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ARISTOTLE ON PLACING GNOMONS ROUND (PH. 3.4, 203a10–15)

Published online by Cambridge University Press:  07 September 2015

Monica Ugaglia
Affiliation:
Scuola Normale Superiore, Pisa
Fabio Acerbi
Affiliation:
CNRS, UMR8560 Centre Alexandre Koyré, Paris
Corresponding

Extract

At Ph. 3.4, Aristotle begins his discussion of the Unlimited. As is customary with him, a preliminary to the investigation proper is set up: he presents and discusses other thinkers' opinions on the subject, from which to draw indications on the meaning—and possibly the existence—of the entity at issue. As for the question of what the Unlimited is, Aristotle points out that there is a complete agreement on two points: the Unlimited pertains to physics, and it presents itself as a principle of sorts. A matter of disagreement concerns the categorial status of the Unlimited: Plato and the Pythagoreans regard it as something existing per se, whereas the Physicists claim that it is only the attribute of something else. A concise exposition of these theories follows.

Type
Research Article
Copyright
Copyright © The Classical Association 2015 

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References

1 The Pythagorean doctrines are expounded in greater detail in Metaph. Α.5.

2 Ph. 3.4, 203a10–15. We shall call the underlined sentence ‘the (mathematical) example’; it is a complex sentence, whose dependent and independent clauses will be referred to as ‘the genitive absolute’ and ‘the principal clause’, without further qualifications; note that the verb of the principal clause is in the infinitive: a verbal form ‘say’ is understood, and this means that Aristotle is relating the contents of the doctrines referred to; as a consequence, the mathematical example is presented by Aristotle as being originally Pythagorean. The text preceding the mathematical example in our quote will be called ‘the immediate context’, the whole quote ‘the (Aristotelian) passage’.

3 At in Ph. 394.17: ἔχει δὲ ἡ λέξις ἀσάφειάν τινα διὰ τὴν συντομίαν. ‘Unfortunately the illustration throws very little light on the problem, being itself scarcely intelligible’, in the words of W.A. Heidel, ‘Πέρας and ἄπειρον in the Pythagorean philosophy’, AGPh 14 (1901), 384–99, at 391.

4 As a matter of fact, our earliest source on Pythagorean σχηματογραφία and on its lexicon is Aristotle himself, who discusses it precisely in this text and in the Eurytus passage at Metaph. Ν.5, 1092b8–15. The noun σχηματογραφία in this sense is first attested in Nicomachus' Introductio arithmetica (2.6.2, 8.1, 8.3, 10.2, 12.1, 13.1, 15.2). Elsewhere it is used as a technical astrological term.

5 M. Timpanaro Cardini, ‘Una dottrina pitagorica nella testimonianza aristotelica’, Physis 3 (1961), 105–12. The quoted clauses can be read at Simpl. in Ph. 457.22 and 457.15–16 respectively.

6 The paraphrase is short and deserves being quoted in full (here and elsewhere terms that are relevant to our discussion are underlined): φέρουσι δὲ καὶ ἄλλο σημεῖον τοῦ πέρατος μὲν εἶναι αἴτιον τὸν περιττόν, ἀπειρίας δὲ τὸν ἄρτιον· λαβόντες γὰρ μονάδα τοὺς ἐφεξῆς αὐτῇ περισσοὺς ἐπισυντιθέασι χωρὶς ἕκαστον, οἶον γ καὶ ε καὶ ζ καὶ θ. καὶ ἑκάστη τοίνυν τούτων ἐπισύνθεσις τὸ συναγόμενον ἀεὶ τετράγωνον διαφυλάττει δ θ ιϛ καὶ κε. διὰ τοῦτο γνώμονας καλοῦσιν οἱ ἀριθμητικοὶ τοὺς περιττούς, ὅτι φυλάττουσι τὸ εἶδος τοῦ τετραγώνου οἱ ἐφεξῆς τοῖς πρώτοις περιτιθέμενοι ὥσπερ οἱ γραμμικοί. πάντως δὲ ὅ τι πότ’ ἐστι γνώμων, ἐν γεωμετρίᾳ γινώσκεις· οὐ γὰρ ἀμαθέσι παντελῶς ταῦτα συγγράφεται. οὕτω μὲν οὖν οἱ περιττοὶ φυλακτικοὶ τοῦ εἴδους εἰσὶ καὶ τὸ ἓν τηροῦσιν, οἱ δὲ ἄρτιοι προστιθέμενοι τῇ μονάδι κατὰ τοὺς ἐφεξῆς ἀεί τι καινὸν εἶδος ποιοῦσι καὶ ἡ διαφορὰ πρόεισιν εἰς ἄπειρον τρίγωνον, εἶτα ἑπτάγωνον, εἶθ’ ὅ τι καὶ τύχοι. οὕτως οὖν τοῖς Πυθαγορείοις ὁ ἄρτιος μόνος ἀριθμὸς ἄπειρος γίνεται. The pentagon is omitted since an even number is added to the result of the previous addition.

7 The exegesis at Simpl. in Ph. 456.16–458.10 can be segmented as follows: 456.16–22, preliminary statement about adding even and odd numbers to numerical εἴδη; 456.22–34, numerical examples clarifying the preliminary statement; 456.34–457.1, resumption of the preliminary statement; 457.1–8, that the Pythagoreans called odd numbers γνώμονες, with an explanation of what a geometric γνώμων is; 457.8–12, paraphrase of the mathematical example; 457.12–25, Alexander's ἐπιβολή, containing two examples of σχηματογραφία; 457.25–458.10, Simplicius' objection to Alexander.

