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Thabit Ibn Qurra Ibn Marwan al-Sabi al-Harrani was born in the year 836 A.D. at Harran (present Turkey). As the name indicates he was basically a member of the Sabian(1) sect, but the great Muslim mathematician Muhammad Ibn Musa Ibn Shakir, impressed by his knowledge of languages, and realizing his potential for a scientific career, selected him to join the scientific group at Baghdad that was being patronized by the Abbasid Caliphs. In Baghdad Thabit received mathematical training and also training in medicine, which was common for scholars of that time. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he leftHarran and was appointed court astronomer inBaghdad. There Thabit's patron was the Caliph, al-Mu'tadid, one of the greatest of the 'Abbasid caliphs.

At this time there were many patrons who employed talented scientists to translate Greek text into Arabic and Thabit, with his great skills in languages as well as great mathematical skills, translated and revised many of the important Greek works. The two earliest translations ofEuclid's Elements were made by al-Hajjaj. These are lost except for some fragments. There are, however, numerous manuscript versions of the third translation into Arabic which was made byHunayn ibn Ishaq and revised by Thabit. Knowledge today of the complex story of the Arabic translations ofEuclid's Elements indicates that all later Arabic versions develop from this revision by Thabit.

Although Thabit contributed to a number of areas the most important of his work was in mathematics where he [Biography in Dictionary of Scientific Biography (New York 1970-1990)]:

... played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, andnon-Euclidean geometry(2). In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.

In "Book on the determination ofamicable numbers (3)" Thabit claims thatPythagoras began the study of perfect andamicable numbers. This claim, probably first made byIamblichus(4) in his biography ofPythagoras written in the third century AD where he gave theamicable numbers 220 and 284, is almost certainly false. However Thabit then states quite correctly that althoughEuclid andNicomachus studied perfect numbers, andEuclid gave a rule for determining them ([R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994)] or [R Rashed, Entre arithmétique et algèbre: Recherches sur l'histoire des mathématiques arabes (Paris, 1984)]):-

... Neither of these authors either mentioned or showed interest in [amicable numbers].

Thabit continues:

Since the matter of [amicable numbers] has occurred to my mind, and since I have derived a proof for them, I did not wish to write the rule without proving it perfectly because they have been neglected by [Euclid andNicomachus]. I shall therefore prove it after introducing the necessary lemmas.

After giving nine lemmas Thabit states and proves his theorem: for n > 1, let pn= 3.2n-1 and qn= 9.22n-1-1. If pn-1, pn, and qn areprime numbers, then a = 2npn-1pn and b = 2nqn areamicable numbers while a isabundant (5) and b isdeficient(6). Note that an abundant number n satisfies S(n) > n, and a deficient number n satisfies S(n) < n. More details are given in [9] where the authors conjecture how Thabit might have discovered the rule. In [J P Hogendijk, Thabit ibn Qurra and the pair of amicable numbers 17296, 18416, Historia Math.

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(3) (1985), 269-273] Hogendijk shows that Thabit was probably the first to discover the pair ofamicable numbers 17296, 18416.Another important aspect of Thabit's work was his book on the composition of ratios. In this Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. The authors of [B A Rozenfel'd and L M Karpova, Remarks on the treatise of Thabit ibn Qurra (Russian), in Phys. Math. Sci. in the East 'Nauka' (Moscow, 1966), 40-41] and [B A Rozenfel'd and L M Karpova, A treatise of Thabit ibn Qurra on composite ratios (Russian), in Phys. Math. Sci. in the East 'Nauka' (Moscow, 1966), 5-8] stress that by introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalization of the number concept.

Thabit generalizedPythagoras's theorem to an arbitrary triangle (as didPappus). He also discussedparabolas(7),angle trisection (8) andmagic squares(9). Thabit's work on parabolas and paraboliods is of particular importance since it is one of the steps taken towards the discovery of the integral calculus. An important consideration here is whether Thabit was familiar with the methods ofArchimedes. Most authors (see for example [A P Yushkevich, Quadrature of the parabola of ibn Qurra (Russian), in History Methodology Natur. Sci.

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(Moscow, 1966), 118-125]) believe that although Thabit was familiar withArchimedes' results on thequadrature of the parabola, he did not have either ofArchimedes' two treatises on the topic. In fact Thabit effectively computed the integral ofx and:The computation is based essentially on the application of upper and lower integral sums, and the proof is done by themethod of exhaustion(10): there, for the first time, the segment of integration is divided into unequal parts.

