Fundamentals of Theoretical and Applied Mathematics: Role of Ancient Arabian and Egyptian Scientists

Fundamentals of Theoretical and Applied Mathematics: Role of Ancient Arabian and Egyptian Scientists

The achievements of Muslims in the field of mathematics are extremely remarkable. A regular study of this science, like all other sciences, was begun during the reign of the second Abbasi Caliph, Al-Mansür in. the second half of the eighth century A.C. During this period the work on mathematics was exclusively done by Muslims.1 Some stimulus came from Indian and Greek works which were later translated into ‘Arabic. The investigations were carried out, and until the end of the fifth century A.H. /the 11th century A.C., nearly all of the original and creative work was done by Muslims, and even the non-Muslims wrote all the works on mathematics in Arabic. In the 12th century the Christians and Jews started the work of translation from Arabic into Latin and Hebrew, and also began to conduct research in this field. But until, the end of the 13th century no mathematical work comparable to that of Muslims could be done by the Christians or Jews.

                                    The Muslims used numerals including zero for counting in contrast with writing the amounts in words, or counting with the letters of alphabet. Thus they made arithmetic simple and applicable to the problems of everyday life in connection with commerce and trade and the division of estates and inheritance. The zero has a great importance in arithmetic. Without zero it is not possible to indicate the figures like tens, hundreds, etc. If zero is not used it becomes necessary to use a table (named abacus) with columns of units, tens, hundreds, etc., to keep each figure in its place.2 The zero was used by the Muslims centuries before it was known in the West. The Latin word ciphra for zero is of Arabic origin; the Arabic word for it being sifr, meaning empty or nil.

                                    The West learnt the use of numerals from the Arabs, and, therefore, called them the ‘Arabic numerals“.3 The Arabs themselves termed them as Hindi numbers‘ (al-A’dad al-Hindi). The word Hindi has been translated as Indian by some writers, for, as they suggest, the numerals are of Indian origin. But this translation of the word Hindi does not seem to be correct, for, the word Hindi is sometimes written in place of Handasi, i.e., what relates to Handasah which means geometry and the art of engineering. Thus the numerals called Hindi may simply mean ‘the mathematical characters‘. There are some such instances in which the word Hindi may be considered to be a substitute for Handasi such as in astronomy ‘wherein a graduated circle called Dä’ira-e-Hindi may better be translated as ‘Mathematical (Geometrical) Circle‘.4

                                    The diffusion of the Arabic numerals in Christian Europe was very slow. The Christian mathematicians either used the old Roman numerals and the abacus, or used the Arabic numerals together with their old system. It was only in the 12th century that after learning from the Muslims the Western scholars were able to produce some literature on the numeral system, without columns, and completed by zero.5 It was Leonardo of Pisa who after traveling in Muslim lands and studying the Arabic system of numerals there, published a work which is mostly responsible for the introduction of that system into Europe. This system was named algorithm (or algorism) which has been derived by the Latin writers from the patronymic al-Khwärizmi (a native of Khwarizm) who was a distinguished Muslim mathematician, astronomer and geographer of the ninth century A.C. He flourished under the Caliph al-Mã’mun. His full name was Abü ‘Abd Allah Muhammad ibn Müsã al-Khawàrizmi. This name gained so much popularity that it was included in many languages of the world. Till the end of the 18th century the science of numbers (1, 2, 3, etc.)was called by theLatin writers as algorism whereas in Spanish it was termed as Guorismo. The English poet Chaucer named zero as augrim.6

                                     The Muslims made advancement in arithmetic. They wrote books on the arithmetic of every day use. They also wrote on commercial arithmetic. Algebra was made by them an exact science. Al-Khwärizmi named his book dealing with this subject as Kitab al-Jabr Wa’l Muqabalah (the book of restitution and comparison). The word Jabr means ‘restitution‘. It is the adding some thing to a given sum or multiplying it so that it becomes equal to another. The word Muqábalah means comparison. This term is applied for the comparison of the two sides of an equation as a + b=5, It seems that the word al-Jabr was originally used for these simple operations i.e. addition and multiplication, but later on it came to mean the whole subject.7

