In many passages of his writings Proclus refers to the theory of "atomoi grammai", attributing it to Plato’s scholar Xenocrates of Chalcedon (i.e. In Tim. II, p. 246, l. 1 ff.; p. 165, l. 8 ff.; In Rem Publ. II, p. 27, l. 4 ff.; p. 48, l. 5 ff.; In Parm., p. 888, ll. 17-36; Instit. Phys. I, sect. 14 ll. 2-17; sect. 17, l. 10 ff.). The most relevant of these passages is to be found in his Commentary on the first Book of Euclid’s Elements (In pr. Eucl., Part I, Prop. X), where he recalls arguments focussing on the «incommensurability among magnitudes» (p. 278, l. 20), that can be read in the Pseudo-aristotelian work On the Indivisibile Lines (968 b 5-22). In this Opusculum (preserved and transmitted inside the Corpus Aristotelicum, but probably spurious) the Author presents a series of “proofs” of the existence of atomoi grammai (968 a 2-b 22), before moving against them “counter-proofs” (968 b 22-969 b 28) and many different arguments, which defend the character of continuity (syneches) of lines and magnitudes (969 b 28-972 34). Proclus’ passage about the same controversy was never thouroghly analysed (for example, only a few number of footnotes are devoted to it by G. Morrow in his translation, Princeton 1970, 19922; stimulating but not systematic comments are made on it by O. Apelt, Beiträge zur Geschichte der griechischen Philosophie, Leipzig 1891, p. 267 ff. and M. Timpanaro-Cardini, Pseudo-Aristotele, De lineis insecabilibus, Milano-Varese 1970, pp. 34-37). I’d like to comment it line by line in order to throw light on two main points: 1. Proclus’ sources and the way he uses them – with particular regard to Geminus (p. 278, l. 12); 2. the position he attributes to Euclid (p. 279, l. 1 ff.) and the one he assumes himself in the ancient debate about continuity or discontinuity of geometrical magnitudes: the divisibility of every continuum (whose definition is given at p. 278, ll. 15-16) is according to Proclus an «axiom» (ivi, l. 24); that every continuum is divisible to infinity is neither an axiom, nor a «hypothesis» (278, l. 2), but it can be demonstrated by geometers «from appropriate principles» (ivi, l. 19).

«An agreed principle in geometry: a magnitude consists of parts infinitely divisibile» (In pr. Eucl., p. 278, ll. 11-12). Proclus’ reception of the Pseudo-aristotelian work "On the Indivisibile Lines"

CATTANEI E
2012

Abstract

In many passages of his writings Proclus refers to the theory of "atomoi grammai", attributing it to Plato’s scholar Xenocrates of Chalcedon (i.e. In Tim. II, p. 246, l. 1 ff.; p. 165, l. 8 ff.; In Rem Publ. II, p. 27, l. 4 ff.; p. 48, l. 5 ff.; In Parm., p. 888, ll. 17-36; Instit. Phys. I, sect. 14 ll. 2-17; sect. 17, l. 10 ff.). The most relevant of these passages is to be found in his Commentary on the first Book of Euclid’s Elements (In pr. Eucl., Part I, Prop. X), where he recalls arguments focussing on the «incommensurability among magnitudes» (p. 278, l. 20), that can be read in the Pseudo-aristotelian work On the Indivisibile Lines (968 b 5-22). In this Opusculum (preserved and transmitted inside the Corpus Aristotelicum, but probably spurious) the Author presents a series of “proofs” of the existence of atomoi grammai (968 a 2-b 22), before moving against them “counter-proofs” (968 b 22-969 b 28) and many different arguments, which defend the character of continuity (syneches) of lines and magnitudes (969 b 28-972 34). Proclus’ passage about the same controversy was never thouroghly analysed (for example, only a few number of footnotes are devoted to it by G. Morrow in his translation, Princeton 1970, 19922; stimulating but not systematic comments are made on it by O. Apelt, Beiträge zur Geschichte der griechischen Philosophie, Leipzig 1891, p. 267 ff. and M. Timpanaro-Cardini, Pseudo-Aristotele, De lineis insecabilibus, Milano-Varese 1970, pp. 34-37). I’d like to comment it line by line in order to throw light on two main points: 1. Proclus’ sources and the way he uses them – with particular regard to Geminus (p. 278, l. 12); 2. the position he attributes to Euclid (p. 279, l. 1 ff.) and the one he assumes himself in the ancient debate about continuity or discontinuity of geometrical magnitudes: the divisibility of every continuum (whose definition is given at p. 278, ll. 15-16) is according to Proclus an «axiom» (ivi, l. 24); that every continuum is divisible to infinity is neither an axiom, nor a «hypothesis» (278, l. 2), but it can be demonstrated by geometers «from appropriate principles» (ivi, l. 19).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/989194
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