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Aristotle, Mathematics, and Colour

  • Richard Sorabji (a1)

Aristotle says in the De Sensu that other colours are produced through the mixture of black bodies with white (440a31–b23). The obvious mixture for him to be referring to is the mixture of the four elements, earth, air, fire, and water, which he describes at such length in the De Generatione et Corruptione. All compound bodies are produced ultimately through the mixture of these elements. The way in which the elements mix is described in i. 10 and 2. 7. They mix in such a way as to produce an entirely new substance, in which the characteristics of the original earth, air, fire, and water survive only in modified form.

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page 293 note 1 D.S. 439b14–18, with 439a18–21 and D.A. 418b9–20. The fact that fire (especially pure, elemental fire) is to a certain degree transparent may also contribute to whiteness, in Aristotle's opinion. And the non-transparency of earth may contribute its blackness. At any rate, on one inter-pretation D.S. 439b6–12 implies that a total lack of transparency would yield black, while a certain degree of transparency increases the whiteness (though, as the next note indicates, too high a degree of trans- parency will instead make something colourless). Cf. also G.A. 779b27–33; 780a27–36 for a connection between non-transparency and blackness.

page 293 note 2 Thus fieriness in air produces light, not white (D.S. 439b14–i8). In general, Aristotle (forgetting the case of wine or coloured gems) speaks as if transparent things (like air or water) have no colour of their own, but only borrowed colour, due to reflection or other causes. This is the best interpretation of D.A. 418b4–6: ‘by transparent I mean what is visible, yet not in itself visible speaking without qualification, but visible through borrowed colour.’ Cf. also D.S. 439b10–14, which, on one interpretation, speaks of ‘transparent things themselves, like water and anything else there may be of this kind, I mean those ones which appear to [sc. merely appear to, but do not really] have a colour of their own’.

On the other hand, water and air can have borrowed colour due to reflection (Meteor, 1. 5 and 3. 2–6; D.S. 439b3–6), or other causes (the water of the eye-jelly takes on colour during the act of vision, and this is not due to reflection). Water displays a colour which gets darker, according as the water gets deeper and less transparent (G.A. 779b27–33; 780b8).

page 293 note 3 Air bubbles give whiteness to foam, semen, oil, hair, and other stuffs. What is more watery, and less full of air, is darker (G.A. 735a30–736a22; 784b15; 786a7–13).

page 294 note 1 For symptoms of these facts in the De Sensu, see 439a18–19 ‘light [i.e. illumination] is the colour of the transparent’; 439b2 ‘sheen is a sort of colour’; 440a11 ‘the sun appears leukos’.

page 294 note 2 Both theories appeal to an illusion. When close up, one would see that there were two colours, black and white, instead of one. The theory of juxtaposition tries to avoid this by making the black and white specks too small to see, but this is a further impossibility. Also against the juxtaposition theory is that it is linked with the untenable idea that effluxes stream from the seen object into one's eyes, and misguidedly postulates a time-lag too short to perceive between the arrival of the black and of the white particles (D.S. 440a15–31, b16–i8).

page 294 note 3 D.S. 440a7–12. Aristotle mentions similar phenomena in the Meteorology (374a3; a27; b10).

page 295 note 1 Goethe (ZurFarbenlehre, 1810, translated with notes by Eastlake as Goethe's Theory of Colours, John Murray, 1840), knew, and wrote about, Aristotle's theory, but he mistakenly counted as Aristotelian the De Coloribus, which talks in ch. 2 of mixing lights, not stuffs.

