## The Persian Mathematicians and Mirrors

After the end of the Alexandrine superiority in 641, a scientific renaissance took place in Persia - mainly in Baghdad - where most of the main Greek works were translated into Arabic during the following centuries (Rouse Ball [9]). Some advances in algebra had already been assimilated by the Persians along with mathematical developments previously achieved in India, such as decimal numbering and the important invention by Bramaguptas in 629 of the "0" symbol for zero. In 820 Al-Khwarizmi published Al-jabr wa'l Muqabala - from which the word algebra is derived - where one of the positive roots of some second degree equations is solved (although symbolic notation did not yet exist). In around 900, Persians geometers were aware of Archimedes's failure to construct the regular heptagon and of the equally fruitless attempts by subsequent Greek geometers; they provided the first constructions of the heptagon around 970. Among their numerous contributions, we will limit our comments to the main works of Alhazen in optics.

• Al-Haytham, known in the West as Alhazen (965-1021), published in 1008 the Treatise on Optics: Kitab ul Manzir, containing seven books. In it he gives a detailed description of the human eye, explaining the function of each part. Here also Alhazen is the first to mention the camera obscura, some of which he built, noticing that the image is inverted. He gives the first explanation of atmospheric refraction. He investigates lenses as well as spherical and paraboloidal mirrors, and is aware of the spherical aberration and of the stigmatic property of a paraboloid already demonstrated by Diocles.

An important question which had been introduced by Ptolemy (85-165 AD, Alexandria) in his famous Almagest is known as Alhazen's Problem: given a spherical mirror of center O, a point source A and another given point B, how can one geometrically construct the intersection point R at the mirror surface where the ray

AR is reflected towards B ? Except for the trivial arrangements where the center O is on AB or on the median plane of AB, the general solution is impossible with the straightedge and compass method. Considering the AOB plane, the solution can be derived from the intersection of the circular section of the mirror with one of the homofocal ellipses of foci A and B, by selecting the ellipse which is tangent to this circle - there are generally two of them - by imposing two conditions: the equality of the ordinates and the equality of slopes. This leads to solving fourth degree equations where the solution is with a double root for the tangency. It seems that Alhazen solved this problem with the marked straightedge [6]. Huygens later gave a solution using the intersection of conics. A construction of point R by the intersection of a circle and a hyperbola is displayed in Fig. 1.2.

Al-Haytham notices that if a point source is at infinity, the image given by a spherical mirror of radius R is located at a distance equal to or a little larger than R/2 from the mirror: this distance is the focal length. In some examples where the source point is at finite distance, he also gives the location of the image point after reflection on a concave mirror: this is the conjugate distance.

^ Although lacking symbolic notation, the Persians knew how to calculate the conjugate distance in ~1000. This may be considered as the prelude to Gaussian optics.

Fig. 1.2 Al-Haytham's Problem: Given two points A, B and a spherical mirror of center O, how can one find the point R where the ray AR is reflected through point B ? In the AOB plane where the mirror intersection is the circle C, one constructs the circles of diameter OA, OB and lines Aa, Ab that intersect these diameters in A',B'. A hyperbola H, and only one, can be constructed passing by O, A',B' with its center O at the middle of A'B' and its asymptotes in the direction of the bisections of lines OA and OB. Among the four points on C and H, the figure shows the two solutions as points R or R' for a convex or a concave mirror, respectively (Arnaudies & Delozoide [6])

Fig. 1.2 Al-Haytham's Problem: Given two points A, B and a spherical mirror of center O, how can one find the point R where the ray AR is reflected through point B ? In the AOB plane where the mirror intersection is the circle C, one constructs the circles of diameter OA, OB and lines Aa, Ab that intersect these diameters in A',B'. A hyperbola H, and only one, can be constructed passing by O, A',B' with its center O at the middle of A'B' and its asymptotes in the direction of the bisections of lines OA and OB. Among the four points on C and H, the figure shows the two solutions as points R or R' for a convex or a concave mirror, respectively (Arnaudies & Delozoide [6])

Al-Haytham engaged in remarkable technological developments, for instance by constructing mirrors of steel and probably of steel-silver alloy and of pure silver, but like others was unable to obtain accurate spheres. He also gave comments on the development of turning lathes.

It is mainly from Spain under Arab domination and not directly from Persia, that Persian scientific writings, rendered into Latin by Adelard, Gherard and many other translators, were introduced into Europe during several centuries after 1150, thus including the Greek heritage also. During this period most of the old European universities were created. This favored the assimilation of these heritages and gave rise to important developments which were materialized by the Renaissance.

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