                                      The Muslims founded analytical geometry as well as plane and spherical trigonometry. The latter which in the early stage of its development was considered to be a part of astronomy, was made a separate branch of mathematics in the 13th century when it became sufficiently advanced. Under the Caliph al-Mansür, the centre of the Muslim Caliphate was transferred from Syria to Persia. The plans of the new capital, Baghdad, were drawn under the direction of the celebrated Minister, Khälid ibn Barmak for making measurements preliminary to the building of Baghdad. A Persian astronomer and engineer, Naubakht, and an astronomer named Masha Allah were appointed for this purpose. To Masha Allah is ascribed a book dealing with the prices of the wares, which is extant in Arabic. It is the first work of its kind in Arabic literature.8 A book on astrological judgments, Kitab al-Ahkam, was written by Naubakht.9

                                       The Caliph gathered around him a number of learned men at his court. His court mathematician, Muhammad Ibn Ibrãhim al-Fazari and Ya’qub Ibn Tariq were the first to introduce Hindu mathematics to Arabs through an Indian astronomical and mathematical work, Siddhanta.10 Al-Hajjaj Ibn Yüsuf who flourished sometimes between 786 and 833 probably in Baghdad. was the first to translate EucZid’s E.emento into Arabic. This work was twice translated, first under the Caliph Hãrün al-Rashid and again under his son al-Ma’mün. The second Arabic version was translated into Latin.11 Abü Sa’id al-Darir al-Jurfani (d. 845), who was a Muslim astronomer and mathematician, wrote a treatise on geometrical problems.12

                                    The Muslim mathematician, astronomer and geographer, Abu ‘Abd Allah Muhammad Ibn Müsä al-Khwãrizmi (d. 850) gathered Greek and Hindu knowledge, and through his arithmetic (later translated into Latin), the Muslims and the Europeans were introduced to the Hindi system of numerals. His influence on mathematical thought has been more than that of any other medieval writer. He wrote an encyclopedic work dealing with arithmetic, geometry, music and astronomy. In one of his books he discusses the question of the origin of numerals.13

                                    In his book on Algebra, al-Khwãrizmi first of all deals with the problems of the second degree equations. After that he describes multiplication and division. He also discusses the measurement of surfaces. A section of the hook deals with the problems of inheritance. The equations of the first degree are illustrated by numerical examples. The author distinguishes the following six cases of the equations of the second degree. Squares equal to roots, ax2=bx; squares equal to numbers, ax2=c; roots equal to numbers, ax=c; squares & roots equal to numbers, ax2+bx=c; squares and numbers equal to roots ax2+c=bx; roots and numbers equal to squares, bx+c= ax2. The author gives an analytical solution of linear and quadratic equations. From his way of treatment of the equations the idea of positive and negative signs seems just to emerge. The author also gives a geometrical solution, illustrated with figures, of quadratic equations as x2+l0x=30. Such equations have been repeated by the later writers.14

                                   In his book, Leonardo of Pisa enumerates the six cases of quadratic equations just as al-Khwärizmi does. He comment that the Arab method is superior to the method of Pythagorus.15 The three brothers, Ahmad, Muhammad and Hasan, called Banü Müsa (the sons of Müsã), who flourished during the reign of al-Ma’mun, wrote many mathematical, mechanical and astronomical works. Ahmad was especially interested in mechanics and Hasan in geometry. The third brother, Abu Ja’far Muhammad, who was also a logician and a student of Euclid and Almagest, was probably the most prominent. Among their works the most important ones are their treatises on the balance (Frastün or Qarastun), on the measurement of the sphere and on the trisection of the angle and the determination of two mean proportional between two given quantities (later translated into Latin). Banü Müs described the kinematical trisection of the angle and the so called gardener’s construction of the ellipse (by means of a string attached to the foci).16

                                      The encyclopedic scientist and the philosopher of the Arabs, Abü Yüsuf Ya’qüb Ibn Isäq, popularly known as al-Kindi (Latinized as Al-Kindus) because he belonged to the Kindah tribe), wrote 270 works, some of which deal with mathematics. He wrote four books on the use of Hindi numerals. The writings of äl-Khwärizmi and ál-Kindi were the main channels through which the numeral system became known to the West.17

                                       In the second, halt of the 9th century A.C., the number of Muslim mathematician increased considerably. Some of them specialized in arithmetic, some in geometry and the others, who were astronomers as well, showed particular interest in trigonometry. During this period the use of numerals became common. It was due to the fact that at that time the trade of the Muslims was flourishing in every part of the world. This active trade accelerated the diffusion of numerals throughout the world. The earliest Muslim documents bearing the numerals date from 874 and 888 A.C.18

