AVICENNA viii. Mathematics and Physical Sciences

referred to, in his encyclopedic work the Šefāʾ, as the mathematical sciences; includes both mathematics and astronomy.

 

AVICENNA

viii. Mathematics and Physical Sciences

Introduction. What is understood by mathematics and physical sciences in this context is what Avicenna himself referred to in his encyclopedic work the Šefāʾ, as the mathematical sciences, which included both mathematics and astronomy, and the physical sciences (ṭabīʿīyāt), which included the usual Aristotelian disciplines designated as the Physics, the Heavens, Generation and Corruption, Meteorologica (the fourth part of which is treated separately under the title al-feʿl wa’l-enfeʿāl, the Soul, the Plants, and the Animals.

A few other works have been attributed to Avicenna in the literature such as: Ḥall-e moškelat-e moʿīnīya (Anawati, p. 227), Taḥrīr al-majesṭī (Mahdawī, p. 263), which can now be definitely determined as being by later authors, the first being the work of Naṣīr-al-dīn Ṭūsī (d. 1274) and the second being the work of Moʿayyad-al-dīn ʿOrżī (d. 1266).

Mathematicalworks. The third treatise (jomla) of the Šefāʾ is devoted to the mathematical sciences. The Avicennian division is that of Aristotle, and thus the mathematical sciences included four parts (fann), namely, Geometry, Arithmetic, Music and Astronomy.

The two major works in this treatise are the one devoted to geometry and that devoted to astronomy. The Arithmetic and the treatise on music are simply renditions of the elementary principles of these two disciplines and need not concern us here, except to say that in the Arithmetic, Avicenna manages, in his eclectic style, to include results reached by earlier mathematicians such as Ṯābet b. Qorra and others. In the same book he also combines material from Euclid’s Elements, Diophantus’ Arithmetic, and the contemporary algebraists. The influence of the latter is especially significant, for it reveals the impact of the Arabic discipline of algebra on Greek arithmetical theory.

Geometry. In the part of Geometry, Avicenna wrote what could be properly called a taḥrīr of Euclid’s Elements, in spite of the fact that he called his work an abridgment (eḵteṣār), thus feeling free to reorganize the material, to supply alternate proofs for theorems, to restate the conditions and the theorems themselves, and to add any “corrections” that he saw fit. The editor of this part of the Šefāʾ has established beyond doubt that Avicenna, although staying close to the order of the Elements, had taken the liberty to rearrange the theorems, to combine proofs, to add new ones, and change the statement of the contents to agree with his own style and taste.

All this deliberate “tampering” with the text of the Elements should, however, be understood in terms of the Arabic translations of the Elements, which, from very early times, included the addition of material that was thought to belong to it, such as the two treatises fourteen and fifteen, attributed to Esqlābes (Hypsicles), but actually authored by Hypsicles and an anonymous author respectively, and not by Euclid. In addition to these translations the Arabic tradition also included commentaries that were sometimes composed of lengthy explanations of the text together with an attempt to put the material in a historical perspective. These commentaries also “took liberties” with the text in the sense that they grouped together theorems and proofs that were seen as closely related, and at times omitted other proofs or parts thereof, because they were thought to be redundant. Avicenna’s work falls within this tradition and is actually closer to a taḥrīr than to an eḵteṣār.

As an example of the free use of the text and the additional remarks appended to it, we note the additional definition of the irrationals given by Avicenna at the beginning of book ten of his version of the Elements. After defining the commensurable and the incommensurable in almost the same terminology as the one used by Euclid, Avicenna goes on to say that “there are no magnitudes that are irrational or rational (aṣamm or monṭaq) by themselves (be-ḏātehe), but are so only in relation to the assumed (unit) magnitude. If it (i.e. the magnitude) is commensurable with that (unit) magnitude then it is rational, otherwise it is irrational. This same irrational could become rational in relation to another (unit) magnitude; and the original (unit) magnitude would then become irrational.” There is no such comment in the Euclidean text or in the Arabic translations of that text, it is obviously an interpolation by Avicenna. It shows the degree to which the Euclidian text was being reorganized and could very well show the indebtedness of Avicenna to the earlier commentators who may have elaborated the text at this point.

