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Abu Jafar alKhazin may have worked on both astronomy and number theory or there may have been two mathematicians both working around the same period, one working on astronomy and one on number theory. As far as this article is concerned we will assume that alKhazin worked on both topics. There seems no way of being certain which position is correct.
AlKhazin's family were from Saba, a kingdom in southwestern Arabia, perhaps better known as Sheba from the biblical story of King Solomon and the Queen of Sheba. In the Fihrist, a tenth century survey of Islamic culture, he is described AlKhurasani which means that he came from Khurasan in eastern Iran.
The Buyid dynasty, ruling in western Iran and Iraq, reach its peak around the time that alKhazin lived. It undertook public schemes such as building hospitals and dams, as well as patronising the arts and sciences. Rayy, situated southeast of present day Tehran, was one of the major cultural centres of the Buyid dynasty. Islamic writers described Rayy as:
... a city of extraordinary beauty, built largely of fired brick and brilliantly ornamented with blue faience (glazed earthenware).
AlKhazin was one of the scientists brought to the court in Rayy by the ruler of the Buyid dynasty, Adud adDawlah, who ruled from 949 to 983. We know that in 959/960 alKhazin was required by the vizier of Rayy, who was appointed by Adud adDawlah, to measure the obliquity of the ecliptic (the angle which the plane in which the sun appears to move makes with the equator of the earth). He is said to have made the measurement:
... using a ring of about 4 meters.
One of alKhazin's works Zij alSafa'ih (Tables of the disks of the astrolabe) was described by his successors as the best work in the field and they make many reference to it. The work describes some astronomical instruments, in particular it describes an astrolabe fitted with plates inscribed with tables and a commentary on the use of these. A copy of this instrument was made but vanished in Germany at the time of World War II. A photograph of this copy was taken and the article [5] examines this.
AlKhazin wrote a commentary on Ptolemy's Almagest which was criticised by alBiruni for being too verbose. Only one fragment of this commentary has survived and a translation of it is given in [6]. The fragment which has survived contains a discussion by alKhazin of Ptolemy's argument that the universe is spherical. Ptolemy wrote [6]:
.. of different figures of equal perimeter, the one with more angles is greater in capacity, and therefore it is necessary that a circle is the greatest of surfaces (i.e. of all plane figures with a constant perimeter) and the sphere the greatest of solids.
AlKhazin gives 19 propositions relating to this statement by Ptolemy. The most interesting results show, with a very ingenious proof, that an equilateral triangle has a greater area than any isosceles or scalene triangle with the same perimeter. When he tries to generalise this result to polygons, however, alKhazin gives incorrect proofs. Other results among the 19 are based on propositions given by Archimedes in On the sphere and cylinder. The author of [6] argues that the ingenious results on triangles are unlikely to be due to alKhazin but are probably taken by him from some unknown source.
The suggestion in [6] that alKhazin is a third rate mathematician is somewhat doubtful given his work on number theory but as we stated at the beginning of this article, it is possible that there were two mathematicians of the same name. The papers [4], [9] and [7] all look at this number theory work by alKhazin (see also [2] and [3]). The work of alKhazin which is described seems to have been motivated by work of a mathematician by the name of alKhujandi.
AlKhujandi claimed to have proved that x^{3} + y^{3} = z^{3} is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem. In a letter alKhazin wrote:
I demonstrate earlier ... that what Abu Muhammad alKhujandi advanced  may God have mercy on him  in his demonstration that the sum of two cubic numbers is not a cube is defective and incorrect.
This seems to have motivated further correspondence on number theory between alKhazin and other Arabic mathematicians. Results by alKhazin here are interesting indeed. His main result is to:
... show how, if we are given a number, to find a square number so that if the given number were added to it or subtracted from it the result would be square.
In modern notation the problem is given a natural number a, find natural numbers x, y, z so that x^{2} + a = y^{2} and x^{2}  a = z^{2}. AlKhazin proves that the existence of x, y, z with these properties is equivalent to the existence of natural numbers u, v with a = 2uv, and u^{2} + v^{2} is a square (in fact u^{2} + v^{2} = x^{2}). The smallest example of a satisfying these conditions is 24 which alKhazin gives
5^{2} + 24 = 7^{2}, 5^{2}  24 = 1^{2}.
He also gives a = 96 with
10^{2} + 96 = 14^{2}, 10^{2}  96 = 2^{2}
although, rather strangely, he seems to discount this case by another of his statements. Rashed suggests this may be because 96 = 2 × 48 = 2 × 6 × 8 and 6^{2} + 8^{2} = 10^{2}is not a primitive Pythagorean triple.
There is a mystery which Rashed notes in [7] (also in [2] and [3]). This relates to the quote above by alKhazin regarding the false proof by alKhujandi of the impossibility of proving x^{3} + y^{3} = z^{3}. Rashed has discovered a manuscript which appears to be by alKhazin, yet contains exactly what he had attributed to alKhujandi. Although alKhazin could have realised the error in alKhujandi's proof and attempted a similar proof himself which he believed correct, there is no really satisfactory explanation of these facts.
Finally we should mention that alKhazin proposed a different solar model from that of Ptolemy. Ptolemy had the sun moving in uniform circular motion about a centre which was not the earth. AlKhazin was unhappy with this model since he claimed that if this were the case then the apparent diameter of the sun would vary throughout the year and observation showed that this were not the case. Of course the apparent diameter of the sun does vary but by too small an amount to be observed by alKhazin. To get round this problem, alKhazin proposed a model in which the sun moved in a circle which was centred on the earth, but its motion was not uniform about the centre, rather it was uniform about another point (called the excentre).
Article by: J J O'Connor and E F Robertson
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