If the four triangles, 1 + 2 + 3 + 4, are taken away, there remains the square on the hypotenuse. But if we take away the two shaded rectangles, which equal the four triangles, there remain the squares on the two sides. Therefore the square on the hypotenuse must equal the sum of these two squares. [Illustration] It has long been thought that the truth of the proposition was first observed by seeing the tiles on the floors of ancient temples. If they were arranged as here shown, the proposition would be evident for the special case of an isosceles right triangle. The Hindus knew the proposition long before Bhaskara, however, and possibly before Pythagoras. It is referred to in the old religious poems of the Brahmans, the "Sulvasutras," but the date of these poems is so uncertain that it is impossible to state that they preceded the sixth century B.C.,[79] in which Pythagoras lived. The "Sulvasutra" of Apastamba has a collection of rules, without proofs, for constructing various figures. Among these is one for constructing right angles by stretching cords of the following lengths: 3, 4, 5; 12, 16, 20; 15, 20, 25 (the two latter being multiples of the first); 5, 12, 13; 15, 36, 39; 8, 15, 17; 12, 35, 37. Whatever the date of these "Sulvasutras," there is no evidence that the Indians had a definite proof of the theorem, even though they, like the early Egyptians, recognized the general fact. It is always interesting to a class to see more than one proof of a famous theorem, and many teachers find it profitable to ask their pupils to work out proofs that are (to them) original, often suggesting the figure. Two of the best known historic proofs are here given. The first makes the Pythagorean Theorem a special case of a proposition due to Pappus (fourth century A.D.), relating to any kind of a triangle. [Illustration] Somewhat simplified, this proposition asserts that if _ABC_ is _any_ kind of triangle, and _MC_, _NC_ are parallelograms on _AC_, _BC_, the opposite sides being produced to meet at _P_; and if _PC_ is produced making _QR_ = _PC_; and if the parallelogram _AT_ is constructed, then _AT_ = _MC_ + _NC_. For _MC_ = _AP_ = _AR_, having equal bases and equal altitudes. Similarly, _NC_ = _QT_. Adding, _MC_ + _NC_ = _AT_. If, now, _ABC_ is a right triangle, and if _MC_ and _NC_ are squares, it is easy to show that _AT_ is a square, and the proposition reduces to the Pythagorean Theorem. The Arab writer, Al-Nair[=i]z[=i] (died about 922 A.D.), attributes to Th[=a]bit ben Qurra (826-901 A.D.) a proof substantially as follows: [Illustration]