Kamal alDin Abu'l Hasan Muhammad AlFarisi
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Tabriz, Iran
Biography
AlFarisi is also known as Kamal aldin. His full name is Kamal aldin Abu'l Hasan Muhammad ibn alHasan alFarisi. He made two major contributions to mathematics, one on light, the other on number theory. His work on light, colour and the rainbow is discussed in [1] but no mention of his work on number theory (nor mention of any other work at all by alFarisi) occurs in that article written by R Rashed. On the other hand his contributions to number theory are discussed the references [2], [3], [4], [7], [8], and [9], most of which are also written by R Rashed but relate to discoveries made after the article [1] was written.AlFarisi was a pupil of the astronomer and mathematician Qutb alDin alShirazi (1236  1311), who in turn was a pupil of Nasir alDin alTusi. His work on light was prompted by a question put to him concerning the refraction of light. AlShirazi advised him to consult the Optics of ibn alHaytham and alFarisi made such a deep study of this treatise that alShirazi suggested that he write what is essentially a revision of that major work. AlShirazi himself was writing a commentary on works of Avicenna at the time.
Then alFarisi went much further, for he undertook a project to study all the optical work of ibn alHaytham. His major work the Tanqih (which means revision) was far more than a commentary on ibn ibn alHaytham's optical writings. AlFarisi does not seek merely to explain the works of a master in a more elementary form, rather he is quite prepared to suggest that some of ibn alHaytham's theories are incorrect and to propose alternative theories himself.
The most important part of this work by alFarisi is his theory of the rainbow. Ibn alHaytham had indeed proposed a theory, but alFarisi considered both this theory and another proposed by Avicenna before giving his own. The theory proposed by alFarisi was the first mathematically satisfactory explanation of the rainbow.
Ibn alHaytham had proposed that light from the sun is reflected by a cloud before reaching the eye. It was a theory which did not allow for a possible experimental verification. AlFarisi, on the other hand, proposed a model where the ray of light from the sun was refracted twice by a water droplet, one or more reflections occurring between the two refractions. This model did allow an experiment to be conducted with a transparent sphere filled with water. Of course this introduced two additional sources for refraction, namely at the surface between the glass container and the water. AlFarisi was able to show that the approximation obtained by his model was good enough to allow him to ignore the effects of the glass container.
In order to explain the colours in the rainbow, however, alFarisi had to produce some new ideas about how colours were formed. The view before alFarisi was that colours were produced a mixing darkness with light. This could not explain the rainbow so, based on the experimental evidence of the colours that he had observed with his transparent sphere experiment, alFarisi proposed that the colours occurred because of the superimposition of different forms of the image on a dark background. He wrote (see for example [1]):
... If the images then interpenetrate, the light is again intensified and produces a bright yellow. Next, the blended image diminishes and becomes a darker and darker red until it disappears when the sun is outside the cone of rays refracted after one reflection.There have been arguments between modern scholars as to whether alFarisi's theory of the rainbow was due to him or whether it was a theory proposed by his teacher alShirazi. Boyer writes in [5]:
... the discovery of the theory should presumably be ascribed to [alShirazi], its elaboration to [alFarisi].Rashed discusses the claims of Boyer and others that the innovation in the theory of the rainbow was from alShirazi, but gives sound arguments for his claim that ascribing the theory to alShirazi is unconvincing.
AlFarisi made a number of important contributions to number theory. He noted the impossibility of giving an integer solution to the equation
$x^{4} + y^{4} = z^{4}$
but he attempted no proof of this case of Fermat's Last Theorem. AlFarisi's most impressive work in number theory is on amicable numbers. Suppose that, in modern notation, $S(n)$ denotes the sum of the aliquot parts of $n$, that is the sum of its proper quotients. The numbers $m$ and $n$ are called amicable if $S(n) = m$, and $S(m) = n$.
In Tadhkira alahbab fi bayan altahabb (Memorandum for friends on the proof of amicability) alFarisi gave a new proof of the following theorem by Thabit ibn Qurra on amicable numbers:
For $n > 1$, let $p_{n} = 3.2^{n}  1$ and $q_{n} = 9.2^{2n1}  1$. If $p_{n1}$, $p_{n}$, and $q_{n}$ are prime numbers, then $a = 2^{n}p_{n1}p_{n}$ and $b = 2^{n}q_{n}$ are amicable numbers.
It was not a simple modification that alFarisi made. Rather he produced a major new approach to a whole area of number theory, introducing ideas concerning factorisation and combinatorial methods. In fact alFarisi's approach is based on the unique factorisation of an integer into powers of prime numbers, and, according to Rashed, he states and attempts to prove this, the socalled fundamental theorem of arithmetic, in this work. Whether alFarisi proved or attempted to prove the fundamental theorem of arithmetic is also discussed in [4].
At the end of his treatise alFarisi gives the pairs of amicable numbers 220, 284 and 17296, 18416, obtained from using Thabit's rule with $n = 2$ and $n = 4$ respectively. To check that Thabit's theorem gives amicable numbers with $n = 4$, alFarisi has to show that $p_{3}, p_{4}$, and $q_{4}$ are prime numbers. Now $p_{3} = 23, p_{4} = 47$ and $q_{4} =1151$ and, to show that 1151 is prime alFarisi uses a number of lemmas including an application of the sieve of Eratosthenes.
The pair of amicable number 17296, 18416 are known as Euler's amicable pair. There is no doubt that alFarisi proved these to be amicable numbers long before Euler. However, alFarisi was probably not the first to discover these amicable numbers. In [6] Hogendijk argues that they were known to Thabit ibn Qurra himself.
AlFarisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early type of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of $n$ objects taken $k$ at a time.
References (show)

R Rashed, Biography in Dictionary of Scientific Biography (New York 19701990).
See THIS LINK.  R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).
 R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984).
 A G Agargün and C R Fletcher, alFarisi and the fundamental theorem of arithmetic, Historia Math. 21 (2) (1994), 162173.
 C Boyer, The rainbow : from myth to mathematics (New York, 1959), 127129.
 J P Hogendijk, Thabit ibn Qurra and the pair of amicable numbers 17296, 18416, Historia Math. 12 (3) (1985), 269273.
 R Rashed, Materials for the study of the history of amicable numbers and combinatorial analysis (Arabic), J. Hist. Arabic Sci. 6 (12) (1982), 278209.
 R Rashed, Nombres amiables, parties aliquotes et nombres figurés aux XIIIème et XIVème siècles, Arch. Hist. Exact Sci. 28 (2) (1983), 107147.
 R Rashed, Le modèle de la sphère transparente et l'explication de l'arcenciel : Ibn alHaytham  alFarisi, Revue d'histoire des sciences 22 (1970), 109140.
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Written by
J J O'Connor and E F Robertson
Last Update November 1999
Last Update November 1999