The Pythagorean Theorem, as it is generally called, has had other names. It is not uncommonly called the pons asinorum (see [page 174]) in France. The Arab writers called it the Figure of the Bride, although the reason for this name is unknown; possibly two being joined in one has something to do with it. It has also been called the Bride's Chair, and the shape of the Euclid figure is not unlike the chair that a slave carries on his back, in which the Eastern bride is sometimes transported to the wedding ceremony. Schopenhauer, the German philosopher, referring to the figure, speaks of it as "a proof walking on stilts," and as "a mouse-trap proof."
An interesting theory suggested by the proposition is that of computing the sides of right triangles so that they shall be represented by rational numbers. Pythagoras seems to have been the first to take up this theory, although such numbers were applied to the right triangle before his time, and Proclus tells us that Plato also contributed to it. The rule of Pythagoras, put in modern symbols, was as follows:
n2 + ((n2 - 1)/2)2 = ((n2 + 1)/2)2,
the sides being n, (n2 - 1)/2, and (n2 + 1)/2. If for n we put 3, we have 3, 4, 5. If we take the various odd numbers, we have
n = 1, 3, 5, 7, 9, ···,
(n2 - 1)/2 = 0, 4, 12, 24, 40, ···,
(n2 + 1)/2 = 1, 5, 13, 25, 41, ···.
Of course n may be even, giving fractional values. Thus, for n = 2 we have for the three sides, 2, 1-1/2, 2-1/2. Other formulas are also known. Plato's, for example, is as follows:
(2n)2 + (n2 - 1)2 = (n2 + 1)2.
If 2n = 2, 4, 6, 8, 10, ···,
then n2 - 1 = 0, 3, 8, 15, 24, ···,
and n2 + 1 = 2, 5, 10, 17, 26, ···.
This formula evidently comes from that of Pythagoras by doubling the sides of the squares.[81]
Theorem. In any triangle the square of the side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides by the projection of the other upon that side.
Theorem. A similar statement for the obtuse triangle.