8 The term ἑτερομήκης is introduced in the initial formulation of the exegesis (note that προστίθημι is used here): οἱ μὲν γὰρ περισσοὶ προστιθέμενοι ἑξῆς τῷ τετραγώνῳ ἀριθμῷ φυλάσσουσιν αὐτὸν τετράγωνον καὶ ἰσάκις ἴσον κατὰ δὲ τὸ ποσὸν μόνον ηὐξημένον, ὁ δὲ ἄρτιος προστιθέμενος ἀεὶ τῷ τετραγώνῳ ἐναλλάσσει τὸ εἶδος ἑτερομήκη ποιῶν, ἄλλοτε κατ’ ἄλλην πλευρὰν παρηυξημένον (in Ph. 456.18–22). A ἑτερομήκης is a (numerical) rectangle, i.e. a number having unequal ‘sides’; we shall return to this term below.

9 Simplicius' paraphrase is quite effective: he makes the double correlative structure explicit (broken underlining)—note the stylistic touch of making it chiastic, as the Aristotelian passage according to this reading; the inclusion of χωρίς in the first correlative structure shows that Simplicius adopts segmentation option (a i ): περιτιθεμένων οὖν περὶ τὸν ἕνα ἀριθμὸν τὸν τετράγωνον ποτὲ μὲν χωρὶς τῶν γνωμόνων, τουτέστι τῶν μονάδων κατὰ τὸ ἑξῆς † περιττὸν ἀριθμῶν (οὗτοι γὰρ ἦσαν γνώμονες), ποτὲ δὲ χωρὶς τῶν ἀρτίων, ὅτε μὲν οὗτοι προστεθῶσιν, ἄλλο ἀεὶ γίνεται τὸ εἶδος παρὰ τὸ ἐξ ἀρχῆς, ὅτε δὲ οἱ γνώμονες, ἓν τὸ αὐτὸ τῷ προτέρῳ (in Ph. 457.8–12). Note, finally, that Simplicius has no qualms about uttering a notional heresy such as τὸν ἕνα ἀριθμόν. Alleged proofs that the unit is not a number are attested in late Neoplatonic writings: see Roueché, M., ‘Why the monad is not a number: John Philoponus and In De Anima 3 ’, JÖByz 52 (2002), 95133 Google Scholar.

10 The result is this nice multilayered conjoined sentence: καὶ ἐν ταῖς γνωμονικαῖς περιθέσεσι καὶ πάλιν χωρὶς ἐν ταῖς ἀριθμητικαῖς συνθέσεσι καὶ τῶν περιττῶν καὶ τῶν ἀρτίων ἔστιν ἰδεῖν τοὺς μὲν περιττοὺς ὅμοια φυλάττοντας, τοὺς δὲ ἀρτίους ἀνόμοια ποιοῦντας (in Ph. 458.2–5). Pairing the first two conjoints (simple underlining) with the second two conjoints (broken underlining) one obtains four combinations, which, however, produce only two outcomes (τοὺς μέν … τοὺς δέ …); the only non-obvious combination (namely, the one on behalf of which Simplicius is trying to argue) is whether the even numbers can be ‘placed round as gnomons’ or not: odd numbers can obviously be both ‘placed round’ suitable numbers ‘as gnomons’ and ‘added’ to any number, even numbers can obviously be ‘added’ to any number. Note Simplicius' resorting to the nomen actionis σύνθεσις (prefix συν-!) to denote addition.

11 Simpl. in Ph. 458.5–7. A final wordplay with ἕν, read either as a synonym of μονάς or as a numeral adjective, allows Simplicius to overcome the residual difficulties in his version of Alexander's ἐπιβολή: περὶ τὸ ἓν δὲ περιτιθεμένων τῶν γνωμόνων εἶπεν ἢ ὅτι περὶ μονάδα ἡ πρώτη περίθεσις γίνεται ἢ ὅτι ὡς περὶ ἕνα ἀριθμὸν γίνεται ἀεὶ ἡ τῶν πλειόνων μονάδων περίθεσις (in Ph. 458.8–10).

12 E. Zeller, Die Philosophie der Griechen in ihrer geschichtlichen Entwicklung (Leipzig, 19237), 1.455 n. 3, and W.D. Ross, Aristotelis Physica (Oxford, 1950), 544 respectively.

13 The exegesis at Philop. in Ph. 392.12–394.30 can be segmented as follows: 392.12–19, preliminary remarks on a logical mismatch between the mathematical example and the immediate context: the odd and the even feature both in the explanandum and in the explanans, yet the latter is said to be only a σημεῖον of the former; 392.19–393.14, what are geometric and numerical γνώμονες; odd numbers are called ‘gnomons’ by ‘the arithmeticians’ since they preserve the geometric shape of numbers; numerical examples of the generation of square numbers by successively adding odd numbers to the unit; 393.15–394.1, construction κατὰ δίαυλον; 394.1–17, general exegesis of the Aristotelian passage; 394.18–30, paraphrase of the mathematical example.

14 The construction of the double-stadium is first attested in Iambl. in Nic. 75.25–77.4 Pistelli (= 4.81–4, 144.20–146.3 Vinel); it surfaces again in Philoponus' exegesis of the Aristotelian passage and in the late Neoplatonic commentaries on Nicomachus' Introductio arithmetica: Philop. in Nic. 2.31; Asclepius, in Nic. 2.11. The way in which Iamblichus refers to the double-stadium suggests that it was not his invention either: κατὰ τὸν λεγόμενον / εἰρημένον δίαυλον at in Nic. 75.25–6, 80.11, 80.21, 88.16 Pistelli (= 144.21, 148.27, 148.34, 158.1 Vinel).

15 In fact, the text has only τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων, the modal adverb referring to the construction obtained by placing odd numbers as gnomons round the unit.

16 Stobaeus' text reads: ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν γνωμόνων περιτιθεμένων ὁ γινόμενος ἀεὶ τετράγωνός ἐστι· τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων ἑτερομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν, ἴσος δὲ ἰσάκις οὐδείς. Note the non-iterative ἀεί (see the lexical remarks towards the end of this paper).

