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Proclus’ division of the mathematical proposition into parts: how and why was it formulated?1

Published online by Cambridge University Press:  11 February 2009

Reviel Netz
Reviel Netz*
Affiliation:
Gonville and Caius College, Cambridge, [email protected]
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Extract

There are a number of ways in which Greek mathematics can be seen to be radically original. First, at the level of mathematical contents: many objects and results were first discovered by Greek mathematicians (e.g. the theory of conic sections). Second, Greek mathematics was original at the level of logical form: it is arguable that no form of mathematics was ever axiomatic independently of the influence of Greek mathematics. Finally, third, Greek mathematics was original at the level of form, of presentation: Greek mathematics is written in its own specific, original style. This style may vary from author to author, as well as within the works of a single author, but it is still always recognizable as the Greek mathematical style. This style is characterized (to mention a few outstanding features) by (i) the use of the lettered diagram, (ii) a specific technical terminology, and (iii) a system of short phrases (‘formulae’). I believe this third aspect of the originality—the style—was responsible, indirectly, for the two other aspects of the originality. The style was a tool, with which Greek mathematicians were able to produce results of a given kind (the first aspect of the originality), and to produce them in a special, compelling way (the second aspect of the originality). This tool, I claim, emerged organically, and reflected the communication-situation in which Greek mathematics was conducted. For all this I have argued elsewhere.

Type
Research Article
Information
The Classical Quarterly , Volume 49 , Issue 1 , May 1999 , pp. 282 - 303
Copyright
Copyright © The Classical Association 1999

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Footnotes

1

I wish to thank Sir Geoffrey Lloyd, as well as an anonymous reader, for helpful comments.

References

2 R. Netz, The Shaping of Deduction in Greek Mathematics (Cambridge, forthcoming).

3 Knorr, W., ‘On the early history of axiomatics: the interaction of mathematics and philosophy in Greek antiquity’, in J., Hintikka et al. (edd.), Theory Change, Ancient Axiomatics, and Galileo's Methodology (Dordrecht, 1981) pp. 145–86Google Scholar, esp. pp. 172ff. The main interest of Knorr there is large-scale style (the organization of treatises), not small-scale style (the organization of individual propositions).

4 It is endorsed, for example, in Mueller, I., ‘Greek mathematics and Greek logic’, in J., Corcoran (ed.), Ancient Logic and its Modern Interpretation (Dordrecht, 1974), p. 67 Google Scholar, n. 4. Fowler, D. notes in The Mathematics of Plato's Academy (Oxford, 1987), p. 367 Google Scholar, n. 54, that it may not have been always strictly followed.

5 For example, 208.4–15 describe the protasis of the first propositon, and explain how it fits the general description of protasis.

6 ‘for the perfect protasis consists of both’ (203.7–8).

7 I have dealt with this in ch. 6. of Netz (n. 2).

8 Here and elsewhere in this article I adapt Morrow's translation with a few changes, mainly in changing the names of the parts where mine are different from his.

9 It is optional in a deep sense. It is very often possible to state the construction within the setting-out itself. For instance, nothing stopped Euclid, in Elements 2.5, from constructing the relevant areas on the line-segments within the setting-out. Then the definition of goal would have referred not to virtual objects, ‘the rectangle contained by the lines…’, but to actual rectangles and squares, and no construction would have been required. Very often, then, this is a purely stylistic decision, how to distribute the burden of construction between the setting-out and the construction (in other cases, of course, the added construction really is very clearly auxiliary to the construction demanded to set out the proposition itself: Pythagoras’ theorem, Elements 1.47, is an obvious case).

10 Proclus sees here the definition of goal as dependent upon the setting-out, and thus he concentrates on whether the setting-out is present or not.

11 As well as others in the Elements. Proclus refers to Books 7–10 in general, and to 10.28 in particular, as furnishing further examples.

12 For example, the division of the definition of goal into several parts, spread through the proposition, is very common in Autolycus: six out of the thirteen propositions of the first book of The Risings and Settings.

13 I cannot do justice in this brief compass to the logical detail of the method; all I need to stress is that the method has a standard stylistic format (which however, again, is not absolutely rigid). On the logical side, see especially the discussion in Hintikka, J. and Remes, U., The Method of Analysis (Dordrecht, 1974).CrossRefGoogle Scholar

14 Printed as Appendix 1.8 in Heiberg's edition of Elements 13 (364.17–376.22).

15 The heading ‘53’ simply does not occur in our main manuscript (it does occur in the margins of another). It was added by Heiberg, who followed the ancient commentator Eutocius—who of course had inserted numbers for the sake of his commentary.

