A history of Greek mathematics - Wilbourhall.org

480 DIOPHANTUS OF ALEXANDRIA
2. If 7Tb 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+ 1), 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 2
+ m + 1)
XY
5
YZ+Y+Z = (2m 2 + 3m+3) 2 , YZ+X
ZX+Z+X = (2m 2 + m + 2) 2 ,
ZX
+ Z = (m 2 + m + 2)\
= (2m 2 + 3m + 2) 2 ,
+ Y = (2m 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 showing
how to make the transformation. The subject of the
transformation of sums and differences of cubes was investigated
by Vieta, 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 + b)} 2 — ab and a 2 (a+ l) 2 + a 2 + (a+ l) 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
PorismSy e.g.
(1) If X z=a 2 x + 2a, Y ={a+\) 2 x+2(a+\) or, in other
words, if xX+ 1 = (ax+ l) 2 and xY "+ 1 = {(a+l)x+l }
2
,
then XY +1 is a square (IV. 20).
(2) If X±a = m 2 , Y±a
In fact
XY+l - {a(a + l)a? + (2a+l)} 2 .
= (m+1) 2 ,
and
Z= 2(X+F)-1,
then YZ±a, ZX±a, XY±a are all squares (V. 3, 4).