Astronomy. Avicenna’s works on astronomy are of the same nature as his work on geometry. In this part of the Šefāʾ, he gives his own version of what he thought were the contents of Ptolemy’s Almagest. The Avicennian text is again called talḵīṣ and, like the eḵteṣār of the Elements, it contains a rearrangement of the material, an abridgment, and several explanatory notes. Treatises nine, ten, and eleven of the Almagest are treated by Avicenna in one chapter, for they all deal with the particulars of the dimensions of the models that Ptolemy proposed for the planets. Many of the lengthy proofs of the Almagest are summarized, and they are all recast in a simplified language, without loosing any of the mathematical rigor of the Almagest. The observations that were used by Ptolemy, for example, to determine the eccentricities of the upper planets Saturn, Jupiter, and Mars are grouped together by Avicenna at the beginning of the fourth chapter of his unified treatise, instead of being spread in X, 7 (Mars), XI, 1 (Jupiter), and XI, 5 (Saturn) as in the Almagest.

On the other hand, the method used by Ptolemy to determine the eccentricities of the models of each of these upper planets is essentially the same for each one of them. It involves a rather sophisticated iteration method. Once this method is mastered for the case of Mars, one needs no longer repeat it for the other two. Ptolemy, however, repeats this iteration method for each one of these planets in essentially the same terms although in somewhat shortened form. Avicenna, on the other hand, must have seen that the theoretical contents of these procedures used for each planet are the same; the only variation being really in the details of the observations leading to the variation in the computed results for each model. As a result, he must have felt that the Ptolemaic exposition contains some redundancy and hence can be better restated. This would then explain why he treated these three treatises, namely, nine, ten, and eleven, in one chapter.

The same method is followed by Avicenna throughout the book, and one could easily follow the Ptolemaic results from the Avicennian exposition of the Almagest. The only parts of the Almagest that are not included here are the numerical tables and the star catalogue.

From another perspective, the theoretical and mainly philosophical foundation of the Almagest is not touched upon by Avicenna throughout the whole book. These theoretical issues had motivated other Islamic astronomers, e.g., the contemporary of Avicenna, Ebn al-Ḥayṯam, to write a full treatise devoted to their analysis and refutation. Ebn al-Ḥayṯam’s treatise al-Šokūk ʿalā Baṭlamyūs managed to isolate what was perceived to be a contradiction in Ptolemaic astronomy in the sense that that astronomy did not harmonize the mathematical and the physical aspects of the world. This does not mean that Avicenna was not interested in these issues or that he did not notice them.

In a treatise written by his student Abū ʿObayd Jūzjānī about the contradiction in Ptolemaic astronomy that came to be known as the problem (eškāl) of the equant, the student says that his teacher Avicenna had even succeeded in solving at least this Ptolemaic contradiction, namely, the problem of the equant. The student further claims that he had asked Avicenna about the veracity of that claim, only to be told by his teacher that it was indeed true, but that he wanted the student to find the solution for himself. The student proceeds to observe that he, i.e., the student, was the first to succeed in obtaining a mathematical solution for the problem of the equant. Here are the student’s own words: “When I asked him (i.e., Avicenna) about this problem (i.e., that of the equant), he said: "I came to understand this problem after great effort and much toil, and I will not teach it to anybody. Apply yourself to it and it may be revealed to you as it was revealed to me." I suspect that I was the first to achieve these results.” However, a closer analysis of the student’s work, which was published recently, shows that he too was not quite successful in establishing a valid solution of the problem.

Should this anecdote be true, then one must admit that by our modern standards the moral attitude of Avicenna left a lot to be desired. But what it also shows is the fact that within the circle of Avicenna, he and his students were at least discussing these same issues, which, as we have seen, had motivated Ebn al-Ḥayṯam to devote a special treatise to them, and thus to formulate the main program of research for later Islamic astronomers for centuries to come, extending well into the fifteenth century as far as we can now tell.

