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Theaetetus and the History of the Theory of Numbers

Published online by Cambridge University Press:  11 February 2009

A. Wasserstein
A. Wasserstein
Affiliation:
Universtiy of Glasgow
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Extract

This famous passage has given rise to much discussion and some perplexity. Theodoras the mathematician is represented by Theaetetus as proving the irrationality of the square roots of the (non-square) numbers from 3 to 17:

‘He took the separate cases up to the root of 17 square feet; and there, for some reason, he stopped.’ (Transl. Cornford.)

The passage is of great importance in the history of Greek mathematics for more than one reason. Theaetetus is said to have generalized the proof of the irrationality of square roots of non-square integers; and thus his connexion with this passage is important because Plato here obviously implies that Theodorus was not giving a generalized proof—otherwise, why should he go up to 17 ? If Theodorus did not know the generalized proof, he clearly had to proceed by enumeration and proof of particular case

Type
Research Article
Information
The Classical Quarterly , Volume 8 , Issue 3-4 , November 1958 , pp. 165 - 179
Copyright
Copyright © The Classical Association 1958

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References

1 See Hardy, and Wright, , Introduction to the Theory of Numbers (3rd ed.), p. 43.Google Scholar See also Appendix below.

2 Their excitement was probably caused not so much by joy at the great discovery, as by its disturbing implications for their metaphysical doctrine: here was a case where no number relationship could be established. The legend invented by the Pythagoreans to the effect that the man who first divulged the discovery died in a shipwreck is a sufficient indication of the disturbance that the ‘irra tional’ caused in the Pythagorean school. See Schol. in Eucl. Elem. 10, p. 417Google Scholar Heiberg: The Scholiast's suggestion concerning the meaning of this story is not without interest: (Instead of read perhaps edv ris ) See also Iambi.de Vit. Pythag. 34. 246–7, p. 132 Deub.Google Scholar

3 See Cantor, , Vorlesungen z. Gesch. d. Mathem. i. 154–5Google Scholar. The claim made by Fritz, K. von (‘The Discovery of Incommensurability by Hippasus of Metapontum’, Annals of Mathematics, xlvi [1945], 245)Google Scholar that ‘the tradition is unanimous in attributing the discovery to a Pythagorean philosopher by the name of Hippasus of Metapontum’ seems to me to be devoid of all foundation. So far from being unanimous, the tradition is, I believe, nonexistent. I know of no single ancient author attributing the discovery to Hippasus. We have indeed the legend mentioned by Iamblichus (Vit. Pyth. 18. 88, p. 52 Deubner; cf. De comm. math. sc. 25, p. 77 Festa) to the effect that Hippasus was drowned at sea as a punishment for divulging the Pythagorean secret of how to inscribe a dodecahedron in a sphere; and this legend is obviously confused with the similar legend connected with the divulgation of the discovery of irrationality (see preceding note). But that confusion does not afford any reason for suggesting that Hippasus, because he is said to have suffered the same fate as the man who published the secret of irrational numbers, must have been identical with that man. And, of course, there is even less justification for identifying Hippasus not only with the divulger but also with the discoverer of irrationals. It may be of interest that this confusion is one of which Iamblichus already knew: see Vit. Pyth. 34. 247, p. 132 Deubner, where we are told the story of the punishment of the man who had divulged the secret of the dodecahedron in scribed in the sphere. It is followed by the remark that ‘some say that it was the man who divulged the secret of irrationality and incommensurability who suffered this (punishment)’. It may perhaps be useful to mention here that what v. Fritz calls an ‘obviously … corrupt reading in some manuscripts’, namely in the Eudemian Summary, in Proclus, in Eucl., p. 65, seems in fact to be not only the right reading but also the unanimous tradition of all manuscripts. There does not seem to be any manuscript evidence at all for which are apparently based on no more than a note in August's Euclid (i. 290): ‘alii ’. See Heath, , The Thirteen Books of Euclid's Elements, i. 351Google Scholar

1 e.g. Zeuthen, ‘Sur la constitution des livres arithmétiques des Éléments d'Euclide et leur rapport à la question de 1'irrationalité, in Oversigt over del Kgl. Danske Videnskabernes Selskabs Forhandlinger, 1915. See also Heath, , Greek Mathematics, i. 206.Google Scholar

1 See also the note by Alexander Aphrodisiensis on An. Pr. i. 41a26. (It is to be observed that Alexander in this note [p. 260, 30, of the Berlin edition] uses the verb =to square, a sense which is unknown to L.S.J.)

1 That the traditional proof is indeed ‘traditional’ (i.e. known before Theaetetus) is accepted by most scholars; see, e.g., Cantor, , Vorlesungen zur Gesch. der Matkematik, i. 155Google Scholar; Heiberg, , ‘Mathematisches zu Aristoteles’, Abhandl. Gesch. Math. Wiss., 1904, p. 24Google Scholar: ‘dieser alte pythagoräische Beweis’. And the case is well argued by Zeuthen (1915, pp. 357–62). The arithmetical character of the proof and its reliance on the distinction between odd and even (in the case of the square root of 2) seem to stamp the proof as Pythagorean. A further argument in favour of its Pythagorean origin is its obvious relationship with the so-called Pythagorean theorem (Euclid 1. 47) which is attested as Pythagorean by many, though admittedly late, authorities. For reference see Heath, , The Thirteen Books of Euclid's Elements, i.351.Google Scholar He also quotes Heron, who ascribes to Pythagoras a general rule for the formation of right-angled triangles with rational integers as sides. And the well-known Eudemian Summary in Proclus (ed. Friedlein, p. 65) attributes the discovery of the theory of irrationals to Pythagoras; the scholiast on Euclid 10. 1, Heiberg, vol. v, pp. 415–16, attributes the discovery of incommensurability to the Pythagoreans. See Heath, , op. cit., p. 353Google Scholar: ‘The investigation (of 1. 47) from the arithmetical point of view would ultimately lead Pythagoras to the other momentous discovery of the irrationality of the length of the diagonal of a square expressed in terms of its side.’

The use of in the Platonic dialogue does not, of course, refer to the geometrical character of the proof. At most it means that Theodorus drew a figure to demonstrate the existence of such lengths; that they were irrational needed a further, arithmetical, proof with which the ‘Existenzbeweis’ to which in that case refers, had nothing to do. Heam in fact gives good reasons for thinking that means no more than ‘he was proving’. See Gk. Math. i. 303.

1 Zeuthen (1915, p. 341) makes the same mistake as Heath: ‘… l'application de la démonstration en question à ces différentes racines ne présenterait que cette seule difltérence qui résulte de la substitution d'un nouveau nombre à celui dont on part.’

1 Zeuthen, 1910, p. 421; he gives examples of terminological parallels between Euclid (7, Def. 16 and 18) and the passage in the Theaetetus. As regards Book 10 we have the statement of the scholiast on 10. 9 who expressly says that that theorem was the discovery of Theaetetus (see Heiberg, , Euclid, v. 450)Google Scholar. On 13. 1 the scholiast tells us (Heiberg, , op. cit. v.654Google Scholar) that part of the subject-matter of that book was derived from Theaetetus. Generally, we are told by Proclus (in Eucl. Elem., ed. Friedlein, p. 68) that … … … ‘Euclid put together the Elements and perfected many of the theorems of Theaetetus’. The practically certain case for pre-Euclidean provenance of part at any rate of the contents of the arithmetical books is supported by a letter of Eratosthenes given by Eutocius (Archimedes, ed. Heiberg, iii. 102 ff.).

1 It is of course 7. 30, not as v. Fritz (in P.W. s.v. Theodoras, c. 1821) says, 7. 27. The latter is the enunciation of the theorem that if two numbers are prime to one another, then their squares will be prime to one another; and so also with their cubes. But it is clear that what we need for our proof is 7. 30, that is to say, the theorem ‘if c divides ab, and c is prime, then c divides either a or b’, with a simple substitution of a square number for ab, that is to say, taking a = 4. Else- where v. Fritz (s.v. Theaetetus, cc. 1356 and 1359) derives what we need here from 7. 25. But that, too, is unnecessary: for there we are told that if two numbers are prime to one anodier the square of one of them will be prime to the other: what we need does not follow from it. Whereas at 7. 30 the enunciation of obviously includes, as a special case, .

2 See on this Hardy, and Wright, , Introduction to the Theory of Numbers, p. 41.Google Scholar

1 The commentator does not seem to accept this answer. He himself suggests, 35, 21–36, 35, that 17 may have been an appropriate place to stop because it is the first number after 16; and 16 is the number of the only square in which the number denoting the sum of the sides is equal to the number denoting the area of the square, since 4+4+4+4 = 4X4.

1 i.e. concerned with musical theory. Cf. Plato, , Theaet. 145 d.Google Scholar

2 I do not know what means here. It has been suggested to me that it may perhaps mean ‘comes in’, ‘enters into the argument’.

3 Cf. Euclid, , Sectio Canonis, Mus. scr. gr. ed. Jan, prop. 3Google Scholar, p. 152; also Archytas ap. Boeth. de mus. iii. 11 = Vorsokratiker6, 47 A 19, p. 429. See Heath, The Thirteen Books of Euclid's Elements, ii. 295.Google Scholar

1 See p. 172, n. 3.

1 It is well known that Greek commentators were in the habit of literally repeating the views of their predecessors when they agreed with them. See, e.g., Minio-Paluello, L. in J.H.S. lxxvii (1957), 100.Google Scholar

1 The numerical ratios of the Octave, the Fourth, the Fifth, and the full tone, i.e. the difference between the Fourth and the Fifth, were, of course known very early as being respectively 2:1; 4:3; 3:2; 9:8. The last ratio is obtained simply by taking the length of the chord producing the fundamental note as 12; the octave will then be 6, the Fourth 9, the Fifth 8, and the ratio of the Fourth to the Fifth, 9:8; this was then defined as the typical full tone. (See on this also Aristides Quintil. iii. 113 Meibom; Nicom. Harm, i. 12 Meibom; Theon Smyrn. p. 66 H.; Ptol. Harm. i. 5.)