Of all Islamic sciences, mathematics has been best served by scholarship, most particularly in the recent decades.[1] Kennedy offers a good survey of the mathematical sciences in Islam.[2] Yet despite this, its due place is not adequately recognised, and only a work of vast proportions, a sort of encyclopaedia of Islamic mathematics, written by those who excel at mathematics, Arabic, Islamic history, and the history of mathematics, could do it justice. Muslim mathematics, like other sciences, has also been a victim of the usual historical distortions, which misattribute, or suppress many of its accomplishments, or demean its importance. The following outline aims at raising some such problems. However, as with all other issues addressed in this work, this author does not have the competence, nor the space, to address all matters as adequately as wished. Others in the future will have to pursue the task.

General Observations

First and foremost, it is important to consider again the issue of the true scale of the Greek legacy on Islamic mathematics. The Greek legacy is decisive in one area, geometry, but is minimal or non-existent in others. It is, however, common for many historians of sciences to widen the impact in one area (geometry) to the whole science of mathematics (just as Islamic mapping relying on aspects of Greek mapping is turned into plagiarism of Greek geography by Muslims). A worse error, however, is the widely held view that nothing happened in the medieval period. O’Connor and Robertson comment on this:

‘There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.’

The authors go on saying:

‘That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem that: ‘Arabic science only reproduced the teachings received from Greek science.’[3]

Rashed also notes how it has been frequently affirmed since Condorcet  and Montucla up to Borbaki, by way of Nesselman, Zeuthen, Tannery and Klein (to cite only a few), that Classical algebra, for instance, is the work of the Italian School perfected by Viete and Descartes.[4] Did not Milhaud (1921) and Dieudonne (1974) trace the early history of Algebraic geometry back to Descartes? [5] The modern mathematician’s text is, in this respect, significant: between the Greek prehistory of algebraic geometry and Descartes, Dieudonne finds only a void, which ‘far from being frightening, is ideologically reassuring.’[6]  

These views are false on two essential grounds:

First, Muslim mathematics was superior in every possible respect to its Greek counterpart, Muslim mathematicians adding a vast new array of findings, and developments to the science.

Secondly, many mathematical accomplishments, thought to be of Western origin, are in fact, of Islamic origin.

On the first point, any comparative work between Greek and Muslim mathematics will prove the immense superiority of Islamic mathematics. Indeed, the medieval period, and the period between the mid 10^th and mid 11^th century, in particular, as noted by Berggren, was a highly creative period for many of the mathematical disciplines in Islam, that saw significant advances in arithmetic and algebra, the development of spherical trigonometry, and brilliant contributions to mechanics, optics, and cartography.[7] This is highlighted by a few instances here. Suter, who studied Islamic mathematics in good detail, could write:

‘In the application of arithmetic and algebra to geometry, and conversely in the solution of algebraic problems by geometrical means, the Arabs far surpassed the Greeks and the Hindus.’[8]

The most famous contributions to this field were Ibrahim Ibn Sina n’s (908-946) writings on the quadrature and the parabola, Abu Al-Wafa’s (b.940-d. 997-8) writings on the construction of regular polygons involving cubic equations, and Abu Kamil’s (850-930) writings on the pentagon and the decagon.[9] Another example is of the depth of the mathematical tradition in mechanics as to be found in the correspondence between al-Kuhi (940-1000) and al-Sabi (fl. 10^th century), part of which is summarised and studied by Sesiano.[10] This correspondence is an example of serious work in Arabic on the determination of centres of gravity, which goes considerably beyond the work of Archimedes, and which contains discussions of interest for geometry and the philosophy of mathematics.[11]  

Rigour and ascertaining findings with precision was, as noted with previous sciences, a real demarcation line between Islamic and Greek science, and Berggren notes how in the 10^th century there was a decided tightening of rigour in Islamic geometry, with the result that methods that had seemed perfectly acceptable to Archimedes were seen as needing further explanation.[12]  

Islamic mathematics, as will be amply discussed in the appropriate heading further on, above all, dwelt on practical problems in a way Greek mathematics never did.

