84 
PYTHAGOREAN ARITHMETIC 
figure, the sides of the triangles differ by unity, and of course 
/ ^n{n— l) + ^n{n + 1) = 7l 2 . 
• • -yi • Another theorem connecting triangular num- 
• ' ' hers and squares, namely that 8 times any 
Y' ’ * ' triangular number + 1 makes a square, may 
easily go back to the early Pythagoreans. It is 
quoted by Plutarch 1 and used by Diophantus, 2 and is equi 
valent to the formula 
8 .\n (n + 1) + 1 = 4n (n+ 1) + 1 = (2n+ l) 2 . 
It may easily have been proved by means of a figure 
made up of dots in the usual way. Two 
equal triangles make up an oblong figure 
of the form n {n + 1), as above. Therefore 
— * ' ’ we have to prove that four equal figures 
. ~. . ■ of this form with one more dot make up 
(2a+ l) 2 . The annexed figure representing 
7 2 shows how it can be divided into four 
‘ oblong ’ figures 3.4 leaving 1 over. 
In addition to Speusippus, Philippus of Opus (fourth 
century), the editor of Plato’s Laws and author of the Epi- 
nomis, is said to have written a work on polygonal numbers." 
Hypsicles, who wrote about 170 b.c., is twice mentioned in 
Diophantus’s Polygonal Numbers as the author of a ‘ defini 
tion ’ of a polygonal number. 
The theory of proportion and means. 
The * summary ’ of Proclus (as to which see the beginning 
of Chapter IV) states (if Friedlein’s reading is right) that 
Pythagoras discovered ‘ the theory of irrationals {rgv tmu 
dXSycov npayyaTelav) and the construction of the cosmic 
figures’ (the five regular solids). 4 We are here concerned 
with the first part of this statement in so far as the reading 
dXoycor (‘ irrationals ’) is disputed. Fabricius seems to have 
been the first to record the variant dvaXbycnv, which is also 
noted by E. F. August 5 ; Mullach adopted this reading from 
1 Plutarch, Plat. Quaest. v. 2. 4, 1003 F. 2 Dioph. IV. 38. 
3 Bt«ypd(/)o(, Vitarum scriptores Graeci minores, ed. Westermann, p. 446. 
4 Proclus on End. I, p. 65. 19. 
5 In his edition of the Greek text of Euclid (1824-9), vol. i, p. 290.