17 See J. Burnet, Early Greek Philosophy (London, 19203), § 48 n. 81; T.L. Heath, A History of Greek Mathematics (Oxford, 1921), 1.82–3; Taylor, A.E., ‘Two Pythagorean philosophemes’, CR 40 (1926), 149–51Google Scholar.

18 The relevant portion of the scholium is χωρὶς περιτιθεμένων τῶν περιττῶν γνωμόνων ἐν τοῖς τετραγώνοις καὶ χωρὶς τῶν ἀρτίων γνωμόνων ἐν τοῖς ἑτερομήκεσιν, ὁτὲ μὲν ἄλλο ἀεὶ γίνεται ἤγουν ἑτερόμηκες, ὁτὲ δὲ τὸ αὐτό, ἤγουν τετράγωνον. We did not have access to the manuscript, nor were we able to find further information on it in the standard databases.

19 G. Milhaud, Les philosophes-géomètres de la Grèce. Platon et ses prédécesseurs (Paris, 1900), 115–7.

20 On the structure of the two commentaries see, most recently, P. Golitsis, Les Commentaires de Simplicius et de Jean Philopon à la Physique d'Aristote (Berlin and New York, 2008).

21 Simplicius lists his sources at in Ph. 918.13–14: Porphyry, Themistius and Alexander.

22 This is borne out by joint references to Ammonius and Alexander at Simpl. in Ph. 59.17–24, 192.14–17, 193.1–6.

23 See the mentions at Simpl. in Ph. 59.23, 183.18, 192.14, 193.1, followed by fairly articulated explanations. In his own commentary, Philoponus also reports the last three explanations, but eliminates the name of Ammonius (see Philop. in Ph. 117.24–118.1, 132.11–14, 129.14–18). That Ammonius held classes on Aristotle's Physics is further confirmed by a text directly pertaining to the Aristotelian passage and revealing Philoponus' habit of treating those of Ammonius' course notes that Philoponus himself published under his own name. Ammonius' course on Nicomachus' Arithmetica was edited and published in several versions in Late Antiquity. We know of at least four different recensions, one of which is ascribed to Philoponus himself (Recension I), while another is attributed to Asclepius of Tralles (Recension III). Now, the introductory sentence to the section describing the formation of square numbers κατὰ δίαυλον (see n. 14 above) reads ἔστι δὲ καὶ ἄλλη μέθοδος τετραγώνων, ἥτις ὀνομάζεται δίαυλος, εἴρηται δὲ ἡμῖν καὶ ἐν ταῖς φυσικαῖς περὶ αὐτοῦ in Philoponus (Rec. I, 2.31.19 Hoche) and ἔστι δὲ καὶ ἄλλη μέθοδος τετραγώνων, ἥτις ὀνομάζεται δίαυλος, εἴρηται δὲ καὶ ἐν ταῖς φυσικαῖς in Asclepius (Rec. III, 2.11.38 Tarán). The conclusions we must draw (compare the analysis in L. Tarán, Asclepius of Tralles, Commentary to Nicomachus’ Introduction to Arithmetic [Philadelphia, 1969], 9–15) are that the quoted sentence is Ammonius', that the construction ‘according to the double-stadium’ was expounded both in his course on Nicomachus' Arithmetica and in that on Aristotle's Physics, and that in both cases Philoponus appropriated it.

24 Cf. Diels's words at Simpl. in Ph. 1447a s.v. Θεμίστιος: saepe celato nomine compilatur a Simplicio. An almost identical remark is made by Vitelli at Philop. in Ph. 992a s.v. Θεμίστιος: sexcenties autem Themistium tacite expilat Philoponus.

25 See n. 23 above. One must bear in mind that for Philoponus' commentary on Aristotle's Physics, unlike in the case of many other commentaries transmitted under his name, we cannot take advantage of the indications provided by the title (the presence or absence of the standard formula ἐκ τῶν συνουσιῶν τοῦ Ἀμμονίου τοῦ Ἑρμείου κτλ.), since no independent manuscript preserves it: see Vitelli's remarks at Philop. in Ph. v n. 1.

26 See again n. 23 above for an example directly pertaining to the Aristotelian passage.

27 Simpl. in Cat. 430.5–431.5, Philop. in Cat. 202.10–203.17, commenting on the passage at Cat. 14, 15a29–31 we shall read in n. 46. Philoponus, in Ph. 392.23–5, explicitly refers to this passage.

28 Mediaeval and Renaissance Latin commentators (Aquinas, Albert of Saxony, Toletus and Pacius) appear to endorse the prevailing ancient interpretation, according to which the continuously different εἴδη are successive polygonal numbers. The only interesting lexical point is made by J. Pacius, Aristotelis Naturalis auscultationis. Libri VIII (Frankfurt, 1596), 510: ‘Et ait apponit seorsum [= χωρίς]: quia non utraque simul adiungitur quadrato: sed modo una, modo altera separatim.’ As for Averroes, he includes the subtraction of odd (even) numbers from squares as an operation that preserves (does not preserve) the original species.

29 See Ar. 1.9.4, 2.11.1, 2.13.6. Let us recall that, given the later authorities he mentions, Stobaeus must be later than the fourth century c.e.

30 To these one should add the reading we have found in the eighth edition of the Liddell-Scott lexicon, s.v. χωρίς: ‘τὸ χ. that which is divisible, τὸ ἓν καὶ χ. Arist. Phys. 3. 4, 4’. This reading has disappeared in the subsequent editions of the LSJ. It was not recorded in Passow's lexicon, and we have not been able to trace its origin.

31 It is, for instance, accepted, with explicit reference to one or both of Ross's discussions (see n. 12 and next footnote), in J.S. Kirk and J.E. Raven, The Presocratic Philosophers (Cambridge, 1960), 243–4; W.K.C. Guthrie, A History of Greek Philosophy (Cambridge, 1962–9), 1.242–3 (but with some words of caution); W. Burkert, Lore and Science in Ancient Pythagoreanism (Cambridge, MA, 1972), 33 n. 27, in which references to Burnet, Kirk-Raven, Guthrie are also provided; and M. Caveing, La figure et le nombre (Lille, 1997), 253–6.