                                      A great mathematician and astronomer of this century was Abü ‘Abd Allah Muhammad Ibn ‘Isa al-Mahãni that is from Mahan, Kerman, Persia (d. 844 A.C.) He wrote commentaries on Euclid and Archimedes, and improved the translation made by Ishäq Ibn Hunain of Menelaos’s Spherics. He tried in vain to solve an Archimedean problem which was to divide a sphere into two segments being in a given ratio. It led to a cubic equation, x3+C2b=cx2, which was called al-Mahani’s equation. This problem became a classical Muslim problem.19

                                     Hilãl al-Himsi translated the first four books of Apollonius into Arabic for Ahmad, one of the three sons of Müsä. Ahmad Ibn Yü’suf who was a secretary of the Tulunis, the rulers of Egypt from 868 to 905, wrote a book as an ode, a commentary on Ptolemy’s Centiloquium, and a book oh proportions. The latter is of special importance because it influenced the medieval thought and through it Western mathematicians became aware of .the theorem of Menelaos (about the triangle cut by a transversal).20

                                     Al-Fadl Ibn Hatim al-Nairizi (Latin Anaritius) who flourished under al-Mu’tadid (d. 922) wrote commentaries on Ptolemy and Euclid, which were translated into Latin. He used the tangent as a genuine trigonometric element.21 Thabit Ibn Qurrah (d. 901) improved the .theory of amicable numbers (if p=3.2n-1; g=3.2n-1-1; r=9.22n-1-1; and if p, q and r are prime together, 2n pq and 2nr are amicable numbers). His mensuration, of parabolas and paraboloids are very remarkable. 22

                                       Al-Battâni (d. 929) devoted the third chapter of his astronomical work to trigonometry. He uses sines considering them to be superior to the Greek chords. He completed the introduction of the functions umbra extensa and umbra versa (whence our cotangents and tangents which are among the fundamental elements of trigonometry), and gave a table of cotangents by degrees. He gave the relation between the sides and angles of a spherical triangle which is expressed by the formula, cos a=cos b cosc+sin b sin c cos A.23

                                       In the 10th century A.C., all the creative work on mathematics was done exclusively by Muslims and all the writings were produced in Arabic. By the end of this century the number of mathematicians increased immensely. Abu Kämil, who was one of the distinguished mathematicians of this period, perfected al-Khwãrizmi‘s work on algebra. He determined and constructed both roots of quadratic equations. He made a special study of pentagon and decagon with algebraic treatment, and of the addition and subtraction of radicals, corresponding to the formula ?a ± ?b= ?(a+b±?2ab). He mentioned the multiplication and division of al-gebraic quantities as well. He resolved systems of equations of up to five unknowns. His work was studied and greatly utilized by al-Karkhi and Leonardo of Pisa.24

                                     The Muslim physician and mathematician Abu Uthmãn Sa’id Ibn Ya’qüb al-Dimashqi who flourished at Baghdad under the Caliphate of Muqtadir (295-320/908-932) translated into Arabic the works of Aristotle, Euclid and Galen on temperament and pulse, etc.‘ The most important translation made by him was that of Book X of Euclid, together with Papus‘ commentary‘ on it. He was the superintendent of the hospitals of Baghdad, Makkah and Madinah.25

                                      The astronomer and mathematician, Abü Ishàq Ibràhim Ihn Sinãn wrote commentaries on the first book of Conics and on Almagest. Many papers on geometrical and astronomical subjects (for example, on sundials) are ascribed to him. His quadrature of parabolas was much simpler than that of Archimedes. In fact, it was the simplest ever made before the invention of the integral calculus.26 Ibràhim’s father Sinãn was also an astronomer and mathematician. Both of them embraced Islam. The Muslim mathematician and astronomer, Ali Ibn Ahmad al-Imrãni (d. 955) wrote a commentary on Abü Kãmil’s algebra.27

