit is commonly assumed that Pythagoras was led *to this theory from the consideration of the isosceles right-angled triangle. It seems to the present writer, however, more probable that the discovery of incommensurable magnitudes was rather owing to the problem: To cut a line in extreme and mean ratio. From the solution of this problem it follows at once that, if on the greater segment of a line so cut a part be taken equal to the less, the greater segment, regarded as a new line, will be cut in a similar manner; and this process can be continued without end. On the other hand, if a similar method be adopted in the case of any two lines which can be represented numerically, the process would end. Hence would arise 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 incommensurable; 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 trian les in numbers, tried to find the sides of an isosceles right-angied 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 absurdum, showing that, if the diagonal be commensurable with the side, it would fo low that the same number would be odd and even*-makes it more probable, however, that this was accomplished 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 Iamblichus of Hippasus is narrated of the person who first published the idea of the irrational, viz. that he suffered shipwreck, &c. Eudemus ascribes the problems concerning the application of figures to the Pythagoreans. The simplest cases of the problems, For this proof, see Euclid X. 117; see also Aristot. Analyf. Pr. i. c. 23 and c. 44.

2 Knoche, Untersuchungen iiber die neuaufgefundenen Scholien des Prokl-us Diadochus zu Euclids Elementen, pp. 20 and 23 fHerford, 1865).

Euclid VI. 28, 29-those, viz. 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 rectilinear 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 rectilinear figures are to each other as the squares on their homologous sides (Euclid VI. 2o); 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 44Q B.C., who was instructed in geometry by the Pythagoreans, possessed this knowledge. We are justified, therefore, in ascribing 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.” That we owe the foundation and development of the doctrine of proportion to Pythagoras and his school is confirmed by the testimony of Nicomachus (14) and Iamblichus (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 harmonica mean, which was then called “ sub contrary." The Pythagoreans were much occupied with the representation of numbers by geometrical figures. These speculations originated with Pythagoras, who was acquainted with t e summation of the natural numbers, the odd numbers and the even numbers, all of which are capable of geometrical representation. See the passage in Lucian (17) and the rule for finding Pythagorean trian les (12) and the observations thereon supra. On the other ham? 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 perimeter the circle isfthe greatest, and that of all solids having the same surface the sphere is the greatest. We must also deny to Pythagoras and his school a knowledge of the conic sections, and 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.

Certain conclusions may be drawn from the foregoing examination of the mathematical Work of Pythagoras and his school, which enable us to 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 (c'i7o'yov) and the construction of the regular solids, the latter requiring the description of certain regular polygons-the foundation, in fact, of the fourth book of Euclid. Secondly, as Ito 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 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 magnitudes only, and were assumed to hold good for all. The Pythagoreans themselves seem to have been aware that their proofs were not rigorous, 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 incommensurable, 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.