squares (Euclid l. 47). Commonly called the “ theorem of Pythagoras, " is attributed to him by many authorities, of whom the oldest is Vitruvius.1 (12) One of the methods of finding right-angled triangles whose sides can be expressed in numbers (Pythagorean triangles)-that setting out from the odd numbers-is referred to Pythagoras by Heron of Alexandria and Proclus? (13) The discovery of irrational quantities is ascribed to Pythagoras by Eudemus (Procl. op. cil. p. 65). (14) The three proportions-arithmetical, geometrical and harmonica-were known to Pythagorasi (15) Iamblichus' says, “ Formerly, in the time of Pythagoras and the mathematicians under him, there were three means only-the arithmetical, the geometrical and the third in order, which was known by the name sub-contrary (bare:/avria), but which Archytas and Hippasus designated the harmonica, since it appeared to include the ratios concerning harmony and melody.” (16) The so-called most perfect or musical proportion, e.g. 6:8::9:12, which comprehends in it all the former ratios, according to Iamblichusf' is said to be an invention of the Babylonians and to have been first brought into Greece by Pythagoras. (17) Arithmetical progressions were treated by the Pythagoreans, and it appears from a passage in Lucian that Pythagoras himself had considered the special case of triangular numbers: Pythagoras asks some one, “How do you count?" He replies, “One, two, three, four." Pythagoras, interrupting, says, “Do you see? what you take to be four, that is ten and a perfect triangle and our oath."' (18) The odd numbers were called by the Pythagoreans “ gnomons, "' and were regarded as generating, inasmuch as by the addition of successive gnomons-consisting each of an odd number of unit squares—to the original square unit or monad the square form was preserved. (19) In like manner, if the simplest oblong (éreponrmes), consisting of two unit squares or monads in juxtaposition, be taken and four unit squares be placed about it after the manner of a gnomon, and then in like manner six, eight unit squares be placed in succession, the oblong form will be preserved. (20) Another of his doctrines was, that of all solid figures the sphere was the most beautiful, and of all plane figures the circle.” (21) According to Iamblichus the Pythagoreans are said to have found the quadrature of the circle.” him as a symbol of health. It is said to have obtained its special name from the letters u, -y 1, 0 (=¢i), a. having been written at its prominent vertices.
De arch. ix.; Praef. 5, 6, 7. Amongst other authorities are Diogenes Laertius (viii. 11), Proclus (op. cit., p. 426), and Plutarch (ut supra, 6). Plutarch, however, attributes to the Egyptians the knowledge of this theorem in the particular case where the sides are 3, 4, and 5 (De Is. et Osir. c. 56),
2 Heron Alex. Geam. et stereom. rel., ed. F. Hultsch, pp. 56, 146; Procl. op. cit. p. 428. The method of Pythagoras is as follows: he took an odd number as the lesser side; then, having squared this number and diminished the square by unity, he took half the remainder as the greater side, and by adding unity to this number he obtained the hypotenuse, e.g. 3, 4, 5; 5, 12, 13. Nicom. Ger. Inlrod. Ar. c. xxii.
In Nicomachi arithmelicam, ed. S. Tennulius, p. 141. 5 Op. cil. p. 168. As an example of this proportion Nicomachus and, after him, lamblichus give the numbers 6, 8, 9, 12, the harmonica and arithmetical means between two numbers forming a geometric proportion with the numbers themselves az-2213-2:-@:b a-H2 2
Iamblichus further relates (loc. cit.) that many Pythagoreans made use of this proportion, as Aristaeus of Crotona, Timaeus of Locri, Philolaus and Archytas of Tarentum and many others, and after them Plato in his Timaeus (see Nicom. Inst. arithm., ed. Ast, p. 153, and Animadversiones, pp. 327-329; and Iambl. op. cit. p. 172 se .).
° Bfwv 1rp&'0'LS, 4, l. 317, Cd. C. I3COlI)ltZ. 7 Fvdipwv means that by which anything is known or “ criterion "; its oldest concrete signification seems to be the carpenter's square (norma) by which a right angle is known. Hence it came to denote a perpendicular, of which, indeed, it was the archaic name (Proclus, op. cit. p. 283). Gnonion is also an instrument for measuring altitudes, by means of which the meridian can be found; it denotes, further, the index or style of a sundial, the shadow of which points out the hours. In geometry it means the square or rectangle about the diagonal of a square or rectangle, together with the two complements, on account of the resemblance of the figure to a carpenter's square; and then, more generally, the similar figure with regard to any parallelogram, as defined by Euclid II. def. 2. Again, in a still more general signification, it means the figure which, being added to any figure, preserves the original form. See Heron, Dejiniliones (59). Vi/hen gnomons are added successively in this manner to a square monad, the first gnomon may be regarded as that consisting of three square monads, and is indeed the constituent of a simple Greek fret; the second of five square monads, &c.; hence we have the gnomonic numbers.