As for the parameters that were reported in the Almagest and were later found to be contradicted by observational facts, we find Avicenna discussing them in an appendix that is usually attached to his talḵīṣ, and which also included what Avicenna thought were defects in Ptolemaic astronomy. In its introduction, he says that one ought to contrast the statements of the Almagest with those of the rational part of natural science. One must further show the method by which the motions of the planets could take place. Thirdly, Avicenna reports some of the observational results reached after the writing of the Almagest and still in agreement with the theoretical statements of the Almagest.

In the body of this appendix, he begins by showing how it is possible for a sphere embedded within another sphere to move by its own motion in spite of the fact that it has to follow the surrounding sphere in the latter’s motion. But if both spheres have the same axis, then it is impossible for the inner sphere to move by its own motion, and to move accidentally with the motion of the surrounding one in such a way that the two motions are opposite in direction. He then analyzes the two cases when the two axes are not identical, namely, (1) when the two axes intersect at the center or (2) when they do not.

Avicenna then takes up the issue of the observational results reached after the writing of the Almagest, which affected the validity of the Almagest statements themselves. In this category, he takes the parameter for the inclination of the obliquity, determined by Ptolemy to have been 23;51 degrees, and reports the results reached by the astronomers working during the reign of the caliph al-Maʾmūn (813-833) as being 23;35. He then claims that it had “decreased” after that by some one minute, and that he himself had observed the inclination and found it in his own days to be less than that by another amount equal to half a minute approximately. These results for the obliquity of the ecliptic were much closer to the true value, as derivable from modern computations, than the value found by Ptolemy.

The next two interrelated parameters of precession and solar apogee were also noted as having been found to be at variance with the results found by Ptolemy. In the first case, Ptolemy found precession to be one degree in one hundred years while the more precise value, which was determined afterwards and was reported by Avicenna, was found to be one degree in every sixty-six years approximately. This also determined the motion of the solar apogee, which was found by Ptolemy to have been stationary at five degrees of Gemini, but was found by later astronomers to have been moving primarily with precession. By Avicenna’s time, the solar apogee must have been around the eighteenth degree of Gemini, instead of the fifth.

The size of the solar disk was also found to be less than the size computed for it, “with some approximation,” by Ptolemy.

Avicenna concludes this appendix by stating that other parameters were also at variance with the results reached by Ptolemy, but that he had no observational results to determine them with any certainty. This means that, although there are claims in Avicenna’s works of direct observations, he did not seem to have had access to a functional observatory, nor did he seem to have had a systematic program of observations that he wished to complete.

Finally, Avicenna’s talḵīṣ of the Almagest includes a very curious note about his alleged observation of the disk of Venus being like “a mole on the face of the sun.” There are several citations in medieval Arabic sources of the transit of Venus, and this would have been just another one of them, had it not been for its crucial importance in the argument for the relative order of the planetary spheres. The problem had already started with Ptolemy when he could not cite a positive proof for the order of the planetary spheres, and finally opted for a proper order that placed the sun in the middle with both Venus and Mercury as inferior planets, while Mars, Jupiter, and Saturn were taken as the superior ones. The planetary distances, which were computed by Ptolemy for each of these spheres included an approximation that allowed someone like Jāber b. Aflaḥ (fl. first half of the 12th century) and later Moʾayyad-al-dīn ʿOrżī (d. 1266) to conclude that according to the Ptolemaic computations Venus should fall above the sun, and hence could not be seen “as a mole on the face of the sun.”

Whether Avicenna had actually seen Venus in transit or not is immaterial, for he was then quoted by later astronomers as confirming Ptolemy’s arrangement of the planets. Among those astronomers who quoted this observation specifically for that purpose were the thirteenth-century astronomer Naṣīr-al-dīn Ṭūsī (d. 1274) and the much later astronomer Cyriacus (ca. 1482), to name only two.

PhysicalScience. We shall confine ourselves to an account of the disciplines treated by Avicenna in the vast field covered by natural science: alchemy, astrology, and theory of vision.

