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Ibrahim Ibn Sinan Ibn Thabit Ibn Qurra

IBRāHīM IBN SINāN IBN THāBIT IBN QURRA

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

mathematics, astronomy. For the original article on Ibrāhīm see DSB, vol 7.

Ibrāhīm Ibn Sinān Ibn Thābit Ibn Qurra composed a survey of the whole field of geometrical analysis, contributed an original quadrature of the parabola, and gave the first general treatment of horizontal sundials. He was a member of a family that had produced distinguished scientists since the time of his grandfather, Thābit ibn Qurra, and Muhammad al-Nadim wrote that “During his time no one appeared who was more brilliant than he was” (alNadim 1970, vol. ii, 649). As he made clear in the preface to his work on the parabola, he saw it as his role to preserve and add to his family’s reputation, and in this he was eminently successful.

Early Life . As the son of the personal physician to the Caliph al-Muqtadir, Ibrāhīm would have been raised in a well-to-do household and would have had the best education possible. Clearly precocious, he wrote in his autobiography that he began his mathematical researches at the age of fifteen and wrote a number of treatises by the time he was eighteen.

Both Arabic biographical sources and Ibrāhīm’s autobiography indicate that at some point his family suffered persecution, probably when Ibrāhīm was in his mid-twenties, at the hand of the Caliph al-Qahir, and were forced to take refuge in Khorasan, a large region, not closely defined, east of Iraq. Ibrāhīm’s father, Sinān, converted from the Sabian sect to Islam under pressure from al-Qáhir, although it appears that his children may have remained Sabian. Ibrāhīm married and had a son named Ishaq.

Dials . Ibrāhīm describes in his autobiography, composed sometime between the age of twenty-five and his death at the age of thirty-eight, the works he had written. Those he characterized as astronomical are On Shadow Instruments; On the Motions of the Sun; and On Ptolemy’s Approximate Methods for the Anomalies of Mars, Saturn, and Jupiter. In the group he calls mathematical are On Tangent Circles; On Analysis and Synthesis; Selected Problems[in Geometry]; Measurement of the Parabola; and Drawing the Three[Conic] Sections.

Of On Shadow Instruments, there survive only Book 1 and a fragment of the beginning of Book 2 of his revision of an original that had been written in his teens. (He wrote later that he detested the prolixity of his teenage work.) Of Shadow Instruments was the first treatise on a general method for the design of plane sun dials, whatever the orientation of the dial’s face. He based it on the idea that a plane dial with arbitrary orientation may be considered a horizontal dial for an appropriate place on the earth’s surface.

The surviving fragment from Book 2 deals with the shape of hour lines on sundials, lines described by the tip of the gnomon’s shadow at a fixed time of day over the course of a year. Ibrāhīm’s grandfather, Thābit, had explicitly stated, but not proved, that some hour lines were not straight. Ibrāhīm proved this, but only part of the proof survives in the existing fragment of Book 2. (Paul Luckey’s restoration of the proof implied that Ibrāhīm’s proof only worked for certain hour lines, and this has been confirmed by a remark of Ibn al-Haytham in his Treatise on the Hour Lines.) Finally, although the surviving sections deal with plane dials, we learn from the autobiography that missing parts of Book 2 as well as the whole of Book 3 dealt with nonplanar dials—concave or convex spherical dials as well as concave conic dials.

In his autobiography, Ibrāhīm stated that after writing about the armillary sphere for his colleagues, he wrote a work on it “in different terms” for a craftsman who was making one for him. This adds further evidence to what is already known from Abū al-Wafā al-Būzjan̄ in his Geometrical Constructions for Artisans about communications between mathematicians and craftsmen in medieval Islam.

Geometry . The first three of Ibrāhīm’s mathematical works are all variations on a single theme: the geometrical method of analysis and synthesis. His complaint about what he called the “abbreviated” argument of his contemporaries in this regard is well-known, and his autobiography makes it plain that he wrote these three works to provide examples to students of three versions of this ancient method. The student would begin by progressing in stages through the (mostly) easy problems in On Tangent Circles (now lost), which contained a careful version of current practice. The student would, next, find in his On Analysis and Synthesis a careful discussion of the full method as found in Apollonius’s Cutting Lines in Ratios. (Here the student would also find a classification of problems according to the number of solutions and the need for further conditions.) Finally, in Selected Problems the student would find (mostly analyses only, omitting the syntheses, of) forty-one geometrical problems to illustrate the current abbreviated practice.

