Pythagorean theorem

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately."

"The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately."

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."

"Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him."

"‘These operations are all founded on a very distinct conception of what happens in the case of an eclipse, and on the knowledge of this theorem, that, in a right-angled triangle, the square on the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of PYTHAGORAS in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as the visionary speculations, with which he delighted to instruct or amuse his disciples.’"

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"If we consider the results obtained together, we will not be able to doubt the conclusion to be drawn from them. The ancient priestly geometry of the Indians not only knew the Pythagorean theorem, but it even played the main role in their calculations; with its help, they constructed elements that the Greeks found in a completely different way; with its help, they also found the irrational quantities. And it was precisely these two things that Pythagoras introduced into the Greek-Italian world; these two things, according to the Greeks, he invented. Indeed, even more! The way in which Pythagoras proved his theorem was also, in all likelihood, the same as that which we find in the Vedic Shulba Sutras. After examining the Shulba Sutras, we could have said: If Pythagoras really was in India, as we previously suggested, and initiated himself into the priestly wisdom of the Brahmins, then he could have brought precisely these theorems of geometric science to Greece; — and history has been telling us for several millennia now that this was indeed the case!"