In his explanation of the theory of metal formation in the Ṭabīʿīyāt section of the Šefāʾ (Cairo ed., part 5, chap. 5, pp. 22f.), Avicenna proposes two different theories: (1) the familiar Aristotelian theory of condensed vapors as being responsible for the formation of the various metals, and which is followed immediately by (2) the mercury-sulphur theory, commonly attributed to Jāber b. Ḥayyān (8th-9th century) (E. J. Holmyard, Alchemy, London, 1957, repr. 1968, pp. 75, 94). In the mercury-sulphur theory, Avicenna argues successfully for the production of all metals through balancing the relationship of the Aristotelian qualities—hot, dry, wet, and cold—in the substances mercury and sulphur. “If,” he says, “the mercury be pure, and if it be commingled with and solidified by the virtue of a white sulphur which neither induces combustion nor is impure, but on the contrary, is more excellent than that prepared by the adepts (i.e., Alchemists), then the product is silver” (Holmyard and Mandeville, Congelatione, p. 39). “If,” he continues, “the sulphur besides being pure is even better than that just described, and whiter, and if in addition it possesses a tinctorial, fiery, subtle, and non-combustive virtue—in short, if it is superior to that which the adepts can prepare—it will solidify the mercury into gold” (ibid.).

One would have expected that once Avicenna had recognized the difference between silver and gold as a difference in the qualities of the two substances—one requiring only a purer sulphur than the other—and not the substances themselves, he would have followed the path of the alchemists who simply held the same opinion. But in a strange twist, Avicenna went on to say: “As to the claims of the alchemists, it must be clearly understood that it is not in their power to bring about any true change of species” (ibid., p. 41), thereby redefining the metals as separate concrete species as distinct as the species horse and dog, as one of his critics best put it (ibid., p. 7).

This generally confused argument of Avicenna against alchemy was duly noted and attacked by Ṭoḡrāʾī (d. 1121). And as much as Ebn Ḵaldūn would have wanted Avicenna’s attack against alchemy to have been successful, he nevertheless found himself obliged to agree with Ṭoḡrāʾī’s criticism of Avicenna’s argument (Moqaddema III, pp. 273-74).

On the subject of astrology, he was not much clearer, in spite of the fact that he had written a treatise especially devoted to the attack of astrology (Resāla fī ebṭāl aḥkām al-nojūm). After classifying astrology as a refutable science, he went on to say that although it was true that each planet had some influence on the earth, it was doubtful whether one could tell of the nature of this effect. The next argument that he put forth was that the astrologers were incapable of determining the exact influence of the stars, although he agreed with them that according to what he called “the scientists” each star did have an influence on the earth.

In essence, Avicenna did not refute astrology, but denied man’s ability to know of the effects of the stars on the sublunar matter. With that, he did not refute the essential dogma of astrology, as someone like the Asḥʿarite Bāqellānī, close to a generation earlier, did, but only refuted our ability to know the principles of that science. If one developed better methods and techniques of gauging the influence of the stars on man’s life and earthly events, then Avicenna would, in principle, have accepted that that would have been possible.

Avicenna’s theory of vision is an explicit restatement of the Aristotelian theory. In terms of intramission versus extramission, this theory occupied the middle grounds. For neither Aristotle nor Avicenna after him could subscribe to the extramission theory by accepting the ability of the eye to issue forth a ray that will have to reach as far as the stars to explain their visibility. For similar reasons, they could not accept the ability of the eye to receive anything from the outside to explain its visibility. The middle ground would therefore be that vision occurs in the medium that separates the eye from the visible object. For vision to take place, the object must have the ability to affect the medium separating it from the eye; and this medium must be transparent; and the perceptive faculty in the eye must sense this change or affectation.

