PYTHAGOREAN See also:

• GEOMETRY

65. 3 See also:

• MOERIS, AELIUS
• MOERIS, LAKE OF

25) states, on the authority of See also:

• FAVORINUS (2nd century A.D.)

Props. 28, 29—are old, and inventions of the Pythagoreans? (ibid. p. 419). (6) This is confirmed by See also:

• PLUTARCH
• PLUTARCH (Gr. llXoi rapxos) (c. A.D. 46–120)

6 We learn from a fragment of Geminus, which has been handed down by Eutocius in his commentary on the Conics of See also:

• APOLLONIUS

2, c. 4. n See also:

• IAMBLICHUS
• IAMBLICHUS (d. c. A.D. 330)

This figure is referred to Pythagoras himself, and in the See also:

• MIDDLE

Amongst other authorities are Diogenes Laertius (viii. u), 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. Geom. et stereom. rel., ed. F. Hultsch, pp. 56, 146; Prod . op. Cit. p. 428. The method of Pythagoras is as follows: he took an odd number as the lesser See also:

• SIDE
• SIDE (mod. Eski Adalia)

3, 4, 5; 5, 12, 13. Nicom. Ger. Intred. Ar. c. xxii. 4 In Nicomachi arithmelicam, ed. S. Tennulius, p. 141. 6 Op. cit. p. 168. As an example of this proportion See also:

• NICOMACHUS

Inst. arithm., ed. See also:

• AST, GEORG ANTON FRIEDRICH (1778-1841)

283). Gnomon is also an See also:

• INSTRUMENT (Lat. instrumentum, from insincere, to build up, furnish, arrange, prepare)

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 See also:

• EGYPTIAN

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 See also:

• UNITS, DIMENSIONS OF
• UNITS, PHYSICAL

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 See also:

• WORLD

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 would 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 See also:

• LEAD
• LEAD (pronounced iced)

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 See also:

• PROCESS

Analyt. Pr. i. c. 23 and c. 44. 2 Knoche, Untersuchungen fiber die neuaufgefundenen Scholien See also:

• DES

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 homologous 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 See also:

• HIPPOCRATES

Certain conclusions may be See also:

• DRAWN

Now it is remark-able 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 See also:

• EUDOXUS