16 In what follows, I speak of the way words are used ‘in our literature’. Greek mathematics is poorly represented by the current CD-ROMs available for Greek literature and, even with computers and indices, categorical statements are never safe. I make categorical statements. I know I may be wrong in each of my statements, and that probably I am wrong in some. I hope my mistakes will be pointed out—and I believe my main argument will still stand.

17 For instance, in the commentary to Planes in Equilibrium, 2.1, Eutocius clearly refers to ‘enunciation’ and to ‘construction’ as two elements of the text, i.e. as stylistic units.

18 Some meanings on offer in LSJ: (verb) ‘expose to danger’, ‘stipulate’, (noun) ‘the earlier part of a dramatic poem’.

19 The Risings and Settings 2.6. This work (late fourth century b.c.?) is considered to be one of the earliest extant mathematical treatises.

20 Elements 11.37—i.e. towards the end of the Elements. There is a similar case in Elements 11.35.

21 See, for example, Prior Analytics 24a16–17 for the general sense, 34a17–19 for the special sense.

22 A reference related to Metaphysics 1089a23 is from the Prior Analytics 49b35. Reading the Metaphysics passage in light of the Prior Analytics, it seems that Aristotle's example of a mathematical protasis would be something like ‘this line is one foot long’. If anything, this is Proclean setting-out, ekthesis.

23 Einarson, B., ‘On certain mathematical terms in Aristotle's logic’, American Journal of Philology 57 (1936), 33–54, 151–172.CrossRefGoogle Scholar

24 It is my impression—but here I am even more cautious than elsewhere—that later commentators on Aristotle do not seem to point this discrepancy between Aristotle's and the mathematicians’ terminology. This is the most natural comment to be made by a late commentator. If proved, therefore, this would be a conclusive negative proof that, even in late antiquity, the Proclean scheme was not seen as ‘the mathematical usage’, but was merely an ad hoc extension of terminology developed elsewhere. (The same, of course, could be said about the further evidence I shall bring forward from Aristotle concerning other parts of the proposition.)

25 See Mignucci, M., ‘Expository proofs in Aristotle's Syllogistic’, Oxford Studies in Ancient Philosophy (1991), suppl., 9–28 Google Scholar, with references to earlier literature.

26 Archimedes, Sphere and Cylinder 2.4 (190.25); Appollonius, Introduction to Conies 4 (2.17). Among later authors are Pappus (7.5), Eutocius (Commentary on Sphere and Cylinder 2.4), and indeed, Proclus himself (Commentary on 1.22).

27 For example, Plato's Timaeus 38c6, Aristotle's On Coming to Being and Passing Away 323a22.

28 Simplicius, In Aristotelis Physica 64.8–11. The best philological analysis of this difficult passage remains Becker, O., ‘Zur Textgestellung des Eudemischen Berichts über die Quadratur der Möndchen durch Hippokrates von Chios’, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 3 (1936), 411–19Google Scholar, which includes the above quotation in Eudemus’ fragment.

29 LSJ offers, for example, ‘fixed assets’, ‘device, trick’, ‘system of gymnastic exercise’.

30 In a passage to which LSJ itself refers, Pappus 174.17, organike kataskeue means ‘solution with a mechanical tool’ (one should ignore however the translation offered by LSJ, probably by Health—LSJ assumes that Proclus’ scheme is a true description of any mathematical practice, and tailors its mathematical examples accordingly).

31 For example, Euclid's Elements 3.14, 204.11.

32 LSJ quote authors such as Euripides (Orestes 802) and Thucydides (4.73).

33 Attested in the fifth century, again, e.g. Aristophanes (Clouds 1334).

34 See Lloyd, G. E. R., Magic, Reason and Experience (Cambridge, 1979), pp. 102–25.Google Scholar

35 LSJ entries, both from Euripides’ Medea (887, 341, respectively).

36 Aristophanes, Frogs 1119, refers to the first part, the prologos. Aristotle, Nicomachean Ethics 1123a21, refers to the second, parodos. Both seem to refer to the stylistic unit.

37 Rosen, C., Sonata Forms (New York, 1988).Google Scholar

38 For example, in the beginning of his commentary to 1.20 (322.4–323.3), Proclus discusses the philosophically interesting question: ‘why prove an obvious result?’.

39 This question arises from Proclus’ definition of the protasis and ekthesis.

40 See Mueler, I., foreword to second edition of Proclus: A Commentary on the First Book of Euclid's Elements, trans. Morrow, G. (Princeton, 1992), pp. xlviii–1.Google Scholar

41 I follow Morrow's translation, from 210.18–25.

42 299.5–11 (paraphrasing): ‘not all the parts are present here, e.g. we do not have the construction—but the proof, which is indispensable, is present’. Are the students expected to check the other four parts?