Islam & Mathematics: A Hidden History

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Islam & Mathematics: A Hidden History

This is the prepared text (and a few of the images) from a PowerPoint presentation given by Randy K. Schwartz [email protected] as a guest speaker at Macomb Community College (Macomb Co., Mich.) on 2 April 2014, funded by a “Let’s Talk About It: Muslim Journeys” grant from the National Endowment for the Humanities and the American Library Association.

oday, I’m going to talk about the historical links between Islam and mathematics. And I’ll show you that during the Middle Ages, Islam was actually a powerful motive force for investigation and discovery in math and science. The fact that Islam calls on believers to study nature as a way to understand Allah has

had world-historic significance— yet this remains unknown to most people living in the West.

Is “Muslim Science” just a fairy tale?

Most of this presentation is about learning from the more distant past, but I actually want to start by recalling an incident that happened just four years ago. It’s a good example of how Arab and Muslim contributions to math and science have been hidden from us in the Western world.

The head of NASA, Charles Bolden, was visiting Cairo, and he mentioned that Pres. Obama had told him that

NASA needed to get scientific contributions from a broader range of nations around the world, including the Muslim nations, whose people have made “an historic contribution to science, math, and engineering.”

But this was met with derision in the Western media:

“NASA Races to Reach the Crescent Moon!” [Oh, I get it. The crescent? Symbol of Islam?] “Islam’s meager contribution to human technological advancement is no accident.” (Washington Times)

We were told that the President’s talk of Muslim contributions was all just fairy tales. “Bedtime Stories for the Islamic World” said the National Review.

“‘Muslim Science’ [is] Fiction” screamed a column in the New York Post. “NASA’s Mad New Mission.”

But you know what? Maybe there’re a few things the President knows that a lot of other Americans don’t.

Because in fact, when you talk about space and astronomy, you have to talk about Arab and Muslim contributions. This is reflected in the fact that if you dig up where the names of key stars came from, you find so many of them that derive from the Arabic language. And that’s because Muslim astronomers during the Middle Ages investigated the heavens in great depth, and applied mathematics to figuring out— and predicting— how the stars and planets move in the sky.

In fact, there are about 2,000 stars visible in the Northern Hemisphere, and of the names that we use for them,

over half come from the Arabic names assigned by medieval Muslim astronomers. This is just a sampling here.

I wonder how this could be if the Muslim contributions to astronomy that Pres. Obama referred to are all just a fairy tale?

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Now of course, astronomy goes way beyond just naming and classifying the objects in the sky. The

motions there are bewildering in their complexity: some objects, the ones we call stars, seem like they’re pasted onto the celestial sphere as it turns around us, while others that we call planets and moons seem to wander all over the place, sometimes moving forward and other times doubling back in a retrograde motion. To try to figure all this out, you do need mathematics.

The Islamic classification of the sciences

In fact, the Arabs, like Plato before them, considered astronomy an actual branch of mathematics, along with geometry, arithmetic, and music, so highly mathematized were these subjects.

Mathematics was aligned with logic and ethics among the mathematical sciences, and these in turn were

classified as “sciences of the intellect”, along with physics with its many branches, as well as psychology and linguistics.

And the sciences of the intellect were distinguished from the sciences of the Qur’ān. This division gave

mathematics a more independent role. It was conceived of as being of service to religion, but not as a part of religion, as had often occurred in ancient India, for example.

Ibn al-Shatir’s model for the motion of the planet Mercury, c. 1350 AD. He was a mathematical astronomer who worked as the timekeeper at the Great Mosque of Damascus, Syria.

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That’s right, mathematics in the service of Islam. For example, Muslim scholars wanted to help determine the correct times of day for the five daily prayers required by the Qur’ān, as well as the correct prayer direction facing Mecca and the correct days of the year for such religious events as the feast of Eid al-Adha, the fasting month of Ramadan, and the season of the pilgrimage (hajj).

For this purpose, they invented the astronomical observatory as we know it today, and they applied

mathematics to the motions in the sky so as to be able— to tell time by the sun to predict the times of sunrise, sunset, and twilight to use the moon and stars to tell time during the night and to use the altitudes and azimuths of stars to gauge one’s location and direction.

