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# Ibrāhīm ibn Sinān: On Analysis and Synthesis

Abstract

This paper is devoted to Ibn Sinān's treatise on analysis and synthesis. Ibn Sinān's text deals with two distinct, though closely related, subjects. First he considers the classification of problems, founded on the logical criteria which are the number and degree of indetermination of the solutions and (though in a less relevant way) the number of hypotheses and their possible independence. This classification does not replace the Hellenistic one, which remains relevant insofar as it purports to solve geometrical problems, but complements it and has a different frame of reference, applying principally to algebra, that new born science. Ibn Sinān's classification will be taken up and used by algebraists. Secondly, the text presents a new way of setting out analysis and synthesis, so that they become exactly reciprocal. This leads Ibn Sinān to bestow more importance on the role played by analysis in the course of the proof, the only role left to synthesis being to check that all implications involved in the course of analysis are in fact equivalences. This method will prove to be productive in algebra too, so much so that some algebraists will explicitly identify algebra with analysis.

Cet article est consacré au traité d'lbn Sinān sur l'analyse et la synthèse; Ibn Sinān y aborde deux sujets distincts quoique étroitement liés:

– une classification des problèmes, fondée sur les critères logiques que sont le nombre et le degré d'indetermination des solutions, ainsi que, mais de façon moins pertinente, le nombre des hypothèses et leur éventuelle indépendance. Cette classification ne se substitue pas à la classification hellénistique qui garde sa pertinence en matière de problèmes de géométrie, mais se développe à côté de celleci dans un cadre radicalement différent, et est directement applicable à l'algebre, science qui se crée aux Xe et XIe siècles. La classification d'Ibn Sinān sera reprise et utilisée par les algébristes;

– une nouvelle façon d'exposer l'analyse et la synthèse d'un problème, afin qu'elles deviennent exactement inverses l'une de l'autre, ce qui conduit Ibn Sinān à accorder dans ses démonstrations une importance accrue à l'analyse, le rôle de la synthèse n'etant plus que de vérifier que toutes les implications mises en œuvre dans l'analyse sont en fait des équivalences. C'est en algèbre également que cette méthode se montrera féconde, au point que certains algébristes n'hésiteront pas à identifier l'algèbre et l'analyse.

References
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1 Ibrāhīm Ibn Sinān Ibn Ṯābit Ibn Qurra (296/909–335/946), son of Sinān, physician and encyclopaedist, friend and companion of the caliph al-Rādī (934–940), grandson of Ṯabit Ibn Qurra, translator and mathematician of genius. Ibn Sinān's family comes from the town of Harrān, on the Upper-Euphrates, crossroads of caravan routes and cultures, and religious centre of the Sabean sect. He was the heir to a prestigious scientific tradition and an intimate of the Caliphal court.

2 Plato, Meno, 86e–87c.

3 d'Alexandrie, Pappus, La Collection mathématique, 2nd ed., trans. Eecke, P. Ver, 2 vols. (Paris, 1982), vol. II, pp. 477–78.

4 de Lycie, Proclus, Les Commentaires sur le premier liure des Eléments d'Euclide, 1st ed., trans. Eecke, P. Ver (Paris, 1948), pp. 5, 15, 37, 63.

5 Greek and Hellenistic geometry, at least in the Elements, is mostly synthetic; nevertheless, although all proofs in the Elements are merely synthetic, examples of analysis can be found in some Greek texts, for instance in Apollonius' On the cutting-off of a ratio (Sectio rationis), as well as in the texts, now mostly lost, that form the so-called Treasury of analysis mentioned by Pappus in his introduction to Book VII of the Collection. To these we can add propositions 54 to 58 of Book III of the Collection (solving the problem of inscribing the five regular solids in a given sphere), as well as propositions 4, 12, 31, 33, 37, 40, 44, of Book IV of the same work. However, in the works of Apollonius and Pappus, proofs are either merely synthetic, or their analysis and synthesis are given, but their analysis is never given alone. The scarcity of theoretical texts dealing with analysis, together with the rare occurrence of analysis in ancient proofs, gave birth in the 16th and 17th centuries to many attempts to uncover the “mystery” of Greek analysis, and to restore it to its past splendour.

6 These examples are the following ones: 1 Construct a triangle equal (i.e. having the same area) to a given triangle, and similar to another given triangle (Elements, Book VI, proposition 25). 2 Find the sides of a triangle having a given area and similar to a given triangle. 3 Let C be a circle, M a point internal to it, then if two straight lines issued from M meet the circle respectively in A, B, A';, B', MA.MB = MA'.MB' (Elements, Book III, proposition 35). 4 Let ABC be an equilateral triangle, A' bisects [BC], M is a point internal to the triangle, a, b, c, its orthogonal projections respectively on (BC), (CA), (AB), then: Ma + Mb + Mc = AA'.

7 See Pappus, La Collection mathématique, introduction to Book VII, vol. II, p. 486, and Proclus, Les Commentaires, second Prologue, pp. 69–73, for attempts to define problems and theorems.

