METHOD OF APPROXIMATION TO LIMITS 479 
therefore y — 2, and 1 /ai 1 — ; and 3-| + J,- = 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 +(|) 2 + (f) 2 . 
Bringing 3,f, | and to a common denominator, we have 
90 18 24 „ Tlr ] 55 
3 0 > 3 0’ 30 3 0 ’ 
and 3 > |§ by |f, 
• 2. < 1S bv Til 
5 ^ 30 V 3 0’ 
4/55 31 
■5 <• "Jo °y -3Ô* 
If now we took 3—If, f +, f + ff as the sides of squares, 
the sum of the squares would be 3 (^g 1 -) 2 or -\ 6 g 3 -, which is >10. 
Accordingly we assume as the sides 3 — 35 x, § +37 x, f + 31 æ, 
where x must therefore be not exactly ^ but near it. 
Solving (3 — 35a;) 2 + (| + 37 x) 2 + (f + 31a;) 2 = 10, 
or 10—116a; + 3555a; 2 = 10, 
we find x = -isW ; 
thus the sides of the required squares are -Vrrn Vtt“ > VttS 
the squares themselves are VoW/rS WsWt“’ 
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 thatand some other pro 
positions 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 (Y. 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 —l = 2mn, or 
m — n = ± 1.]