In his treatise On The Soul, Avicenna states the problem in the following terms: “Among them (i.e. the doctrines of vision) is the doctrine of the one who thinks that, like other sensed objects, which are not perceived (edrāk) due to anything coming out toward them from the senses and touching them or by sending a messenger to them, so is vision. It takes place, not by the issuing of any ray whatsoever that meets the seen (object), but rather by the transmission of the form (ṣūra) of the seen (object) to the eye (baṣar) through the transparency (al-šaffāf) that delivers it” (Šefāʾ, Ṭabīʿīyāt, Nafs III, 5, p. 102).

Later on in the same treatise, he elaborates further the nature of the visibility of the lit objects and the media that separate them from the eye, by saying: “It is characteristic of the body that is bright by itself, or the one that is lit and colored, that it imprints (yafʿal) upon the body facing it—if it were capable of receiving the form (šabaḥ) in the same way the eye is, and having a colorless body between them—an effect, that is, a form (ṣūra) similar to its own form, without having any effect on the intermediary, for it (the intermediary) is transparent and is incapable of reception” (ibid., p. l28).

This terminology is indeed very reminiscent of the Aristotelian version of the theory of vision as it was expressed in so many words in De anima, and in Parva naturalia. In De anima, “Seeing is due to an affection or change of what has the perceptive faculty, and it can not be affected by the seen color itself; it remains that it must be affected by what comes between” (419a.19). In the Parva naturalia, “vision is caused by a process through this medium (that separates the object from the eye)” (438b.4).

In Themistius’ commentary on De anima, which was available in Arabic, vision is explained as “the ability to accept the essence (maʿānī) of the colors that are in the transparent medium that is separate from it (i.e., vision)” (Themistius, De anima 98.12).

With this understanding of Avicenna’s explanation of the theory of vision, it is not surprising to find Roger Bacon classifying Avicenna together with Ebn al-Ḥayṯam (Alhazen) and Ebn Rošd (Averroes) as the Arab philosophers who were opposed to the theory of the extramission of light (Opus maius V, 1, Distinction 7, chap. 3-4).

 

Bibliography:

The major bibliographical works dealing with the Avicennian corpus are that of G. Anawati, Essai de bibliographie avicennienne, Ligue Arabe, Direction Culturelle, Cairo, 1950, and that of Y. Mahdawī, Fehrest-e nosḵahā-ye moṣannafāt-e Ebn Sīnā, Tehran, 1333 Š./1955.

A critical edition of the various parts of Avicenna’s al-Šefāʾ was published in Cairo from 1952 to 1983 under the general editorship of Ebrāhīm Madkūr. The critical editions of the mathematical and the physical works used in the present survey are those produced by this project.

For elementary treatises on music and arithmetic, the reader is referred to the French translation of the Dāneš-nāma, Le livre de science II, by M. Achena and H. Massé, Paris, 1958. See also the study of this treatise presented by R. Rashed, “Mathématiques et philosophie chez Avicenne,” presented at the Millenniary Colloquium of Avicenna in New Delhi, 1981.

For the work of Avicenna’s student Abū ʿObayd Jūzjānī on the problem of the Ptolemaic equant, see the edition, translation and commentary by G. Saliba, “Ibn Sīnā and Abū ʿUbayd al-Jūzjānī: The Problem of the Ptolemaic Equant,” Journal for the History of Arabic Science 4, 1980, pp. 376-403.

Avicenna’s note on alchemy is in the fifth chapter of the first treatise of his Meteorologica, which itself is the fifth part of the Ṭabīʿīyāt. It was translated by E. J. Holmyard and D. C. Mandeville, in Avicennae De Congelatione et Conglutinatione Lapidum, Paris, 1927.

The treatise on astrology was published by H. Z. Ülken, Ibn Sīnā Risāleleri 2, Istanbul Üniversitesi Edebiyat Fakültesi Yayınlarından, no. 552, 1953, pp. 49-67, but was already studied by M. A. F. Mehren, “Vue d’Avicenne sur l’astrologie . . . ,” Le Muséon 3, 1884, pp. 383-403.

(G. Saliba)

Originally Published: December 15, 1987

Last Updated: August 17, 2011

This article is available in print.
Vol. III, Fasc. 1, pp. 88-92