Islamic Civilization

Other Branches, and the Practical Nature  of Islamic Mathematics

When the traveller Leo the African came to Fes , he noted something remarkable there: a chemical society, and he wrote of them:

‘A stupid set of men who contaminated themselves with sulphur and other horrible stinks.’[1]

One of the greatest reproaches made to Muslim science and mathematics by Western historians of science, and mathematics, in particular, was their practical aspect. Whereas Western science, in its origins as well as in the era of classical modernity, paid particular attention to theoretical foundations, Oriental  science is defined essentially by its practical aims.[2] Oriental science, and Muslim science in particular, is said to allow itself to be carried away by empirical rules and practical aims.[3]

Indeed, as the following will serve to highlight the principal point made in this work that by its very nature, Islamic science differs fundamentally from its Greek predecessor mainly due to its practical nature. This practical nature, thus, denotes that the real source of Islamic mathematics, just like all Islamic science, was fundamentally not Greek science. The foundation for it, once more, was the faith, which imposed upon Muslim scholars the need to research solutions for practical problems of the community. Sciences and learning in Islam, as I and L Al-Faruqi correctly point out, have to have a purpose, and  it is the practical knowledge that produces results and leads to virtue, the object of the Muslim's prayer: ‘O God grant us knowledge that is useful and beneficial.'[4] This, as has been seen in the many preceding sciences, is also the case in mathematics in its diverse branches.

As Sabra puts it, ‘no mystery’ surrounds Islamic geometry; ‘it was Greek in origin, methods and terminology.'[5] But two major remarks can be added here: one on the additional findings and studies by Muslim mathematicians, and the other, of course, the practical end of their work.

On the first point, profiting by the researches of Ibn al-Haytham (965-1040) on a theorem not proved by Archimedes in his On the Sphere and the Cylinder, al-Kuhi (940-1000) constructed a segment of a sphere equal in volume to the segment of a given sphere and in its surface area to another segment of the same sphere.[6] He resolved the problem very ingenuously with the help of two auxiliary cones and the intersection of two auxiliary conic sections-a hyperbola and a parabola.[7] Ibn al-Haytham, himself elaborated upon Euclid's fifth postulate, using a triangular quadrilateral (called in the West, Lambert's Quadrangle.)[8] Euclid also attracted the focus of Al-Hajaj ibn Yusuf who was the first translator of his elements into Arabic. Al-Abbas wrote commentaries upon them, and Abu Said al-Darir wrote a treatise on Euclidian geometrical problems.[9] Regard for Euclid did not stop Islamic mathematicians from questioning and refuting him when necessary, though.[10] Muslim mathematicians also used Greek techniques to formulate and solve new problems, making rearrangements or extensions of Euclid’s theorems. Attempts to prove Euclid's parallels postulate have been found in their writings dating from the 9^th to the 13^th  century. Rather than acquiesce in a ready made solution, they pursued the search for ever better solutions, formulating and proving some non Euclidean theorems; one of their attempts to prove the postulate later became known to European mathematicians such as Wallis and Saccheri.[11] 

An aspect of geometry of special interest to Muslim authors was its use in making calculations.[12] Amongst these are Ibrahim Ibn Sina n’s treatise on the quadrature of the parabola, Abu’l Wafa on the construction of regular polygons, which led to the equations of third degree, and Abu Kamil on the construction of the pentagon and the decagon, also by means of equations.[13]

With regard to the second development, i.e the practical nature of geometry in Muslim hands, this is where lies the fundamental Islamic contribution to the science. Geometry was applied to many practical problems such as surveying, studies in mechanical tools, the construction of improved mills, of norias (wheels with scoops for the continuous drawing of water,) mangonels, tractors etc.[14] Ozdural notes how mathematicians, who taught practical geometry to artisans, played a decisive role in the creation of patterns in Islamic art, and also in designing buildings.[15] Mathematicians gave instructions to artisans on certain principles and practices of geometry.[16] They also worked on geometric constructions of two or three dimensional ornamental patterns or gave advice on the application of geometry to architectural construction.[17] In his Book On What is Necessary From Geometric Construction for the Artisan, Abu’l Wafa (940-998) gave solutions of two and three dimensional problems ‘A form of geometry that would have upset the Greeks, to whom geometry was solely theoretical art,' says Ronan.[18] In this work, Abu’l Wafa displays knowledge of pure geometry, familiarity with practical applications, and skill in teaching theoretical subjects to practical minded people.[19] Abu’l Wafa also provides us with insight into the collaboration of mathematicians and artisans in the Islamic world, telling how he attended meetings between mathematicians and artisans in Baghdad .[20] Omar  Khayyam (1048-1131), likewise, shows the same practical concern, solving a right triangle with the aid of cubic equations in one of his treatise following a question at a meeting attended by geometers and artisans.[21] For his part, in referring to carpentry, Ibn Khaldun  says that:

‘It needs a good deal of geometry…. In order to bring the form (of things) from potentiality into actuality in the proper manner.[22]

It is not just geometry, but all aspects of Islamic mathematics, which adopt and respond to practical problems. Al-Farabi’s (d.950) words, in fact captures this concern with perfection. In his Classification of the Sciences, we find, it is ‘to determine the means by which those things whose existence is demonstrated in the various mathematical sciences can be applied to physical bodies.’[23] He explains that

‘In order to produce the truths of mathematics artificially in material objects, the latter may have to be subtly altered and adapted. In this sense, the ‘science of devices' is a general art which includes algebra (on this account a kind of applied arithmetic that seeks to determine unknown numerical quantities) as well as building, surveying, the manufacture of astronomical, musical and optical instruments, and the design of wondrous devices. All these and similar arts are principles of the practical crafts of civilisation.’[24]

Al-Farabi offers us an exciting passage on the science of mechanics and other devices, and of which Saliba[25] provides an excellent English translation unavailable anywhere else, and surely a translation that cannot be matched (due to Saliba’s proficiency in both Arabic and English):

‘The science of mechanics (Hiyal) is the knowledge of the procedure by which one applies all that was proven to exist in the mathematical sciences that were mentioned above in statements and proofs to the natural bodies, and the act of locating all that, and establishing it in actuality. The reason for that is that these mathematical sciences concern themselves with lines, surfaces, volumes, numbers, and bodies. When one wants to locate these ideas that form the subject matter of the mathematical sciences and wilfully exhibit them, by means of a craft, in the natural bodies that are perceptible to the senses, one needs  a force through which he proceeds to establish them in these bodies and to apply these ideas to these bodies. For the material and perceptible bodies have special conditions that prohibit them from accepting the ideas that were demonstrated by proofs from being located in them as one pleases to do. On the contrary, these natural bodies have to be prepared to accept what one seeks to establish in them, and one has to contrive to remove the obstructions.

The sciences of mechanics are therefore those that supply the knowledge of the methods and the procedures by which one can contrive to find this applicability and to demonstrate it in actuality in the natural bodies that are perceptible to the senses.

Of these mechanical sciences are the many arithmetical ones including the science known to the people of our times as the science of algebra, for it partakes of arithmetic and geometry...

Among them (i.e the mechanical sciences) also, are the many geometric (or engineering, handasiya) mechanical sciences, such as:

            -The art of overseeing construction.

            -The devices for determining the areas of bodies.

-The devices used in the production of astronomical and musical instruments and in the preparation of instruments for many practical crafts such as bows and arrows and various weapons.

-The optical devices used in the production of instruments that direct the sight in order to discern the reality of the distant objects, and in the production of mirrors upon which one determines the points that reverse the rays by deflecting them or by reflection or refraction. With this, one can also determine the points that reverse the sun’s rays into other bodies, thus producing the burning mirrors and the devices connected with them.

-The devices used in the production or marvellous objects, and the instruments for the several crafts.

These and their likes are the mechanical sciences which in turn are the principles of the civil and practical crafts that are applicable to bodies, shapes, positions, order, and assessments such as in the crafts of masonry, carpentry and others.