3 Diag. Laért. De vit Pyth. viii. 19.
Simplicius, In Arislotelis physicorum libro.: quattuor priores comment aria, ed. H. Diels, p. 60.
On examining the purely geometrical work of Pythagoras and his early disciples, as given in the preceding extracts, we observe that it is much concerned with the geometry of areas, and we are indeed struck with its E yptian character. This appears in the theorem (3) concerning the Elling up a plane with regular figures for floors or walls covered with tiles of various colours were common in Egypt; in the construction of the regular solids (8), for some of them are found in Egyptian architecture; in the problems concerning the application of areas (5); and lastly, in the theorem of Pythagoras (II), coupled with his rule for the construction of rightanglcd triangles in numbers (12). We learn from Plutarch that the Egyptians were acquainted with the geometrical fact that a. triangle whose sides contain three, four and five parts is right angled, and that the square of the greatest side is equal to the squares of the sides containing the right angle. It is probable too that this theorem was known to them in the simple case where the right-angled triangle is isosceles, inasmuch as it would be at once suggested by the contemplation of a floor covered with square tiles -the square on the diagonal and the sum of the squares on the sides contain each four of the right-angled triangles into which one of the squares is divided by its diagonal. It is easy now to see how the problem to construct a square which shall be equal to the sum of two squares could, in some cases, be solved numerically. From the observation of a chequered board it would be perceived that the element in the successive formation of squares is the gnomon or carpenter's square. Each gnomon consists of an odd number of squares, and the successive gnomons correspond to the successive odd numbers, and include, therefore, all odd squares. Suppose, now, two squares are given, one consisting of sixteen and the other of nine unit squares, and that it is proposed to form from them another square. It is evident that the square consisting of nine unit squares can take the form of the fourth gnomon, which, being placed round the former square, will generate a new square containing twenty-five unit squares. Similarly it may have been observed that the twelfth gnomon, consisting of twenty-five unit squares, could be transformed into a square each of whose sides contains five units, and thus it may have been seen conversely that the latter s uare, by taking the gnomonic or generating form with respect to flue square on twelve units as base, would produce the square of thirteen units, and so on. This method required only to be generalized in order to enable Pythagoras to arrive at his rule for finding right-angled triangles whose sides can be expressed in numbers, which, we are told, sets out from the odd numbers. The nth square together with the nth gnomon forms the (n+1)th square; if the nth gnomon contains m' unit squares, rn being an odd number, we have 2n-I-1 =m2, . '.n=§ (m2-1), which gives the rule of Pythagoras.
The general proof of Euclid I. 47 is attributed to Pythagoras, but we have the express statement of Proclus (op. cil. p. 426) that this theorem was not proved in the first instance as it is in the Elements. The following simple and natural way of arriving at the theorem is suggested by Bretschneider after Camerer.1° A square can be dissected into the sum of two squares and two equal rectangles, as in Euclid II. 4; these two rectangles can, by drawing their diagonals, be decomposed into four equal right-angled triangles, the sum of the sides of each being equal to the side of the square; again, these four right-angled triangles can be placed so that a vertex of each shall be in one of the corners of the square in such a way that a greater and less side are in continuation. The original square is thus dissected into the four triangles as before and the figure within, which is the square on the hypotenuse. This square, therefore, must be equal to the sum of the squares on the sides of the right-angled triangle.
It is well known that the Pythagoreans were much occupied with the construction of regular polygons and solids, which in their cosmology played an essential part as the fundamental forms of the elements of the universe. We can trace the origin of these mathematical speculations in the theorem (3) that “the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons." Plato also makes the Pythagorean Timaeus explain—“ Each straight-lined figure consists of triangles, but all triangles can be dissected into rectangular ones which are either isosceles or scalene. Among the latter the most beautiful is that out of the doubling of which an equilateral arises, or in which the square of the greater perpendicular is three times that of the smaller, or in which the smaller perpendicular is half the hypotenuse. But two or four right-angled isosceles triangles, properly put together, form the square; two or six of the most beautiful scalene right-angled triangles form the equilateral triangle; and out of these two figures arise the solids which correspond with the four elements of the real world, the tetrahedron, octahedron, icosahedron and the cube ” 11 (Timaeus, 53, 54, 55). The construction of the regular solids is distinctly ascribed to Pythagoras himself by Eudemus (8). Of these five 1° See Bretsch. Die Georn. 'vor Enklides, p. 82; Camerer, Euclidis elem. i. 444, and the references given there.
“ The dodecahedron was assigned to the fifth element, quinta pars, aether, or, as some think, to the universe. (See PHILOLAUS.)