480 
DIOPHANTUS OF ALEXANDRIA 
2. If m 2 , (m + l) 2 be consecutive squares and a third number 
be taken equal to 2 {m 2 + (m+l) 2 } + 2, or 4(m 2 + m+l), the 
three numbers have the property that the product of any two 
plus either the sum of those two or the remaining number 
gives a square (V. 5). 
[In fact, if X, Y, Z denote the numbers respectively, 
XY+ X +Y = (m 2 + m + 1 ) 2 , X Y + Z = (m 2 + in + 2) 2 , 
YZ +Y + Z = (2m 2 + 3m + 3) 2 , YZ+ X = (2m 2 + 3m + 2) 2 , 
ZX + Z + X = (2m 2 + m+ 2) 2 , ZX + Y = (2 m 2 + m+ l) 2 .] 
3. The difference of any two cubes is also the sum of two 
cubes, i.e. can be transformed into the sum of two cubes 
(V. 16). 
[Diophantus merely states this without proving it or show 
ing how to make the transformation. The subject of the 
transformation of sums and differences of cubes was investi 
gated by Yieta, Bachet and Fermat.] 
II. Of the many other propositions assumed or implied by 
Diophantus which are not referred to the Porisms we may 
distinguish two classes. 
1. The first class are of two sorts ; some are more or less 
of the nature of identical formulae, e.g. the facts that the 
expressions {(a -(- h)} 2 — ah and a 2 (a + 1 ) 2 + a 2 + {a + 1 ) 2 are 
respectively squares, that a (a 2 — a) + a + (a 2 — a) is always a 
cube, and that 8 times a triangular number plus 1 gives 
a square, i.e. 8 ,^x{x+ 1) + 1 = (2x + l) 2 . Others are of the 
same kind as the first two propositions quoted from the 
Poi'isms, e.g. 
(1) If X=a 2 x + 2a, Y = {a+ 1) 2 îc + 2(a+ 1) or, in other 
words, if xX + 1 = {ax + l) 2 and xY + 1 = {(a+ l)æ+ 1 } 2 , 
then XY + 1 is a square (IV. 20). In fact 
AT -f-1 = { cl (ci + 1)îc + [2 a + 1 )} 2 . 
(2) If X±a = m 2 , Y±a = (m+ l) 2 , and Z — 2(A+ F) — 1, 
then YZ±a, ZX±a, XY±a are all squares (V. 3, 4).