516 
DIOPHANTUS OF ALEXANDRIA 
4. The vertical columns are in arithmetical progression, the 
common difference being the triangle in the preceding column. 
Plutarch, a contemporary of Nicomachus, mentions another 
method of transforming triangles into squares- Every tri 
angular number taken eight times and then increased by 1 
gives a square. 
In fact, 8 . %n{n + 1) + 1 = .{2n+ l) 2 . 
Only a fragment of Diophantus’s treatise On Polygonal 
Numbers survives. Its character is entirely different from 
that of the Arithmetica. The method of proof is strictly 
geometrical, and has the disadvantage, therefore, of being long 
and involved. He begins with some preliminary propositions 
of which two may be mentioned. Prop. 3 proves that, if a be 
the first and l the last term in an arithmetical progression 
of n terms, and if s is the sum of the terms, 2s — n(l +a). 
Prop. 4 proves that, if 1, 1 +b, 1 + 2b,... 1 + (n— 1)6 be an 
A. P., and s the sum of the terms, 
2s = n {2 + {n— 1)6}. 
The main result obtained in the fragment as we have it 
is a generalization of the formula 8 . \n(n + 1) + 1 = (2 n + l) 2 . 
Prop. 5 proves the fact stated in Hypsicles’s definition and also 
(the generalization referred to) that 
8 P (a — 2) + (a — 4) 2 = a square, 
where P is any polygonal number with a angles. 
It is also proved that, if P be the nth a-gonal number 
(1 being the first), 
8P(a-2) + (a-4) 2 = {2 + (2n- 1) (a-2)} 2 . 
Diophantus deduces rules as follows. 
]. To find the number from its side. 
p - ' 2 + ( 2n ~ 1) (ft-2)} 2 -(q —4) 2 _ 
~ ‘ ’ 8 (a — 2) 
2. To find the side from the number.