Islamic Mathematics

Historical background and cultural context

Muhammad ibn Abdullah was born in Mecca (modern day Saudi Arabia) in 570 CE. Around 610 he began preaching his updated vision of the Abrahamic religions (Judaism and Christianity) calling it Islam (meaning submission to the will of God). After several years of persecution and exile, he returned with an army and conquered his hometown a few years before his death in 632. Muhammad’s successors expanded the empire (caliphate) through military conquest. The speed of the Muslim conquests is shown in the picture below. At his death, the Arab peninsula was Islamic. Over the next 30 years the pink area was conquered, and by 750 the caliphate extended from Spain across north Africa through the Middle East, Persia, Afganistan and Pakistan. In later times there were serious schisms1 and several successor empires emerged, the longest-lasting of which was the Ottoman Empire (c.1300–1922). Even though centralized Islamic political control has dwindled, the religion is still dominant over the entire region shown in the map (with the notable exception of Spain and Portugal) and over a greater region of Africa and south-east Asia.

Islam had several important features when it came to education and learning. Somewhat like with the Romans, early conquered peoples were often allowed to maintain their own culture (especially if they were Jews or Christians, ‘people of the book’) albeit while acknowledging their new overlords and paying extra taxes. It was relatively easy for indigenous people to become Islamic and thereby play a full role in the empire. Many of the great Islamic thinkers were just such people, born on the periphery and travelling to the great centers of learning. Generally speaking, the Islamic conquerors did not view the heritage of the conquered as inferior or ﬂawed, and so welcomed such knowledge as they found. In particular, the knowledge that had been developed in Alexandria (and spread around the Mediterranean) and western India (modern Pakistan) was accepted and integrated in to their learning. In the mid-700’s paper-making came to the caliphate from China, greatly facilitating the dissemination and consolidation of knowledge. Schools (madrassas) were founded, in reﬂection of a strong cultural focus on learning. During the European Dark Ages (c.500–1200) between the fall of the Romans and the Renaissance, European philosophical development stagnated. Among chauvinistic European scholars, particu- larly during the colonial period, it was fashionable to credit Islam merely with the preservation of

1In particular between the Sunni and Shia branches of the faith. Much of the modern-day tension between Saudi Arabia and Iran stems from this rupture. ancient European knowledge. This is fanciful on many levels. Islamic scholars certainly played an important part in gathering and consolidating the learning of earlier civilizations: in particular much of the modern understanding of ancient Greek and Indian learning was transmitted via Islam.2 How- ever, to suggest that this is all they did is to deny their massive contributions to many areas of knowledge, and in particular, mathematics.

Mathematics To modern mathematicians, perhaps the two most important innovations of Islamic mathematics are algebra and algorithms.3

• Algebra (al-˘gabr means restoring) refers to moving a negative (deﬁcient) quantity from one side of an equation to another. For example,

x2 + 7x = 4 − 2x2 =⇒ 5x2 + 7x = 4

The word was preserved intact when later Latin scholars absorbed the idea, though its meaning has expanded enormously! It is important to note that Islamic scholars did not use symbols or equations in our sense of the words. Instead these statements were written out in sentences.

• Al-muqabala (comparing or balancing) refers to subtracting the same positive quantity from both sides of an equation. For example,

x2 + 7x = 4 + 5x =⇒ x2 + 2x = 4

This term was not preserved by Latin scholars, its meaning instead being absorbed into that of algebra.

• The concepts of proof and axiomatics were learned from Greek texts such as the Elements. Like the Greeks, they believed that only geometry could provide legitimate proofs and so set about trying to prove algebraic relations in a geometric way. Recognizing Ptolemy’s work on astronomy and trigonometry as being of the highest caliber, the Islamic scholars gave it the name by which it is now known, the Almagest (Great Work).

• While respecting the primacy of geometric proof, Islamic scholars made great advances in the ability to calculate and to solve practical mathematical problems. This in large part came from their willingness to take the best of the ideas from wherever they found them and to synthesize improvements, such as a complete development of the decimal place system.

Al-Khw¯arizm¯ıand Algorithms Perhaps the most famous and inﬂuential Islamic mathematician was Muhammad ibn Mus¯ a¯ Al- Khwarizm¯ ¯ı (780–850). He was born near the Aral Sea in modern-day Uzbekistan and tavelled to

2Indeed much of the knowledge of Chinese learning also passed through Muslim lands, even once direct communi- cation and trade had developed between China and Europe. Deciding who, or which culture, should get credit for any particular advance is extremely difﬁcult! 3When a word in English begins al- you might suspect an Arabic origin: e.g. alkali, albatross. . . Many other Arabic terms have been latinized so the etymology is less clear, e.g. elixir.

