COMBINATIONS OF PYTHAGOREAN TRIANGLES AS 
GIVING EXERCISES IN COMPUTATION. 
BY TBUMAN HENRY SAEFORD. 
It has been the author’s habit to practice his pupils in compu¬ 
tation, in order to test and increase their numerical skill, by- 
teaching them to form and calculate triangles whose sides are ex¬ 
pressed in whole numbers and whose areas are also so expressed. 
The simplest way to obtain such triangles is to combine Pytha¬ 
gorean triangles after the example of Hero of Alexandria, emi¬ 
nent as an engineer as well as a mathematician. Hero was 
the inventor of the Hioptra, an instrument containing the germ 
of the theodolite. He was also the inventor of the Aeolopile, 
a precursor of the steam engine, and in all probability also the 
inventor of the well known formula for the area of a triangle 
whose sides are given. 
Among his works is found the remarkable triangle whose 
sides are 13, 14, and 15, and whose area is 84. Hero put to¬ 
gether this triangle by combining the two Pythagorean triangles 
whose sides are 5, 12, and 13 and 9, 12, and 15 respectively. 
The area of the combined triangle 84 is the sum of 30 and 54, 
the areas of the two Pythagoreans. A combination of the same 
two Pythagoreans can be made in another way, giving the tri¬ 
angle whose sides are 4, 13, and 15 and whose area is 24, the 
difference of the areas 30 and 54. These methods can be em¬ 
ployed with any pair of Pythagorean triangles which have a 
common leg; in the case of Hero’s pair 12 is the common leg. 
We may apply Hero’s method to the two Pythagorean tri¬ 
angles whose sides are 10, 8, and 6 and 17, 15, and 8 respect¬ 
ively. We thus get the triangles whose sides are 10, 17, and 
21 and 9, 10, and 17 respectively. The areas by Hero’s formula 
