Abu al-Hasan Thabit ibn Qurra al-Harrani al-Sabi (born in Harran, now in southern Turkey, in 836 and died in Baghdad on 18 February 901) was a prolific scientist of the ninth century....

**Abu al-Hasan Thabit ibn Qurra al-Harrani al-Sabi **

**Born in Harran, now in southern Turkey, in 836.**

**Died in Baghdad on 18 February 901.**

Thabit was a prolific scientist of the ninth century. He belonged to the Sabian community, mentioned in the Qur’an; the Sabians descended from the Babylonian star worshippers. Because their beliefs were related to the stars, they produced many astronomers and mathematicians. During the Hellenistic era they spoke Greek and took Greek names; and after the Islamic conquest they spoke Arabic and began to assume Arabic names. Thābit, whose native language was Syriac, also knew Greek and Arabic. Most of his scientific works were written in Arabic, but some were in Syriac; he translated many Greek works into Arabic. When he was a distinguished scientist and physician in Baghdad, mainly under the reign of the Abbasid Caliph al-Mu‘tadid (892–902), Thābit was a leader of the Sabian community in Iraq. His son Sinān and his grandsons Ibrāhīm and Thābit were well–known scholars in mathematics, medicine and astronomy. Sinan ibn Thabit converted to Islam, thus putting an end to the Sabian tradition of the prestigious family his father had founded in the capital of the Muslim empire, and strengthened its integration in the Muslim society (see here).

In his youth, Thābit worked in Harrān as a money-changer. Muhammad, one of three Banū Mūsā brothers, who was traveling through Harrān, was impressed by his intellectual skills and invited him to Baghdad, where he became a great scientist. His son Sinān and his grandsons Ibrāhīm and Thābit were well-known scholars. Thābit, whose native language was Syriac, also knew Greek and Arabic. Most of his scientific works were written in Arabic, but some were in Syriac. His mathematical writings, the most studied of his works, played an important role in preparing the way for the extension of the concept of number to real numbers, integral calculus, theorems in spherical trigonometry, and non-Euclidean geometry. In astronomy Thābit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of the Arabic tradition of the science of weights.

In mathematics, Thābit ibn Qurra translated or edited in full or in part many Greek mathematical works of Euclid, Archimedes, Apollonius, Theodosius, and Menelaus. He also wrote commentaries on Euclid’s *Elements* and Ptolemy’s *Almagest.*

His *Kitāb al-Mafrūdāt* (Book of Data), consisting of thirty-six propositions in geometry and geometrical algebra, was very popular in Medieval times. *Maqāla fī Istikhrāj al-A’dād al-Mutahābba* (On the Determination of Amicable Numbers) contains ten propositions in number theory, including the problem, first solved by Thābit, of the construction of “amicable” numbers (pairs of numbers of which the sum of the divisors of each is equal to the other). In *Kitāb fī Ta’līf al-Nisab* (Book on the Composition of Ratios) Thābit studied the theory of compound ratios. This theory later led to the notion of real numbers and to the discovery of differential calculus.

*Risāla fī al-Shakl al-Qattā‘* (Treatise on the Secant Figure) presents a simple and elegant proof of Menelaus theorem, the first theorem of spherical geometry. *Misāhat al-Ashkāl al-Musattaha wa-‘l-Mujassama* contains rules for computing the areas of plane figures and the surfaces and volumes of solids.

Thābit produced proofs of the Pythagorean Theorem and its generalization in *Risāla fi ‘l–Hujja al-Mansūba ilā Suqrāt fi al-Murabba’ wa-Qutrihi* (On the Proof Attributed to Socrates on the Square and its Diagonal). In *Kitāb fī Misāhat Qat’ al-Makhrūt alladhī Yusammā al-Mukāfi’* Thābit computed the area of the segment of a parabola. Historians of mathematics consider that this computation, different from that done by Archimedes in *Quadrature of the Parabola*, is equivalent to that of the integral. The computation is based essentially on the application of upper and lower integral sums, and the proof is done by the method of exhaustion.

Another area where Thābit exerted his mathematical genius is the classical problem of proving Euclid’s fifth postulate, which he investigated in two treatises: *Maqāla fī Burhān al-Musādara al-Mašhūra min Uqlīdis* (The Proof of the Well-known Postulate of Euclid) and *Maqāla fī anna al–khattayn Idhā Ukhrijā ‘alā Zāwiyatayn Aqal min Qā’imatayn Iltaqayā* (Two Lines Drawn at Angles Less Than Two Right Angles Will Meet). The first attempt is based on the unclear assumption that if two straight lines intersected by a third move closer together or farther apart on one side of it, then they must move farther apart or closer together on the other side. The “proof” consists of five propositions, the most important of which is the third, in which Thābit proves the existence of a parallelogram by means of which Euclid’s fifth postulate is proved in the fifth proposition. The second attempt is based on kinematic considerations. Criticizing the approach of Euclid, who banned the use motion in geometry, Thābit asserted the necessity of its use. He postulates that in the parallel motion of a body, all its points describe straight lines. The “proof” consists of seven propositions, in the first of which, based on the necessity of using motion, he concludes that equidistant straight lines exist; in the fourth proposition he proves the existence of a rectangle that is used in the seventh proposition to prove the Euclidean postulate. Thābit’s procedure and results were further developed by Ibn al-Haytham (d. c. 1050), ‘Umar al-Khayyām (d. 1131) and Nasīr al-Dīn al-Tūsī (d. 1274) and later led to the discovery of non-Euclidean geometry.

