Ibr?h?m Ibn Sin?n Ibn Th?bit Ibn Qurra

(b. Baghdad [?], 908; d. Baghdad, 946)

mathematics, astronomy.

Born into a family of celebrated scholars, Ibn Sinan was the son of Sinan ibn Thabit, a physician, astronomer, and mathematician, and the grandson of Thabit ibn Qurra. Although his scientific career was brief—he died at the age of thirty-eight-he left a notable body of work, the force and perspicuity of which have often been underlined by biographers and historians. This work covers several areas, such as tangents of circles, and geometry in general; the apparent motions of the sun, including an important optical study on shadows; the solar hours; and the astrolabe and other astronomical instruments.

Since it would hardly be feasible to give even a summary sketch of Ibn Sin?n’s entire work in a brief article, the best course will be to concentrate attention on two important contributions: his discussions of the quadrature of the parabola and of the relations between analysis and synthesis.

His study of the parabola followed directly out of the treatment given the problem in the work of his grandfather. Th?bit ibn Qurra had already resolved this problem in a different way from that of Archimedes. Although his method may have been equivalent to that of summing integrals, the approach was

more general than that of Archimedes in that the intervals of integration were no longer divided into equal subintervals. Th?bit’s demonstration was lengthy, however, containing twenty propositions. Another mathematician, one al-Mah?nni, had given a briefer one but Ibn Sin?n felt it to be unacceptable that (as he wrote) “al Mah?nni’s study should remain more advanced than my grandfather’s unless someone of our family (the Ibn Qurra) can excel him” (R?s’ilu Ibn-i-Sin?n, p. 69). He therefore sought to give an even more economical demonstration, one that did not depend upon reduction to the absurd. The proposition on which Ibn Sin?n founded his demonstration, and which he took care to prove beforehand, is that the proportionality of the areas is invariant under affine transformation.

His method considers the polygon a_n to be composed of 2ⁿ ? 1 triangles and inscribed in the area a of the parabola. The polygon a₁ is the triangle EOE?, a₂ is the polygon ECOC?E? etc. Ibn Sin?n demonstrated that if a_n and a?_n are two polygons inscribed, respectively, in the two areas a and a? of the parabola, then

Actually, he derived an expression equivalent to

from which he obtained

and finally derived

Ibn Sin?n’s originality in this investigation is manifest. It was with that same independence of mind that he intended to revive classical geometric analysis in order to develop it in a separate treatise. By virtue of that study, the author may be considered one of the foremost Arab mathematicians to treat problems of mathematical philosophy. His attempt has the form of a critique of the practical geometry in his own times. “I have found,” he wrote, “that contemporary geometers have neglected the method of Apollonius in analysis and synthesis, as they have in most of the things I have brought forward, and that they have limited themselves to analysis alone in so restrictive a manner that they have led people to believe that this analysis did not correspond to the synthesis effected” (ibid., p. 66)

In this work, Ibn Sin?n proposed two tasks simultaneously, the one technical and the other epistemological. On the one hand, the purpose was to provide those learning geometry with a method (tar?q) which could furnish what they needed in order to solve geometrical problems. On the other hand, it was equally important to think about the procedures of geometrical analysis itself and to develop a classification of geometrical problems according to the number of the hypotheses to be verified, explaining the bearing, respectively, of analysis and synthesis on each class of problems.

Considering both the problem of infinitesimal determinations and the history of mathematical philosophy, it is obvious that the work of Ibn Sin?n is important in showing how the Arab mathematicians pursued the mathematics they had inherited from the Hellenistic period and developed it with independent minds. That is the dominant impression left by his work.

BIBLIOGRAPHY

I. Original Works. The R?sa’ilu Ibn-i-Sin?n (Hyderabed, 1948) comprises Fi’l astrol?b (“On the Astrolabe”), Al-Tahlil wa’l-Tarkib (“Analysis and Synthesis”, Fi Harakati’š-Šams (“On Solar Movements”), Rasm al-qut? attal?ta (“Outline of Three Sections”), Fi mis?hat qat‘al-Mahr? al-muk?fi (“Measurement of the Parabola”), and Al-Handasa wa’n-Nuj?m (“Geometry and Astronomy”).

II. Secondary Literature. See Ibn al-Qifti, Ta’r?h al-Hukam?’, J. Lippert, ed. (Leipzig, 1903); Ibn al-Nad?m, Kit?b al-Fihrist, Flugel, ed. (Leipzig, 1871-1872), p. 272; C. Brockelmann, Geschichte der arabischen Literatur, I (Leiden, 1943), 245; H. Suter, Die Mathematiker und Astronomer der Araber und ihre Werke (Leipzig, 1900), pp. 53-54; “Abhandlung uber die Ausmessung der Parabel von Ibrahim ben Sinan ben Thabit ben Kurra,” in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich,63 (1918), 214 ff.; A. P. Youschkevitch, “Note sur les determinations infinitesimales chez Thbit ibn Qurra, in Ar-chives internationales d’histoire des sciences, no. 66 (January-March 1964), pp. 37-45.

Roshdi Rashed