In On Drawing the Three Sections, Ibrāhīm wrote that there was no instrument by which one could draw the conic sections. Consequently, he said, he wrote this treatise on how to find as many points as one wishes on a given section. (Abū al-Wafā’ al-Būzjan̄, in his treatment of burning mirrors in his Geometrical Constructions, reproduced Ibrāhīm’s method for drawing a parabola.) However, Abū Sahl al-Qūhī wrote a late-tenth-century treatise, On the Complete Compass, which describes just the kind of instrument that Ibrāhīm said did not exist in his time. Thus, Ibrāhīm’s remark provides a fairly narrow window, around the mid-tenth century, for the introduction of the complete compass in medieval Islam.

In the preface to his Measurement of the Parabola, Ibrāhīm referred to two earlier versions, both of which were lost. He also called special attention to the fact that he used only three theorems and was able to give a direct proof rather than—as in earlier treatments—a proof by contradiction.

Additionally, Ibrāhīm wrote a commentary on the Conics of Apollonius (lost), a Description of the Notions/Theorems Used in Astronomy and Geometry, and a “Letter of Ibrāhīm b. Sinān to Abu Yusuf al-Hasan b. Isra il on the Astrolabe.” (This last work is, however, disputed.) Finally, he stated in his autobiography that he gathered his miscellaneous papers into a volume of about three hundred pages, of which he had the sole copy.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY IBRAHIM IBN SINAN

Die Schrift des Ibrāhīm b. Sinān b. Thābit über die

Schatteninstrumente. Translation and commentary by Paul Luckey. Edited by Jan P. Hogendijk. Frankfurt am Main, Germany: Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, 1999.

The Works of Ibrāhīm Ibn Sinān. Edited by A. S. Saidan. Kuwait

City: National Cultural Council, 1983.

OTHER SOURCES

Al-Nadim, Muhammad. The Fihrist. Translated by Bayard Dodge. New York and London: Columbia University Press, 1970.

Rashed, Roshdi. Les mathématiques infinitésimales du IXe au XIe siècle. Vol. 1, Fondateurs et commentateurs. London: Al-Furqan Islamic Heritage Foundation, 1996.

———, and Hèléne Bellosta. Ibrāhīm b. Sinān: Logique et géométrie au Xe siècle. Leiden, The Netherlands; Boston: Brill, 2000.

Rosenfeld, Boris A. “Geometrical Transformations in the Medieval East.” XIIe Congrès International d’Histoire des Sciences 3, series A (1971): 129–131.

Sezgin, Fuat. Geschichte des Arabischen Schrifttums. Vols. 5–7. Leiden, The Netherlands, E. J. Brill, 1974, 1978–1979.

J. L. Berggren

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Ibrāh

Ibrāhīm Ibn Sinān Ibn Thābit 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 Sinān’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. Thābit 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. Thābit’s demonstration was lengthy, however, containing twenty propositions. Another mathematician, one al-Mahānni, had given a briefer one but Ibn Sinān felt it to be unacceptable that (as he wrote) “al Mahānni’s study should remain more advanced than my grandfather’s unless someone of our family (the Ibn Qurra) can excel him” (Rās’ilu Ibn-i-Sinān, 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 Sinān 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 an 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 Sinān demonstrated that if an and an 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 Sinān’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 Sinān 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 (tarīq) 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 Sinān 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 Rāsa’ilu Ibn-i-Sinān (Hyderabed, 1948) comprises Fi’l astrolāb (“On the Astrolabe”), Al-Tahlil wa’l-Tarkib (“Analysis and Synthesis”, Fi Harakati’š-Šams (“On Solar Movements”), Rasm al-qutū attalāta (“Outline of Three Sections”), Fi misāhat qat‘al-Mahrū al-mukāfi (“Measurement of the Parabola”), and Al-Handasa wa’n-Nujūm (“Geometry and Astronomy”).

II. Secondary Literature. See Ibn al-Qifti, Ta’rīh al-Hukamā’, J. Lippert, ed. (Leipzig, 1903); Ibn al-Nadīm, Kitāb 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

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