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EUCLID 
9. a 2 + h 2 = 2[{|(a + 6)} 2 + {£ {a + b)—6} 2 ], 
or (a + /3) 2 -f-(a-/3) 2 = 2(a 2 + /3 2 ), 
10. (2a + h) 2 +b 2 = 2 (a 2 + (a + 6) 2 }, 
or (a + /3) 2 + (/3 — a) 2 = 2 (a 2 + /3 2 ), 
As we have seen (pp. 151-3), Propositions 5 and 6 enable us 
to solve the quadratic equations 
z,v 2 7 9 x + y = a) 
(1) ax — x 2 — b- or u 
w xy = 6 2 ) 
and (2) ax + x 2 = b 2 or ^ * ~ f l. 
' xy = 6 2 ) 
The procedure is geometrical throughout; the areas in all 
the Propositions 1-8 are actually shown in the figures. 
Propositions 9 and 10 were really intended to solve a problem 
in numbers, that of finding any number of successive pairs 
of integral numbers (‘ side- ’ and ‘ diameter- ’ numbers) satisfy 
ing the equations 
2 x 2 —y 2 = + 1 
(see p. 93, above). 
Of the remaining propositions, II. 11 and II. 14 give the 
geometrical equivalent of solving the quadratic equations 
x 2 + ax = a 2 
and x 2 = ab, 
while the intervening propositions 12 and 13 prove, for any 
triangle with sides a, b, c, the equivalent of the formula 
a 2 = b 2 + c 2 — 2bc cos A., 
It is worth noting that, while I. 47 and its converse con 
clude Book I as if that Book was designed to lead up to the 
great proposition of PjThagoras, the last propositions but one 
of Book II give the generalization of the same proposition 
with any triangle substituted for a right-angled triangle. 
The subject of Book III is the geometry of the circle, 
including the relations between circles cutting or touching 
each other. It begins with some definitions, which are