¹ Republic, 528b. Cf. 528d.Google Scholar

² e.g. Alexander Polyhistor apud Diogenes Laertius VIII.25. The subject is discussed at length in Guthrie, W. K. C., A history of Greek philosophy, Cambridge, 1971, i, 239ffGoogle Scholar, and is criticized by Burkert, W. in Lore and science in early Pythagoreanism, Cambridge, Mass., 1972, p. 66ffGoogle Scholar. Aristotle asserts the priority of arithmetic to geometry on these grounds at Posterior Analytics, I.27, 87a34–7. Hereafter cited as An. Post.

³ The principal sources for this are Aristotle, , Nicomachean ethics, I.2Google Scholar; Plato, , Euthydemus, 290b–dGoogle Scholar, Cratylus, 390bGoogle Scholar and Politicus, 259e–260b, 281d–3a (cf. 303e–5e)Google Scholar. It is discussed briefly and well by Cooper, J., Reason and human good in Aristotle, Cambridge, Mass., 1975, p. 14f.Google Scholar

⁴ Cf. Plato, , Republic, VII, 526dGoogle Scholar, for an example of the subordination of a theoretical science (geometry) to a practical one (strategy). Along similar lines, carpentry may be called subordinate to geometry insofar as it produces triangles, compasses, etc., for the use of the geometer.

⁵ Euthydemus 290cGoogle Scholar. The idea is developed in Republic, VI, 510–11 and VII, 521ff.Google Scholar

⁶ For the phrase see An. Post., I.7, 75b15.

⁷ The reservations and qualifications will be made below, in sections V ff.

⁸ These can be found at An. Post., I.9, 76a22–5; I.13, 78b37–9, b40–79a2, 79a10–11.

⁹ I shall refer to pairs of sciences like geometry and optics as subalternate sciences. Geometry is the superior and optics the subordinate science.

¹⁰ Lejeune, A., Euclide et Ptolémée: deux stades de l'optique géometrique grecque, Louvain, 1948, p. 94.Google Scholar

¹¹ Although this view of the relation between Aristotle and the geometry of Euclid's Elements is disputed (See contra Solmsen, F., Die Entwicklung der aristotelischen Logik und Rhetorik, Berlin, 1929, pp. 109–35Google Scholar. I hope to argue for my interpretation in a work to follow) all that matters for the present discussion is that (for whatever historical reasons) Aristotle's account in fact does fit the Elements and Optics, so that these works can reasonably be used as models to interpret Aristotle's words and to suggest ways to fill gaps in his account.

¹² For Aristotle's use of the visual ray, see for example Meteorologica, III, 2–6Google Scholar. His rejection of this view is clear enough in De anima, II, 7 (especially 418b14ff.)Google Scholar and De sensu 2, 437b11ffGoogle Scholar. Alexander, , Commentaria in meteorologica, Hayduck, , ed., Commentaria in Aristotelem Graeca, Berlin, 1899, iii, 2, 141.20Google Scholar, points out that from the mathematical point of view, it makes little difference which way the situation is described. Euclid's use of the visual ray is primarily to facilitate geometrical treatment of his topic. However, it has been claimed (Lejeune, , op. cit. (10), p. 78Google Scholar) that Euclid tended to conceive the visual ray as a physical factor in vision, and that this is shown by difficulties he encountered in representing it geometrically. Since visual rays were material, vision could not be continuous, and there had to be a distance between the rays in order to prevent physical collisions as they approached the eye. But might not the explanation for his treating vision as non-continuous be due instead to the desire to ‘save the phenomena’? It provides a way unavailable to a continuous theory of vision to account for the fact that when something is removed a sufficiently great distance, even though there is nothing between it and our eye, we nevertheless are no longer able to see it (the point of Optics, proposition 3). Also, it provides an explanation of the phenomenon (in fact due to the unevenness of the sensitivity of the retina) that we do not view simultaneously all the parts of an object before us (cf. Optics, proposition 1). I owe the above physiological explanation to Eecke, P. Ver, Euclide, l'Optique et la Catoptrique, Paris & Bruges, 1938, p. xvi.Google Scholar

¹³ Lejeune, , op. cit. (10), p. 174.Google Scholar

¹⁴ Ibid., p. 122.

¹⁵ I supply these numbers to indicate the different formal divisions of the proof. See note 17.

¹⁶ I have supplied references to the relevant theorems, etc. The Optics does not contain this justification of its reasoning.

