A history of Greek mathematics - Wilbourhall.org

PORISMS AND PROPOSITIONS ASSUMED 481
In fact YZ±a = {(m+l)(2m+l) + 2a} 2 ,
ZX±a = {m(2m+ l) + 2a} 2 ,
XY±a= {.m(m+l) + a} 2 .
(3) If
X = m 2 + 2, F=(m+l)*+2, Z = 2{m 2 + (m + 1) 2 + 1 ]
+
then the six expressions
F^_(F+if), #X-(#+X), XY-(X+Y)
}
YZ-X, ZX-Y, XY-Z
are all squares (V. 6).
In fact
2,
YZ- (Y + Z) = {2m 1 + 3m + 3)
2
, F^-I=(2m
2 + 3m + 4) 2 ;
&c.
2. The second class is much more important, consisting of
propositions in the Theory of Numbers which we find first
stated or assumed in the Arithmetica. It was in explanation
or extension of these that Fermat's most famous notes
were written. How far Diophantus possessed scientific proofs
of the theorems which he assumes must remain largely a
matter of speculation.
(a)
Theorems on the composition of numbers as the sum
of two squares.
(1) Any square number can be resolved into two squares in
any number of ways (II. 8).
(2) Any number which is the sum of two squares can be
resolved into two other squares in any number of ways (II. 9).
(It is implied throughout that the squares may be fractional
as well as integral.)
(3) If there are two whole numbers each of which is the
sum of two squares, the product of the numbers can be
resolved into the sum of two squares in two ways.
In fact (a 2 + b 2 ) (c2 + d 2 )
= (ac ± bd) 2 + {ad + be) 2 .
This proposition is used in III. 19, where the problem is
to find four rational right-angled triangles with the same
1523.2 I i