Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.Template:Sfn

Quotes

• 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.
  • (Baudhayana sutra 1.48; Apastambd sutra 1.4; Katyayana sutra 2.11). References from Seidenberg 1983, 98. ** in Bryant, E. F. (2001). The Quest for the Origins of Vedic Culture : the Indo-Aryan migration debate. Oxford University Press. chapter 12

• The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately.
  • Baudhayana Sulbasutra:दीर््घचतुरस्रस््ययाक्षष्णया रज््जजुुः पार्शश्वमानी तिर््यङ्मानी च यत््पपृथग््भभूते कुरुतस््तदभु यंकरोति॥१.१२॥
  • Apastamba Sulbasutra: दीर््घस््ययाक्षष्णयारज््जजुुः पार्शश्वमानी तिर््यङ्मानी च यत् पृथग््भभूते कुरुतस््तदभु यं करोति॥१.४॥
  • Amma, T. A. Sarasvati. Geometry in Ancient and Medieval India. Delhi: Motilal Banarsidass, 2007. Thibaut, G. ‘On the Sulvasutras’. Journal of the Asiatic Society of Bengal, Vol. XLIV, (1875): pp. 227. quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• 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)
  • Einige von den Dingen, welche ganz besonders der Geschichte der Mathematik angehören, werden wir allerdings nicht verzichten Pythagoras selbst zuzuschreiben. Dazu gehört der pythagoräische Lehrsatz, den wir unter allen Umständen ihm erhalten wissen wollen. (Cantor 1880: 129)
  • Cantor, Moritz. Vorlesungen über Geschichte der Mathematik, Leipzig, 1880. in The Sulbasutras and the 'Pythagoras Theorem'

• 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.”
  • Elst, Koenraad (1999). Update on the Aryan invasion debate New Delhi: Aditya Prakashan.

• 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.
  • Der erste Schriftsteller, der diesen Satz als von Pythagoras gefunden bezeichnet, ist Vitruv, schwerlich ein zuverslässiger Zeuge; von da an verbreitete sich diese Angabe allgemeiner, aber immer in Verbindung mit jener bekannten Hekatombe, welche Pythagoras vor Freude über die Entdeckung jenes Lehrsatzes dargebracht haben soll – eine Anekdote, welche die Glaubwürdigkeit der ganzen Nachricht stark beeinträchtigt. Denn dies Opfer verträgt sich nicht mit dem strengen Verbote alles blutigen Opfers, welches uns aus den pythagorischen Ritualgesetz die Schriftsteller derselben Zeit, ja oft dieselben überliefert haben, die anderswo von der Hekatombe erzählen. Schon Cicero nahm daher an jener Anekdote Anstoss und in der spätesten Tradition der Neupythagoriker wird das blutige Opfer durch das eines ‘aus Mehl geformten Ochsen’ ersetzt.Proklus, ein einsichtiger Schriftsteller, drückt sich auffalend unbestimmt so aus: ‘Wenn wir die, welche alte Geschichten erzählen wollen, hören, so finden wir, dass sie dieses Theorem auf Pythagoras zurückführen.’ Auch ihm war, wie hieraus hervorgeht, keine sichere Quelle bekannt.
  • Hermann Hankel (1839-1873) Hankel 1874: 97-98) Hankel, Hermann. Zur Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig 1874.

• 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.
  • Heath, Thomas. A History of Greek Mathematics, Vol. 1. Oxford at the Clarendon Press, 1921 , quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189 (Heath 1921: 144)

• ‘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.’
  • Playfair, John. ‘Remarks on the astronomy of the Brahmins’. Transactions of the Royal Society of Edinburgh, Vol. II. (1790): pp.135-192. quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• 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.
  • The fifth century mathematician Proclus quoted in (Morrow 1970: 337). Morrow, Glenn R. Proclus, A Commentary on the First Book of Euclid's Elements: Translated, with Introduction and Notes. Princeton University Press, 1970.

• 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.
  • Professor H. G. Rawlinson writes: (Legacy of India 1937, p. 5). attributed at

• 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!
  • Leopold von Schroeder’s Pythagoras und die Inder, 1884 quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189