                                       Abü Ja’far Al-Khàzin (d. 961) wrote a commentary‘ on the Tenth Book of Euclid. Other mathematical and astronomical writings are also ascribed to him. He solved by means of conic sections, the cubic equation, called Al-Mãhãni’s equation which Al-Mãhãni himself could not solve. 28 Al -Kühi, who was the author of many mathematical and astronomical, works, made investigations on the Archimedean and Apollonian problems leading to the equations of a higher degree than the second. He solved some of these equations and discussed the conditions of solvability. These investigations are among the best ones made on geometry by Muslims.29

                                        The mathematician Abü Sa ‘id Ahmad al-Sijzi (951—1024) made a special study of the intersection of conic sections and circles. He replaced the old kinematical trisection of an angle by a purely geometric solution. (Intersection of a circle and an equilateral hyperbola).30Al-Sãghàni (d. 990) also made a study of the trisections of the angle.31

                                        Abu’l Wafã Muhammad al-Büzjani, who was an astronomer and one of the greatest Muslim mathematicians, flourished in Baghdad where he died in 997 or 998. He was one of the last Arabic translators and commentators of Greek works. He wrote commentaries on Euclid, Diophantos, and al-Khwãrizmi, and a practical arithmetic entitled Kitab al-Kãmil (the complete book). A book on applied geometry named Kitâb al-Handasah is also ascribed to him; He gave the solution of geometrical problems with one opening of the compass, He described the construction of a square equivalent to other squares, and approximate construction of regular heptagon (taking for its side half the side of the equilateral triangle inscribed in the same circle). He described regular polyhedron, and gave the construction of parabola by points. He gave the geometrical solution of X4=a  and X4+ax3=b.

                                        Abu’l-Wafá’s contribution to the development of trigonometry is remarkable. He was the first to show the generality of the sine theorem relative to spherical triangles. He gave a new method of constructing sine tables, the value of sin 300 being correct to the eight decimal places. He knew relations for sin (a±b) and also the relation equivalent to 2sin2?/2=1-cos ? sin?=2sin?/2 cos ?/2. He introduced the secant and cosecant which are among the six fundamental elements of trigonometry. He made the study of tangent, and compiled a table of tangents. He had the knowledge of those simple relations between the six trigonometric lines, which are now often used to define them.32

                                           Abü Mahmüd Hãmid Al-Khujandi Cd. 1000 A.C.) proved imperfectly that the sum of two cubic numbers cannot be a cubic number. He may be considered to be the discoverer of the sine theorem relating to spherical triangles.33 Abü Nasr Mansür Ibn Ali, the teacher of al-Birüni, was one of the three to whom goes the credit of the discovery of this theorem. He gave an improved edition of Menelaos’s spherics. He is the author of many trigonometrical and astronomical works.34 Maslamah Ibn Ahmad wrote on commercial arithmetic a book entitled Al -Muâmalât. He spoke of the erotic power of the amicable numbers.35

                                           Let us now discuss the progress made in mathematics in the 11th century. Giving air account of the development of this branch of science in the Latin and Muslim world, George Sarton writes: „A little stream of mathematical thought may be detected in the Latin writings, a stream which will gradually increase, but which will not become truly significant until the 13th century when a sufficient amount of Arabic water will have flowed into it“. He further writes:“ Let us pass on to Islam. It is almost like passing from the shade to the open sun and from a sleepy world into one tremendously active“.36 In the East the work on mathematics was done in Egypt, and in the West in Spain, Morocco, Tunis and other places.

                                            Ibn Yünus (d. 1009) introduced the first of those formulae which were indispensable before the invention of logarithms, namely, the equivalent of cos? cos? = ½ [ cos (?-?) + cos (?+?)]. He gave the approximate value of sin10= (1/3)*(8/9)*sin (9/8)0+ (2/3)*(16/15)*sin (15/16)0.37 The Persian mathematician Küshyar lbn Labbän made valuable contribution to the development of trigonometry. He continued the investigations which Abü Ja’far started on the tangent and compiled a table of tangents.38

                                             Abu Ja’far Muhammad Ibn al-Husain wrote a memoir (translated into French) on rational right angled triangles, and another on the determination of two mean proportionals between two lines by a geometrical method vs. kinematic method, that is by the use of what the Muslims called  Al-Handsah al-thabit, „fixed geometry“. He gave the solution of the equation X2+a=Y2.39

