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Al-Kindī's Commentary on Archimedes' ‘The Measurement of the Circle’*

Published online by Cambridge University Press:  24 October 2008

Roshdi Rashed
Roshdi Rashed
Affiliation:
Centre d'Histoire des Sciences et des Philosophies Arabes et Médiévales, CNRS, 27 rue Damesme, 75013 Paris, France
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Abstract

The author examines the relationship between mathematics and philosophy in the works of al-Kindī, and suggests that the real character of his contribution will become clear only when we restore to mathematics their proper role in his philosophy. The recently discovered treatise of al-Kindī on the approximation of π, of which the author gives the editio princeps here, throws important new light on al-Kindī's knowledge of mathematics, and on the history of the transmission of The Measurement of the Circle of Archimedes. The author shows that al-Kindī's commentary on the third proposition of the Measurement of the Circle was written before 857, at the same time if not before that of the Banū Mūsā, and that it was one of the sources of the Florence Versions, the Latin commentary on the same proposition.

Pour mieux comprendre le projet philosophique d'al-Kindī, l'auteur a été amené à s'interroger sur les rapports entre les mathématiques et la philosophie dans l'œuvre de celui-ci. Cette interrogation a exigé que soient examinés les écrits mathématiques d'al-Kindī. La découverte de l'epître de ce dernier sur l'approximation de π – jusqu'ici inconnue et qui a été rédigée avant 857 – permet non seulement d'engager cette interrogation, mais aussi de reprendre l'histoire de la transmission de La mesure du cercle d'Archimède. L'auteur montre qu'en même temps que Banū Mūsā, et peut-être avant, al-Kindī a donné un commentaire de la troisième proposition du texte d'Archimède, qui constitue l'une des sources du commentaire latin de cette dernière, connu sous le titre des Versions de Florence.

Type
Research Article
Information
Arabic Sciences and Philosophy , Volume 3 , Issue 1 , March 1993 , pp. 7 - 53
Copyright
Copyright © Cambridge University Press 1993

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References

1 Al-Nadīm, , Kitāb al-fihrist, ed. Tağaddud, R. (Tehran, 1971), pp. 315–20.Google Scholar

2 For the ancient bibliographers, see esp. al-Qiftī, , Ta'rīh al-ḥukamā’, ed. Lippert, J. (Leipzig, 1903), pp. 366–78;Google ScholarIbn, Abī Usaybi'a, ‘Uyūn al-anbā' fi tabaqāt al-atibbā’, ed. Ridā, N. (Beirut, 1965), pp. 285–93.Google Scholar For the modern bibliographers, cf. esp. McCarthy, R.J., al-Taṣānīf al-mansūba ilā faylasūf al-'arab (Baghdad, 1963);Google ScholarRescher, N., Al-Kindī, An Annotated Bibliography (Pittsburgh, 1964).Google Scholar

3 Al-Kindī's letter to the poet al-Ğahm, 'Alī ibn, On the Uniqueness of God and on the Finitude of the Universe;Google Scholar cf. Rasā'il al-Kindī al-falsafiyya, ed. Rīda, M.A. Abū, 2 vols. (Cairo, 1950–1953), vol. I, pp. 201–7 (hence forward Rasā'il).Google Scholar

4 This project had been taken up by al-Kindī several times for philosophy as well as for the sciences, something which has already been pointed out [see Jolivet, J. and Rashed, R., ‘Al-Kindī’, Dictionary of Scientific Biography (New York, 1978), vol. XV, Suppl. I, pp. 261–7].Google Scholar By way of example, what he writes in First Philosophy may be cited: ‘It is good…that we start off this book, according to our practice whatever the subject, by recalling what the Ancients said everything about, by means of the shortest and easiest methods for those of us who are researching into it, and by completing anything that the Ancients did not say everything about, according to the usage of the (Arabic) language and the custom of the time, to the extent that we are able to do so’; cf.Rasā'il, , I, 103.Google Scholar Let us also quote an example from his scientific writings, such as Optics: ‘In our desire to embrace the mathematical sciences, to explain the first results that the Ancients have left on this subject, to develop what they have begun and the points where they have allowed us to harvest the full fruits of the spirit… (et augere quod inceperunt et in quibus fuerunt nobis occasiones adipiscendi universas bonitates animales)’; cf.Björnbo, V.A. and Vogl, S., ‘Al-Kindī, Tideus und Pseudo-Euclid: Drei optische Werke’, Abhandlungen zur Geschichte der mathematischen Wissenschaften, XXVI, 3 (1912): 3–41, p. 3.Google Scholar

