The evidence has been carefully studied by Heath in his "Euclid," who concludes with these words: "On the whole, therefore, I see no sufficient reason to question the tradition that, so far as Greek geometry is concerned ..., Pythagoras was the first to introduce the theorem ... and to give a general proof of it." That the fact was known earlier, probably without the general proof, is recognized by all modern writers. [Illustration] Pythagoras had studied in Egypt and possibly in the East before he established his school at Crotona, in southern Italy. In Egypt, at any rate, he could easily have found that a triangle with the sides 3, 4, 5, is a right triangle, and Vitruvius (first century B.C.) tells us that he taught this fact. The Egyptian _harpedonaptae_ (rope stretchers) stretched ropes about pegs so as to make such a triangle for the purpose of laying out a right angle in their surveying, just as our surveyors do to-day. The great pyramids have an angle of slope such as is given by this triangle. Indeed, a papyrus of the twelfth dynasty, lately discovered at Kahun, in Egypt, refers to four of these triangles, such as 1^2 + (3/4)^2 = (1 1/4)^2. This property seems to have been a matter of common knowledge long before Pythagoras, even as far east as China. He was, therefore, naturally led to attempt to prove the general property which had already been recognized for special cases, and in particular for the isosceles right triangle. How Pythagoras proved the proposition is not known. It has been thought that he used a proof by proportion, because Proclus says that Euclid gave a new style of proof, and Euclid does not use proportion for this purpose, while the subject, in incomplete form, was highly esteemed by the Pythagoreans. Heath suggests that this is among the possibilities: [Illustration] [triangles]_ABC_ and _APC_ are similar. [therefore] _AB_ x _AP_ = (_AC_)^2. Similarly, _AB_ x _PB_ = (_BC_)^2. [therefore] _AB_(_AP_ + _PB_) = (_AC_)^2 + (_BC_)^2, or (_AB_)^2 = (_AC_)^2 + (_BC_)^2. Others have thought that Pythagoras derived his proof from dissecting a square and showing that the square on the hypotenuse must equal the sum of the squares on the other two sides, in some such manner as this: [Illustration: FIG. 1] [Illustration: FIG. 2] Here Fig. 1 is evidently _h_^2 + 4 [triangles]. Fig. 2 is evidently _a_^2 + _b_^2 + 4 [triangles]. [therefore] _h_^2 + 4 [triangles] = _a_^2 + _b_^2 + 4 [triangles], the [triangles] all being congruent. [therefore] _h_^2 = _a_^2 + _b_^2. The great Hindu mathematician, Bhaskara (born 1114 A.D.), proceeds in a somewhat similar manner. He draws this figure, but gives no proof. It is evident that he had in mind this relation: [Illustration] _h_^2 = 4 . _ab_/2 + (_b_ - _a_)^2 = _a_^2 + _b_^2. A somewhat similar proof can be based upon the following figure: [Illustration]