A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

METHOD OF APPROXIMATION TO LIMITS 479
therefore y = 2, and 1 /x 2 = 3%
; and Sj + ^g = ±jg-, a square.
We have now, says Diophantus, to divide 10 into three
squares with sides as near as may be to -^.
Now 10 = 9 + 1 =3 2 + (f)
2
+ (f)
2.
Bringing 3, f- , f and ^ to a common denominator, we have
90 18 24 onrl 55
30 > 3~0> 3~0 d/11U 3 '
and 3 > || by §f
5 ^ 3 u J 30'
5" < "30 °y 30'
If now we took 3 — §J, f + |J , § + f§ as the sides of squares,
the sum of the squares would be 3 (V) 2 or " 3 3-
3%
>
which is > 10.
Accordingly we assume as the sides 3 — 35 #, § + 3 7 &•, f + 3 1 #,
where a? must therefore be not exactly 3V but near it.
Solving (3-35 l£) 2 + (! + 37£) 2 + (f
+ 31^) 2 = 10,
or 10-116^ + 3555^2 = 10,
we find x = ^A 5
thus the sides of the required squares are -VtiN ^ttt* VttS
the squares themselves are WA%
4
iS T5r£rr> VSfiflRtf-
Other instances of the application of the method will be
found in V. 10, 12, 13, 14.
Porisms and propositions in the Theory of Numbers.
I. Three propositions are quoted as occurring in the Porisms
(' We have it in the Porisms that ...'); and some other propositions
assumed without proof may very likely have come
from the same collection. The three propositions from the
Porisms are to the following effect.
1. If a is a given number and x, y numbers such that
x + a = m 2 , y + a = n 2 , then, if xy + a is also a square, m and n
differ by unity (V. 3).
[From the first two equations we obtain easily
xy + a = m 2 n 2 — a (m 2 + n 2 — 1) + a 2 ,
and this is obviously a square if m 2 + n 2 — 1 = 2 mn, or
m — n = ±1-]
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).