5 This treatise was mentioned in the ancient bibliographies; for example in that of al-Nadīm, , al-Fihrist, p. 316.Google Scholar

6 Rasā'il, I, 363–84.Google Scholar

7 Al-Kindī writes, having given his various groupings of the books of Aristotle: ‘These, then, are the books [of Aristotle] which we have previously mentioned; and these are the books of which the perfect philosopher must have knowledge after the study of mathematics, that is to say, those which I have specified by their names; for anyone without the knowledge of mathematics – and by this I include arithmetic, geometry, astronomy and music – trying to use these books throughout his life, will be unable to perfect his understanding of them, and all his efforts will only lead to his being able to repeat those of their contents he can recount from memoryGoogle Scholar. As far as acquiring a deep knowledge and understanding of them, this is simply unattainable if he does not have a sound basis in mathematics’, Ibid., I, 369–70.

8 This is in fact the case for the letters about theoretical philosophy, such as First Philosophy, On the Explanation of the Finitude of the Universe, etc. To take the case of this latter text, al-Kindī proceeds by the ‘ordered’ method to demonstrate the inconsistency in the concept of a body of infinite size. He begins by defining the basic terms ‘magnitude’ and ‘homogeneous magnitudes’. He then introduces what he calls the ‘truth proposition’ (qadiyya haqq) [Rasā'il, I, 188], or, as he explains it, ‘the true premises conceived directly’ (al-muqaddamāt al-uwwal) [‘First Philosophy’, Rasā'il, I, 114], or ‘the self-evident premises conceived directly’ [‘Regarding the Essence of That Which Cannot Be Infinite’, Rasā'il, I,202]; that is to say, tautological propositions. These are formulated in terms of basic concepts, of relations of order on themselves, of operations of reunion and separation on themselves, and also in terms of the predications ‘finite’ and ‘infinite’. It is a question of propositions such as this one: homogeneous magnitudes of which the ones which are not greater than the others are equal; or, such as this one: if one adds to one of the equal homogeneous magnitudes another magnitude which is homogeneous to it, then the magnitudes will be unequal [Rasā'il, I, 188]. Finally, al-Kindī continues with a demonstration, with the help of reductio ad absurdum, in using the following hypothesis: the part of an infinite magnitude is necessarily finite.Google Scholar

9 Some extracts of which had been translated into Arabic. See the edition of Badawī, 'A. in al-Aflātūniyya al-muhdata 'inda al-'arab (Cairo, 1955), pp. 4–33.Google Scholar

10 Al-Nadīm, and others after him, mention a work by al-Kindī: ‘The sphere is the greatest of the solid figures, and the circle is the greatest of the plane figures’, al-Fihrist, p. 316.Google Scholar Further, in his Great Art, al-Kindī reminds himself that he has written a book ‘on the sphere, and the solids, knowledge of which is connected with that of the sphere…’. Cf.al-Kindī, , Fī al-ṣinā'a al-'uzmā, ed. Aḥmad, 'Azmī Taha al-Sayyid (Cyprus, 1987), p. 120.Google Scholar