2 Baghdad to be a scholar in the House of Wisdom, the great school of learning of the time, where he eventually became the chief librarian. His treatise Al-kit¯abal-mukhtasar f¯ıhis¯abal-˘gabrwa’l-muq¯abala (the Compendious book on the calculation by restoring and balancing), published in 820, provided an instruction manual on computation. It contained algorithms demonstrating how to solve a given problem, followed by a geometry proof of each result. While much of the content is a synthesis of Babylonian methods with the Euclidean axiomatic approach its inﬂuence is hard to underestimate. Upon its translation into Latin in the 1100’s it became a standard textbook in European mathematics, displacing Euclid in places due to its greater emphasis on practical calculation. The importance of this book is preserved in the word algorithm, the Latin phrase dixit algorismi meaning Al-Kw¯arizm¯ısays.

Here is an example of how Al-Khwarizm¯ ¯ı described the solution to a quadratic equation.

Question What must be the square which, when increased by four of its own roots, amounts to 60? That is, we wish to solve the equation x2 + 4x = 60.

Solution The solution is presented as an algorithm. Observe how it may be applied to any equation of the form x2 + ax = b where a, b > 0: here a is the number of ‘roots,’ and b is the total ‘amount.’

1 • Halve the number of roots 2 = 2 a 2 1 2 • Multiply by itself = 4 a

1 2 • Add to 60 64 = 4 a + b q = 1 2 + • Take the root of this 8 4 a b x q 1 2 a • Subtract half the number of roots 6 = 4 a + b − 2

x 2 The algorithm essentially√ describes the solution using the −a+ a2+4b quadratic formula: = 2 . Proof The picture justiﬁes the result in the style of Euclid: indeed this result is Book II, Proposition 4 of the Elements. No symbols were used, but the geometry is still obvious: the square has been increased by four of its roots and the algorithm is simply ‘completing the square.’

Other algorithms were given in order to solve every type of quadratic. Al-Khwarizm¯ ¯ı is far from the only Islamic mathematician of note. For instance, Abu¯ Kamil¯ (Egypt 850–930) generalized Euclid’s geometric arguments from Book II of the Elements to allow him to substitute, as long as the resulting equation was quadratic, e.g.

1 + x 2 1 + x √ + = 1 =⇒ x = 5 3 + x 3 + x In practice, he combined several algorithms of Al-Khwarizm¯ ¯ı, each of which was based on geometry, but when combined could not straightforwardly be justiﬁed geometrically. This method of substitution was an early step towards establishing the modern primacy of number and algebra over geometry and length.

3 Perhaps the most famous Islamic mathematician of the next several centuries was Omar Khayyam (1048–1131) who produced ground-breaking work on the solution of cubic equations, astronomy, the binomial theorem, and irrational numbers. He did early work which laid the foundations for non-Euclidean geometry and the theory of irrational numbers.

Islamic Trigonometry and the Qibla Late 8th century Indian work on trigonometry, linked back to Hipparchus, was known in Baghdad, as was the work of Ptolemy. Islamic scholars were very interested in trigonometry, and not just for astronomical purposes. A primary requirement in Islam is to face the Ka’aba in the Great Mosque at Mecca when at prayer: this is known as the qibla which simply means direction in Arabic. A mosque is often built so that one wall faces Mecca for convenience. When the Islamic empire was small, determining the qibla was relatively easy. For Muslims further aﬁeld however, the curvature of the Earth starts to make determination more difﬁcult. The resolution of the qibla problem motivated scores of Islamic mathematicians for centuries and spurred much work on spherical trigonometry.

Islamic developments in trigonometry

• Borrowed the concept of half-chord (modern sine) from India and worked with circles of various radii.

• Al-Battan¯ ¯ı (c.858–929) introduced an early version of cosine as the complementary half-chord for angles less than 90°. He also worked with the function versine:4 in a circle of radius 1, this is

versin θ = 1 − cos θ

• Al-B¯ırun¯ ¯ı (973–1048) eventually deﬁned tangent, cotangent, secant and cosecant (though with- out using these Latinized words) by projecting from a gnomon (sundial) onto either a horizontal or vertical plane. In the picture, the gnomon is the vertical stick of length 1. In this deﬁnition, al-B¯ırun¯ ¯ı is moving towards the modern consideration of trigonometry which is entirely based on triangles rather than circles.

sin α crd(2α) α

1 sec α = csc β versin α 1

cos α β α α tan α = cot β

4Versed sine refers to the measurement of a length in a reversed direction (perpendicular) to that of sine

4 • Created trigonometric tables with improved accuracy over Ptolemy. By applying the half-angle formula many times and using fact that, sin(α + β) − sin α < sin α − sin(α − β) whenever5 0° < α − β < α + β < 90°, Abu¯ al-Wafa¯ (940–998) and his descendents were able to compute sine + tangent values for every minute of arc accurate to 5 sexagesimal places!

Spherical Trigonometry Ptolemy Almagest contained several calculations on spherical trigonometry and the Indians had done similar work. Ptolemy’s approach relies on Menelaus’ Theorem (c.100 AD).