32 See Milhaud (n. 19), 115–17, who states (115): ‘Les mots καὶ χωρὶς nous semblent signifier : «et autrement», c'est-à-dire «autrement qu'autour de l'unité»’; Burnet (n. 17), § 48 n. 81, who cites Milhaud and Stobaeus' passage and writes: ‘The words καὶ χωρὶς apparently mean χωρὶς τοῦ ἑνός, i.e. starting from 2, not from 1’; Heath (n. 17), 1.82–3, who mentions no previous literature as is customary in this book of his, and asserts (1.82) ‘the words καὶ χωρίς (‘and in the separate case’, as we may perhaps translate) imperfectly describe the second case, since in that case even numbers are put round 2, not 1, but the meaning seems clear’, and cites Stobaeus' passage to corroborate his statement; again T.L. Heath, Mathematics in Aristotle (Oxford, 1949), 101–2, where Ross (n. 12), 542–5 is approvingly referred to and Stobaeus' passage is cited again; F.M. Cornford, Plato and Parmenides. Parmenides’ Way of Truth and Plato's Parmenides. Translated with an Introduction and a Running Commentary (New York, 1951), 9–10, who states: ‘The oblong figures obtained by putting gnomons round 2 must be meant, however the words καὶ χωρίς be interpreted’; Ross (n. 12), 542–5, and Aristotelis Metaphysica (Oxford, 1924), 1.148–9, commenting on Metaph. 986a18—Ross (n. 12), 543, cites Milhaud, Burnet and Heath as proposing one and the same interpretation and approves of it.

33 But note that this is a doctrine that Aristotle assigns to ἕτεροι τῶν αὐτῶν τούτων (sc. other Pythagoreans) and that he expounds it after a simplified, and quite likely more authoritative, ontology: numbers themselves are the principle and matter of everything, their elements being the even and the odd; the elements of the latter are in their turn the Limiting and the Unlimited (the correspondence is chiastic), while the ‘one’ (which is both even and odd) proceeds from both of these (Metaph. Α.5, 986a15–21).

34 Besnier, R., ‘Le rôle des nombres figurés dans la cosmologie pythagoricienne, d'après Aristote’, RPhilos 183 (1993), 301–54Google Scholar, at 341, modifies Milhaud's interpretation in only one respect: the εἶδος that ἄλλο ἀεὶ γίγνεσθαι in the second construction is the necessarily different parity (namely, the fact of being even or odd) of the factors (i.e. in ancient geometrical parlance, the ‘sides’) of each resulting ἑτερομήκης. We shall see that there is some truth to this alteration, even though it is not corroborated by any ancient sources and is quite arbitrary.

35 Ross (n. 12), 544: in Nic. 58.19 and again 73.15ff. Pistelli; Heath (n. 32), 102: 73.15ff. Pistelli, in particular 73.22–6 and 77.4–8 (= 4.12, 128.5–8; 4.73, 142.14ff.; 4.74, 142.19–21; 4.85, 146.4–6 Vinel respectively). As a matter of fact, Heath (n. 32), 101 n. 1 depends on Ross with regard to the passage at 73.15ff. Pistelli.

36 This was pretty clear to ancient writers: see Iambl. in Nic. 75.20–5 and 77.4–8 Pistelli (= 4.80, 144.18–20 and 4.85, 146.4–6 Vinel respectively).

37 The introduction of the dyad in place of the Unlimited treated as ‘one’ was referred to by Aristotle, Metaph. A.5, 987b25–7, as a feature that distinguishes Plato's doctrines from those of the Pythagoreans. A few lines later (987b33–988a1), Aristotle asserts that, in Plato's system, the dyad is necessary for the generation of numbers. The idea that the dyad is the principle of numbers became standard Neo-Pythagorean doctrine: Anatolius, Dec. 30.18 Heiberg; see also the discussion in Burkert (n. 31), 57–9, and the sources therein adduced: Diog. Laert. 8.25, Aët. 1.3.8 (= Doxographi graeci, 281a.6–12), Sext. Emp. Math. 10.261 and 10.276, the Πυθαγόρου βίος excerpted in Phot. Bibl. codex 249, 438b33–439a7. See also nn. 71 and 72 below.

38 The remark is made in Heath (n. 17), 1.83, and in N. Vinel, ‘L'In Nicomachi arithmeticam de Jamblique. Introduction, édition critique, traduction et commentaire’ (Ph.D. Diss., Université Blaise Pascal, Clermont-Ferrand II, 2008), 1.xliv. The mistake is particularly blatant with Ross (n. 12), 544, who conflates the Pythagorean list of opposed principles with Middle-Platonic and Neo-Pythagorean accounts of the ἑτερομήκης as a number of the form n(n + 1): Theon of Smyrna, Exp. 26.21–2 Hiller; Nic. Ar. 2.17.1 and 2.18.2; Iambl. in Nic. 74.19–23 Pistelli (= 4.76, 142.33–5 Vinel).

39 In other words: excluding prime numbers, the numerical dichotomy square/ἑτερομήκης is exclusive and exhaustive.

40 Taylor (n. 17), 150.

41 Taylor (n. 17), 151. On account of the above-noted fact that in antiquity the second segmentation option never developed into a full-fledged interpretation of the mathematical example, it is surprising that Milhaud (n. 19), 116 n. 1 and Taylor (n. 17), 150–1 respectively take the scholium and Stobaeus' text (the latter being in fact no more than stock material about the formation of squares and ἑτερομήκεις) as supporting their exegeses.

42 See Vinel (n. 38), 1.xl–lviii. This Ph.D. thesis has been published as N. Vinel, Jamblique, Sur l'arithmétique de Nicomaque (Pisa-Roma, 2014), but the interpretation here outlined has not been included in the volume.