11 Al-Nadīm, , al-Fihrist, pp. 316–7.Google Scholar

12 In his treatise on The Plane Projection of the Figures <of the Constellations> and of the Spheres (Tasṭīh al-ṣuwar wa tabṭīh al-kuwar), al-Bīrūnī writes: ‘It is possible to transfer what belongs to a sphere onto a plane by another method attributed by Abū al-'Abbās al-Farġānī in several copies of his book entitled al-Kāmil to Ya'qūb ibn Ishāq al-Kindī and in several other copies to Hālid ibn ‘Abd al-Malik alMarwarrūdī, which is called the astrolabe in the shape of a melon’; Rashed, cf.R., Géométrie et dioptrique au Xe siècle: Ibn Sahl, al-Qūhī, Ibn al-Haytham (Paris, 1993), p. CIV n. 16.Google Scholar This projection, and the type of astrolabe which it allows one to construct, are due to al-Kindī or at the very least, were improved by him. Moreover, al-Bīrūnī comes back to this in his book Istī'āb al-wugūh al-mumkina fi san'at alasturlāb (Concerning all the Possible Ways of Making an Astrolabe), [MS Leiden 1066, fol. 89v–90r], and recalls some criticisms made by Muhammad ibn Mūsā ibn Šākir. Al-Bīrūnī does not support this criticism which he finds at the very least without import or, as he puts it: ‘Muhammad ibn Mūsā ibn Šākir has in this matter done no more than attack the person who made it and slander the person who invented it’. According to al-Bīrūnī, Muhammad's criticism was a consequence of the hostility between the Banū Mūsā and al-Kindī. The evidence of al-Farġānī and of al-Bīrūnī, as well as the attack by Muhammad ibn Mūsā, are further indications that contemporary mathematicians and those who came after al-Kindī did not regard him simply as a philosopher, but also as one of their own.

13 This refers to the astrolabe in the shape of a melon. See note 12 above.

14 This opinion has been supported by Rosenthal, F., ‘Al-Kindī and Ptolemy’, Studi Orientalistici in onore di Giorgio Levi della Vida, 2 vols. (Rome, 1956), vol. II, pp. 436–56.Google Scholar

15 The history of these commentaries in Arabic has not yet been written. For mathematical texts in Latin, see Clagett, M., Archimedes in the Middle Ages, 5 vols. (Madison, 1964–1984), vol. I.Google Scholar

16 This text of al-Kindī, hitherto unknown, is part of MS n°n 7073 in the Tehran University Library. The colophon shows us that al-Kindī's letter was copied in 1036 A.H. (1626 A.D.). The folios are not numbered; the copy is in nasta'līq, and there is nothing in the margin which indicates that the copyist compared his copy with the original master manuscript.Google Scholar

17 The Great Art, p. 174.Google Scholar

18 Ibid., p. 175.

19 Ibid., pp. 175–6. In his demonstration, al-Kindī begins with this statement: ‘It has been said that the circumference of the circle is approximately the triple of its diameter plus a seventh again’.

20 On the life of Ibn Māsawayh and his activities, see particularly al-Nadīm, al-Fihrist, pp. 295–6; Uṣaybi'a, Ibn Abī, ‘Uyūn al-anbā’, pp. 246–55.Google Scholar See also the article by Sournia, J.C. and Troupeau, G., ‘Médecine arabe: biographies critiques de Jean Mésué (VIIIe siècle) et du prétendu “Mésué le Jeune” (Xe siècle)’, Clio Medica, 3 (1968): 109–17.Google Scholar

21 These are: Risāla fi al-nafs wa af'ālihā, and Risāla fi 'ilm al-katif.Google Scholar

22 Cf. the Supplementary notes to the text.

23 This thesis has been presented by Endress, G., Proclus Arabus: Zwanzig Abschnitte aus der Institutio Theologica in arabischer Übersetzung, Beiruter Texte und Studien, 10 (Wiesbaden, 1973), pp. 66–193; note particularly pp. 101–5, 192 and 242–5.Google Scholar

24 By way of example, al-Kindī revised the translation by Ibn Lūqā of Anaphorikos (Kitāb al-matāli) of Hypsicles. Al-Nadīm, moreover, attributes other revisions to al-Kindī.Google Scholar

25 Ibn Abī Uṣaybi'a in fact attributes On the Shape of the Sphere and the Cylinder to Ibn Lūqā himself (cf. ‘Uyūn al-anbā’, p. 330). It is most probable, however, that he is referring to Archimedes' treatise and a translation of it, and not to an independent work. To this we must add the testimony of the Hebrew translator of the Arabic version of The Sphere and the Cylinder of Archimedes, Kalonymos b. Kalonymos, who attributes the Arabic translation to Qustā ibn Lūqā.Google Scholar See Steinschneider, M., Die arabischen Übersetzungen aus dem Griechischen, repr. (Graz, 1960), p. 174.Google Scholar

26 Rashed, R., ‘Archimède dans les mathématiques arabes’Google Scholar in Idem, Optique et mathématiques. Recherches sur l'histoire de la pensée scientifique arabe, Variorum, Collected Studies Series, 378 (London, 1992), IX.

