**Al-Farisi** was an Islamic mathematician who made two major contributions to mathematics, one on light, the other on number theory.

- His full name is Kamal al-din Abu'l Hasan Muhammad ibn al-Hasan al-Farisi.
- Al-Farisi was a pupil of the astronomer and mathematician Qutb al-Din al-Shirazi (1236 - 1311), who in turn was a pupil of Nasir al-Din al-Tusi.
- Al-Shirazi advised him to consult the Optics of ibn al-Haytham and al-Farisi made such a deep study of this treatise that al-Shirazi suggested that he write what is essentially a revision of that major work.
- Then al-Farisi went much further, for he undertook a project to study all the optical work of ibn al-Haytham.
- Al-Farisi 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 al-Haytham's theories are incorrect and to propose alternative theories himself.
- The most important part of this work by al-Farisi is his theory of the rainbow.
- Ibn al-Haytham had indeed proposed a theory, but al-Farisi considered both this theory and another proposed by Avicenna before giving his own.
- The theory proposed by al-Farisi was the first mathematically satisfactory explanation of the rainbow.
- Al-Farisi, 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.
- Al-Farisi 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, al-Farisi had to produce some new ideas about how colours were formed.
- The view before al-Farisi 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, al-Farisi proposed that the colours occurred because of the superimposition of different forms of the image on a dark background.
- There have been arguments between modern scholars as to whether al-Farisi's theory of the rainbow was due to him or whether it was a theory proposed by his teacher al-Shirazi.
- Al-Farisi made a number of important contributions to number theory.
- Al-Farisi's most impressive work in number theory is on amicable numbers.
- It was not a simple modification that al-Farisi made.
- Al-Farisi'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 so-called fundamental theorem of arithmetic, in this work.
- There is no doubt that al-Farisi proved these to be amicable numbers long before Euler.
- However, al-Farisi was probably not the first to discover these amicable numbers.
- Al-Farisi 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 nnn objects taken kkk at a time.

Born about 1260. Died about 1320, Tabriz, Iran.

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Ancient Arab, Origin Iran

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive