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A NOTE ON THE CONSTRUCTION OF THE EQUILATERAL TRIANGLE WITH SCALENE ELEMENTARY TRIANGLES IN PLATO'S TIMAEUS: PL. TI. 54A-B*

Published online by Cambridge University Press:  01 October 2015

Ernesto Paparazzo
Ernesto Paparazzo*
Affiliation:
Istituto di Struttura della MateriaCNR, Rome, Italy
*
paparazzo@ism.cnr.it
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Extract

In the Timaeus Plato says that, among the infinite number of right-angled scalene elementary triangles, the best (τὸ κάλλιστον) is that

ἐξ οὗ τὸ ἰσόπλευρον ἐκ τρίτου συνέστηκε.

Apart from few exceptions to be mentioned shortly, the translations of the Timaeus, which I am aware of spanning the period from the second half of the nineteenth century up to recent times, have usually rendered this passage as meaning that such an elementary triangle is that which, when two are combined, the equilateral triangle forms as a third figure. For instance, Bury and Zeyl respectively translate:

out of which, when two are conjoined, the equilateral triangle is constructed as a third.

and

from [a pair of] which the equilateral triangle is constructed as a third figure.

I shall refer to this sort of translation as the Prevailing Translation (hereafter PT).

Type
Research Article
Information
The Classical Quarterly , Volume 65 , Issue 2 , December 2015 , pp. 552 - 558
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Copyright
Copyright © The Classical Association 2015 

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Footnotes

*

The author is affiliated to the research group ‘Power Structuralism in Ancient Ontologies’, directed by Dr Anna Marmodoro and based in the Faculty of Philosophy at Oxford, with funding from the European Research Council.

References

1 Pl. Ti. 54a-b.

2 The equilateral triangle is the face bounding three of the four best bodies (τὰ κάλλιστα σώματα, Pl. Ti. 53e) of the universe, which shape the genera fire, air and water respectively: pyramid (tetrahedron), Pl. Ti. 54e–55a; octahedron, Pl. Ti. 54a-b; and icosahedron, Pl. Ti. 54b. Cf. Plut. Quaest. Plat. 5.1003C.

3 (Information in square brackets denotes that the year in which a given translation first appeared differs from the year of the edition I consulted.) F. Acri, Timeo (Turin, 2007 [1870]), 461; R.D. Archer-Hind, The Timaeus of Plato (London, 1888), 193; B. Jowett, Plato Timaeus (Teddington, 2009 [1892]), 27; O. Apelt, Platons Dialoge, Timaios und Kritias (Leipzig, 1922), 82; A. Rivaud, Platon. Timée. Critias (Paris, 1925), 173; A.T. Taylor, A Commentary on Plato's Timaeus (Oxford, 1928), 370; R.G. Bury, Plato. Timaeus, Critias, Cleitophon, Menexenus, Epistles (Cambridge, MA, 2005 [1929]), 129; F.M. Cornford, Plato's Cosmology: The Timaeus of Plato Translated with a Running Commentary (London, 1937), 214; R. Rufener, Platon Spätdialoge. Philebos. Parmenides. Timaios. Kritias (Zürich und Stuttgart, 1965), 245; D. Lee, Plato. Timaeus and Critias (London, 1977 [1971]), 74; G. Lozza, Platone. Timeo (Milan, 1994), 73; L. Brisson, Le mème et l'autre dans la structure ontologique du Timée de Platon. Un commentaire systématique du Timée au Platon (Sankt Augustin, 1994), 362; D.J. Zeyl, Plato. Timaeus (Indianapolis, 2000), 44; F. Fronterotta, Platone. Timeo (Milan, 2003), 285; R. Waterfield and A. Gregory, Plato Timaeus and Critias (Oxford, 2008), 47.

4 Bury (n. 3).

5 Zeyl (n. 3).

6 G. Fraccaroli, Platone. Il Timeo (Milan, 1906), 265; G. Giarratano, Platone. Opere complete. Clitofonte, La Repubblica, Timeo, Crizia, vol. 6 (Bari, 1986 [1928]), 173; G. Reale, Platone. Timeo (Milan, 2000 [1955]), 157.

7 Pl. Ti. 54d-e.

8 See LSJ s.v.

9 See n. 6.

10 See, for instance, Euc. 1.7.

11 Taylor (n. 3), 374; Cornford (n. 3), 210; I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (Cambridge, MA, 1981), 207 and 251–5; J.J. Cleary, Aristotle and Mathematics: Aporetic Method in Cosmology and Metaphysics (Leiden, 1995), 13, 56; M.F. Burnyeat, ‘Plato on why mathematics is good for the soul’, in T. Smiley (ed.), Mathematics and Necessity: Essays in the History of Philosophy (Oxford, 2000), 1–81, at 24; M.J. White, ‘Plato and mathematics’, in H.H. Benson (ed.), A Companion to Plato (Oxford, 2006), 228–43, at 229.

12 In LSJ s.v. συνίστημι there is a misleading reference to the use of this verb in Euc. 1.7. In fact, Euclid explicitly refers the ‘two’ to the number of lines, and does not implicitly take it as being contained in, and integral to, the meaning of the verb συνίστημι.

13 Pl. Ti. 53c. Translation after Bury (n. 3), 127 with a modification: ‘rectilineal’ instead of ‘rectilinear’.

14 See Pl. Ti. 54d-e, that is, the detailed construction of the equilateral triangle.

15 Pl. Ti. 54d-e.

16 Cornford (n. 3), 237.

17 Pl. Ti. 53d.

18 Ibid.

19 Pl. Ti. 53b.

20 Pl. Ti. 54d-e.

21 Pl. Resp. 511d. Burnyeat (n. 11), 41.

22 Pl. Resp. 510c-d. Burnyeat (n. 11), 27.

23 Pl. Resp. 511b-c. Plato discusses other shortcomings of mathematical reasoning in, for instance, Pl. Resp. 511d and 533b-d.

24 Plato gives the geometrical properties of the scalene elementary triangle in Pl. Ti. 54b and Pl. Ti. 54d, whereas in Pl. Ti. 54d-e he specifies what the number is of such triangles that produces the equilateral triangle.

25 A given equilateral triangle of area S is obtained by a number (n) of elementary triangles which scales inversely with their area expressed as a fraction of S, that is to say: $n \times \displaystyle{S \over n} = {n \hskip-5pt / \over n \hskip-5pt /} \times S = S$ .

26 For instance, Taylor (n. 3), 370 says: ‘T[imaeus] selects as the fairest of them all [sc. the scalene triangles] that from a pair of which the equilateral triangle is composed as a third figure (ἐκ τρίτου), i.e. the right-angled triangle got by dropping a perpendicular from one vertex of an equilateral  triangle on the opposite side.’ See Fig. 1, panel I, triangle A’.

27 Euc. 1, Def. 20 lists the geometrical properties of the equilateral triangle, the isosceles triangle and the scalene triangle.

28 Pl. Ti. 53d.

29 Ibid.

30 Pl. Ti. 54a.

31 Taylor (n. 3), 369–70.

32 P. Lang (ed.), ‘Pythagorean numbers according to the doctrines of Philolaus’, Theologumena Arithmetica Speusippi Fragmenta (Bonn, 1911), fr. 4.

33 Taylor (n. 3), 370.

34 Proclus, Euclid 1.383. See Taylor (n. 3), 371.

35 Pl. Ti. 54a. Euclid 1.32 states that the sum of the interior angles of a triangle is equal to two right angles, that is, 180°. In the case of a right-angled triangle, an angle of which is equal to 90°, the sum of the non-right angles will be 180°−90° = 90°. Now, there is only one type of isosceles triangle, because the non-right two angles, say α1 and α2 (0°<αi<90°; i = 1,2), are equal to one another, that is, α1 = α2 = αi 2×αi = 90° αi = 45°. Conversely, the right-angled scalene triangle, which involves α1 ≠ α2, can be of as many types as are the combinations that fulfil the following conditions: α1+α2 = 90°, and 0°<αi<90°.

36 Pl. Ti. 55b.

37 Pl. Ti. 55b-c.

38 Ibid.

39 See n. 18.

40 I. Mueller, ‘Mathematical method and philosophical truth’, in R. Kraut (ed.), The Cambridge Companion to Plato (Cambridge, 1992), 170–99, at 172 and 175.

41 Cleary (n. 11), 1–2; Burnyeat (n. 11), 21–42.

42 J. Dillon, ‘The Timaeus in the Old Academy’, in G.J. Reydams-Schils (ed.), Plato's Timaeus as Cultural Icon (Notre Dame, IN, 2003), 80–94, at 81. According to Burnyeat (n. 11), 17, Speusippus ‘rejected the Theory of Forms entirely’.

43 Taylor (n. 3), 370.

44 Pl. Ti. 28c–29a. T.K. Johansen, ‘The Timaeus on the principles of cosmology’, in G. Fine (ed.), The Oxford Handbook of Plato (Oxford, 2008), 463–83, at 464–5 and 471.

45 In the Republic Plato discusses the distinction between mathematics and philosophy in the last two segments of the image of the Line: Pl. Resp. 510c–511c.

46 J. Annas, An Introduction to Plato's Republic (Oxford, 1981), at 273–93, especially 284, 288 and 290; Burnyeat (n. 11), 41.

47 Pl. Ti. 54d-e.

48 Pl. Ti. 55b.

49 V. Harte, Plato on Parts and Wholes. The Metaphysics of Structure (Oxford, 2002), 240.

50 In a forthcoming article (E. Paparazzo, ‘It's a world made of triangles: Plato's Timaeus 53B-55C’, AGPh [2015]), I propose that only the constructions of the equilateral triangle and of the square with six and four elementary triangles respectively comply with Plato's theory of Forms, whereas those involving just two ἡμι-triangles do not.