^{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. 32–37.
^{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. 198–202.
^{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 A^{2}/B^{2} = 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 = A^{2}/B^{2}, 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 A^{2}/B^{2} 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 A^{2}/B^{2} = 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. 3–93. — In Rasā'il Ibn Sinān, Saidan, A.S. (Kuwait, 1983), pp. 77–143. 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.