                                             Abü Bakr Muhammad Ibn al-Hasan (or Husain) al-Hãsib al-Karkhi, flourished in Baghdad during the Ministership of Abü Ghälib Muhammad Ibn Khalaf Fakhr al-Mulk (d. 1016). He was one of the greatest Muslim mathematicians. He wrote a book on arithmetic entitled al-Kafi fi’l-Hisâb (the sufficient on calculation). This book is largely based on Greek knowledge. Instead of using the numerals, the author wrote the name of numbers in full. He wrote a book on algebra entitled Al- Fakhri which he dedicated to the Minister. He gave with proofs the complete solution of quadratic equations and mentioned the reduction of equations of the type ax2p + bxp= c to quadratic equations.. He also described the addition and subtraction of radicals, ?8 + ?18=?50; 3?54 – 3?2=3?16 and summation of series: ?1n i2 = (?1n i) (2n+1)/3

                                                                    ?1n i2 = (?1n i) 2 (with geometrical proof).
He gave the solution of Diophantine equations including twenty five problems which are not found in Diophantos.40

                                             The Persian mathematician Abu’l Hasan ‘Ali al-Nasawi, who flourished under the Buwayhid Sultan, Majd al-Dawlah (d.. 1029) and under his successor, wrote a practical arithmetic in Persian, and translated it into Arabic. In his arithmetic called Al-Mughni-fi’l-Hisäb al-Hindi, he explains the division of fractions and the extraction of square and cubic roots. His replacement of hexadecimal by decimal fractions is remarkable, e.g. ?17° = 1/100 ?l70.0000 = (1/l00) 4120 =40 7′ 12?. 41

                                               Ibn al-Haitham conducted research in catoptrics which contains Alhazen’s problem. it is as follows: „Draw lines from two points in the plane of the circle which meet the circumference at a point and make equal angles with the normal  at the point“. This problem leads to the equation of the fourth degree. Ibn al-Haitham solved it by the help of a hyperbola intersecting a circle. He also solved al-Mãhãni’s (cubic) equation in the same way.42

                                              The encyclopaedist Abü Raihãn Muhammad ibn Ahmad al-Biruni (d. 1045) who was a mathematician, astronomer, philosopher, geographer and traveller, made a good contribution to mathematics. He was one of the greatest scientists of all times. He spent a considerable time in India. He translated a number of works from Sanskrit into Arabic and also transmitted Muslim knowledge to the people of India. Many works on various scientific subjects are ascribed to him. He gave a very clear account of Arabic numerals and a method for the trisection of the angle. He solved many problems (later called Al-Birunic problems) which cannot be solved with ruler and compass alone.43

                                      Now we come to the mathematical work of ‘Umar Ibn ibràhim al-Khayyãm who was one of the greatest Muslim mathematicians and astronomers of the middle Ages. His algebra marks a considerable advance on the work of the Greeks and on that of al-Khwarizmi and other Muslim algebraists, and indicates an advanced stage of the development of this science. While al-Khwarizmi deals only with quadratics, ‘Umar al-Khayyäm mostly discusses the cubic equations. He makes a remarkable classification of the equations, which are based on the complexity of the equation, i.e. on the number of different terms which they contain. Since the beginning of the 17th century the modern classification has been made on the basis of the degree of the equation. But it may be noted that the higher the degree of an equation, the more different terms it can have. The author classified the cubic equations into 27 classes which are subdivided into four categories. He tried to solve them and discussed the limits of the solution. He gave a partial geometrical solution of a number of them. He also investigated Euclid’s postulates and generalities. The algebra of Umar al-Khayyäm was edited and translated into French by F. Woepeke in Paris in 1857. 44

                                    A great astronomer al-Zarqàli (Latin Arzachel) explained the construction of the trigonometrical tables.45 Jabir ibn Aflah, in his book on astronomy named Kitdb al-Hai’ah or Islah al-Majisti wrote an important introduction on trigonometry. He gave the equivalent of the formula: cos‘ B=cos a, sin B for a spherical triangle recangu1ar in C.46 Ibn al-Yâsmini wrote a short poem on algebra, and Muhammad al-Hassär, who flourished in the 12th or 13th century, wrote a treatise on arithmetic and algebra, which was translated into Hebrew in 1271.47

