144 
PYTHAGOREAN GEOMETRY 
namely (1) that, if n be the number of the sides or angles, the the 
interior angles of the polygon are together equal to 2n~ 4 at 
right angles, and (2) that the exterior angles of the polygon the 
(being the supplements of the interior angles respectively) fine] 
are together equal to four right angles. The propositions are But 
interdependent, and Aristotle twice quotes the latter. 1 The sacr 
Pythagoreans also discovered that the only three regular in o: 
polygons the angles of which, if placed together round a com- men 
mon point as vertex, just fill up the space (four right angles) the 
round the point are the equilateral triangle, the square, and who 
the regular hexagon. 
” ® { j 
(f3) The '‘Theorem of Pythagoras’ (= End. I. 47). we r 
gora 
Though this is the proposition universally associated by ( p gC( 
tradition with the name of Pythagoras, no really trustworthy obse: 
evidence exists that it was actually discovered by him. The writ 
comparatively late writers who attribute it to him add the mos l 
story that he sacrificed an ox to celebrate his discovery. ^\ e 8 
Plutarch 2 (born about a.d. 46), Athenaeus 3 (about a.d. 200), 
and Diogenes Laertius 4 (a.d. 200 or later) all quote the verses Jt 
of Apollodorus the ‘calculator’ already referred to (p. 133). the i 
But Apollodorus speaks of the ‘ famous theorem ’, or perhaps i n 
‘ figure ’ (ypdfifj.a), the discovery of which was the occa- 
sion of the sacrifice, without saying what the theorem was. the 1 
Apollodorus is otherwise unknown; he may have been earlier expr 
than Cicero, for Cicero 5 tells the .story in the same form then 
without specifying what geometrical discovery was meant, accej 
and merely adds that he does not believe in the sacrifice, squa 
because the Pythagorean ritual forbade sacrifices in which 0 f ar 
blood was shed, Vitruvius 6 (first century b.c.) connects the to de 
sacrifice with the discovery of the property of the particular posit 
triangle 3, 4, 5. Plutarch, in quoting Apollodorus, questions it wa 
whether the theorem about the square of the hypotenuse was Xr 
meant, or the problem of the application of an area, while in reliec 
another place 7 he says that the occasion of the sacrifice was put a 
1 An. Post. i. 24, 85 b 88 ; ih. ii. 17, 99 a 19. 18 
2 Plutarch, Non posse suaviter vivi secundum Epicurum, c. 11, p. 1094 b. 
3 Athenaeus x. 418 f. 4 Diog. L. viii. 12, i. 25. 1 P< 
5 Cicero, De not. deor. iii. 36, 88. 2 H 
6 Vitruvius, De architecturei, ix. pref. 3 B 
7 Plutarch, Quaest. conviv. viii. 2, 4, p. 720 a. pp. 54 
152;