Everyone knows that the altitude of a star is the angle it makes above the horizon, measured in degrees. But does anyone know what its azimuth is? It’s the compass direction of the star, relative to either North or South on the horizon. So on the celestial sphere, altitude and azimuth are like the latitude and longitude, respectively.

Much of mathematical astronomy would be distilled into a zij, which was a handbook consisting of dozens and dozens of trigonometric tables that were of use for such practical calculations. It would also include explanations of the mathematical techniques that were used to calculate the entries in the tables, which included not only basic arithmetic but things like trigonometry, methods of estimation, linear and quadratic interpolation, etc.

The ancient Greek models of the planets were based on the idea that all heavenly bodies must move in

perfect circles. The Muslim astronomers investigated to see whether this theory was adequate to account for the complexities of planetary movements. In this manuscript, the Persian mathematician Nasīr al-Dīn al-Tūsī proved that linear (straight-line) motion can be produced by a couple of spheres, one sphere rolling inside a larger sphere of exactly twice the size, a device now known as the Tūsī couple or a hypocycloid of two

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cusps. See how, if the red point is imagined as a body lying on the surface of the blue sphere, it can be made to move along a straight line as a result of an addition, or composition, of two different circular motions: the blue sphere is rotating around its own center, and its center is revolving around the center of the black sphere. This was an important building block that was incorporated into the geocentric model of Ibn al-Shatir in Damascus and eventually into Copernicus’s work on a heliocentric model.

So those are a few of the Muslim contributions to mathematical astronomy. As I see it, there are two main reasons why a low awareness of these contributions has persisted in

America and Europe. First, there’s been a shortage of knowledge and scholarship about the global history of mathematics. Many historical works have been lost or not fully studied, so that even the Arabs have not been fully aware of their own history. Second, because of a climate of Eurocentrism, many scholars in the West have adopted a state of willful ignorance concerning what has been known for a long time about the relationship between Islam and science.

The algorithmic approach

What I want to do now is to delve into some of these breakthroughs in more depth. Let’s look at six examples from some of the most basic areas of mathematics: arithmetic, algebra, combinatorics, geometry, the theory of polynomials, and spherical trigonometry.

This page from a zīj of Ibn al-Shatir is from a table showing positions of the Sun, moon, and planets as viewed from Cairo. Bibliothèque nationale Française, Paris, MS Arabe 2522, folio 66v

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When it comes to Hindu-Arabic numerals, a lot of people make a big deal about the invention of zero, but they’re missing the much more important innovation of place value. The idea is to “code” a number by letting position represent a value like 10 or 100; that way, a small number of symbols could be used to concisely write any number. The zero wasn’t even necessary or important except for its use in place-value numeration, which the Arabs borrowed from the Babylonians (base 60) and the Indians (base 10).

Once numbers were coded into columns in that way, it opened up a whole world of algorithmic

calculation that would have been too difficult to carry out without it. For example, al-Kashi invented this algorithm for extracting the square root of any number. Until electronic calculators came on the scene in the 1970s, this method was still being taught to many middle-school pupils in the United States, although where the technique came from was rarely known or mentioned by teachers. Al-Kashi was a native Persian speaker born in 1379 in a desert town in what is now Iran. After learning Arabic and making a name for himself in astronomy, he moved northward to Samarqand, in what is now Uzbekistan. Besides square roots, al-Kashi devised algorithms for extracting cube roots as well as fourth and fifth roots, too.

By the way, the reason we call a root a “root,” and speak of “extracting it,” is because the Arabs called it

al-jathr, their word for a carrot or other root. In a metaphor that they borrowed from India, they thought of the number as something useful that was hidden underground, and it had to be pulled up or “extracted” by difficult methods like this one. This was translated into Latin as radix, which is cognate with our English words “radish,” “radical,” and “root” itself. So all of these words share a common— “root”!

