## "Arabic" Numerals or Roman Numerals?

By the time the Islamic Empire was fully formed, during the Umayyad period (661‒750), there were only three major forms of numerals for calculators and record keepers to employ. There was of course the old Roman numeral system that is still familiar to us from the prefatory pagination of books that usually employ those numerals on the pages preceding the actual contents of the book, and on monumental inscribed dates and the like that appear on institutional buildings, governmental or otherwise. This Roman numeral system used a mixed arrangement of so-called positional and non-positional place notation, and it used more than one symbol to represent the same numeral. For the symbols themselves, the more common convention was to use capital letters, such as V, X, L, C, D, and M, to denote the numbers 5, 10, 50, 100, 500, and 1,000 respectively, and their values depended more on the letters themselves than on the place those letters occupied. The letters D and M were sometimes represented by "I" followed by an inverted "C" to designate "500" and "CI" followed by the same inverted "C" to designate 1,000 respectively. But the shapes of the letters themselves, although they had a numerical value of their own, nevertheless did not determine the actual value of the number. That value depended also on the place the letter/number occupied with respect to the others surrounding it. In sequences such as LX, the number "10" is added to get 50+10 = 60, while in XL the same number "10" is subtracted to make 50-10 = 40. Thus the positional meaning of the number "X" = "10" makes a great difference to the actual value of the number, depending on whether the letter/number "X" was placed to the right or to the left of the number "L". This is analogous, but not identical, to our more familiar place-value notation that allows us to designate "36" by adding "6" to "30" when "6" is placed to the right of "30," and to produce a completely different number "63", when the "6" is placed to the left of the "3."

Although this rule of adding or subtracting Roman numerals according to their placement looks simple enough, it was not followed in all instances. Moreover, there were also different conventions for representing the same number, as with the number "90," for example, in the form LXXXX, or XC, with no strict rule as to which was preferred.

Furthermore, this Roman numeral system did not have a concept or symbol for "zero," which is indeed the place marker par excellence. Think of the difference between such numbers as "01" and "10" in order to appreciate the importance of the placement of the number "1" with respect to "0." This omission holds even though both the concept as well as the marker for a "zero" were already developed in Mesopotamia around the third century BC, and could have been used by the Romans had they wished to do so. The Mesopotamian concept of the "zero" seems to have traveled south though, instead of west, and reached the Indian subcontinent, before it came back to the Euphrates area to be incorporated together with the Indian numerical system in what is now called the system of "Arabic numerals."

Then there was the alphabetically numerical system that was used in the Hellenistic world, namely, the system that assigned various values to the letters of the alphabet starting with α = 1, β = 2, and so on, and rendering numbers such as κδ = 24, without any place-value notation. That is, if the letters in the last number switch around to δκ, the result is not a new number, but a meaningless succession of alphabetic letters. The Hellenistic system, however, had a marker for the zero position, and in some of the surviving papyrus fragments that symbol looks very similar to our modern-day common division sign "÷." And once that zero symbol was developed in Hellenistic times, one can even argue that its form persisted, with slight modifications and was transmitted to the Islamic culture when the latter came into being. The forms for the zero symbol in the analogous Arabic alphabetic numerical system, which were used mainly in astronomical texts, could be traced back to forms that were direct descendants of the zero symbol that was used in the papyri.

The Arabic alphabetic sign for zero seems to have developed from a quickly drawn line with a circle underneath it, exactly as it was drawn in the papyri. Other variations of this symbol were also known in the relatively rare astronomical tables.

The Indian civilization also developed a numerical system of its own. It seems to have borrowed the concept of a zero place holder from the ancient Mesopotamian civilization that had also developed, in addition to the zero concept and sign, its own numerical sexagesimal system written in cuneiform signs. Vestiges of the latter are still with us to this day, as we count every 60 seconds to make a minute and every 60 minutes to make an hour, or a degree, and so on. The Indian numerical system was not in cuneiform and seems to have developed independently in India without any outside influences. This Indian system appears to have migrated north, and there are written documents that testify to its presence in the upper Euphrates area at least as early as 661, the time when the Islamic Umayyad dynasty assumed power.

The essence of the Indian numerical system is to employ only nine signs, and to attach to those signs a place value, meaning that a sign gains significance both by virtue of its own shape, and by the position it occupies with respect to the other surrounding signs. Thus although "31" is greater than "23", when the numbers "1" and "3" switch place, the result is "13," and no longer greater than "23."