            These are the mathematical sciences and their divisions.’[26]

The Muslim scientists involved in the design, construction and use of such devices: i.e the Banu Musa, al Biruni, al Karaji, Omar  Khayyam, Ibn al Haytham etc, were not just distinguished mathematicians, but were also men of practical skills, their science aiming at meeting practical needs.[27] And so did Al-Kindi, whose medical treatise established posology on a mathematical basis.[28] A century later, Abu’l-Wafa, better known for his works on geometry, also wrote A Book on What is Necessary From the Science of Arithmetic for Scribes and Businessmen, And the Arithmetic for Government Officials.[29] Ibn Al-Samh who lived at Granada (d.1035) wrote on mental calculus (hisab al-hawai) on the nature of numbers and geometry, but most of all treatises on commercial arithmetic (Al-mu'amalat).[30] Abu Muslim Ibn Khaldun  (d. 1057) wrote mathematical works dealing with business arithmetic for sale of merchandise, measurement of land, calculating charity taxes, and other business transactions.[31] From North Africa , the author of Kitab maqayis al-Juruh [Book of the Measures of Wounds] describes the method for measuring all types of wounds and the way to calculate the legal indemnities demanded by victims.[32] Ibn al-Banna (b. Marrakech  1256) wrote Tanbih al-albab, which gives precise mathematical answers to the composition of medicaments, the calculation of drop irrigation of canals, arithmetical explanations of Qura’nic law of inheritance, explanation of frauds related to instruments of measurement, the exact calculation of legal tax in the case of a delayed payment, etc.[33] Furthermore, as Arnaldez and Massignon note, practical problems also brought the Muslims face to face with such progressions as the sum of even or odd numbers, though they failed to prove their generality.[34] Muslim arithmetical texts also refer to weights and measures, to the purity of coins and to the methods of counting.[35] In the 15^th century, Al-Kashi wrote a treatise entitled Miftah al-Hisab (The Key to Arithmetic,) which was intended for the use of merchants, clerks and surveyors, as well as theoretical astronomers.[36] Al-Kashi also solved a problem about a triangular levelling instrument at the construction site of the observatory of Samarkand  during a meeting of artisans, mathematicians, and dignitaries.[37]

Mathematical geography developed largely out of the need to find distances to,  and directions of, Makkah. The fundamental Muslim source of determining latitudes and longitudes of cities, the distance between them, and the azimuth of one relative to another is al-Biruni’s Kitab Tahdid al-Amakin (Book of Demarcation) which has been edited in Arabic by Bulgakov[38] and translated into English by Ali.[39] Al Biruni also wrote Kitab Tastih al-Kuwar which is devoted to map projections, has been edited into Arabic by Saidan,[40] and translated into English by Berggren, in 1982.[41] In this treatise, al-Biruni mentions al-Saghani’s generalisation of the astrolabic projection (stereographic projection), which is studied by Lorch 1985.[42] Map and astrolabe projection belonged in the Islamic classification of sciences to the science of instruments, and the publications by King[43] and Lorch contain expositions of the theory and history of several instruments designed to produce solutions to a large number of problems arising in applications of mathematics.[44]

Trigonometry has extensive links with astronomy; trigonometrical applications serving to solve astronomical problems, and also problems of a practical nature.[45] Al-Battani  (850-929) was the first to use the expressions "sine" and "cosine"; and was very much aware of the superiority of his ‘sines’ over the Greek chords.[46] He computed to a very high degree of accuracy the first complete tables of sines, tangents and co-tangents, and established the fundamental trigonometrical relations by introducing the notion of trigonometrical ratios. Al-Battani also applied algebraic operations to trigonometric identities, and his methods were widely used by Regiomontanus.[47] The Muslims innovated in the invention of plane and spherical trigonometry.[48] Abu'l-Wafa, making a special study of the tangent, tabulated its values, and introduced the secant and the cosecant.[49] He knew the simple relationships between these six basic trigonometric functions, which are often used even today to define them.[50] Carra de Vaux has demonstrated, following Moritz Cantor, that it was Abu'l-Wafa and not Copernicus who invented the secant; he called it the 'diameter of the shadow' and set out explicitly the ratio in its modern form.[51] Ibn Hamzah al-Maghribi (16^th century) from Algeria, surpassed Ibn Yunus (950-1009) towards logarithmic operations in his work on geometric progression. He established the following theorem:

‘The order of any given term of a geometric progression, starting with unity, equals the sum minus unity of the powers of the common ratio of the two terms whose product equals the given term.’[52]

In the Christian West, the subject only made its beginnings in the 14^th century at Merton College, Oxford.[53]

Again, trigonometry arose as means to respond to the requirements of the faith, and it progressed in order to resolve such practical problems. Abu’l Wafa, for instance, used trigonometry in order to calculate the distance between Baghdad  and Makkah .[54] A standard application of trigonometric methods was in calculating both the times of prayers and the direction of Makkah for the same purpose.[55] Over the centuries, numerous Muslim scientists discussed the Qibla problem, presenting solutions by spherical trigonometry, or reducing the three dimensional situation to two dimensions, and solving it by geometry or plane trigonometry.[56]

There are also methods, which, although non trigonometric, developed side by side with trigonometric methods and interacted with them in a way whose history is yet to be written.[57] One such method, involving rotation and orthogonal projection of arcs on a sphere into one plane, is that of the anlemma, and aspects of its history as relating to the problem of finding the direction of prayer are discussed by Berggren.[58]

All these mathematical subjects and their applications travelled to the Christian West. The ideas and methods elaborated or perfected in the Islamic countries took root and led to new results in the most advanced European countries.[59] Yushkevitch notes how this is the case in the books of Leonardo of Pisa (Fibonacci) (fl 1202), who emphasises the significance of Islamic science for the Latin  people, just as others before him  did.[60] The significance of Islamic mathematics (and astronomy) had been earliest understood in  Lotharingia, modern day Lorraine in France, which pioneered in the transmission of Muslim mathematics (and early astronomy) to the Christian West. This followed the visit of John of Gorze (970-4) to Spain, and his taking back Muslim manuscripts north of the Pyrenees.[61] Following John’s trip, mathematics thrived in the schools of Lorraine, and from that region scientific learning ‘radiated' to other countries: Germany, France and England.[62] Not long after John, came the famed trip to Spain by the future pope, Gerbert of Aurillac, who despite hostile reception of his ‘Saracen’ mathematics, still impacted greatly on the sciences of the Christian West.[63] Toledo ’s 12^th century role is famed, its school of translation playing a leading part in the introduction of Islamic learning into Europe. To Toledo went Gerard of Cremona, John of Seville , Robert of Chester and all the great translators of the age. Plato of Tivoli's translation of Al-Battani  in 1145 as Liber enbadorum, for instance, introduced Muslim trigonometry and mensuration into the West, and had great impact on the geometry of Leonardo of Pisa.[64] Gerard of Cremona translated mathematical treatises by the Banu Musa, al-Khwarizmi, al-Farghani, Thabit Ibn Qurra, Abu Kamil, al-Zarqali, and Jabir Ibn Aflah. The translations of Robert of Chester and Adelard of Bath of al-Khwarizmi were also decisive for the advance of European mathematics.[65] Muslim works on geometry and trigonometry became known to the Latins through the translation of the Liber trium fratrum (On the Measurement of Plane and Spherical Surfaces) written by the three Banu Musa brothers.[66] The early 13^th century revolution in European arithmetic was due to Fibonacci, whose father (a Pisan merchant), whilst trading with the Muslims, discovered the superiority of Arabic numerals, and so sent his son to study in the city of Bejaia  (Algeria) with a Muslim teacher.[67] The resulting outcome in 1202 was Leonardo’s treatise: Liber abaci, which according to Sarton, was ‘the first monument of European mathematics.'[68] Despite all such transfers, as  Hogendijk points out, the  transmission of Muslim mathematics to the Christian West was far from complete, and until 1450 the level of mathematics in Europe was below that in the Islamic world.[69] A later transfer concerns Al-Kashi’s treatise ‘The Key to Arithmetic', whose use of decimal fractions was noted in a Byzantine document which made its way to Vienna in 1562.[70] All the discoveries of Muslim mathematicians did not become known to Europeans, but those that did, Yushkevitch insists, considerably enriched mathematical knowledge,[71] and it would be a mistake to underestimate the role of Muslim mathematics in the progress of sciences in Europe.[72]