The algebra of Qur’ānic inheritance

The fourth chapter of the Qur’ān sets forth some elaborate rules regarding the inheritance of wealth, rules that tended to safeguard the status of women and children. When a man died, first his debts were to be paid, along with any bequests to “strangers,” meaning nonrelatives. The remainder of the estate was then to be divided up among the surviving brothers, sisters, widows and children according to certain prescribed ratios. For example, if a man died leaving two sisters and a wife, each of the 3 women was to receive 1/3 of the remainder of the estate after payment of debts and bequests.

In what your wives leave, your share is a half, if they leave no child; but if they leave a child, ye get a fourth; after payment of legacies and debts. In what ye leave, their share is a fourth, if ye leave no child; but if ye leave a child, they get an eighth; after payment of legacies and debts. If the man or woman whose inheritance is in question, has left neither ascendants nor descendants, but has left a brother or a sister, each one of the two gets a sixth; but if more than two, they share in a third; after payment of legacies and debts; so that no loss is caused (to any one). Thus is it ordained by Allah; and Allah is All-knowing, Most Forbearing.

— Qur’ān, surah 4, verse 12

In a very famous book on algebra written by Muhammad al-Khuwārizmi in the early 800s, roughly half of it is devoted to a discussion of “story problems” based on such division of estates. Al-Khuwārizmi, whose name is where we got the word “algorithm,” was from a town in what is now Uzbekistan. He had moved to Baghdād, which was the capital of the most important Islamic kingdom, or caliphate, in the Near East. He worked at one of the Houses of Wisdom, which were libraries and research centers supported by the caliph’s treasury and located in what is now Iraq. Scholars speaking many different languages arrived there from throughout the Middle East, communicating with each other in Arabic.

Suppose, writes al-Khuwārizmi, that a man dies leaving two sisters and a wife and has willed that a

certain stranger be given a share equal to the difference between each woman’s share and 1/8 of their total share. He asks, What should be the stranger’s share?

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You can see that it’s a circular problem, because the stranger’s share is specified to depend on the

women’s shares, but the women’s shares are formed from what’s left over after the stranger gets his share. This is a problem that really lends itself to algebra, because neither share is known, but we can use symbols to express the relation between the unknowns. Using modern notation, we would write the unknown quantity as x. The Arabs simply wrote this with the word shai, meaning “thing.”

At one point, we have to combine like terms by joining x to 5/24ths of x. Notice the name that al-

Khuwārizmi uses for that operation, al-jabr, which literally meant restoring a broken bone by joining its two pieces together. The novelty of this step is seen in the need to adapt an older word for it dealing with bone-setting, and also in the fact that it became the name for an entire branch of mathematics, “algebra.” The ancient Greeks had never done anything like this: setting up equations containing unknown quantities, and manipulating them with the same operations as the known quantities, which is the most direct, efficient way to solve a circular problem like this. This essential method of algebra was invented by Muslims, probably in Baghdād in the 700s.

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1 share s'each woman share sstranger'

So the stranger’s share x works out to be 5/29ths of the estate. And we can check this… But again, what’s more important than the answer is the fact that whole new techniques of arithmetic and algebra were put to use in solving such problems.

The sacred language

Since a lot of you in the audience are students taking statistics, I wanted to be sure to include the example of combinatorics, which is an important component of probability theory. Combinatorics figures out how many different ways there are to build a structure from component parts. Scholars in Islam made great advances in this field. One of the things that motivated them was that they asked, How many different words (of various lengths) can be formed by combining any of the 28 letters of the Arabic alphabet?

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Because the Qur’ān is written in Arabic, that language, and especially the letters of script needed to write it, took on a sacred character for Muslims. And this was a big motivation for their research on combinatorics. It’s also the main reason that calligraphy became a leading art in the Middle East.

So it’s funny that in Europe, winning at gambling— in games of dice and cards— would be the main

motivation for developing combinatorics and probability, especially by French aristocrats, whereas in the Middle East and India the motivation was not gambling but religion!

Now the phonetic and linguistic rules for combining Arabic letters into words are complex, so

mathematicians like Ibn Mun‘im started by solving easier problems, which they used to discover and explain the basic formulas. Ibn Mun‘im lived in Marrakech, Morocco, around the year 1200. He asked, How many different types of tassel can be made from 3 out of 10 colors of silk thread?