Place shifts also occurred every ten digits, giving rise to the system being called the decimal system. Starting with the regular counting procedure, 1, 2, 3, and so on, the nine signs are soon exhausted. At that point, the number "1" is moved by one place and a "zero" is placed in the empty place just vacated thus marking the number ten "10." The process is repeated until all the tens are exhausted, and then the first digit is moved again, this time two places, and the other places are left empty and marked with "zeros" to mark one hundred "100," and so on. To echo the words of the seventh-century Syriac-writing scientist from the Upper Euphrates area, and from whose work we know of the spread of the Indian numerical system outside India around the year 661, the date of the document written by this scientist, with this system the Indians were capable of writing any number they pleased by using this finite number of nine signs and zero.

In contrast to the previous systems that were known in the ancient Middle East, around the time when the Islamic civilization was taking its formative steps, the Indian system quickly gained ascendancy, chiefly on account of its simplicity. Since then, and thanks to the catalytic role played by the Islamic civilization in propagating this system, its reach has now become almost universal.

But the advantages of the Indian numerical system were not only limited to its representational features of any number―it was also easier to manipulate in actual computational situations, because of its reliance on place value. It could also represent fractions with equal simplicity, a feature not as easily achievable in the Roman system just mentioned.

Having encountered all three systems in the same area where the Islamic civilization was to define its geographic domain, scientists of that nascent civilization seem to have opted to disregard the Roman numerical system, already discarded by the earlier Hellenistic scientists, and adopted instead both the Indian numerical system as well as an Arabized version of the Hellenistic system. And of the two adopted systems, it was natural that the class of merchants, shopkeepers, and grocers would opt for the simpler Indian numerical system. The more advanced scientists, such as astronomers, astrologers, and their ilk, adopted the alphabetized Arabized version of the Hellenistic system. As early as the ninth century, when advanced Greek scientific texts began to be translated into Arabic, their Arabic versions always included the Arabized version of the Hellenistic system.

The Indian numerical system remained much the most commonly used system among lower government bureaucrats, shopkeepers, and so on. As a result, a new genre of writing began to appear and was quickly designated as belonging to al-hisab al-hindi (Indian arithmetic). In books of this genre, there is not one set of nine signs but two, exactly as was the case in the Indian sources. One of those sets is commonly called "Arabic numerals" nowadays, which is mostly documented in sources originating in the western domain of the Islamic world and in Latin sources, while the other set is the one still used to this day in the eastern Islamic world and properly called "Indian numerals."

One of the most famous Arabic works that was written in the genre of Indian arithmetic by a famous early ninth-century mathematician, geographer, and astronomer by the name of Muhammad ibn Musa al-Khuwarizmi (active 813‒46) was later translated into Latin during the Middle Ages. It was this particular work, now no longer extant in the original Arabic as far as we know, which played a crucial role in transmitting the Indian numerical system to the Latin West, but in the form that is now called the "Arabic numeral" form. Once it appeared in one Latin book, it was easy to see why the Indian numerical system would quickly spread to cover the rest of Europe. But we can document that this spread did not come without struggle. There were edicts in European centers forbidding the use of the new Indian numerical system, especially in areas where people were already used to the Roman numerical system. And that should not be surprising, as cultures hold tenaciously sometimes to ideas and concepts that have already outlived their utility. This truth is evident in the persistence of the use of the old English system of inches and feet in a few countries, concurrently at times and to the exclusion, at others, of the much simpler metric system that is now used almost exclusively in scientific circles.

The fact that the Latin world went through a period of resisting the introduction of the Arabic numerals can be beautifully illustrated in the following example. It is well known that Arabic, like most other Semitic languages, is written and read from right to left, while the Latin language and its derivatives are written and read in the opposite left-to-right direction. Thus in Arabic manuscripts containing descriptions of the Indian numerals, in both their forms, the Indian form as they are called in the eastern Islamic world and the western form that is called "Arabic numerals" in Europe, those numerals are written in the same direction as the language, that is from right to left.

Similarly, the arithmetical operations in works written in Arabic are also carried out from right to left, arranging the ascending powers of ten again from right to left as is still universally done up to this day. In contrast, in early Latin, English, and French manuscripts of about the 12th century that dealt with the art of calculation, or what we would now call books on arithmetic, one notes the appearance of the numerical listings of the same nine signs of "Arabic numerals" still written from right to left, despite the fact that the rest of the prose in the text is supposed to read from left to right as the western languages dictate. Only in later centuries, and in some instances as late as 1475, one begins to find Arabic numerals listed from left to right in manuscripts and printed texts, a clear indication that "Arabic numerals" were finally assimilated by the Latin West.