                                     ‘Abd al-Malik al-Shiräzi abreviated Apollonius treatise, and Fakhr al-Din al-Rãzi, scholar and philosopher, wrote on Euclid’s Postulates. Muhammad ‘Abd Allah Ibn al-Hassär composed, with the help of Ibn Yünus a treatise entitled Risãlah al-Birkàr al-Tamam, (the treatise on the perfect compass). The ‘perfect compass‘ is an instrument by which every conic could be drawn.48 The Hispano-Muslim mathematician, lbn Badr, composed a compendium of algebra, called Ikhtisar al-Jabr wali-Muqäbalah. This compendium includes the theory as well as numerical examples. It deals with quadratic equations, surds, multiplication of, polynomials, arithmetical theory of prort.1on, linear Diophantine equations and similar problems.49

                                       The encyclopaedist Kamãl al-Din Ibn Yunus (d. 1242) wrote a treatise on arithmetic, algebra, square numbers, and regular heptagon and similar topics. He solved one of the questions asked by the Emperor Frederick It, which were submitted to him by the Ayyübi al-Kãmil, (the ruler of Egypt from 1218 to 1238 and of Damascus from 1234 to 1235). The question he solved was how to construct a square equivalent to a circular segment. The proof of the solution was given by one of his pupils, Al-Mufaddal Ibn ‘Umar al-‘Abhari, who wrote an essay on it.50

                                        The Moroccan astronomer, mathematician and geographer al-Hasan al-Marakãshi, who florished until. c. 1262, wrote various works on astronomy. His main work named Jami‘ al-Mabadi wa’l-Ghäyãt (The uniter of the beginnings and the ends; i.e., principles and results) is a good compilation of practical knowledge on astronomical instruments and methods, trigonometry and gnomonic. In this work he mentions not only the sine and versed sine (sahm arrow), but also what he called complementary sine(Jaib tamãm) sin (90°-a) = cos a and exceeding sin (jaib fadl), sin (a-90°) = -cos a. He compiled a table of sines for each half degree and the tables of versed sines and arc sines.51

                                         Another Moroccan mathematician and astronomer of the 13th century, Abu’l-‚Abbãs Ahmad Ibn Muhammad was a very popular Muslim writer. He is the author of about 74 works, most of which ate on mathematics and astronomy. The most popular of his works is the book Al-Tälkhis ‘an al-Hisab. This book was studied for at least two centuries, and many commentaries were written on it. It was highly admired by Ibn Khaldün. A french translation of it appeared in 164. It is an arithmetical summary which contains many interesting features; improved treatment of fractions; constant use of Hindi numerals, in their western form of course (ghubar). It also contains the summation of squares and cubes; casting out of nines, eights, and sevens; rule of double false position. ? (a2+r) ? a + (r/2a), if r?a

                                    ? a + {r/(2a+1)}, if r>a

                                         Beside Talkhis the author composed four treatises on calculations which deal respectively with integers, fractions, roots and proportions. He also wrote a treatise on binomials and epitomes i.e., quantities of the form a±?b or ?a±?b; on inheritance problems, and on geometry. He wrote an introduction to Euclid and R treatise on the measurement of surfaces. A book on algebra is also ascribed to him.52

                                         Now we come to the mathematical work of Nasir al-Din al-Tüsi who was a Persian philosopher, mathematician, astronomer, physician and scientist. He wrote in Arabic and Persian. He was one of the greatest Muslim mathematicians and scientists. He was born in 1201 in Tus, Khurasan. After the end of the ‘Abbãsi Caliphate in 1258, he became minister to Hulagu. Sixty-four writings on many subjects are ascribed to him. He wrote a number of treatises or commentaries on geometry and a treatise on algebra. Some arithmetical works are also ascribed to him. He wrote on inheritance, problems and a treatise on the proof that the sum of two odd squares cannot be a square. He proved that if a circle internally touches another circle, of double diameter and if the two circles turn or roll uniformly in opposite direction, remaining tangents and the speed of he smaller being twice greater than that of the other, then the original point of contact of the smaller circle will move along a diameter of the greater circle.