He already knew the formula for 2 out of n colors. You take the big number, n, times its predecessor,

n – 1, and divide by 2. So for example, for 2 out of 10 colors, the number of different combinations is 10 times 9 divided by 2, or 45.

Now here’s where he got really clever! He said, to pick three colors out of 10, I’ll always name first

which of my three colors is highest in the list of 10, and then I’ll chose any 2 of the colors below it. This gives us all the different ways of picking 3 out of 10 colors. It allowed Ibn Mun‘im to derive a formula for 10C3 by adding up a bunch of nC2 calculations.

And using the same logic, he showed that the pattern continues. Ibn Mun‘im calculated the different ways

of picking four out of 10 colors by adding up a bunch of nC3 calculations, and so on and so forth. In this way he developed general rules for how to calculate combinations.

And he summarized these results in an arithmetical triangle, equivalent to what we call Pascal’s Triangle,

which was developed in France over four centuries later. I translated his triangle into Western script for use by students in two of our courses. So, that’s an example of what the Muslims did in combinatorics.

The arithmetical triangle of Ibn Mun‘im

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Al-Kashi as “the first modern mathematician”

In geometry, the Arabs built firmly on the great foundations laid by Euclid, Archimedes, and the other ancient Greek geometers. But they carried things further. Part of their motivation was for the design and ornamentation of mosques and other public structures. A single word, al-hindasah, came to mean both “geometry,” “architecture,” and “engineering.”

One of the masters of al-hindasah was al-Kashi in Samarqand, whom I’ve mentioned before with his

square root algorithm. Samarqand was the capital of a Mongol dynasty, and the emperor there, who was himself a renowned astronomer, appointed al-Kashi as one of several full-time astronomers and mathematicians in his court.

Among al-Kashi’s specialties was the design of archways, vaults, and domes, so prominent in Islamic

architecture. He developed highly accurate ways to compute the surface area and other dimensions of various types of domes. Now let me ask you this: Can anyone think of a practical reason why it would’ve been important to know the surface area of a dome? Well, for example, to calculate the amount of materials needed, or to calculate the pay for craftsmen. If the dome were covered with, say, intricate tilework, the artisan needed to be paid accordingly.

Al-Kashi was a genius in the use of the muqarnaṣ, the design element that became known as the stalactite

vault in the West. It allows a ceiling or overhang, such as underneath a dome, to be transformed in its cross-sectional shape as it rises from lower to higher levels. Look at these other examples.

At a conference that I attended in Marrakech, the mathematician Yvonne Dold-Samplonius described how she and her colleagues at the University of Heidelberg, in Germany, have spent several years studying al-Kashi’s manuscripts and reconstructing the geometry that he used to design the muqarnaṣ. His first step was to break the dome interior into constituent parts. This determines what cross-sectional shape the dome interior will have at various levels. In this example from Iran, a dome lined with tiers of muqarnaṣ in eight-fold symmetry allows a square floor to be gracefully capped with a circular dome. The handful of carefully designed constituent parts are combined in various ways to create the three-dimensional design for the dome interior.

muqarnaṣ

at Masjid-i-Shykh, Isfahan, Iran

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Prof. Dold-Samplonius has called al-Kashi “the first modern mathematician,” because he was the first to

make a concerted effort to optimize his solutions to problems, for example by inventing iterative algorithms where he calculated and controlled the maximum amount of error at each step. Certain types of equations, such as cubic polynomials and trigonometric equations, were too difficult to solve exactly, but he figured out how to solve them by a method of approximation now known as fixed-point iteration, which is still a very important approach used today.

The muqarnaṣ technique arose in the 10th Century in northeastern Iran, and in the early 11th Century in

Algeria. The stalactite design is believed to be a symbol for the Cave of Ḥirā’ where Muhammad received his first revelation.

In the construction of the 14th-Century palace at the Alhambra in Granada, Spain, designers found

ingenious ways to tile the palace walls with colorful mosaics having pleasing symmetries of various kinds, like rotations, reflections, and repetitions. The mosaic in this alcove combines horizontal and vertical repetition with a rotational symmetry of 60°.