¹⁷ (1) πρότασις (2) ἔκθεσις (3) διορισμός (4) κατασκευή (5) ⋯πόδειξις (6) συμπέρασμα. For the terminology and discussion of these formal parts of theorems, see Heath, T. L., A history of Greek mathematics, 2 vols., Oxford, 1921, i, 370.Google Scholar

¹⁸ The work begins with seven ὅροι which are indemonstrable principles of optics. From the word ‘ὅροι’ we might expect Euclid to be following his practice in the Elements, where the ὅροι are definitions. But they are not stated in the form of definitions, and not all of them can be interpreted as such, though some can. They are stated as follows: (1) Let it be hypothesized that straight lines coming out from the eye are carried a distance of great magnitudes; and (2) that the figure contained by the visual rays is a cone with its vertex in the eye and its base at the limits of what is seen; and (3) that those things are seen on which the visual rays fall, and those things on which they do not fall are not seen; and (4) things seen under greater angles appear greater, things seen under lesser angles appear lesser, and things appear equal which are seen under equal angles; and (5) things seen under higher rays appear higher, and things seen under lower ones appear lower; and (6) similarly things seen under rays which are more to the right side appear more to the right, and things seen under rays more to the left appear more to the left; (7) things seen under more angles are seen more precisely. Of these, (4) to (7) can be considered definitions—statements of how terms like ‘appear larger’ and ‘appear to the right of’ will be used. In fact, in the theorem given as an example, ὅρος (4) is used at the end to enable us to infer from the fact that one angle is greater than another to the conclusion that it appears larger. This suggests that the function of the ὅροι is to provide the fundamental liaison between optics and geometry which will allow certain statements in geometry to be interpreted or translated into statements in optics, thus preparing the way for the geometrical treatment of the subject. The same is true of (1) through (3), which do not so much introduce terms as provide a geometrical basis for optics, asserting that there are visual rays which move in straight lines from the eye, etc. We are told in effect that the study of optics will be a kind of projective geometry. For this reason I have preferred to translate ‘ὅρος’ as ‘determination’ instead of ‘definition’.

¹⁹ Cf. De partibus animalium I.1, 639b 7–10Google Scholar: καθάπερ οί μαθηματικο⋯ τ⋯ περỉ τήν άστρολογίαν δεικνύουσιν οűτω δεĩ καί τόν φυσικόν τ⋯ φαινόμενα πρ⋯τον τ⋯ περί τ⋯ ζῷα θεωρήσαντα καί τ⋯ μέρη τ⋯ περί ἕκαστον, ἔπειθ' οűτω λέγειν τό διά τί καἱ τ⋯ς αỉτίας. The φυσικός is like the optician in investigating the φαινόμενα concerning his subject; the δι⋯ τί, corresponding to the διότι, as revealed in demonstrations, requires the further knowledge of the mathematicians.

²⁰ See Barnes, J., Aristotle's Posterior analytics, Oxford, 1975, p. 152Google Scholar, for a handy and detailed list. Barnes's detailed discussion pp. 151–5, is useful and in broad agreement with the present interpretation.

²¹ Barnes says that some of Aristotle's statements on this head are ‘too vague to be helpful’ (op. cit. (20), p. 153), and that another is ‘not very precise’ (ibid.). The intention of the present work is to use non-Aristotelian material to obtain a clearer picture of what may be behind these unsatisfying remarks of the An. Post.

²² An. Post., I. 7, 75b10–12; I. 9, 76a8–9.

²³ An. Post., I. 9, 76a15: το⋯των αἱ ⋯ρχα⋯ ἔχουσι τ⋯ κοινόν.

²⁴ This topic has been discussed best and most thoroughly by Zabarella, , in (i) De tribus praecognitis, chapters XII–XIVGoogle Scholar (in Zabarella, , Opera logica, Venice, 1586Google Scholar) and in (ii) his commentary on the Posterior analytics (Lyons, 1587)Google Scholar. In what follows I make use of his treatment.