In astronomy, Thābit was the author of many treatises on the movement of the sun and moon, sundials, visibility of the new moon, and celestial spheres. In a well-known treatise extant only in a Latin version *(De motu octave spere)*, he added an eighth sphere, that of the fixed stars, to Ptolemy’s spheres (those of the sun, moon, and five planets) and proposed the theory of “trepidation” to explain the precession of the equinoxes. This theory first appeared in Islamic astronomy in connection with Thābit’s name.

Thābit studied the uneven apparent motion of the sun according to Ptolemy’s eccentricity hypothesis in *Kitāb fi Ibtā’ al-Haraka fī Falak al-Burūj* (The Deceleration of the Motion on the Ecliptic). He also investigated the apparent motion of the sun in *Kitāb fī Sanat al-Shams* (On the Solar Year). *Kitāb fi Alāt al-Sā‘āt allatī Tusammā Rukhāmāt* is his treatise on the sundial in which Thābit used propositions of trigonometry equivalent to the spherical theorems of cosines and sines for spherical triangles of general forms to solve concrete problems in spherical astronomy. In another treatise on the sundial, *Maqāla fī Sifat al-Ashkāl allatī Tahduthu bi-Mamarr Taraf Zill al-Miqyās*, Thābit examines conic sections described by the end of a shadow of the gnomon on the horizontal plane and determines the diameters and centers of these sections for various positions of the sun.

Two of Thābit’s treatises on balances and weights, *Kitāb fi Sifat al-Wazn wa-Ikhtilāfihi* (On the Properties of Weight and Nonequilibrium) and *Kitāb fī al-Qarastūn* (On the Steelyard [Balance]), are devoted to problems of practical and theoretical mechanics. But it is in *Kitāb fī al-Qarastūn* that Thābit presents a systematic theory of the steelyard with geometrical proofs. The main topic of the treatise is the determination of the weight which must be applied to the extremity of a homogeneous beam in order to maintain it in equilibrium, when this same beam is suspended from one of its points. To solve this problem, which he proved in the final part of the treatise, Thābit had to demonstrate the law of the lever and to determinate the statical moment of the beam.

Recent studies proved that *Kitāb fī al-Qarastūn* was the founding text of the Arabic tradition of the science of weights, which was continued and developed in the Latin tradition of *Scientia de ponderibus.*

**Bibliography**

**1. Articles on**www.MuslimHeritage.com

- Mahbub Ghani, Hail the Queen of Mathematics! (published 2 September 2005).
- Mahbub Ghani, The Science of Restoring and Balancing – The Science of Algebra (published 5 January 2007).
- Muslim Founders of Mathematics
- Mohammed Abattouy, A New Arabic Text of Mechanics: Sinan ibn Thabit on the Theory of Simple Machines (published 7 February 2012).

**2. General references**

- Abattouy, Mohammed (2001): “Greek Mechanics in Arabic Context: Thābit Ibn Qurra, al-Isfizārī and the Arabic Traditions of Aristotelian and Euclidean Mechanics,”
*Science in Context*, vol. 14: 179-247. - Dold Samplonius, Yvonne (1996): “The
*Book of Assumptions*by Thābit Ibn Qurra (836-901).” In:*History of Mathematics: The States of the Art.*Edited by J. Dauben*et al.*San Diego/Boston: Academic Press, 207-222. - Lorch, Richard (2001):
*Thābit Ibn Qurra: On the Sector-Figure and Related Texts.*Edited with Translation and Commentary. Frankfurt, 2001. - Morelon, Régis (1994). “Thābit b. Qurra and Arab Astronomy in the 9th Century.”
*Arabic Sciences and Philosophy*, vol. 4: 111–139. - Rosenfeld, B. A. and A. T. Grigorian (1976): “Thābit ibn Qurra.” In
*Dictionary of Scientific Biography*, edited by Charles Coulson Gillispie. New York: C. Scribner’s Sons, vol. 13, 288-295. - Thābit ibn Qurra (1987):
*Oeuvres d’astronomie.*Texte établi et traduit par R. Morélon. Paris: Les Belles Lettres.

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