                                        Nasir al-Din is well-known mostly for his high achievements in trigonometry. He made a translation of Menelaos’s spherics, and himself wrote on the subject a separate treatise called Shakl al-Qatta‘ which means the figure of sector. This phrase refers to Menelaos‚ theorem about the triangle cut by a transversal; the Latin words cata or catta in the Latin phrases figura cata or regula catta for Shakl al-Qatta‘ is the corruption of the Arabic word al-Qatta‘ . Out of the five books of the Shakl al-Qattä‘ the books 3 and 4 are devoted to plane and spherical trigonometry respectively. It was the first book wherein trigonometry was treated as a separate branch of science and independently from astronomy. It was the greatest work of its kind in the middle Ages. It contains the first explicit formulation of the sine law relative to plane triangles with two proofs of it. It also contains the six fundamental formulae for the solution of spherical right angled triangles. It also gives the method for the solution of other triangles replacing, if necessary, the consideration of angles by that of sides and vice versa by means of polar triangles. 53

                                       After the Shakl al-Qattä of Nasir al-Din, a book bearing the same title was written by Muhiyy al-Din al-Maghribi who was a Hispano-Muslim mathematician ind astronomer. The latter is partly based on the former, yet it contains important original developments; for example, it gives two proofs of the sine theorem for right angled spherical triangles. One of the proofs is different from those given by Nasir al-Din. This theorem is then generalized for other triangles. The author has also produced editions of Greek classics as Euclid’s Elements, Apollonios’s Conies, with a brief preface, the Odosios and Menelaos‘ Spherics arid other works.54


1. Sarton ,George, Introduction to the History of Science, Washington, 1953, Vol I. p. 521

2. Arnold and Guillaume, The Legacy of Islam,Oxford, 1974 p. 385.

3. Ibid., p. 384.

4.IbId., pp. 384-385.

5. Ibid., p. 386.

6. Al-Khawarizmi, Muhammad Ibn Müsa, Kitàb al-Jabr wa’l Muqabalah, Cairo, 1939, preface by Ali Mustafa and Muhammad Müsä, p. 13.

7. Arnold & Guillaume, op. cit., p. 382.

8. Sarton, op. cit., Vol. I, p. 531.

9. Ibid.

10. Ibid., p. 530.

11.  Ibid. p. 562.

12.  Ibid.

13. Ibn Nadeem, Al-Fihrist, Cairo, p. 383.  Sarton, op. cit., p. 563.

14. Al-Khawarizmi, op. cit., pp. 15-66 , 67-106.

15. Arnold & Guillaume, op. cit., p. 384.

16.  lbn Nadeem, op. cit., pp. 378—379. Sarton, op. cit., p. 561.

17. Ibid., p. 357.

18.  Sarton, op. civ., p. 585.

19. Ibid., p. 597.

20. Ibid., p. 598.

21. Al-Qifti., Tarikh al-Hukamä, Leipzig, 1903, p. 254

22. Ibid. p. 599.

23. Ibid., p. 602.

24. Ibid., p. 630.

25. Ibid., p. 631.

26. Ibid.; Al-Qifti,‘ op. cit., p. 57.

27. Al-Qifti, Ibid., p. 233.

28. Sarton, op. cit., p. 664.

29. Al-Qifti, op. cit., p. 351.

30. Sarton, op. cit., p. 665.

31. Ibid., p. 666.

32. Ibid., p. 666. ; Ibn Nadeem, op.cit., p. 394.

33. Ibid., p. 667.

34. Ibid., p. 668.

35. Ibid.

36. Ibid., p. 695.

37. Ibid., p. 716. ; A1-Qifti, op. cit., p. 230

38. Al-Baghdádi, Ismail Basha, Hadiyyat al-‘Arifin, Istanbul, Vol. I, 1951, p. 838.

39. Sarton, op. cit., p. 666.

40. Ibid., p. 718.

41. Ibid., p. 719.

42. Al-Baghdádi, op. cit., Vol. II, p. 66. ;  Sarton, op. cit., p. 721.

43. Sarton, Ibid., p 707.

44. Ibid., p. 759.

45. Ibid., p. 758.

46. Ibid.,Vol. II, part I,  p. 206.

47.  Ibid., p. 400

48. Ibid., p. 401

49. Ibid.,Vol. II, part II,  p. 622.

50. Ibid., p. 600.

51. Ibid., p. 621; Al-Baghdádi, op. cit., Vol. I, p. 286.

52.Sarton, Ibid., p. 998.

53.  Ibid., p. 1001. ; Al-Baghdádi, op. cit., Vol. II, p. 131.

54. Sarton, op. cit., p. 1051. ; Al-Baghdádi, op. cit., p. 516.

Source by Md. Wasim Aktar