43 Metaph. Ν.3, 1091a15–18: φανερῶς γὰρ λέγουσιν ὡς τοῦ ἑνὸς συσταθέντος […] εὐθὺς τὸ ἔγγιστα τοῦ ἀπείρου ὅτι εἵλκετο καὶ ἐπεραίνετο ὑπὸ τοῦ πέρατος.

44 These are in Nic. 59.2–4, 59.17–21 and 62.10–18 Pistelli (= 4.14, 128.12–13; 4.16, 128.21–4; 4.29, 130.35–9 Vinel respectively); we shall discuss them later. As remarked above, Iamblichus simply expounds a doctrine that is an evolution of the Pythagorean material partly presented by Aristotle: we are not entitled to project this derived material back in its entirety as if it were a genuine early Pythagorean doctrine and use it to back up an interpretation of the mathematical passage.

45 ‘[…] l'orthodoxie pythagoricienne exige que ὁτὲ μὲν … ὁτὲ δέ ne décrive pas deux figures distinctes, manifestant séparément le πέρας et l'ἄπειρον, mais deux aspects d'une seule et unique figure, illustrant l'action conjointe du πέρας et de l'ἄπειρον’: Vinel (n. 38), 1.xlvii.

46 At Cat. 14, 15a29–31, in particular, Aristotle explains the difference between increase and alteration by resorting to the example of the gnomon placed round a square: ἀλλ’ ἔστι τινὰ αὐξανόμενα ἃ οὐκ ἀλλοιοῦται· οἷον τὸ τετράγωνον γνώμονος περιτεθέντος ηὔξηται μέν, ἀλλοιότερον δὲ οὐδὲν γεγένηται.

47 See the detailed analysis in Burkert (n. 31), 253–5. As we shall see, it is more appropriate to speak of ‘Limiting’ than of ‘Limited’.

48 Fr. 44 B 1 DK = Diog. Laert. 8.85 (Huffman's translation, with one slight modification). The same opposition is expounded in fragments 2 and 6: ἀνάγκα τὰ ἐόντα εἶμεν πάντα ἢ περαίνονταἄπειραπεραίνοντά τε καὶ ἄπειρα, ἄπειρα δὲ μόνον οὔ κα εἴη. ἐπεὶ τοίνυν φαίνεται οὔτ’ ἐκ περαινόντων πάντων ἐόντα οὔτ’ ἐξ ἀπείρων πάντων, δῆλόν τἆρα ὅτι ἐκ περαινόντων τε καὶ ἀπείρων ὅ τε κόσμος καὶ τὰ ἐν αὐτῷ συναρμόχθη […] (44 B 2 DK = Stob. Ecl. 1.21.7a); περὶ δὲ φύσιος καὶ ἁρμονίας ὧδε ἔχει· ἁ μὲν ἐστὼ τῶν πραγμάτων, ἀΐδιος ἔσσα καὶ αὐτὰ μὰν ἁ φύσις, θεία ἐντὶ καὶ οὐκ ἀνθρωπίναν ἐνδέχεται γνῶσιν, πλάν γα ἢ ὅτι οὐχ οἷόν τ’ ἦς οὐδὲν τῶν ἐόντων καὶ γιγνωσκόμενον ὑφ’ ἁμῶν γενέσθαι, μὴ ὑπαρχοίσας τᾶς ἐστοῦς τῶν πραγμάτων ἐξ ὧν συνέστα ὁ κόσμος καὶ τῶν περαινόντων καὶ τῶν ἀπείρων […] (44 B 6 DK = Stob. Ecl. 1.21.7d).

49 Metaph. A.5, 986b6–8: ἐοίκασι δ’ ὡς ἐν ὕλης εἴδει τὰ στοιχεῖα τάττειν· ἐκ τούτων γὰρ ὡς ἐνυπαρχόντων συνεστάναι καὶ πεπλάσθαι φασὶ τὴν οὐσίαν. To regard numerical principles, and numbers in general, as inherent to things or not is, according to Aristotle, one of the main points of disagreement between Pythagoreans and Platonists, as the latter claim that numbers and the real world are separate. A detailed discussion of the question is contained in Metaph. A, Μ and Ν (see in particular A.6, 987b29–33; Μ.1, 1076a32–7; Μ.6, 1080a37–b23; Μ.8, 1083a20–b19; N.3, 1090a20–b5).

50 On this point see in particular Burkert (n. 31), 48 and 468.

51 Metaph. A.5, 986a16–20: καὶ οὗτοι τὸν ἀριθμὸν νομίζοντες ἀρχὴν εἶναι καὶ ὡς ὕλην τοῖς οὖσι καὶ ὡς πάθη τε καὶ ἕξεις, τοῦ δὲ ἀριθμοῦ στοιχεῖα τό τε ἄρτιον καὶ τὸ περιττόν, τούτων δὲ τὸ μὲν πεπερασμένον τὸ δὲ ἄπειρον, τὸ δ’ ἓν ἐξ ἀμφοτέρων εἶναι τούτων (καὶ γὰρ ἄρτιον εἶναι καὶ περιττόν). For a discussion, see Burkert (n. 31), 264–5; Cornford (n. 32), 10–11.

52 In the mathematical lexicon, the few occurrences of the verb are concentrated in Archimedes (6 in Con. sph. and 1 in Spir.), an additional occurrence being in Apoll. Con. 1.4. The construction involved is always that of a segment of a straight line (Archimedes) or a part of a plane (Apollonius) ‘cut off and enclosed’ by other surfaces or lines.

53 The passages are in Nic. 59.2–4 and partly 62.10–18 Pistelli (= 4.14, 128.12–13 and 4.29, 130.35–9 Vinel resepectively); they are discussed in Vinel (n. 38), 1.xlviii–l.

