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Indivisible Lines

Published online by Cambridge University Press:  11 February 2009

A. T. Nicol
Affiliation:
Newnham College, Cambridge

Extract

The name of Democritus can claim a place in any discussion of indivisibles. Yet its introduction in this paper seems to depend on the lucus a non lucendo principle; for Democritus did not believe in the existence of indivisible lines. Nowhere is the belief ascribed to him and in at least one place it is implicitly denied, the scholion on De Caelo 268a 1, which says he made his elements indivisible solids, as contrasted with lines or surfaces. Two passages, one from Plutarch, the other from Simplicius, will show why he could not believe in indivisible lines.

Type
Research Article
Copyright
Copyright © The Classical Association 1936

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References

page 121 note 1 Since I wrote this paper, W. D. Ross's edition of the Physics has appeared. In a note on this passage he states his reasons for referring ἒνιοι to the atomists: firstly that as Simplicius points out, Plato did not believe in a ἁπλῶς μὴ ὂν but in a μὴ ὂν τι secondly that atomism was the most important theory of ἃτομα μεγέθη, and is in fact described (De Gen. et Corr. 324b 35–25a 32) as giving in to the Eleatic argument about μὴ ὂν by asserting the existence of τὸ κενόν. He compares ὁμολογήσας (sc. Λεύκιππος) in De Gen. et Corr. with ἐνέδοσαν in the Physics (Ross, , Physics pp. 479–81Google Scholar).

But Simplicius' objection would apply equally to Metaphysics 1089a 2–6, which has ἀνάγκη εῑναι τὸ μὴ ὂν δεῖξαι ὂτι ἒστιν, and which certainly refers to the Platonists. Again, in the De Gen. et Corr. passage the dichotomy is not mentioned. The atomists are said to admit the more general Eleatic argument that the nature of the one is inconsistent with the appearances of motion and plurality. At Physics VI, 9, 239b 18 τῷ διχοτομεῖν is used of Zeno's first paradox, the division of a line, and many of the arguments cited by Simplicius on 187a I are most readily understood if referred to points and lines, e.g. between any two points there must be another point. Hence it is probable that the ἂτομα μεγέθη posited by those who gave in to the dichotomy were indivisible lines.

page 122 note 1 βουλόμενοι δὲ τὰς οὐσίας ἁνάγειν είς τὰς ἀρχὰς μήκη μὲν τίθεμεν ὲκ βραχέος καὶ μακροῦ, ἒκ τινοςμικροῦ καὶ μεγάλου, καὶ ἐπίπεδον ἐκ πλατέος καὶ στενοῦ, σῶμα δ' ἐκ βαθέος καὶ ταπεινοῦ. καίτοι πῶς ἒξει ἣ τὸ ἐπίπεδον γραμμὴν ἣ τὸ στερεὸν γραμμὴν καὶ ἐπίπεδον; ἂλλο γὰρ γένος τὸ πλατὐ καὶ στενόν καὶ βαθὐ καὶ ταπεινόν. ὢσπερ οὖν οὐδ' ἀριθμὸς ὑπάρχει ἐν αὐτοῖς, ὂτι τὸ πολὺ καὶ όλίγον ἒτερον τοὺτων, δῆλον ὂτι οὐδ' ἂλλο οὐθὲν τῶν ἂνω ὑπάρξει τοῖς κάτω. ἀλλὰ μὴν οὐδὲ γένος τὸ πλατὺ τοῦ βαθέος. ἦν γὰρ ἂν ἐπίπεδόν τι τὸ σῶμα. ἒτι αί στιγμαί κτλ.

page 123 note 1 Ζητῶν γὰρ τὰς ἀρχὰς τῶν ὂντων ὁ Πλάτων, ἐπεὶ πρῶτος ὁ ἀριθμὸ ἐδόκει αὐτῷ τῇ ϕύσει εῑναι τῶν ἂλλων (καὶ γὰρ τῆς γραμμῆς τὰ πέρατα σημεῖα, τὰ δὲ σημεῖα εῖναι μονάδας θέσιν ἒχουσας, ἂνευ τε γραμμῆς μήτε ἐπιϕάνειαν εῖναι μήτε στερεόν, τὸν δὲ ἀριθμὸν καὶ χωρὶς τούτων εῖναι δύνασθαι) ἐπεὶ τοίνυν πρῶτος τῶν ἂλλων τῇ ϕύσει ὁ ἀριθμός, ἂρχὴν τοῦτου ὴγεῖτο εῖναι, καὶ τὰς τοῦ πρώτου ἀριθμοῦ ἀρχὰς καὶ παντὸς ἀριθμοῦ ἀρχάς….

page 123 note 2 ‘Zahl und Gestalt’ p. 78; ‘Platon sucht ein geordnetes System von Grenzen, an denen immer in eine andere Art binübergeschritten werden darf und muss.’

page 124 note 1 In De Caelo III, 1. Aristotle objects that there is more than one way of putting surfaces together, just as there is more than one way of putting lines together. Did he have in mind, whether consciously or not the idea that Plato talked outside the dialogues as if surfaces could be laid one on top of the other to form solids? This idea might arise from a misunderstanding of the indivisible solid, which had no actual depth and yet held the possibility of depth. I do not suggest that Aristotle made this mistake, but it may have been made, and it may have had something to do with Aristotle's criticism of the Timaeus.

page 125 note 1 See Zeno of Elea: a Text with Introduction and Notes,’ by Lee, H. D. P. (Cambridge Classical studies)Google Scholar.