The four triangles _T_ can be proved congruent. Then if we take from the whole figure _T_ and _T'_, we have left the squares on the two sides of the right angle. If we take away the other two triangles instead, we have left the square on the hypotenuse. Therefore the former is equivalent to the latter. A proof attributed to the great artist, Leonardo da Vinci (1452-1519), is as follows: [Illustration] The construction of the following figure is evident. It is easily shown that the four quadrilaterals _ABMX_, _XNCA_, _SBCP_, and _SRQP_ are congruent. [therefore] _ABMXNCA_ equals _SBCPQRS_ but is not congruent to it, the congruent quadrilaterals being differently arranged. Subtract the congruent triangles _MXN_, _ABC_, _RAQ_, and the proposition is proved.[80] The following is an interesting proof of the proposition: Let _ABC_ be the original triangle, with _AB_ < _BC_. Turn the triangle about _B_, through 90 deg., until it comes into the position _A'BC'_. Then because it has been turned through 90 deg., _C'A'P_ will be perpendicular to _AC_. Then 1/2(_AB_)^2 = [triangle]_ABA'_, and 1/2(_BC'_)^2 = [triangle]_BC'C_, because _BC_ = _BC'_. [therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_ABA'_ + [triangle]_BC'C_. [therefore] 1/2((_AB_)^2 + (_BC_)^2) = [triangle]_AC'A'_ + [triangle]_A'C'C_ [Illustration] (For [triangle]_ABA'_ + [triangle]_BC'A'_ + [triangle]_A'C'C_ is the second member of both equations.) = 1/2_A'C'_ . _AP_ + 1/2_A'C'_ . _PC_ = 1/2_A'C'_ . _AC_ = 1/2(_AC_)^2. [therefore] (_AB_)^2 + (_BC_)^2 = (_AC_)^2. 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: _n_^2 + ((_n_^2 - 1)/2)^2 = ((_n_^2 + 1)/2)^2,