Theorem (Menelaus). For the given conﬁguration of spherical triangles on a sphere of radius 1,

sin CE sin CF sin BD = · sin AE sin DF sin AB Computations are difﬁcult, as they often depend on creating many spherical triangles. Al-Wafa¯ simpliﬁed this result, providing a new starting point for the study of spherical triangles. Theorem (al-Wafa)¯ . If 4ABC and 4ADE are spherical triangles with right angles at B, D and a common acute angle α at A, then sin BC sin DE (sin α =) = sin AC sin AE We’ve written al-Wafa’s¯ result in modern language, where the sphere has radius 1. The arc-lengths BC, AC, etc., therefore equal the central angle of the corresponding arc in radians: BC = ∠BOC. Al-Wafa’s¯ version didn’t mention the sin α, hence the parentheses. Proof. Let O be the center of the sphere. Project C orthogonally onto the plane containing O, A, B to produce K. Project K orthogonally onto OA to produce L. Now consider the red planar triangle. Since α is the angle between two planes, we have α = ∠CLK. We also have

CK = sin BCCL = sin CA

The usual sine formula says

CK sin BC sin α = = CL sin AC The same ratio is obtained for triangle 4ADE.

5This is simply the downwards concavity of the sine function.

5 Corollary (Sine Rule). If a, b, c are the sides of a spherical triangle with angles A, B, C, then

sin a sin b sin c = = sin A sin B sin C Proof. Drop a perpendicular to H from C. By al-Wafa’s¯ result we have sin h sin h sin B = sin A = sin a sin b eliminating the sin h terms completes the ﬁrst equality. The rest follows by symmetry.

Al-Wafa’s¯ proof was a little more complicated, but essentially the same. He extended AB and BC until they both make quarter circles. We have right angles at D and E and have that DE is part of a great circle on the sphere with central angle B. It follows that the arc DE is sin B. Al-Wafa’s¯ theorem implies that

sin h sin B = =⇒ sin h = sin a sin B sin a sin 90° Repeating by extending AB past B and equating the sin h terms yields the result.

Finding the qibla Armed with these results, al-Wafa¯ could solve spherical triangles, though his method was rather complicated and required several auxiliary triangles.

Al-B¯ırun¯ ¯ı simplified matters with the addition of a version of the cosine rule. Here is a highly simplified version of his method applied to find the qibla from a location L on the Earth’s surface. Let M be Mecca and N the north pole. We wish to compute the angle β (the initial bearing to Mecca from L). It is assumed that we know the latitude and longitude of both L and M: for the picture, we assume that L is north of M, though the picture is symmetric so ultimately this does not matter. The difference in the longitudes of M and L is α. Since latitude is measured by angle, we lose nothing by working on a sphere of radius 1 and equating these angles with the arcs b and c: our initial data is therefore colored blue and green in the picture.6 We extend NL to Q with the same latitude as M. Similarly P is chosen to have the same latitude as L. By symmetry, the points L, P, Q, M are coplanar. The quadrilateral LPQM lies on the intersection of a plane and a sphere: a circle! Ptolemy’s Theorem applies: measured as straight lines (chords) and

6Strictly b and c are colatitudes, since latidude is measured positively north of the equator.

6 using symmetry (|PQ| = |LM|), we have

|LM|2 = |LQ|2 + |LP| |QM|

However, the arc-lengths on the sphere are related to the straight distances via the usual chord relation, for instance LM |LM| = crd LM = 2 sin 2 Putting this together, we obtain the following relation for arc-length LM b − c LP QM sin2 = sin2 + sin sin 2 2 2 2 By bisecting the angle α we obtain two pairs of right-triangles: al-Wafa tells us that

α sin LP sin QM sin = 2 = 2 2 sin c sin b whence LM b − c α sin2 = sin2 + sin2 sin c sin b (∗) 2 2 2 2 1 Corollary (Cosine Rule). Using the double- and multiple-angle formulæ for sine and cosine (sin x = 2 (1 − cos 2x) and cos(b − c) = cos b cos c + sin b sin c) quickly turns (∗) into the cosine rule for spherical triangles

cos LM = cos b cos c + cos α sin b sin c

One may now compute LM and use the sine rule sin b sin LM = sin β sin α to compute the angle β. Whew!

For fun, here is some real data: the co-ordinates of Mecca and London are 21°25’N 39°49’E and 51°30’N 8’W respectively. This corresponds to

α = 39°570, b = 68°350, c = 38°300

We therefore have

cos LM = cos 68°350 cos 38°300 + cos 39°570 sin 68°350 sin 38°300 =⇒ LM = 43.110°

With the Earth having circumference 24,900 miles, this gives a distance London → Mecca of 2981 miles. Finally we ﬁnd the qibla sin α sin b β = 180° − sin−1 = 118°590 sin LM where we subtracted from 180° since London is north of Mecca. Check it yourself at the Great Circle Mapper (the result is a little out due to slightly different initial data).