For details of modern colour theory, consult Ralph Evans M., An Introduction to Colour, John Wiley and Sons, 1948, esp. pp. 65–6, 68–9.

page 295 note 2 Archytas' followers, ap. Porphyry's commentary on Ptolemy's Harmonics, ed. Wallis, p. 277; Plato, Timaeus 8OA–B; Aristotle, D.A. 426b3–6, Metaph. 1043a10 (cf. D.S. 447a12–448a19); pseudo-Aristotle, Problems 19. 27, 38; Euclid, Sectio Canonis, introd., ed. Meibom, p. 24; pseudo-Euclid, Isagoge § 5, ed. Meibom, p. 8; Nicomachus, Enchiridion § 12, ed. Meibom, p. 25; Gaudentius, Isagoge § 8, ed. Meibom, p. 11; Aelian, ap. Porphyry, loc. cit., pp. 218, 265, 270; Boethius, De Institutione Musica, 1. 3, 1. 8, 1. 28.-In general, see Gevaert F.A., Histoire et Théorie de la Musique de l'Antiquite, Ghent, 1875.

page 296 note 1 The word logos sometimes implies uncomplicated ratios (An. Post. a19; 20; D.S. 439b27-440a3; 440a14–i5; Probl. 19. 41). But sometimes it has a wider sense, covering all sorts of mathematical relations (An. Post. 90a22; D.S. 440b19; 442a15; 448a8; a10), even incommensurable ones.

page 296 note 2 A major problem about 440a3–6 is that it starts off by suggesting that all colours are (expressible) in (rational) numbers, but appears to finish up by talking of a sub-class which are not in numbers. If we retain the traditional text, the best way to remove this appearance of contradiction is probably that of J. I. Beare, in the Oxford translation. Instead of ‘not in numbers’, we must understand Aristotle to mean ‘not such (i.e. not pure) in their numbers’. I would translate: ‘Or one can also suppose that all colours are in numbers, but some are regular, others irregular, and these latter are produced when the colours are not pure through not being pure in their numbers.’ This, admittedly, is a strain on the Greek word order. In this translation, I take it that the impure colours are not a sub-class of, but are identical with, the irregular colours. It is for one and the same reason that they are called ‘irregular’ and ‘impure’. For the meaning of ‘impure’, see below, p. 297.

page 296 note 3 442a14–16 says ‘according to a logos [or ?] in a relation of more and less, whether according to certain numbers in the mixture and interaction, or also in an indefinite way’. I prefer to stick to the MSS. reading, which omits ‘or’. In that case, the opening phrase, ‘according to a logos in a relation of more and less’, will be a perfectly non-committal one, which does not specify the particular relationships available. It need mean no more than ‘in a quantitative relationship’. The insertion of ‘or’ has appealed to those who over-hastily connected the word logos with the logoi, or uncomplicated ratios, of chapter 3 (439b29–30; 440a14–15), and the words ‘more and less’ (mallon kai hêtton) with what in chapter 3 is described as merely (monon) an incommensurable (asummetron) relation of predominance and subordination (huperochê kai elleipsis, 439b30; 440b20).

On our interpretation, it is left to the following words to specify what the possible relationships are. And the following words can be taken in either of two ways. Perhaps the phrase ‘whether according to certain numbers in the mixture and interaction’ introduces the second and less usual alternative of 440a3–6, according to which all the relationships are commensurable and rational. The last phrase (‘or also in an indefinite way’) will then revert to the more usual alternative, according to which some relationships are commensurable, but there are ‘also’ incommensurable ones.

Alternatively, the remaining words confine themselves to the more usual alternative, and simply spell out the choice it offers between being in rational numbers and not being in rational numbers. In that case, the less usual alternative is never alluded to again after its original mention.

page 297 note 1 One would suppose, however, that a tertiary colour, such as pink, could be produced not only by mixing two secondary colours, but also by mixing one secondary colour (red) with white, or with colourless water, or again by mixing two tertiary colours.

page 297 note 2 ‘Eudoxos-Studien V’, in Quellen und Studien zur Geschichte der Mathematik, Abt. B. 3, 1936, p. 403.

page 297 note 3 ‘Platons Farbenlehre’, in Synusia, Festgabe für Wolfgang Schadewaldt, edd. Hellmut Flashar and Konrad Gaiser, Pfullingen, 1965.