²⁵ Lejeune, , op. cit. (10), p. 22; cf. p. 31.Google Scholar

²⁶ Cf. Metaphysics, M. 3, 1078a 14–17Google Scholar: ⋯ δ' αὐτ⋯ς λ⋯γος κα⋯ περ⋯ περι ⋯ρμονικς κα⋯ ⋯πτικ⋯ς· ούδέτερα γ⋯ρ ή ⋯ψις ἤ ή φωνή θεωρεί, ⋯λλ' ή γραμμαί καί ⋯ριθμοί (οίκεĩα μέντοι ταũτα π⋯θη έκείνων), καί ⋯ μηχανικ⋯ δ⋯ ώσα⋯τως. (Although Aristotle refers here only to lines, his view applies equally well to the rest. He is not attempting to be exhaustive in his list of the objects of optics any more than he is at Physics, 194a10 when he refers to geometry as examining a line.) This puts the case better than the more familiar Physics, 2.2, 194a7–12: δηλοῖ δ⋯ καί τ⋯ φυσικώτερα τ⋯ν μαθημ⋯των, οῑον ⋯πτικ⋯ καί ⋯ρμονικ⋯ κα⋯ ⋯στρολογ⋯α· ⋯ν⋯παλιν γ⋯ρ τρόπον τιν’ ἒχουσιν τ⋯ γεωμετρία. ή μ⋯ν γ⋯ρ γεωμετρία περί γραμμ⋯ς φυσικ⋯ς σκοπεῖ, ⋯λλ’ οὐχ ῄ φυσική, ή δ’ ⋯πτική μαθηματικ⋯ν μ⋯ν γραμμήν, ⋯λλ’ οὺχ ή μαθηματική ⋯λλ’ ῄ φυσική.

²⁷ Cf. An Post., I.9, 76a12: τ⋯ γ⋯ρ ὑποκείμενον γ⋯νος ἓτερον.

²⁸ For arguments see Zabarella, , op. cit. (24, i), chapter XIV.Google Scholar

²⁹ Ibid.

³⁰ To use Zabarella's phrase.

³¹ That is, irrelevant to the main part of the proof. The use of the ‘determination’ in optics after the geometrical conclusion has been reached returns us to the subject matter of optics, but aside from this ‘application’ of the result, the treatment is wholly geometrical.

³² This is an overstatement. As Philoponus points out (Commentario in An. Post., Wallies, , ed., Commentaria in Aristotelem Graeca, Berlin, 1909, xiii, 3, 100.6ff.Google Scholar), the optician does not use all the principles of geometry, only those that are found relevant to the properties of visual perspective.

³³ Cf. Metaphysics, M.3, 1078a14–17, cited above, note 26.

³⁴ This accounts for Aristotle's statement in An. Post., I.9, 76a13 that the per se attributes in such demonstrations belong to the subject genus of the superior science.

³⁵ Cf. De partibus animalium, I.1, 639b7–10, quoted above, note 19.

³⁶ An. Post., I.9, 76a12–13: τ⋯ δ⋯ δι⋯τι τ⋯ς ⋯νω (sc. ⋯πιστ⋯μης), ⋯ς καθ’ αντ⋯ τ⋯ π⋯θη ⋯στ⋯ν.

³⁷ This is Philoponus’ view (op. cit. (32), 178.20ff.) and one which I think is right. It is worth pointing out, however, that in practice the optician is very likely to wear the geometer's hat a good deal of the time, to judge from the extent of the ‘geometrical’ parts of Euclid's ‘optical’ proofs.

³⁸ See An. Post., I.13, 79a8–10 and pp. 213–14 below for this proviso.Google Scholar

³⁹ Zabarella, , op. cit. (24, i), XII, 213a26ff.Google Scholar

⁴⁰ Cf. An. Post., I.9, 76a23ff; I.13, 78b37f. For the remaining examples, see the discussion below, p. 210 ff.Google Scholar

⁴¹ As in the example given by Zabarella: it is possible for a person to know that mules are sterile and yet upon seeing a mule with a large belly, to fail to recognize that it is a mule, and to think that the animal is pregnant; Op. cit. (24, ii), 92a1. This example is better suited to the Metaphysics passage than that in the An. Post.

⁴² This is my rendering of ‘⋯ρμονικἠ … ⋯ κατ⋯ τἠν ⋯κο⋯ν’ literally ‘harmonics in respect of hearing’.

⁴³ For the case of mechanics see below, note 79. For the pair geometry/medicine, which is mentioned at 79a14–16 as an example of sciences which are not subalternate, but where nevertheless in a given example geometry provides the explanation (διότι) of a fact (ὅτι) known by the physician, see Barnes, , op. cit. (20), p. 153Google Scholar. The example is a weak one and shows Aristotle yielding to the temptation to stretch his theory beyond its proper bounds.