It turns out that there are exactly 17 different ways to combine different types of symmetry in tiling a wall. At a conference that I attended in Granada, Rafael Pérez-Gómez showed us that the walls of the Alhambra contain examples of all 17 types. This was an unprecedented intellectual achievement. At no other cultural site in the world are more than a handful of these design symmetries to be found.

Images by Rafael Pérez-Gómez, Univ. of Granada

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The impact of al-tawhīd

Although they got help from mathematicians, these artists used geometric patterns not to elaborate a mathematical theory but as a way to express the central Islamic belief in al-tawhīd, which means “oneness,” “unification,” or “unity within multiplicity.” According to this monotheistic doctrine, the entire universe in all its multiplicity is stamped by a single God that permeates everything. They thought that studying any aspect of the universe was, quite literally, a way to contemplate God. And it was the duty of every Muslim to pursue such knowledge.

In addition, Islam taught that to discover the laws of the universe, you have to go beyond visible

appearances in order to grasp the underlying, more abstract structure of the world. Geometry and mathematics were viewed as crucial in training the intellect to do this by moving freely between the concrete and the abstract. Mathematics became known as “the first study” following the spread of al-Kindi’s 9th-Century treatise (I love this title:) “In that Philosophy Cannot Be Attained Except by Way of Mathematics.” In modern times we try to boast that “Mathematics is the Queen of the Sciences”, but according to this, even philosophy takes its leadership from mathematics!

The 14th-Century historian Ibn Khaldūn summed up the Islamic view of geometry by writing, “It should

be known that geometry enlightens the intellect and sets one’s mind right. All its proofs are very clear and orderly… Our teachers used to say that one’s application to geometry does to the mind what soap does to a garment. It washes off stains and cleanses it of grease and dirt.”

The doctrine of unity also affected art and design. Muslims believed that if an artist portrays a human or

an animal, it’s almost a sacrilege because it usurps the role of God as Creator; and even portraying a single inanimate object is frowned upon because it downplays the interconnectedness of all things. Instead, designers used the abstract geometry of repeated and linked figures to suggest this interconnectedness. This impulse toward geometry affected everything, including art and architecture.

The doctrine of al-tawhīd, the underlying unity of all things, was a comprehensive outlook on society and

nature, and a powerful impulse to uncover knowledge from wherever it was hidden. It encouraged the embrace of all knowledge and all people.

Relation between theory and practice

Another important way that things were taken far beyond Greek science and mathematics was in the realm of experiment, and the interplay between theory and practice.

Let’s take an example. Here’s a typical debate from ancient Greece: Everyone knew that light is

necessary for vision. But when we see an object, is it because light has traveled from the object to our eyes, or from our eyes to the object? They didn’t know in ancient times. Euclid, Ptolemy, and Aristotle were all part of this debate, and the starting point was always axioms and arguments written on papyrus, as opposed to physical experimentation. Finally in Cairo around the year 1000, the mathematician Ibn al-Haytham did a whole series of experiments, with darkrooms, candles, pinholes, and other devices, to prove the point that light rays enter the eye from the outside. And he went on to work out a whole theory of light refraction in the eyeball, as well as his more famous theory of reflection, which he used to solve a now-classic problem known by his Latinized name, Alhazen. Ibn al-Haytham’s works based on actual observation represent the very birth of the experimental method in science. His approach was then translated into Latin and taken up centuries later by Roger Bacon, Francis Bacon, and Galileo.

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The rainbow is another great example of how Muslims combined mathematics and physical experiment to make discoveries in science. Where do rainbows come from? and why is a second rainbow sometimes visible above the first, with its colors in reverse order, from blue above to red below? Ibn al-Haytham had shown the reflective and refractive properties of spheres. Building on this, the scientist Kamaladdin al-Farisi carried out some experiments in the early 1300s in which he turned his back to the sun and held up a spherical glass vessel filled with water, to represent a single raindrop. As he raised the vessel, he observed light beams, in a succession of colors from blue through red, emerging from the vessel at an angle roughly 42º from the incoming sunlight. The same thing occurred again at about 52º, but with less brightness and with the colors in reverse order.

