**Ibrahim ibn Sinan** was a son of Sinan ibn Thabit and a grandson of Thabit ibn Qurra and studied geometry and in particular tangents to circles. He also studied the apparent motion of the Sun and the geometry of shadows. There is no doubt that had he not died at the young age of thirty-eight, he would have achieved a degree of fame for his mathematical works going even beyond the opinion of Sezgin (see [5] and [6]) that he was:-

... one of the most important mathematicians in the medieval Islamic world.

Perhaps his early death robbed him of the chance to make a contribution even more important than that of his famous grandfather.

Ibrahim's most important work was on the quadrature of the parabola where he introduced a method of integration more general than that of Archimedes. His grandfather Thabit ibn Qurra had started to view integration in a different way to Archimedes but Ibrahim realised that al-Mahani had made improvements on what his father had achieved. To Ibrahim it was unacceptable that (see for example [1]):-

... al-Mahani's study should remain more advanced than my grandfather's unless someone of our family can excel him.

Ibrahim is also considered the foremost Arab mathematician to treat mathematical philosophy. He wrote (see for example [1]):-

I have found 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.

We know of Ibrahim's works through his own work *Letter on the description of the notions Ibrahim derived in geometry and astronomy* in which Ibrahim lists his own works. This is one of seven treatises by Ibrahim given with full Arabic text and English summaries in [2]. Among the works published in [2] are *On drawing the three conic sections* in which Ibrahim give a pointwise construction for the ellipse, the parabola and the hyperbola. Although based on ideas due to Apollonius there are aspects of this work which illustrate the changed point of view of Arabic mathematicians. For example Ibrahim uses an arithmetical term to denote the product of two geometrical lines.

In *On the measurement of the parabola* Ibrahim ibn Sinan gives a beautiful proof that the area of a segment of the parabola is four-thirds of the area of the inscribed triangle. Another work is *On the method of analysis and synthesis, and the other procedures in geometrical problems* which contains a systematic exposition of analysis, synthesis and related subjects, with many easy examples. This is in contrast to *The selected problems* in which 41 difficult geometrical problems are solved, usually by analysis only, without a discussion of the number of solutions or conditions which make the solutions possible.

*On the motions of the sun* is an astronomical work which discusses of the motion of the solar apogee. It also provides a critical analysis of the observations underlying Ptolemy's solar theory, and Ibrahim ibn Sinan provides his own theory of the sun. The work *On the astrolabe* includes work on map projections. Ibrahim proves in this work that the stereographic projection maps circles which do not pass through the pole of projection onto circles.

In fact geometric transformations figure a great deal in Ibrahim's works and this interesting aspect is discussed in detail in [4]. Examples are given which illustrate how Ibrahim applied an orthogonal compression to transform a circle into an ellipse, and an oblique compression to map a hyperbola into a second hyperbola. In a different work Ibrahim uses a transformation which maps figures keeping invariant the ratio between their areas.

Ibrahim's contribution is summed up in [1] as follows:-

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

**Article by:** *J J O'Connor* and *E F Robertson*

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