⁴⁴ I shall refer to these groups of sciences as subalternate triples, for the sake of convenience. Subalternation is still a relation between two sciences: optics remains subordinate to geometry, and the science of the rainbow is subordinate to optics. Within this subordinate triple, I shall refer to geometry as the highest science, to optics as the intermediate science, and to the science of the rainbow as the lowest science.

⁴⁵ Stereometry/mathematical astronomy/nautical astronomy might appear to be another subalternate triple, but in fact it is not. This is clear from the Republic, where Plato places astronomy after stereometry in the Guardians' curriculum for the same sort of reason which led him to place stereometry after geometry (528a–b). The main principle governing the order in which subjects are studied is what I referred to above (p. 197) as complexity of subject matter. So we are told to take the second dimension first (i.e. plane geometry), then the third (stereometry), and then solids in revolution (i.e. astronomy). In Aristotle's sense of subordination, which is the subject of the present paper, stereometry is not subordinate to geometry, nor does he anywhere say that astronomy is subordinate to stereometry. (Plato treats harmonics as having the same relation to arithmetic as astronomy does to stereometry at 530d, 531c, but here he seems misled by the analogy with astronomy. The subordination here is not this kind, but Aristotle's.) The same conclusion emerges from an examination of two extant ancient works on mathematical astronomy from a generation or two after Aristotle. Autolycus, Both' On the moving sphereGoogle Scholar and Euclid, 's PhaenomenaGoogle Scholar deal with the geometry of the moving sphere, and although this requires some knowledge of the geometry of the stationary sphere, it has as much a right to be considered a separate branch of mathematics as stereometry does to be distinguished from geometry, despite the fact that it presupposes many of the results of geometry. For Autolycus, cf. Heath, T. L., op. cit. (17), i, 348ff.Google Scholar On the other hand, there is a kind of subalternation between mathematical and nautical astronomy. It will appear that they correspond to the intermediate and lowest sciences of a subalternate triple.

⁴⁶ 79a10ff.: ⋯χει δ⋯ καί πρός τ⋯ν ⋯πτικ⋯ν, ὡς αὔτη πρ⋯ς τ⋯ν γεωμετρίαν, ἂλλη πρός ταύτην, οίον τό περί τ⋯ς ἴριδος.

⁴⁷ I take it that the expression ἢ ⋯πλ⋯ς ἢ το⋯ κατ⋯ τ⋯ μάθημα (79a12f.) refers to the optician mentioned immediately before and that the two alternatives are different ways of describing him: if he is being contrasted with someone who studies physical optics [For the distinction between ‘mathematical’ and ‘physical’ or ‘visual’ optics, see below, p. 211.], or a part of it, such as the science of the rainbow, he can be called an optician who deals with mathematics, but unless this contrast is being made, it is sufficient to call him simply an optician. Somewhat similarly, Ross, , Aristotle's Prior and Posterior analytics, Oxford, 1949, p. 555Google Scholar ad loc. For another interpretation, see Zabarella, , op. cit. (24, ii), 92b52–93a5.Google Scholar

⁴⁸ The text is printed in volume viii of the Heiberg-Menge edition of Euclid's works (Leipzig, , 1916Google Scholar). For comments on the work's authenticity, see the prolegomena to that volume, pp. xxxviii–xlii.

⁴⁹ Other ancient evidence for the study of harmonics is discussed by Burkert, W., op. cit. (2), chapter V.Google Scholar In some points of detail, the present discussion of Aristoxenus makes use of Burkert's conclusions.

⁵⁰ The ideal harmonics which Plato champions (531c) is not relevant to this investigation. Significantly he does not try to represent it as an actually existing science. What is of value to the present study are his reflections on the nature of the two kinds of harmonics known to him.

⁵¹ In Elements VII–IX the principles are of only one kind: definitions; in book I there are of course three kinds: definitions, postulates, and common notions.

⁵² The thirteen books of Euclid's Elements, translated by Heath, T. L., 3 volumes, Cambridge, 1925, ii, 113.Google Scholar

⁵³ See Aristoxenus, , Elements of harmonicsGoogle Scholar, II.36 for this verdict on the Pythagoreans.

⁵⁴ See Macran, H. S., The Harmonics of Aristoxenus, Oxford, 1902, introduction pp. 87ffGoogle Scholar. In the quotations below, I use Macran's translation, slightly modified where necessary.