8 Chasles, M., Les Trois liures de porismes d'Euclide (Paris 1860), pp. 3237.

9 Proclus, Les Commentaires, second Prologue, p. 71.

10 Because Ibn Sinān aimed his treatise at beginners, and because his purpose was to classify mathematical propositions rather than to solve problems, he deliberately chose plain examples, mostly stemming from the works of Euclid, Apollonius, and Theodosius (the only mathematicians mentioned in the treatise), or from his own geometrical works, and we may consider them part of the mathematical knowledge of the learned people of his time.

11 Mentioned by Itard, Dedron et, Mathématiques et mathématiciens (Paris, 1959), pp. 198202.

12 Pappus, , La Collection mathématique, Book IV, vol. I, pp. 206–07. The Hellenistic classification of problems, by Pappus for example, is as follows: — plane problems, that can be solved by means of straight lines and circles, — solid problems, which are solved by means of conic sections, — linear problems, requiring the use of other curves.

13 The importance of the notion of “known” or “given”, closely connected with the fundamental question of the nature and existence of mathematical objects, is well known. Ibn Sinān, for his part, accepts as his own the definition given by Euclid in the Data (definitions 1 and 2): “an object is known if one can find others equal to it.” For the links between analysis and the concept of given, see: “L'analyse et la synthèse selon Ibn al-Haytham,” in Rashed, R. (ed.), Mathématiques et philosophie de l'Antiquité à l'Age classique. Hommage à Jules Vuillemin (Paris, 1991), pp. 131–62.

14 See the passage of Ibn Sinān's text translated as an appendix.

15 i.e. as in every classical text, segments.

16 This is exactly the same problem as problem 9 of Vieta's Zeteticorum, Book II, which is stated in the following terms: “the rectangle on the sides and the difference of the squares being given, find the sides.” Vieta only gives the analysis of this problem, and proves that it amounts to a problem already solved: “find two sides, the sum and the difference of their squares being given,” (Zeteticorum, Book I, problem 1);cf. Vauzelard, , La nouvelle algèbre de M. Viète, Corpus des œuvres de philosophie en langue française (Paris, 1986).

17 The beginning of synthesis I (without being part of the analysis) is not, according to Ibn Sinān, part of the synthesis strictly speaking: the question is now to draw the straight lines, in practice, the existence of which has been proved in the course of analysis I. Without any justification, and without clearly stating any connection between analysis I and these constructions, all these constructions rest on definition 9, Book V of the Elements (if A/B = B/C then A2/B2 = A/C), used and used again in a masterly manner in two different ways: — if a ratio A/C is known and if we want to write it as a ratio of two squares, (in other terms, if we want to find its square root), we take B, mean proportional to A and C, and then A/C = A2/B2, A/B is then the square root of A/C (see Elements, Book VI, proposition 13 for the construction of the mean proportional, and see also example 4 supra); — if a ratio A2/B2 is known, and if we want to write it as a ratio of two lengths, (in other terms if we want to find the square of A/B), we take C third proportional to A and B, and then A2/B2 = A/C (see Elements, Book VI, proposition 11 for the construction of the third proportional).

18 See the passage of Ibn Sinān's text translated as an appendix.

19 Itard, J., Essais d'histoire des mathématiques, textes réunis et introduits par Rashed, R. (Paris, 1984), pp. 209–11.

20 Ibn Sinān's classification is present embryonically, but not explicitly, in Pappus's, work (La Collection mathématique, introduction to Book III, vol. I, p. 21): “On the other hand, he who propounds a problem…and bids us to do something which is impossible to construct, is pardonable and does not lay himself open to criticism, for it is part of the solver's duty to determine the conditions under which the problem is possible or impossible, and, if possible, when, how, and in how many ways it is possible.”

21 Fermat has taken up again, almost in the same terms, the notion of “abundant” or “over-abundant” problems, i.e. problems in which more equations than unknowns are found, the concept of extra hypotheses giving place to the one of extra equations. Mentioned in Itard, Essais d'histoire des mathématiques, p. 220.

22 As-Samaw'al, , al-Bāhir en algèbre d'as-Samaw'al, édition, notes et introduction par Ahmad, S. et Rashed, R. (Damas, 1972), pp. 227–51 (Arabic text).

23 For another treatment of the problem of analysis and synthesis, connected with a new idea of geometry, see: Rashed, “L'analyse selon al-Haytham.”

24 As-Samaw'al, al-Bāhir en algèbre, p. 74.

25 Rashed, R., Entre arithmétique et algèbre. Recherches sur l'histoire des mathématiques arabes, Sciences et Philosophie Arabes (Paris, 1984), p. 44.

26 There have been two non critical editions of this text: — In Rasā'il Ibn Sinān (Hyderabad, 1948), pp. 393.In Rasā'il Ibn Sinān, Saidan, A.S. (Kuwait, 1983), pp. 77143. We are preparing a critical edition of this text, based on the extant manuscripts (Paris, Bibliothèque Nationale, MS 2457; Cairo, Dār al Kutub, MSS Riyāda 40 and Taymūr 323; and Bankipore, MS 2468) with a French translation and analysis to be available soon.

* I wish to thank Professor Rashed, who suggested that I study this text. His generosity with help and advice have resulted in many improvements.

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