Based on these observations, al-Farisi worked out the geometry that explains the double rainbow. The

incoming ray is first refracted, or bent at a predictable angle, at the surface of each raindrop, much as a glass prism refracts sunlight to make the color spectrum visible. The rays then reflect, or bounce, off the far wall of the drop, again at a predictable angle. Unlike refraction, however, reflection reverses the order of the colors. The rays refract one last time as they emerge from the raindrop. Other sunbeams are able to ricochet twice before emerging from the drop. The extra reflection reverses the order of the colors one last time. In this way, al-Farisi was able to explain why there’s a second, fainter rainbow at a higher angle in the sky than the first. This also explains why the color order is reversed between the primary and secondary rainbow.

Primary rainbow

42º

Secondary rainbow 52º

Angle discrepancies are exaggerated here for the sake of clarity.

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Steps toward calculus Advances were also made in solid geometry, the geometry of three-dimensional space. Within Greek culture,

mathematicians like Pappus, who lived in the Greek colony of Alexandria, Egypt, around the year 300, had studied a whole series of geometry problems of this type: Take a line segment 6 feet long, and cut it into two pieces in such a way that the length of the first piece times the square of the length of the second piece equals a specified volume.

See how, as you cut the line at different points, the volume fluctuates. If you make the first piece 4 feet and

the second piece 2 feet long, you would get a box 4 by 2 by 2 feet, which is 16 cubic feet. But if you cut it into pieces both 3 feet long, you can make a box 3 by 3 by 3 feet, which is 27 cubic feet, much bigger.

There’s enough flexibility in the problem that you can achieve a wide range of volumes by cutting the line at

the right place, and people like Pappus figured out how to do this. Of course, you can’t get every volume out of it: clearly there’s no way you can get a volume of 1000 cubic feet using a line that’s only 6 feet long! So the type of question that was asked by Middle Eastern scholars, and the way they took things much further, is that they wondered: What’s the biggest volume you can get like this from a line 6 feet long? The way they figured this out was that they translated the geometry problem into the language of algebra that they’d invented. This allowed them to write the volume of the box as a function of the length of the first piece— I’ve written the function in red there. If you graph that function, you can literally see the high point that gives you the highest volume, but even that’s useless to us unless we can calculate its coordinates.

That brilliant stroke came in the year 1209 in Baghdād by a mathematician from Persia, Sharaf al-Dīn al-

Tūsī— he’s a different Tūsī than the Nasīr al-Dīn al-Tūsī that we met earlier; from the same hometown, but a different century. Sharaf al-Dīn al-Tūsī discovered that to find the coordinates of the high point, or optimal point, all we have to do is solve an easier equation derived from the original one, what we today call the derivative. Solving this equation tells us to make the first piece 2 feet long and the second one 4 feet, giving the optimal volume of 2 by 4 by 4 feet, or 32 cubic feet.

But more important than any single answer like this is the new method used to figure it out. The Muslims

built on Greek geometry, but in doing so they invented whole new methods, even whole new branches of mathematics.

Praying toward Mecca

Another example of this was the question of determining the qibla, or prayer direction to Mecca from any point on the earth’s surface. As you know, Muslims everywhere on the planet are supposed to pray five times daily toward the holy city. The urge to determine the qibla precisely was a major stimulus for Muslim scholars to make an intensive study of spherical trigonometry, since they knew that the Earth’s surface basically is a sphere.

Although it might seem trivial at first, this is not a trivial problem, and its solution is still used in airplane

navigation today. Let’s take Macomb Community College as an example. Let’s say you want to pray facing toward Mecca. Mecca lies south and east of us. What would you say: is the shortest, most direct path going to be toward the south and east?

If you’ve ever flown by plane across the Atlantic, you know that the shortest, most direct path between the

two airports is along a great circle of the Earth, which actually takes the plane well to the north of either airport. Although Mecca does lie south and east of us, the correct prayer direction is not toward the southeast but toward the northeast. The correct direction from Macomb turns out to be about 38 degrees North of due East.