54 But it also shows an obvious affinity with the passage at Cat. 14, 15a29–31, quoted above (n. 46). Note that Iamblichus redacted a commentary on Aristotle's Categorias, as is borne out by the several references to it which Simplicius inserted in his own commentary. To show that Iamblichus is, here as elsewhere, in full command of the language he uses, it suffices to note the hapax legomenon ἐπιπέδωσις in the first quote and, in the second, the nomen actionis περίθεσις, further attested at in Nic. 75.23, 77.7 Pistelli (= 4.80, 144.19 and 4.85, 146.5 Vinel respectively), but nowhere else in the Greek mathematical corpus.

55 Cf. Rh. 1.2, 1357b1–21. As a matter of fact, Aristotle introduces two kinds of σημεῖον, depending on whether U is a sufficient or a necessary condition for P. In the latter case, the ‘indication’ is called a τεκμήριον, in the former, the kind bears no specific name and will be referred to as σημεῖον tout court. The Aristotelian example of ‘indication’ of the first kind in this passage reads thus: ἔστιν δὲ τῶν σημείων τὸ μὲν ὡς τὸ καθ’ ἕκαστον πρὸς τὸ καθόλου ὧδε, εἴ τις εἴπειεν σημεῖον εἶναι ὅτι οἱ σοφοὶ δίκαιοι, Σωκράτης γὰρ σοφὸς ἦν καὶ δίκαιος (1357b10–13).

56 The formula σημεῖον δέ … is typical of Aristotle. In addition to our passage, there are 5 occurrences of the subformula σημεῖον δὲ τὸ συμβαῖνον ἐπί …· … γάρ …: they are at Gen. an. 1.21, 729b33–a4; Gen. an. 5.3, 783a18–20 (U: the cold congeals and dries, and this makes things hard; P: this is what happens with sea-urchins: for they live in cold seawater and have long, hard spines); Part. an. 4.5, 680a29–35 (P involves sea-urchins again!); Part. an. 4.10, 688b6–14; Poet. 4.4, 1448b5–13 (U: it is natural for all to delight in works of imitation; P: this is what happens with artwork: for we delight in viewing the realistic representations of objects that are painful to see in themselves). Note that, even if the σημεῖον is properly the relation between P and U (by its very definition: τῶν δὲ σημείων τὸ μὲν οὕτως ἔχει ὡς τῶν καθ’ ἕκαστόν τι πρὸς τὸ καθόλου at Rh. 1.2, 1357b1–2), Aristotle often abridges the relational structure and refers to P itself as the σημεῖον.

57 This reading is directly contradicted by the very passage Cat. 14, 15a30–1, quoted in note 46 above: οἷον τὸ τετράγωνον γνώμονος περιτεθέντος ηὔξηται μέν, ἀλλοιότερον δὲ οὐδὲν γεγένηται (note the passive aorist: the construction places just one gnomon round a square and thereby reaches a final result). The fact that only one gnomon is placed round the square and the very definition of γνώμων rule out the meaning ‘to circumscribe’ for περιτίθημι in this context. This does not exclude, however, that in other contexts in Aristotle's writing the verb can mean ‘to circumscribe (completely)’, and this is indeed what happens: see e.g. the technical passage at Cael. 2.2, 285a31–b5.

58 At in Nic. 58.22, 59.15, 60.12, 77.5 Pistelli (= 4.12, 128.6 app., corrected to προστεθεῖσα; 4.16, 128.20; 4.20, 128.38; and 4.85, 146.4 Vinel respectively), always in the context of placing gnomons round. An isolated occurrence is at Theon, in Alm. 1.10, 472.11 Rome, referring to the gnomon placed round a square during the procedure of extraction of an approximate square root.

59 Most occurrences in the Greek mathematical corpus actually refer to a γνώμων as the pointer of a sundial. The standard definition of a geometrical γνώμων in Greek mathematics is Euc. El. 2.def.2: παντὸς δὲ παραλληλογράμμου χωρίου τῶν περὶ τὴν διάμετρον αὐτοῦ παραλληλογράμμων ἓν ὁποιονοῦν σὺν τοῖς δυσὶ παραπληρώμασι γνώμων καλείσθω. A definition of the numerical γνώμων can be found in Iambl. in Nic. 58.19–21 Pistelli (= 4.12, 128.5–6 Vinel): εἴρηται δὲ γνώμων ὁ αὐξητικὸς ἑκάστου εἴδους τῶν πολυγώνων κατὰ πρόσθεσιν τὸ αὐτὸ εἶδος διαφυλάττων. We also read a definition, quite contrivedly adapted to both the geometrical and the numerical case, in the pseudo-Heronian Definitiones; see Def. 58, 44.13–14 Heiberg: καθόλου δὲ γνώμων ἐστὶν πᾶν, ὃ προσλαβὸν ὁτιοῦν, ἀριθμὸς ἢ σχῆμα, ποιεῖ τὸ ὅλον ὅμοιον, ᾧ προσείληφεν. All sources, from Iamblichus to Philoponus, agree that a γνώμων ‘derives metaphorically its name from those employed in geometry’ (μετῆκται δὲ ἀπὸ τῶν ἐν γεωμετρίᾳ τὸ ὄνομα, again in the words of Iamblichus, in Nic. 58.23–4 Pistelli = 4.12, 128.7 Vinel). The only attestation of the term γνώμων among early Pythagorean authors is in Philolaus, fr. 44 B 11 DK: νῦν δὲ οὗτος καττὰν ψυχὰν ἁρμόζων αἰσθήσει πάντα γνωστὰ καὶ ποτάγορα ἀλλάλοις κατὰ γνώμονος φύσιν ἀπεργάζεται σωμάτων καὶ σχίζων τοὺς λόγους χωρὶς ἑκάστους τῶν πραγμάτων τῶν τε ἀπείρων καὶ τῶν περαινόντων (= Stob. Ecl. 1.prooemium.3, 1.16.8–12). However, the entire fragment is surely spurious: C.A. Huffman, Philolaus of Croton, Pythagorean and Presocratic (Cambridge, 1993), 347–50. To the arguments expounded by Huffman, we may add that this quote has an obvious lexical affinity with the Aristotelian passage: note in particular the adverb χωρίς.

