# 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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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  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 . That a “geometric algebra” interpretation should be reinstated as a viable historical hypothesis is argued in .

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

1. O. Becker. Warum haben die Griechen die Existenz der vierten Proportionale angenommen? (Eudoxos-Studien II). Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2(B) (1933), 369–387.

2. V. Blåsjö. In defence of geometrical algebra. Archive for History of Exact Sciences 70:3 (2016), 325–359.

3. K. Borsuk and W. Szmielew. Foundations of Geometry. North-Holland, 1960.

4. Diogenes Laërtius. Lives of the Eminent Philosophers, volumes I–II. The Loeb Classical Library, Harvard University Press, 1925.

5. T. L. Heath. The Thirteen Books of Euclid’s Elements, vols. I–III. Dover, 1956.

6. T. L. Heath. A History of Greek Mathematics, vols. I–II. Dover, 1981.

7. D. Hilbert. Grundlagen der Geometrie, 10th ed., B. G. Teubner, 1968. English translation by L. Unger. The Foundations of Geometry, 2nd ed., Open Court, 1971.

8. J. Høyrup. What is “geometric algebra” and what has it been in historiography? AIMS Mathematics 2:1 (2017), 128–160.

9. J. Høyrup. Hippocrates of Chios—his elements and his lunes, AIMS Mathematics 5:1 (2019), 158–184.

10. V. J. Katz and K. H. Parshall. Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press, 2014.

11. W. Knorr. The Evolution of The Euclidean Elements. Reidel, 1975.

12. E. E. Moise. Elementary Geometry from an Advanced Standpoint. Addison-Wesley, 1974.

13. I. Mueller. Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Dover, 2006.

14. Proclus. A Commentary on the First Book of Euclid’s Elements. Translated by Glen R. Morrow. Princeton University Press, 1970.

15. I. Thomas. Selections Illustrating the History of Greek Mathematics, vols. I–II. The Loeb Classical Library. Harvard University Press, 1998.

16. S. Unguru. On the need to rewrite the history of Greek mathematics. Archive for History of Exact Sciences 15:1 (1975), 67–114.

17. B. L. van der Waerden. Geometry and Algebra in Ancient Civilizations. Springer, 1983.

18. L. Zhmud. Pythagoras and the Early Pythagoreans. Oxford University Press, 2012.

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Correspondence to Zoran Lučić.