When it comes to mathematics, perhaps no pre-modern culture has influenced the development of the subject more than that of the Islamic world. While many of the revered ancient Greek mathematicians we know today (such as Archimedes and Euclid) are known to have come up with the foundational concepts of geometry, they only did so in basic arithmetic terms, and never dealt with algebra, for example as a separate subject. There is also the significant case of misattribution of mathematical concepts to the Greeks (such as Pythagoras’ theorem, which was used centuries before by the ancient Babylonians and Indians [1]) which has resulted in a significant focus on the Greeks in popular culture as the founding fathers of mathematics.
The most famous of the Islamic mathematicians was the Persian polymath Muhammad ibn-Musa al Khorazmiy (commonly misspelt as Khwarizmi, 780-850) from modern day Uzbekistan. Born in Khorazm, al-Khorazmiy moved to Baghdad as a scholar in the famous House of Wisdom, where he was involved with the study and translation of ancient Sanskrit and Greek manuscripts concerning mathematics and the sciences.
Fig.1 – a page of Al-Jabr concerning the method of completing the square.
Within the Western world, al-Khorazmiy is known as the father of algebra, especially with his famous publication Hisab al-jabr wa’al-muqabala (The Concise Book of Balancing and Restoration), with the transliteration of title being the ultimate etymology of the word itself (al-Jabr becoming algebra). Within his treatise, he initially provides a lengthy definition of the set of natural numbers using abstract terms – a completely new concept at the time which opens with the comical ‘When I consider what people want in calculating, it is generally a number’ [2] before explaining conditions for such. In terms of what we consider modern algebra, he identifies different cases of linear and quadratic equations and demonstrates the solutions using the quadratic formula (assuming the format In his book he invents the idea of rearranging the subject through ‘balancing’ and ‘restoring’ – the process of taking certain quantities away or adding certain quantities to both sides of an equation. While a seemingly elementary concept now, it was a huge innovation and has led to the publication becoming a landmark work in mathematical history.
However, al-Khorazmiy’s work had some problems – namely due to it being limited to natural numbers. He intended the algebra within al-Jabr to be used concerning practical applications considering property tax and inheritance [3], so failed to consider more complex cases such as fractions, and negative numbers. As a result, the Egyptian mathematician Abu Kamil Shuja’ (850-930) developed his work, being the first mathematician to include irrational numbers and surds as solutions to quadratic equations. He also worked with polynomials of a much wider range than al-Khorazmiy (up to x8 rather than just x2 [4]) and his publication Kitab al-jabr wa’al muqabala (which unhelpfully has almost the exact same name as Khorazmiy’s work) provided a much-needed expansion of Khorazmiy’s ideas into different cases of numbers. Abu Kamil also provided one of the earliest methods of finding the integral of an indeterminate equation given limits, using the definition of equations given in Diophantus’s Arithmetica.
Fig.2 – the Sphere and Cylinder proof which Archimedes considered his greatest work. Ibn Sinan derived the proof without any prior knowledge of Archimedes’ works.
Ibrahim ibn Sinan (908-946) was a mathematician concerned primarily with geometry who managed to develop many of Archimedes’ proofs (such as the famous sphere and cylinder proof) independently [6]. He also developed the first general formula of integration which, unlike Archimedes’ method of exhaustion, applied to different cases of parabolas. His early death at the age of 37 meant that he was unable to develop his formula into a publication, leaving the method unfinished until the time of Leibniz and Newton.
Iraqi polymath Hasan Ibn al-Haytham (commonly misspelt from romanisation as Alhazen, 965-1030) also provided important contributions to number theory concerning prime and perfect numbers. He discovered what the Western world would later come to know as Wilson’s Theorem, which states that any number n is only prime if the product of all positive integers less than it are 1 less than a multiple of n (i.e. (n-1)! = kn-1 for kR) and also was the first to define perfect numbers in the form 2n-1(2n-1). He was well-versed in integral calculus, and used a method of finding the integral sum of the fourth power (applicable to any polynomial without providing a general formula) in order to determine the volume of a paraboloid. In geometry, ibn al-Haytham used a proof by contradiction to prove the Euclidean parallel postulate [7] (no.5 of Euclid’s Elements, assuming Euclidean geometry) and was the first to formulate a definition for the Lambert quadrilateral as a result. He also used a geometric proof in order to find the sum of the first 100 natural numbers, and provided the proof for what is known as Alhazen’s Lunes, which shows that the lunes of the two shorter sides of a right-angled triangle have an equal area to that of the triangle itself [8].
Fig.3 – Alhazen’s lunes, where the area of triangle ACB is equal to the darker shaded area of the two lunes around AC and CB.
Finally, we have the famous Persian polymath Omar Khayyam (better known for his poetry, 1048-1131), who is credited with being the first mathematician to conceive a systematic method of solving cubic equations. Using methods of geometric construction to find intersections of conics, he found solutions for cubic equations of all 14 different cases (which he self-determined by observation) using prerequisite lemmas such as Euclid’s Elements. While the work is no longer extant, he is also known to have fully developed and invented a general binomial formula (for polynomials of order n) [10] as well as what Europeans later came to know as Pascal’s Triangle.
Fig.4 – one of al-Khayyam’s proofs for a simple case of a ‘depressed cubic’ equation in form x3+a2x=b obtained by plotting the parabola y=x2/a and a circle with centre (b/2a2,0) and radius b/2a2
Considering so much significant contribution to mathematics, why is the origin of such discoveries unknown then? It can be attributed to a number of reasons, such as the 1258 Mongol sack of Baghdad, where much of the core of Islamic knowledge and research regarding the sciences was completely lost. Moreover, further European research following the Renaissance may have also had a part to play in downsizing the importance of the discoveries of the Muslims, who were viewed into the Industrial Era as uncivilised and rowdy following the decline of its empires.
References
[1] – Kahn, Charles H. (2001). “Pythagoras and the Pythagoreans: A Brief History.”
[2] – Rosen, Friedrich August (1831). “The Mathematics of Mohammed ben Musa.”
[3] – David A. King (2003). "Mathematics applied to aspects of religious ritual in Islam".
[4] – Levey, Martin (1970). "Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujāʿ"
[5] – Hartner, W. (1960). "ABŪ KĀMIL SHUDJĀʿ". Encyclopaedia of Islam. Vol. 1 (2nd ed.).
[6] – Van Brummelen, Glen (2007). "Ibrāhīm ibn Sinān ibn Thābit ibn Qurra"
[7] – Eder, Michelle (2000), “Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam”
[8] – Alsina, Claudi; Nelsen, Roger B. (2010), “Charming Proofs: A Journey into Elegant Mathematics.”
[9] – Eves, H. (1958). "Omar Khayyam's Solution of Cubic Equations"
[10] – Struik, D.J. (1958). "Omar Khayyam, mathematician"