In relation to the second point, as Islamic mathematics is examined more thoroughly, most particularly due to the works of scholars such as Djebbar, Rashed, Berggren, Saidan, etc, it comes to be realised that many former accomplishments thought to be Western, in fact, date from centuries earlier, back to the Islamic period. Recent research, as O’Connor and Robertson insist, is, indeed:

‘Painting a very different picture of the debt owed to Islamic mathematics.. that modern mathematics is closer to Muslim mathematics than the Greek,… many of the ideas which were previously thought to have been brilliant new conceptions of European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Islamic mathematicians centuries earlier.’[13]

O’Connor and Robertson provide a good outline of the main Islamic mathematicians and their works to prove the point.[14]

The same point was also made by Berggren, who insists on the mathematical autonomy and originality of Islamic civilisation, and how innovations in arithmetic and algebra that once seemed due to outside influence have emerged as integral parts of the corpus of Islamic mathematics.[15]

Rashed, most particularly, gives instances, of how Muslim mathematicians of the 11^th-12^th centuries achieved results still incorrectly attributed to mathematicians of the 15^th-16^th centuries, for, instance, the method attributed to Viete for the resolution of numerical equations; the method ascribed to Ruffini-Horner; general method of approximation, in particular the one Whiteside designated as the Al-Kashi-Newton method, and lastly the theory of decimal fractions.[16] In addition to methods which were to be reiterative and capable of leading to in a recursive way to approximation, 11^th and 12^th century Muslim mathematicians also formulated new procedures of demonstration such as mathematical induction still found in the 17^th century.[17] Similarly, they engaged in new logico-philosophical-debate: for example, the classification of algebraic propositions and the status of algebra in relation to geometry. It was their successors who were to tackle the problem of symbolism.[18] All of this is to say, Rashed concludes, that certain concepts, methods and results attributed to Chuquet, Stifel, Faulhaber, Scheubel, Viete, Stevin, etc, were actually the work of this tradition of al-Karaji’s (953-1029) school, known furthermore to the Latin  and Hebrew mathematicians.[19]

Recent research, such as by Rashed himself and Djebbar, Berggren dates,[20] has also shown that the study of number theory formed a continuous tradition and led to the discovery of theorems or problems usually ascribed to Western mathematicians several centuries later, such as the appearance of ‘Wilson’s’ theorem in the work of Ibn al-Haytham, ‘Bachet’s problem of the weights in al-Khazini (fl.1115-1130), or the summation of the fourth powers of the integrers 1,2…., n in the work of the 10^th century mathematician al-Qabisi.[21]

In geometry, the Muslim pioneering role is noted by Yushkevitch, who observes how the birth of the theory of construction using a compass with only one opening was due to Abu Al-Wafa (940-998), a study of the problem only beginning in the time of the European ‘Renaissance.’[22] The theory of parallels was undertaken by al-Gauhari (ca 800-860), and followed up by Thabit Ibn Qurra (826-901), Ibn al-Haytham, Omar  Khayyam (1048-1131), and others. The Muslims put forward ideas which were later developed by Saccheri and Lambert, and in fact the first propositions of non-Euclidian geometry dates from this time.[23] 

This previous misattribution of modern mathematics to the Greeks, and to post ‘Renaissance’, modern Western scholars, just as other accomplishments are also wrongly attributed to them, had a particular colonial purpose. Joseph makes excellent comments on this, how such Eurocentric scholarship aimed at devaluing the contribution of the people under Western colonial rule so as to ease their subjugation and domination.[24] The classical Eurocentric trajectory, as he calls it, deliberately passes scientific knowledge from the Greeks into a period of Dark ages, then a re-discovery of Greek learning leading to the Renaissance, itself leading to European hegemony over cultural dependencies.[25] One such Eurocentric ‘scholar’, highly in vogue in Western history of science, Tannery, recalls that:

‘The more one examines the Hindu and Arabic scholars, the more they appear dependent upon the Greeks….. (and) quite inferior to their predecessors in all respects.’[26]

 The same ‘historian’ also notes how Muslim algebra, for instance, ‘in no way superseded the level attained by Diophantus.’[27] 

Demeaning the contribution of colonised people, generally the Muslims, was used to justify the colonial argument that the Westerner had taken over their country so as to civilise them, because, they (the Muslims) have an inferior civilisation.[28]

The colonial argument was not alone in demeaning the Islamic role, though. As will be amply expanded in the final part of this work, hostility to Islam and thence to its civilisation date from the medieval period, and went on throughout the centuries, the argument remaining the same, only the form altering with time. The Muslim achievements such as in mathematics during the medieval period were, thus, deemed the work of the devil, and Muslim mathematics taken north of the Pyrenees by the future pope Sylvester II (pope 999-1003) was deemed ‘Saracen magic.’ It was said that he had bartered his soul with Satan whilst in Muslim Spain.[29] This hostility took a turn for the worse in the later medieval and Renaissance period,[30] This hostility, as noted by Joseph, was stirred by colonial scholarship of the late 18^th-19^th century, and still prevails today amongst a scholarship that generally fails to dissociate itself from past antecedents.

It is certain that the Muslims, in mathematics, as in other sciences, relied on other traditions, the Chinese, above all, and also the Hindu and pre-Islamic cultures of the region. Hill, however, notes that Muslim mathematicians, in general, felt free to investigate aspects of the subject that appealed to their tastes and met their requirements and to pursue their investigations in any way they chose.[31] They did not feel constrained to use any particular method of approach; no method was considered so superior to all others that it became dominant. Hill pointed to al-Karaji (fl 11^th century), for instance, who totally neglected Indian mathematics.[32] Yushkevitch maintains that in the case where Chinese and Indian scientists gave only a few rules for calculation, the Islamic mathematicians completed scientific theories.[33] Compared with the works of Indian and Chinese scientists, Muslim mathematical works were more systematic, more complete, and distinguished by the greater use of proofs, even in books of a purely practical character.[34]

Other civilisations provided the means, but the real force behind Muslim mathematics,  as with the rest of the sciences and accomplishments in civilisation, remains the faith itself.[35] Islamic practice demanded precise calculations in inheritance matters, commercial transactions etc.[36] Arithmetic (al-hisab) was, as Ibn Khaldun  observes in his Prolegomena, the first of the mathematical sciences to be used by the Muslims, being a means of solving such material problems that present themselves in daily life as assessment of taxes, reckoning of legal compensation, and division of inheritances according to Qur’anic law.[37] Al-Baghdadi’s Takmila, for instance, contains an example of arithmetical problems arising from Muslim religious requirements, namely the calculation of the community’s share of personal wealth: the zakat.[38] The surveys in the Encyclopaedia of Islam under the headings ‘Mirat’ (Inheritance) and ‘Faraid’ (Obligations) contain the basic stipulations of Islamic law on the topic.[39] Al-Khwarizmi  also says in the opening flourish of his book on equations:

‘Numbers and shapes which man must understand thanks to the beneficence of God the creator who has bestowed upon man the power to discover the significance of numbers.'[40]

The Islamic impact is also seized on by Wickens, who observes that:

‘Mathematics is par excellence the great field of pure theory, of elegant and economical demonstration in the abstract,' all so close to Islam’s view ‘of the rational recognition of beauty in the ordering of things.'[41]

Arabic, equally, played its part. Arabic, Arnaldez and Massignon observe, by facilitating ‘the interiorisation of thought, is particularly suited to expressing exact scientific concepts and to developing them, in much the same way as mathematical concepts have evolved historically.’[42] Stock, too, notes that Arabic is ‘inherently a good language for expressing scientific constructions-a fact of considerable importance for mathematisation.'[43]

The following outline serves to reinforce most of the points just made, especially with regard to the originality of Islamic mathematics, its pioneering aspects, and how it impacted on the Christian West and modern mathematics.