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Who Proved Pythagoras’s Theorem?

Socrates: Try to tell me then how long a line you say it is. — Three feet.

—Plato, Meno, 83 e.

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Notes

  1. According to a scholium attributed to Proclus, proportion theory from Euclid’s Book V is the work of Eudoxus [5, vol. 2, p. 112].

  2. For the concept of geometric algebra, see [8] and [10, pp. 34–48].

  3. The case of rational side lengths is easily reduced to the case of sides of integer length. It is merely a matter of the selection of a segment of unit length.

  4. In lemma 1 to proposition X.28 of the Elements, Euclid records a method for generating Pythagorean triples. Proclus discusses this issue in [14, p. 340].

  5. In Plato’s Meno (82b–85b), Socrates leads young Meno’s slave to the conclusion that the square whose side is a diagonal of a given square is twice as large as that square.

  6. Book X is about such triples. Judging by Euclid’s definitions X. def. III.1–6, an apotome is the difference of the hypotenuse and a leg of a right triangle determined by a triple that is of the form \((\sqrt{p},\sqrt{q},\sqrt{r})\). Binomials (X. def. II.1–6) are their sums.

  7. For many examples of such triangles, see [11, pp. 314–327].

  8. For the definition of the measure of a segment, see [3, pp. 167–172].

  9. Van der Waerden believed that in the Neolithic age there must have been an established doctrine of Pythagorean triples and their ritual applications [17, p. 25].

  10. Because if a, b, c are the lengths of the sides, then \(a^2\), \(b^2\), \(c^2\) are the areas of the squares on them [12, p. 166].

  11. For the notions of equidecomposability and equicomplementability, see [7, §18].

  12. In some manuscripts of the Elements, among Common Notions there is one that could be applied directly at this place. That is: “Things that are halves of the same thing are equal to each other” [5, vol. I, p. 223].

  13. For a proof that two triangles are equicomplementable if they have equal bases and equal altitudes, see [7, §19, Th. 27].

  14. Propositions VI.14–17 are about the relation between equality of parallelograms or triangles and the sameness of ratios of their sides.

  15. For Eudemus’s report on the answer to the question, see [15, vol. I, pp. 235–239], [6, vol. I, pp. 191–192], and [9, pp. 163–167].

  16. This notion was far beyond the reach of Greek mathematics. For the definition of the area function and areas of polygonal regions, see [12, pp. 153–175].

  17. For many different visually obvious proofs of the theorem, see [5, vol. I, pp. 364–366] and [6, vol. I, p. 149].

  18. That “geometric algebra” is a “monstrous concept” that “leads to absurdities” is argued in [16]. That a “geometric algebra” interpretation should be reinstated as a viable historical hypothesis is argued in [2].

  19. Since nobody else has ever been mentioned in ancient literature as a potential author of the theorem, there is no reason to doubt that the early Pythagorean proof is Pythagoras’s.

  20. For a complete proof, see [3, pp. 167–176, 273].

  21. Diogenes Laërtius informs us that Eudoxus was associated with Plato’s Academy [4, 8.86–88].

  22. We could come to the conclusion that the proof was possible only in the period between Eudoxus and Euclid even without Proclus’s explanation. Proclus provides further information about the authorship of the proposition.

  23. Such a proof is attributed by an-Nairizi to Thābit ibn Qurra [5, vol. I, pp. 364–365].

References

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Lučić, Z. Who Proved Pythagoras’s Theorem?. Math Intelligencer (2022). https://doi.org/10.1007/s00283-022-10205-x

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