⁵⁵ This view presupposes that Aristoxenus wrote after the An. Post., but assuming the latter to have been an early work of Aristotle, the scanty details of Aristoxenus' chronology make this most plausible. See Macran, , op. cit. (54), p. 86Google Scholar, which refers to Westphal, , Aristoxenus von Tarent, Melik und Rhythmik des classischen Alterthums, Leipzig, 1893, ii, pp. i–xiiGoogle Scholar for the ancient evidence.

⁵⁶ As Burkert points out (op. cit. (2), p. 380 n. 46, 370f.Google Scholar) the venom of this criticism is due to the fact that the Pythagoreans' theory was a serious rival of Aristoxenus'. When account is taken of Aristoxenus' hostility and of Plato's approval (though this was limited; cf. Republic, 531cGoogle Scholar), there is no difficulty in seeing that the same people are being referred to here in Aristoxenus and at Republic, 530d and 531cGoogle Scholar. It is extremely difficult to be certain of the precise nature of the Pythagoreans' activities and accomplishments in musical theory in and before the time of Plato and Aristotle. See Burkert, op. cit. (2), chapter V for a careful treatment, and especially his conclusion (p. 399f).Google Scholar

⁵⁷ A good example of his consciously non-mathematical approach is found in his definition of the interval of a tone as the difference in magnitude between the first two concords (i.e. between the fifth and the fourth), whereas the Pythagoreans had defined it as the ratio of 9:8 (which is the result of dividing the ratios of the fifth and the fourth, 3:2 and 4:3; cf. Euclid, , Data 8Google Scholar for dividing ratios). See Aristoxenus, , ElementsGoogle Scholar, I.21 and Macran, , op. cit. (54) ad loc.Google Scholar While Aristoxenus' work is not so highly mathematicized as the Sectio canonis, it is worth mentioning that in book III he finds it convenient to present his material in the form of a series of propositions requiring proof and comment. He is also concerned with the different scales, or tunings of the tetrachord, and his approach here could be called mathematical in that it consists largely of finding possible combinations of musical intervals whose sum is a musical fourth, and he does this by treating the fourth as equal to two and one-half tones and by looking for combinations of tones, semitones, etc. whose sum equals two and one-half tones (see II.46ff.). Despite these ingredients of his work, the contrast between that and the mathematical Sectio is marked.

⁵⁸ Above, p. 206.

⁵⁹ Pp. 205f.

⁶⁰ This is not to suggest that ‘acoustical’ harmonics had no other function or interests, much less that there was an entire branch of study devoted to producing data for mathematical harmonics to work on. There is a wide variety of topics in Greek musical theory which mathematical harmonics did not attempt to treat. (See Aristoxenus, I.3ff. for an indication of the range of the subject matter, and indications that some of it at least had been taken up before his own efforts). Moreover, it is reasonable to concede that those who pursued this subject had at least a small degree of interest in theory. The ‘isolated dogmatic statements’ which Aristoxenus found these people making (see above, p. 208) are presumably general statements (‘the least audible interval is the smallest enharmonic diesis’) rather than particular judgments (‘the smallest interval I can detect on this monochord is the quarter tone’), and even if they were ‘isolated’ and not made into a unified theory, they would still most likely have been intended as universally valid truths.

⁶¹ This is of course not to claim any great antiquity for the serious and systematic pursuit of it as a science.

⁶² Cf. Meteorologica, III.2, 371b26–372a10. Also note the beginning of that chapter, where Aristotle sets out the topics of the next several chapters. He will consider what haloes, rainbows, etc. are and what are their causes; 371b18f.

⁶³ Cf. An. Post., I.13, 79a11f.

⁶⁴ See Meteorologica, III.4.Google Scholar

⁶⁵ Thus it appears that when the three-level view of subalternation is introduced, optics as found in Euclid's work will turn out to be the intermediate science. In this case, for the reason just given, there was in fact no distinct lowest science of ‘visual’ optics. (The treatises we have on ‘catoptrics’ would suggest that that science, like optics, would occupy the intermediate position and would likewise be subordinate to geometry.)