Thabit also wrote on astronomy, writing Concerning the Motion of the Eighth Sphere. He believed (wrongly) that the motion of theequinoxes(11) oscillates. He also published observations of the Sun. In fact eight complete treatises by Thabit on astronomy have survived and the article [R Morelon, Tabit b. Qurra and Arab astronomy in the 9th century, Arabic Sci. Philos. 4 (1) (1994), 6; 111-139] describes these. The author of [ibid] writes:

When we consider this body of work in the context of the beginnings of the scientific movement in ninth-century Baghdad, we see that Thabit played a very important role in the establishment of astronomy as an exact science (method, topics and program), which developed along three lines: the theorization of the relation between observation and theory, the 'mathematisation' of astronomy, and the focus on the conflicting relationship between 'mathematical' astronomy and 'physical' astronomy.

An important work Kitab fi'l-qarastun (The book on the beam balance) by Thabit is on mechanics. It was translated into Latin byGherard of Cremona and became a popular work on mechanics. In this work Thabit proves the principle of equilibrium of levers. He demonstrates that two equal loads, balancing a third, can be replaced by their sum placed at a point halfway between the two without destroying the equilibrium. After giving a generalization Thabit then considers the case of equally distributed continuous loads and finds the conditions for the equilibrium of a heavy beam. Of courseArchimedes considered a theory of centres of gravity, but in [K Jaouiche, Le livre du qarastun de Tabit ibn Qurra. étude sur l'origine de la notion de travail et du calcul du moment statique d'une barre homogène, Arch. History Exact Sci.

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(1974), 325-347] the author argues that Thabit's work is not based onArchimedes' theory.Thabit had a student Abu Musa Isa ibn Usayyid who was a Christian from Iraq. Ibn Usayyid asked various questions of his teacher Thabit and a manuscript exists of the answers given by Thabit, this manuscript being discussed in [S Pines, Thabit Qurra's conception of number and theory of the mathematical infinite, in 1968 Actes du Onzième Congrès International d'Histoire des Sciences Sect. III : Histoire des Sciences Exactes (Astronomie, Mathématiques, Physique) (Wroclaw, 1963), 160-166]. Thabit's concept of number follows that ofPlato and he argues that numbers exist, whether someone knows them or not, and they are separate from numerable things. In other respects Thabit is critical of the ideas ofPlato andAristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments.

Thabit also wrote on:

... logic, psychology,ethics, the classification of sciences, the grammar of the Syriac language, politics, the symbolism ofPlato's Republic ... religion and the customs of the Sabians.

His son,Sinan ibn Thabit, and his grandsonIbrahim ibn Sinan ibn Thabit, both were eminent scholars who contributed to the development of mathematics. Neither, however, reached the mathematical heights of Thabit.

## Notes:

1-The Sabian religious sect were star worshippers from Harran often confused with the Mandaeans (as they are in Biography in "Dictionary of Scientific Biography", New York 1970-1990).

2- A non-Euclidean geometry is a geometry in which Euclid'sparallel postulate fails, so that there is not a unique line failing to meet a given line through a point not on the line.

If there is more than one such "parallel" the geometry is called hyperbolic; if there is no line which fails to meet the given one, the geometry is called elliptic.

3-

For example, 220 and 284 are amicable numbers.

4- Iamblichus (born

## :

about 250 AD in Chalcis in Coele Syria [now in Lebanon], died## :

about 330) was a Syrian philosopher who played a large part in the development of Neoplatonism. His writings which have survived include On the Pythagorean Life; The Exhortation to Philosophy; On the General Science of Mathematics; On the Arithmetic of Nicomachus; and Theological Principles of Arithmetic. He was a major influence on Proclus.5-An abundant number is an integer for which the sum of its proper divisors is greater than the number itself.

For example, the proper divisors of 12 are 1, 2, 3, 4, 6 which sum to 16.

6-A deficient number is an integer for which the sum of its proper divisors is less than the number itself.

For example, the proper divisors of 8 are 1, 2, 4, which sum to 7.

7- A parabola is one of the 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

8-

One of the classic problems of Greek mathematics was to find aruler and compass construction to divide any angle into three equal pieces.The other classic problems weresquaring the circle andduplicating the cube.

9- A magic square is a set of integers (often 1, 2, ... , n2 ) arranged in a square in such away that each row, each column (and often the two diagonals as well) sum to the same number.

For example: is a 3

3 magic square10- The method of exhaustion is calculating an area by approximating it by the areas of a sequence of polygons.

For example, filling up the interior of a circle by inscribing polygons with more and more sides.

11-

Equinoxes occur about 21st March and 23rd September.