27 Archimedis Opera Omnia cum Commentariis Eutocii, ed. Heiberg, J.L., 3 vols. (Leipzig, 1880), vol. I, pp. 257–71 and vol. III (1881) (Eutocius' commentary), pp. 263–303.Google Scholar

28 We have established the text of this version from the two existing manuscripts; it is this edition which we translate here.Google Scholar

29 Cf. below.

30 We have examined all the extant Arabic versions of The Measurement of the Circle, and a separate paper will present the results of this examination.Google Scholar

31 Cf. al-Kindī' text.

32 Nothing then allows us to confirm that al-Kindī knew Eutocius' commentary.Google Scholar

33 If we assume we find that from which

34 Clagett, Archimedes in the Middle Ages, pp. 91–142.Google Scholar

35 Ibid., pp. 40–58.

36 Ibid., p. 106.

37 Ibid., p. 92.

38 Ibid., p. 113.

39 Ibid., p. 95.

40 Ibid., p. 112.

41 Ibid., p. 112.

42 Ibid., p. 112.

43 W. Knorr proposes the following suggestion for the identity of the text's author: Johannes de Tinemue. He also sees in The Florence Versions ‘an interrelated ensemble, not a chance assembly of unconnected treatments’. Cf. Textual Studies in Ancient and Medieval Geometry (Birkhaūser, 1989), pp. 618–24 (note especially p. 620). This conclusion will doubtless have to be revised in the light of the results of our own researches into al-Kindī's letter.Google Scholar

44 At least seven of al-Kindī's works have been identified in Latin translations: including two in philosophy, one on sleep and visions, and one on astrology. Cf. Rescher, Al-Kindī, An Annotated Bibliography, and d'Alverny, M.T. and Hudry, F., ‘Al-Kindī, De radiis’, Archives d'histoire doctrinale et littéraire du moyen âge, 41 (1974): 139–260.Google Scholar

45 Cf. Garro, I., ‘Al-Kindī and mathematical logic’, Proceedings of the First International Symposium for the History of Arabic Science (Aleppo, 1976), vol. II, pp. 36–40.Google Scholar

46 Lit.: On the approximation of the circumference as related to the chord, cf. Supplementary note [1].Google Scholar

47 See Supplementary note [2].

48 That is to say, the Elements, cf. Supplementary note [3].Google Scholar

49 Euclid, Book VI, prop. 33.

50 Euclid, Book IV, corollary of proposition 15.

51 Euclid, Book XIII, prop. 12.

52 Lit.: the ‘preceding books’ means, as is always the case, the Elements, cf. Supplementary note [3].Google Scholar

53 MS: FD.

54 Immediate consequence of Euclid, Book VI, prop. 3.

55 Lit.: which has twelve bases.

56 Lit.: bases, which we translate as ‘sides’ throughout the text.Google Scholar

57 The writer does not give his proof here, but he subsequently brings to mind the correspondence between inscribed angles and intercepted arcs. Here use is made of proposition 26 of Book I of Euclid and, further on, proposition 27.Google Scholar

58 Cf. note 52.

59 Euclid, Book IV, corollary of proposition 15.

60 The sum of the two angles KAL and LAB is the angle KAB inscribed in the circleGoogle Scholar

61 Cf. note 52.

62 Euclid, Book III, prop. 27.

63 Euclid, Book VI, prop. 4.

64 Here, he uses the word dil' for side.Google Scholar

65 *…* See Supplementary note [4].

66 Cf. Supplementary note [5].