506 Safford—Combinations of Pythagorean Triangles. 
are 84 and 36, the sum and difference of the two areas of the 
right angled triangles 60 and 24 respectively. 
This series of processes can be extended to as many pairs of 
Pythagorean triangles as we please, provided always that the 
common leg required be brought about by multiplying the parts 
of one triangle by a whole number. 
In Table I, I give a series of Pythagorean triangles, computed 
by Sir George B. Airy, the late astronomer royal at Greenwich, 
(Nature, 33, 532.) This table seems to me to give a sufficient 
selection of Pythagorean triangles for practical purposes. It 
can be extended by the formulae 
x — 2 fab 
y =/ (a?—b 2 ) 
z=/(a 2 +& 2 ) 
in which a and b are two numbers relatively prime, and / is 
any whole number or, if both a and b are odd, is the half of 
any whole number. Thus if a— 7, 6=1, f—\ we shall have 
a = 21 
y = 72 
2 = 75 
In Table II, I give a selection of triangles whose areas are 
whole numbers derived from the combination of those given in 
Table I. 
They can be readily tested, and to indicate the method I 
give the calculation of Triangle No. 14, whose sides are 41, 51, 
and 58. 
a — 41 
(s-a) = 34 
tan y 2 A = f 
y 2 A =21° 48' 5' 
6 = 51 
(s-b) = 24 
tan^^ = ^ 
= 32 19 
c= 58 
(s-c) = 17 
tan y 2 C = £ 
y 2 c = 38 29 35 
8 = 75 
Hence r = 
4/34.24.17 
75 & 
Sum of half angles, 89° 59' 59' 
In computing the angles I used a five figure table of natural 
tangents, which is usually sufficient to give the sum of the 