page 297 note 4 A Commentary on Plato's Timaeus, Oxford, 1928, pp. 485, 489, 491. More extravagantly, J. Zürcher claims that the theory is not Aristotle's, but was added later by Theophrastus, under the influence of Aristoxenus (Aristoteles' Werk und Geist, Paderborn, 1952, pp. 302–5). A comparatively extensive contribution by Aristotle seems to be allowed in the account of Kucharski P., ‘Sur la théorie des couleurs et des saveurs dans le “De Sensu” aristotélicien’, Revue des études grecques lxvii (1954), 355, and Cornford F.M., ‘Mysticism and Science in the Pythagorean Tradition’, Classical Quarterly xvi (1922), 144.

page 298 note 1 Empedocles, in Aëtius 1. 15. 3; Democritus in Theophrastus, De Sensibus 7382, and in Aëtius 1. 15. 8; Plato in the Timaeus 67C–68D.

page 298 note 2 See 327b22–31; 328a29–31; 334b8–30, with Joachim Harold H.'s useful commentary, Aristotle on Coming-to-be and Passing away, Oxford, 1922.

page 299 note 1 D.S. 445b31–446a20. Tiny variations cannot be perceived on their own, but are perceived only through being part of, and through contributing to, a larger variation.

page 299 note 2 ‘Not in numbers’ (440a2–3) must mean ‘not expressible in (rational) numbers at all’. We cannot rescue Aristotle by taking it to mean ‘not in simple numbers’. The next line (440a3–4) rules out this interpretation, by putting forward the alternative that all colours are ‘in numbers’. This cannot mean ‘in simple numbers’, for there are not enough simple ratios to go round all the colours.

page 300 note 1 440a1–2.

page 301 note 1 ‘Mixed’ is a confusing word to use, since in one sense, black and white are mixed colours, i.e. colours that mix with each other to form other colours. But Aristotle clearly means to refer to colours like red and purple, which are mixed in the sense that each is composed of black and white.

‘Mixed things’ cannot refer to black and white. For (a) The case of black and white has already been dealt with in 448a1–5. (b) Lines 11–13 mention not two terms, but four (much: little and little: much). These four terms must correspond to the black and white that enters into a purple and the black and white that enters into a red. (c) If it were black and white that Aristotle was describing as ‘mixed‘, he would be implying that they were already mixed with each other to form a single intermediate colour. He could not then go on to say (448a8) that it would be impossible to perceive them simultaneously.

I believe there will be no obstacle to giving ‘mixed’ the required reference, provided we make a small emendation of the text at 448a8–9 from inline-graphic to inline-graphic. With this emendation, the argument will be using the opposition between red (which is one ratio) and purple (which is an opposed ratio), to explain why one can't perceive red and purple simultaneously. Without the emendation, Aristotle will be irrelevantly emphasizing the opposition within red and within purple.

page 302 note 1 Translated from Études sur le r⊚cle de la pensée médiévale dans la formation du systéme eartésien, Paris, 1930, p. 199.

page 303 note 1 In Introduction à la physique aristotélicienne, Louvain and Paris [1st edition, 1913], 2nd edition, 1946, esp. pp. 188, 225; and in ‘La physique aristotélicienne et la philosophic’, printed in Philosophie et Sciences, Journées d'études de la Société thomiste, 1936.

page 303 page 2 La Pensée grecque, Paris, 1923, p. 332.

page 303 note 3 Aristotle's System of the Physical World, Ithaca, New York, 1960, pp. 259–62.

page 303 note 4 Commentary on Aristotle's De Caelo, p. 642.

page 303 note 5 Aristotle's Criticism of Plato and the Academy, Baltimore, 1944, e.g. pp. 123, 124, 130, 161, 164. According to Cherniss, Aristotle also believes that different sounds are irreducible qualities, not to be explained as quantitative relations (p. 158 note). Because of this belief, says Cherniss, Aristotle denies in the De Anima that high notes are identical with swift movements (420a31–3). But in fact, the De Anima passage is very guarded. And whatever it says, it does not disagree with the view of the De Generation Animalium (786b25–787b20) that pitch varies with variation in speed.