60 See e.g. the discussion of the mathematical number at Metaph. M.6–7.

61 See Theon of Smyrna, Exp. 20.19–20: Ἀρχύτας δὲ καὶ Φιλόλαος ἀδιαφόρως τὸ ἓν καὶ μονάδα καλοῦσι καὶ τὴν μονάδα ἕν; Alex. in Metaph. 39.14–15: [οἱ Πυθαγόρειοι] διὰ τὸ μόνιμον δὲ καὶ τὸ ὅμοιον πάντῃ καὶ ἀρχικὸν τὸν νοῦν μονάδα τε καὶ ἓν ἔλεγον.

62 See Acerbi, F., ‘Ones’, GRBS 53 (2013), 708–25Google Scholar, at 713–21.

63 At in Ph. 457.12 and in Ph. 394.21 respectively. See also Cornford (n. 32), 10–11.

64 See again Cat. 14, 15a30–1: οἷον τὸ τετράγωνον γνώμονος περιτεθέντος ηὔξηται μέν, ἀλλοιότερον δὲ οὐδὲν γεγένηται.

65 See e.g. the text at Ph. 8.9, 265a32–b2: τῆς δὲ περιφεροῦς ἀόριστα· τί γὰρ μᾶλλον ὁποιονοῦν πέρας τῶν ἐπὶ τῆς γραμμῆς; ὁμοίως γὰρ ἕκαστον καὶ ἀρχὴ καὶ μέσον καὶ τέλος, ὥστ’ ἀεί τε εἶναι ἐν ἀρχῇ καὶ ἐν τέλει καὶ μηδέποτε. διὸ κινεῖταί τε καὶ ἠρεμεῖ πως ἡ σφαῖρα· τὸν αὐτὸν γὰρ κατέχει τόπον: the same point in a circle can be the beginning or the end of any motion along the circle itself.

66 On the exclusively iterative meaning of ἀεί in mathematics see Ch. Mugler, Dictionnaire historique de la terminologie géométrique des Grecs (Paris, 1959), 43–4; Federspiel, M., ‘Sur les emplois et les sens de l'adverbe ἈΕΙ dans les mathématiques grecques’, LEC 72 (2004), 289311 Google Scholar. On iterative ἀεί in Aristotle see Ugaglia, M., ‘Boundlessness and iteration: some observations about the meaning of ἀεί in Aristotle’, Rhizai 6 (2009), 193213 Google Scholar.

67 See, in particular, Ph. 6.2, 232b26–233a10; 6.4, 235a18–37; 6.5, 236a28–35, 236b10–15; 6.6, 237a20–34; 6.7, 238a36–b9; 8.8, 263a4–11; 8.10, 266a23–b24, on the transitivity of unlimited divisibility; Ph. 6.9, 240a29–b7, on circular motion; Ph. 6.10, 241b14–20; Gen. corr. 2.11, passim, on the chain of natural changes; Ph. 7.1, 242a49–57; Gen. corr. 2.5, 332b31–333a7; Metaph. Θ.8, 1050b4–6, on the chain of productive principles; Ph. 2.7, 198a33–5; Metaph. Λ.10, 1075b24–6, on causal chains; Ph. 1.4, 187b30–188a4; Cael. 3.7, 305b20–6, on the divisibility of the elements; An. post. 1.23, 84b32–5 and 85a1–12; 2.14, 98a1–12, on explicative chains; Metaph. Θ.12, 1037b32–1038a16; Top. 4.2, 122a3–5; An. pr. 1.31, 46a39–b22; de An. 1.3, 407a23–30; Part. an. 1.3, 644a2–3, on constructing definitions; An. post. 2.12, 95b13–24, on inserting middle terms; Metaph. Ζ.7, 1032b6–9; Eth. Nic. 3.3, 1113a2, on the deliberative process. The emphasis is always on the fact that a step is necessarily followed by another similar to it: this is further confirmed by the fact that, when describing iterative processes, Aristotle frequently associates the adverb συνεχῶς (or its derivates) with ἀεί (see e.g. Gen. corr. 2.10, 336a34–b2; 2.11, 338a11–13). Sometimes, συνεχῶς directly replaces ἀεί (Mete. 1.8, 346b8; Metaph. Δ.28, 1024a29–31).

68 In Greek arithmetic, the sexagesimal orders are numerical εἴδη; they correspond to the orders of magnitude in the decimal system: hundreds and thousands are numerical εἴδη, since they are squares (100 is the square of 10) and cubes (1000 is the cube of 10) respectively. They are numerical species but not numbers: indeed, 300 is not a square, but 3 items of the ‘square’ εἶδος 100. This also holds true for fractional numbers: the ‘seconds’ of the sexagesimal system belong to the species ‘square’, in so far as 1/3600 is the square of 1/60. In number theory, the system of abstract numerical species is fully developed in Diophantus' Arithmetica (1.2.14–6.21 Tannery). The species there introduced are assigned a denomination and a conventional sign, made of the initial letter of each component of the denomination: to the square or δύναμις corresponds the sign ΔΥ, to the κύβος the sign ΚΥ, to the ‘fourth power’ or δυναμοδύναμις the sign ΔΥΔ, etc.

69 Philolaus, fr. 44 B 5 DK = Stob. Ecl. 1.21.7c, 1.188.9–12. Note the genus-species relation formulated in terms of the εἶδος-μορφή opposition. For a valuable discussion of the fragment see Huffman (n. 59), 177–85, from which the translation is taken, with one slight modification.