So, the Muslims defined the problem of calculating the qibla as a matter of finding the compass direction

from any point Z to any point M, measuring that direction from due south. The four values that are available for the calculation are the latitudes and longitudes of the two points.

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The trigonometry of the sphere turns out to be much more difficult than that of the plane. Even if you impose the simplifying assumption that all three “sides” of a triangle lie on great circles, weird things can happen: for instance, the three interior angles do not in general add up to 180 degrees like they do on a flat plane. Instead, they can add up to almost anything! Consider this triangle on the earth’s surface with vertices at the North Pole, the point on the Equator at 0 degrees West, and the point 90 degrees due West of that. What do the angles add up to? That’s right, 90 + 90 + 90 is 270 degrees. That’s a lot more than 180, isn’t it!

One of the ways to deal with these difficulties is that instead of measuring the length of an arc, you measure

what angle the arc represents. Like, in this example, each side takes up 90 degrees of the circle that it lies on. Since the sides represent angles, too, we see formulae where trig functions are applied to both the interior angles and sides of triangles.

To solve problems like that of the qibla, the Muslims had to discover and prove laws for spherical triangles

that corresponded to the known laws for planar triangles. For example, they figured out spherical versions of the Pythagorean Theorem, the Law of Sines, and the Law of Cosines. This went way beyond what had ever been done in Greece or in India. The Greeks and Hindus had certainly made advances in trigonometry, but they had never developed a theory of angles on the surface of a sphere, and this was another big breakthrough.

Entire treatises were written to address the qibla problem. Several different types of solutions were found. In

Afghanistan, for example, al-Bīrūnī developed a solution based on the Spherical Law of Sines and the other theorems of spherical trigonometry. By the way, al-Bīrūnī was famous for his appreciation of other religions and cultures, including those of Hindus and Jews. He begins his discussion of the qibla problem by noting that the same mathematics will be useful to people of religious faiths other than Islam; for example, he writes, Jews are instructed to pray toward Jerusalem.

These solutions are way too complex to go through here in any detail, but I just want to point out how clever

these solutions were. The Muslims reformulated the problem as a problem in astronomy. They imagined a star situated directly over Mecca; the problem of finding the direction to Mecca itself is equivalent to finding the direction, or azimuth, to this imaginary star located at the zenith of Mecca. By reformulating the qibla problem as a problem on the celestial sphere, al-Bīrūnī and others were able to make use of techniques that Muslims and others had already developed to solve problems in astronomy.

Al-Bīrūnī’s strategy consisted of three uses of the Spherical Law of Sines and one use of the Spherical Rule

of Four Quantities. This resulted in a four-step algorithm, corresponding to the four triangles that are shaded here. So that was al-Bīrūnī’s strategy, which was one of several types of solution to this difficult problem of the qibla.

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Al-Bīrūnī is one of dozens of examples of important mathematicians and scientists who lived under Islam but were either non-Arab, non-Muslim, or both. This chart that I made shows a few other examples, representing several different native languages and religions. It’s important to note that the parts of the Muslim world that had the greatest cultural mixtures were the ones that were most fruitful in math and science.

Conclusion

Let me sum up now the three main points that I’m hoping you got from my presentation. The first is that people from Middle Eastern cultures have made huge contributions in mathematics. They

broke whole new ground in fields like higher arithmetic (such as the root-extraction algorithms), algebra, geometry, trigonometry, combinatorics, and the scientific method. Far from simply preserving and transmitting ancient Greek and Indian learning, the Arabs and Muslims richly extended these and pushed mathematics in whole new directions.

The second point is that Islam itself was actually a facilitator, a motive force, an engine, in driving

forward this technical creativity. Its doctrine of al-tawhīd, or unity within multiplicity, encouraged people to study every facet of the universe as a way to better know God and to submit to His design and will.

The third point is that we should value cultural diversity as an actual treasure, because knowledge and

creativity are advanced best when the ideas and contributions of all people are brought together and allowed to cross-pollinate.

“The history of science is a long river. No people can claim all the waters for themselves.” — Franz Gnädinger, scholar of ancient Egyptian mathematics, Zürich, Switzerland