page 304 note 1 The stages in his thought are traced out by Mansion A., Introduction à la physique aristotélicienne, 2nd edition, Louvain and Paris, 1946, pp. 190–5.

page 304 note 2 See Metaph. 993a17–22; 1092b17; D.A. 408a14; 410a1–6; 429b16; P.A. 642a18–23; G.A. 734b33.

page 304 note 3 For further evidence of Aristotle's willingness to use mathematics, see Owen G.E.L., ‘Aristotle’, in the Dictionary of Scientific Biography, ed. Gillispie C.C., vol. i (1970).

page 304 note 4 See p. 300, where it is pointed out that there are perhaps ten simple ratios available, and that Aristotle has not told us which of the ten correspond to the four or five pleasantest colours.

page 304 note 5 Op. cit., esp. pp. 193–5.

page 305 note 1 Review of Cornford's Principium Sapientiae in Gnomon xxvii (1955), 65.

page 305 note 2 He thinks theories about the observable world must be rejected if they are not consistent with the observable facts (see above, p. 303), but this is not yet to think of framing theories so as to fit them for empirical testing.

page 306 note 1 The Logic of Scientific Discovery, Hutchinson, 1959.

page 306 note 2 ‘Aristote et les mathématiques‘, Archiv für Geschichte der Philosophie 1903. Also Les Philosophes–Géometres de la Grèce, Paris, 1900, PP.358365.

page 306 note 3 Jaeger, Aristotle, Fundamentals of the History of his Development, Oxford, 2nd edition (translated from the German of 1923): ‘Aristotle lacked the temperament and the ability for anything more than an elementary acquaintance with the Academy's chief preoccupation, mathematics’ (p. 21). Robin, La Pensée grecque, Paris, 1923, says (p. 332) that it does not seem that Aristotle had the same mastery of mathematics as Plato. Blond Le, Logique et méthode chez Aristote, Paris, 1939, p. 192, calls Aristotle a mediocre mathematician. In contrast to Plato, he was not fundamentally a mathematician.

page 306 note 4 Aristotle, Physics, a revised text with introduction and commentary, Oxford, 1936, p. 29. Cf. p. 31: the need for more complexity should have been apparent also from the fact that Aristotle is forced to admit a certain exception to his proportionalities.

page 307 note 1 Carteron concludes that Aristotle was little versed in the mathematical sciences (Budé edition of Aristotle's Physics, 1926, vol. i, p. 16), Mansion that he neglected them (‘La physique aristotélicienne et la philosophic’, op. cit. (1936), pp. 26–7; cf. Introduction a la physique aristotélicienne, 2nd edition, 1946, p. 188).

page 307 note 2 Carteron H., La Notion de force dans le système d'Aristote, Paris, 1924;Owen G.E.L., ‘Aristotle’, in Dictionary of Scientific Biography, ed. Gillispie C.C., vol. i (1970).

page 307 note 3 See the definition of dialectic, at Top. 100a29–30, as reasoning that starts from endoxa.

page 307 note 4 For appeal to these facts, but as providing an exception, not confirmation, see Phys. 250a17–19.

page 307 note 5 See Phys. 250a3–4, ‘for in this way there will be a proportion.’.

page 308 note 1 440a3 442a22; see p. 296 n. 3 on 442a14–16.

page 308 note 2 I read earlier drafts of this paper in three places, and received many helpful comments. I have responded to, or made use of, those by Professor J. L. Ackrill, by Jonathan Barnes, by Willie Charlton, and by Professor H. Post and his colleagues at the Chelsea College of Science and Technology. The donkey-work on Aristotle's De Sensu was done while I held a Howard Foundation Fellowship from Brown University, and a project grant (no. H68–0–95) from the National Endowment for the Humanities.

I have used the following abbreviations:

An. Post. Analytica Posteriora



Cael.De Caelo

G. et C.De Generatione et Corruptions


D.A.De Anima

D.S.De Sensu

G.A.De Generatione Animalium



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