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# Ibrahim ibn Sinan ibn Thabit ibn Qurra

(908 in Baghdad- 946 in Baghdad)

Ibrahim ibn Sinan was a grandson ofThabit ibn Qurra and studied geometry and in particulartangents(1) 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 [2] and [3]) 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 thequadrature(4) of theparabola(5) where he introduced a method of integration more general than that ofArchimedes. His grandfatherThabit ibn Qurra had started to view integration in a different way toArchimedes but Ibrahim realised thatal-Mahani had made improvements on what his father had achieved. To Ibrahim it was unacceptable that (see for example [6]):

...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 [6]):

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 [7]. Among the works published in [7] are On drawing the threeconic sections(8) in which Ibrahim give a pointwise construction for theellipse(9), the parabola and thehyperbola(10). Although based on ideas due toApollonius 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 theinscribed(11) 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 solarapogee (12). It also provides a critical analysis of the observations underlyingPtolemy's solar theory, and Ibrahim ibn Sinan provides his own theory of the sun. The work On theastrolabe(13) includes work on map projections. Ibrahim proves in this work that thestereographic projection (14) 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 [15]. 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 [16] as follows:

Considering both the problem ofinfinitesimal(17) 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

## Notes:

1-A tangent to a curve at a point p is the best linear approximation to the curve near that point. It can be regarded as the limit of the chords from the point p to other points close to p.

If two curves have a common tangent at a point of intersection they are said to touch or be tangent.

2- F Sezgin, History of Arabic literature (German) Vol. 5 (Leiden, 1974), 292-295

3- ibid, Vol. 6 (Leiden, 1978), 193-195

4-Quadrature means calculating the area of a figure or the area under the graph of a function. Literally, finding a square with the same area.

5-A parabola is one of theconic sections.

It may be defined using thefocus directrix property as thelocus of points which are equidistant from a fixed line and a fixed point.

or via Cartesian coordinates as the set of points in a plane satisfying the equation :

y = x2

6- Biography in Dictionary of Scientific Biography (New York 1970-1990)

7-A S Saidan (ed.), Ibrahim ibn Sinan, The works of Ibrahim ibn Sinan (Arabic) (Kuwait, 1983)

8-A conic or conic section is one of the three curves: aparabola,hyperbola orellipse which one can obtain by intersecting a plane with a (double sided) cone.

9-A ellipse is one of theconic sections.

It may be defined using thefocus directrix property as thelocus of points whose distance from a fixed point is a fixed multiple e < 1 from a fixed line.

Or via Cartesian coordinates as the set of points in a plane satisfying the equation:

ax2 + by2 = 1

10-A hyperbola is one of theconic sections.

It may be defined using thefocus directrix property as thelocus of points whose distance from a fixed point is a fixed multiple e > 1 from a fixed line.

or via Cartesian coordinates as the set of points in a plane satisfying the equation :

ax2 - by2 = 1

11- A circle is said to be inscribed to a triangle or other polygon if the edges are tangents to the circle.

The polygon is than said to becircumscribed to the circle.

The (unique) circle inscribed to a triangle is called the incircle and its centre is the incentre.

12-The apogee is the point where a heavenly body is furthest away from the centre of its orbit.

The nearest point is called the perigee.

13-An astrolabe is an early instrument for measuring the angle between the horizon and a star or planet. It was superceded by the octant and sextant.

14-Stereographic projection is projection of the points of a sphere from the North Pole of the sphere along straight lines on to the equatorial plane or onto the tangent plane at the South Pole.

15- B A Rozenfel'd and M M Rozanskaja, Geometric transformations and change of variables in the writings of Ibrahim ibn Sinan (Russian), in History and Methodology of Natural Sciences IX : Mechanics, Mathematics (Moscow, 1970), 178-181.

16- Biography in Dictionary of Scientific Biography (New York 1970-1990).

17-An infinitesimal is an arbitrarily small quantity which early mathematicians found it necessary to incorporate into their theories in the absence of a proper theory of limits.

Infinitesimal calculus is the Differential and Integral Calculus.

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