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Geometrical Method and Aristotle's Account of First Principles1

Published online by Cambridge University Press:  11 February 2009

H. D. P. Lee
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Extract

The object of this paper is to show the predominance of the influence of geometrical ideas in Aristotle's account of first principles in the Posterior Analytics— to show that his analysis of first principles is in its essentials an analysis of the first principles of geometry as he conceived them. My proof of this falls into two parts. I. A consideration of the parallel between Aristotle's and Euclid's account of first principles. II. A comparison between the general movement of thought in the Aristotelian ⋯παγωγ⋯ and ⋯π⋯δειξις and in the δι⋯νοια and ν⋯ησις of the Platonic dialectic.

Type
Research Article
Information
The Classical Quarterly , Volume 29 , Issue 2 , April 1935 , pp. 113 - 124
Copyright
Copyright © The Classical Association 1935

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References

page 113 note 2 Anal. Post. Ch. 2.

page 113 note 3 This characteristic is not actually stated in Ch. 2, in the enumeratiion of the characteristics of the ⋯ρχα⋯ of science. But it is implied by by 71b 13 and stated in Ch. 6; and I have therefore included it.

page 114 note 1 Cf. Ross, , Aristotle, p. 43Google Scholar. Dr. Ross (ibid. p. 44) notices the similarity between Aristotle's and Euclid's account of first principles: but does not work it or its implications out in any detail.

page 115 note 1 According to Aristotle the geometer must assume the existence of points and lines (76b 5): Euclid's Postulates 1 to 3 may in eflect be said to do this, declaring the possibility of drawing a but straight line between any two points, of producing a straight line indefinitely, and of describing a circle, ‘the only “lines” beside straight lines used in the elements’ (Heath, E. 146). Postulates 4 and 5 are also closely connected with constructions (Heath, , G.M. P. 375Google Scholar, q.v.): but v. below sect. 7.

page 116 note 1 Heath, E. p. 119 does not seem to keep these two senses distinct: at any rate he gives no indication that it is with a subsidiary sense only of hypothesis that Aristotle is dealing in this passage.

He remarks also (p. 119) that 76b 32, ἔστιναἴτημα τ⋯ ὐπεναντ⋯ον τοὐ μανθ⋯νοντος τῇ δοξῇ, ἤ ⋯ ἄντις ⋯ποδεικτ⋯ν ὂν λαμβ⋯νῃ κα⋯ χρ⋯ται μ⋯ δε⋯ξας, ‘gives a wider meaning to “postulate” which would cover “hypothesis” as well.’ If the reading given is correct, it is still only the subsidiary sense of hypothesis that would be covered. I prefer to leave out the ἤ with the Oxford Translation, thus making the sentence a definition postulate simply.

page 116 note 2 There seems to be a still further sense of hypothesis defined in 76b 39–77a 3: ‘assumptions which serve to bring the truth of the conclusion home to the student but whose truth is not required by the proof; e.g., the geometer's assumption that the line he draws is a foot long or that it is straight.’ (Ross, , Aristotle, p. 44.)Google Scholar

page 116 note 3 Cf. αἰτεῖν=to beg the question. αἴτημα scarcely occurs often enought in Aristotle to be a technical term. The Index Aristotelicus gives five references apart from this passage: 97a 21, where it again has the meaning illegitimate assumption: 86a 34, and 418b 26, where it seems neutral, meaning simply assumption: 1433b 17 ff. where a rhetorical postulate (referred to also at 1428a 8) is defined—a different sense entirely from the logical.

page 118 note 1 Used as a text-book by the Academy (Heath, l.c.).

page 118 note 2 Cf. Procl, . Eucl. I, p. 66Google Scholar, quoted Heath, , G.M. 320Google Scholar.

page 120 note 1 ‘αἴσθησις … is the most general term for immediate perception whether sensuous or intellectual’ (Burnet, , Ethics, Introd. p. xxxviii)Google Scholar.

page 120 note 2 I agree with Grant that it is to the intuitive perception of νο⋯ς that Aristotle, is here referring (Ethics, p. 453Google Scholar: note ad loc), rather than with Stewart, who says he refers to sense perception as the basis of ⋯παγωγ⋯ (Notes on N.E. Vol. I, p. 112).

page 120 note 3 Cf. Plato's ἄψασθαι.

page 120 note 4 Cf. ‘It is by perception or intuition that the first principles of mathematics, for instance, are apprehended’ (Burnet, , Ethics, Introd. p. xxxvii)Google Scholar.

page 121 note 1 Cf. Burnet, , Ethics, p. 39Google Scholar, note on ⋯θισμῷ 1098b 4: Stewart, Notes on the N.E. Vol I, pp. 112–3: Grant, , Ethics, Vol. I, p. 453Google Scholar.

page 122 note 1 The ‘particulars’ involved in this induction are species, e.g., man, horse and mule in the example Aristotle gives: v. Ross, , Aristotle, p. 39Google Scholar.

page 123 note 1 Cf. Burnet, : Ethics, Introd. §§ 24Google Scholar and 25 for the dialectical method in ethics.