70 Burkert (n. 31), 264–5 rightly argues that the one is here alluded to.

71 To the sources mentioned in n. 37 above, one may add Alex. in Metaph. 56.18–35 and 85.15–18, Simpl. in Ph. 151.6–8 and 453.25–8, all referring to Aristotle's exposition of Platonic doctrines in his Περὶ τἀγαθοῦ. For a discussion, focussing on whether the notion of Indefinite Dyad can be ascribed to Plato or not, see Burkert (n. 31), 21–2 and n. 28, and L. Tarán, Speusippus of Athens. A Critical Study with a Collection of the Related Texts and Commentary (Leiden, 1981), 225–6 and 351–6. See, finally, [Iambl.] Theol. 8.20–9.2 De Falco, and cf. with ibid. 9.12–10.8, where the constructions of squares by placing odd numbers round the unit and of ἑτερομήκεις by placing even numbers round the dyad are opposed, as indications that τῶν μὲν πάντη ὁμοίων καὶ ταὐτῶν καὶ μονίμων, ὅ ἐστι τετραγώνων, ἡ μονὰς αἰτία, whereas τῶν δὲ πάντη ἀνομοίων, ὅ ἐστιν ἑτερομήκων, ἡ δυὰς πάλιν αἰτία.

72 For Neo-Pythagorean arithmologic writings see Nic. Theol., apud Phot. Bibl., codex 187, 143b1–2, and cf. [Iambl.]. Theol. 8.8–10 and 12.10–13.2, and the sources mentioned in n. 71. For the ascription of this doctrine to Pythagoras himself, see the sources mentioned in n. 37 above.

73 But this correlative nexus is poorly attested before Aristotle, where we gather about 140 occurrences of it. If we keep to original writings, we find it only at Thuc. 7.27.4, and, with some 34 occurrences, in several treatises of the Corpus Hippocraticum. In all cases, the usage conforms with the one in Aristotle.

74 Here are some examples [note non-iterative ἀεί (broken underlining)]: τοῦτο δὲ οὐχ ὁτὲ μὲν ἔδει γίγνεσθαι ὁτὲ δὲ οὔ, ἀλλ’ ἀεί (Mete. 1.6, 343b27–8); ἢ γὰρ ἁπλῶς ἔχει τὸ φύσει, καὶ οὐχ ὁτὲ μὲν οὕτως ὁτὲ δ’ ἄλλως, οἷον τὸ πῦρ ἄνω φύσει φέρεται καὶ οὐχ ὁτὲ μὲν ὁτὲ δ' οὔ (Ph. 8.1, 252a17–19); οὐ γὰρ ὁτὲ μὲν ἄλλοις ὁτὲ δὲ ἄλλοις μέμεικται ταῖς ψυχαῖς ὁ παρὰ τοῦ θεοῦ χρυσός, ἀλλ’ ἀεὶ τοῖς αὐτοῖς (Pol. 2.5, 1264b11–13); τὸ γὰρ ‘τοῦτο’ σημαίνει ὁτὲ μὲν ‘οὗτος’ ὁτὲ δὲ ‘τοῦτον’ (Soph. el. 14, 173b36).

75 But, on the other hand, ὁτὲ μέν … ὁτὲ δέ … is quite an effective syntactical device if the ἀπειρία of a process has to be emphasized.

76 See e.g. de An. 2.7, 418b31–419a1: ἡ γὰρ αὐτὴ φύσις ὁτὲ μὲν σκότος ὁτὲ δὲ φῶς ἐστιν.

77 See e.g. Metaph. B.5, 1002a34–b2: ὅταν γὰρ ἅπτηται ἢ διαιρῆται τὰ σώματα, ἅμα ὁτὲ μὲν μία ἁπτομένων ὁτὲ δὲ δύο διαιρουμένων γίγνονται.

78 Cael. 1.10, 279b12–16: γενόμενον μὲν οὖν ἅπαντες εἶναί φασιν, ἀλλὰ γενόμενον οἱ μὲν ἀΐδιον, οἱ δὲ φθαρτὸν ὥσπερ ὁτιοῦν ἄλλο τῶν συνισταμένων, οἱ δ’ ἐναλλὰξ ὁτὲ μὲν οὕτως ὁτὲ δὲ ἄλλως ἔχειν [φθειρόμενον], καὶ τοῦτο αἰεὶ διατελεῖν οὕτως (cf. 280a11–19).

79 Ph. 4.9, 217a26–31: ἔστι δὲ καὶ σώματος ὕλη καὶ μεγάλου καὶ μικροῦ ἡ αὐτή. δῆλον δέ· ὅταν γὰρ ἐξ ὕδατος ἀὴρ γένηται, ἡ αὐτὴ ὕλη οὐ προσλαβοῦσά τι ἄλλο ἐγένετο, ἀλλ’ ὃ ἦν δυνάμει, ἐνεργείᾳ ἐγένετο, καὶ πάλιν ὕδωρ ἐξ ἀέρος ὡσαύτως, ὁτὲ μὲν εἰς μέγεθος ἐκ μικρότητος, ὁτὲ δ’ εἰς μικρότητα ἐκ μεγέθους.

80 Taking the noun εἶδος according to two different meanings is a linguistic trick. But Aristotle is fond of this argumentative strategy, and one might surmise that he presents the example both to implicitly show its inconsistency (after all, Aristotle does not believe that the Pythagorean doctrine is true) and in admiration of its clever conception. A case in point is the adverb ἀεί at Ph. 3.5, 203a16, immediately responding to the ἀεί at the end of the Aristotelian passage and unconvincingly athetized by Ross on the basis of contradictory manuscript evidence: οἱ δὲ περὶ φύσεως πάντες ἀεὶ ὑποτιθέασιν ἑτέραν τινὰ φύσιν τῷ ἀπείρῳ τῶν λεγομένων στοιχείων. There is a supreme irony in lexically correlating the endless process of production of continuously different numerical species (i.e. of a continuously different parity: logical iterative ἀεί) and the endless process of production of continuously different theories (i.e. of a continuously different quality: temporal iterative ἀεί) on the nature of the Unlimited. Note, also, as a further stylistic touch correlating the two passages, the verbs in variation περιτίθημι and ὑποτίθημι.

81 The first square must be excluded, since it is obtained by placing a gnomon round the one, which is neither even nor odd.

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