A history of Greek mathematics - Wilbourhall.org

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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-
,
90 18 24 onrl 55
30 > 3~0> 3~0 d/11U 3 '
f and ^ to a common denominator, we have
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-]