⁶⁶ σχεδόν συνὡνυμοι: 78b39. The point is about the similarity of the names, not synonymy in the technical sense of Categories, 1Google Scholar. Cf. Themistius, , Analyticorum posteriorum paraphrasis, Wallies, , ed., Commentaria in Aristotelem Graeca, Berlin, 1900, v, 1, 29.9Google Scholar, and Zabarella, , op. cit. (24, ii), 91a22ff.Google Scholar, as opposed to Philoponus, , op. cit. (32), 179.25.Google Scholar

⁶⁷ Cf. above, p. 201.

⁶⁸ Moreover, the second and third of these points appear again in An. Post., I.27, where he briefly discusses an issue that is not unrelated to the subalternate sciences. In discussing ways in which one science can be more precise (άκριβέστερα) than and prior to another, two of his three criteria are that the science which treats both διότι and ὃτι holds sway over one which treats only the latter (87a31–33; with Ross, 's interpretation, op. cit. (44), p. 596 ad loc.Google Scholar) and that an abstract science (⋯ μ⋯ καθ΄ ὑποκειμ⋯νου [sc. ⋯πιστ⋯μη]) is more precise than and prior to the corresponding concrete science (τ⋯ς καθ' ὑποκειμένου); 87a33.

⁶⁹ An. Post., I.13, 79a10f.

⁷⁰ Above, pp. 205f.

⁷¹ In keeping with my terminology for the sciences in a subalternate triple, I shall refer to the highest and the intermediate sciences as the ‘higher couple’ and to the intermediate and lowest sciences as the ‘lower couple’ in the following discussion.

⁷² Barnes also notices the difficulties (op. cit. (20), p. 154).Google Scholar

⁷³ Aristotle's remarks at 79a6ff. apply equally to the highest and to the intermediate sciences.

⁷⁴ See below, p. 214.

⁷⁵ Pp. 210ff.

⁷⁶ Barnes, (op. cit. (20), p. 154)Google Scholar chides Aristotle for failing to notice that subordinate sciences need starting points in addition to those which are specifications of the starting points of the superior sciences. In the terms of the present discussion this amounts to an assertion that he failed to notice the need for ‘determinations’. Indeed, he does not explicitly recognize the need for them, and his statement that optical results are proved from the same things as geometry (I.12, 77b1–2) may easily be taken as evidence that he missed this point. However, as Barnes, points out (op. cit. (20), p. 153)Google Scholar, there are reasons to believe that Aristotle meant merely that some of the starting points of the subordinate science are the same as those of the superior science. In any case, since the subordinate science (here, the intermediate science) has an enmattered subject genus, and one which though in a way identical with that of the highest science (I.7, 75b9), is different (I.9, 76a12), Aristotle ought to hold that it has in addition starting points of its own. (Barnes, ibid., makes this point.) And the additional starting points demanded by the subject genus will play precisely the role of the ‘determinations’ in Euclid, 's OpticsGoogle Scholar: they will provide the means of interpreting the mathematical conclusions of the highest science so as to apply to the physical subjects of the other.

⁷⁷ Where harmonics was conceived broadly, as including what Aristotle would later distinguish as both mathematical and acoustical harmonics.

⁷⁸ In view of the previous discussion, Barnes, 's contention (op. cit. (20), p. 154)Google Scholar that ‘the fact that [the higher and lower pair] do not share all their relations does not show that they do not share some: and in particular, it does not show that the relation of subordination does not hold for [the lower pair of a subalternate triple] as it does for [Aristotle's primary examples of sub-alternate sciences]’ must be challenged. What is at stake is the very nature of the subordination relation, and an analysis must go beyond Aristotle's assertions that certain pairs of sciences have this relation, and must investigate the grounds for these assertions.

⁷⁹ The pseudo-Aristotelian Mechanica would favour the subordination of mechanics to plane, not to solid geometry. The relation it has to solid geometry may be suggested in statements that an important area of mechanics concerned the construction of spheres to represent the motions of the sun, moon, and planets; cf. Pappus, , Collectio, VIII, Praefatio, 2Google Scholar, and Proclus, , In primum Euclidis Elementorum librum commentaria, Friedlein, , ed., Leipzig, 1873, 41.3ffGoogle Scholar. Archimedes is said to have constructed such an orrery and to have written a treatise (now lost) On sphere making, but I have found no evidence for the practical use of the science as early as Aristotle. See Solmsen, , op. cit. (11), pp. 130–5Google Scholar for the use of mechanical principles by Democritus in discovering theorems of solid geometry. For a different interpretation, see StThomas, , Commentary on the Posterior analytics of Aristotle, lectio 25 on book I.Google Scholar