One should note however that this assimilation was not complete, or rather too complete, depending on the direction of the assimilation. Every time we embark on balancing our checkbook at the end of each month we still arrange the units to the right of the tens, and the tens to the right of the hundreds, and so on as we stated above. In essence, we still follow the same order that was followed in medieval Arabic manuscripts and arrange the ascending powers of ten from right to left, just as Arabic writers had done for centuries and still do today. Any mathematician could easily teach us how to rearrange those arithmetical operations so that the ascending powers of ten would be lined up from left to right, and yet this rearrangement has never been done to this day.

This phenomenon indicates not only the persistence of those Arabic numerals and the arithmetic operations involving them, but also that their very simplicity must have been mainly responsible for their longevity. Once those numerals were assimilated into the Latin culture they became widely used and must have become highly appreciated even in areas that were reserved for alphabetical numbers in the Arabic texts. In the Islamic tradition most Arabic astronomical tables either used only alphabetic numerals throughout the long history of their civilization or allowed only a limited number of tables to contain Indian numerals. The other non-astronomical texts and documents, commercial and otherwise, would systematically use the "Indian numerals." But in the West we find that even the astronomical tables used only the "Arabic numerals" to the exclusion of all other forms.

The role of Islamic civilization in the history of the "Arabic numerals" was not strictly speaking restricted to propagating those numerals to the West and from there to the rest of the world. It went beyond that. The historical evidence demonstrates through a manuscript that was written in Damascus in the year 952 that a mathematician by the name of al-Uqlidisi (i.e., someone who probably taught the book of Uqlidis, the Arabized form of Euclid's name) had gone beyond the Greek and Indian traditions of numbers. While both of these earlier traditions did not consider fractions as numbers per se, but thought of them as ratios between two geometric magnitudes as the Greeks certainly did, Ahmad ibn Ibrahim al-Uqlidisi seems to have made the big conceptual jump and incorporated the fractions within the real number system―to be manipulated in cases of addition, subtraction, multiplication, and division like any other numbers. And in order to do that, he had to invent a technique that would still separate the fractional parts of a number from the units, which he did with a small super-linear marker.

That marker shows al-Uqlidisi to have been the inventor of the decimal point. Of course, the concept of decimal fractions and the development point marker became more sophisticated during the 11th and 12th centuries within the Islamic world and were then bequeathed to Latin Europe together with other concepts that also spread widely. Thanks to this particular development we can now watch even young school children carry out the operation of multiplying 2.36 x 3.74 = 8.8264, without experiencing any particular difficulty, thus performing an arithmetical feat the most sophisticated minds of classical Greece could not achieve.

The introduction of the numerals, then, together with the arithmetical and new conceptual techniques of manipulating numbers, constituted a major Islamic bequest to the Latin West, and even to the present day, of unparalleled proportion. The use of those numerals is so ubiquitous nowadays that no one could even stop to think of asking the question: what would a modern scientist do if she or he were to record all of her or his lab results in Roman numerals, or Greek alphabetic numbers? Some would even argue that without the appropriation of those ideas from the Islamic world, and their incorporation especially in such areas as commercial arithmetic, double-entry bookkeeping, and the like, the very concept of Western capitalism and free markets would not have arisen in Renaissance Europe, or at least would not have reached the proportions it has reached today.

Algebra or Arithmetic?

The very English word "algebra" derives from Arabic al-jabr (compulsion), which means, among other things, something like "compelling," "forcing out," or “coercing." The word designating the mathematical discipline with the same name was first coined in ninth-century Baghdad by the famous mathematician, astronomer, and geographer, Muhammad ibn Musa al-Khuwarizmi, whose name gave us the other English word, algorithm. The world al-jabr was introduced by Khuwarizmi in his own book, described in the introduction as a concise book in the hisab (arithmetic) of al-jabr wa al-muqabala (compulsion and comparison). The intent of the book was to teach the reader how to compel unknown quantities to acquire known value, and, once that was done, how to compare the calculated value (for some problems could yield more than one correct solution) with the actual conditions of the problem in order to choose the value that best suited the conditions. In a marginal note that appeared on one of the manuscript copies of this book, the author of the note described the book in the following terms:

And in these Latin translations, it continued to be of interest as late as the sixteenth and seventeenth centuries and thereafter, and certainly became the subject of study for various European practicing mathematicians who most of the time started their algebraic works with comments of discussions of Khuwarizmi's book. It was those extensive engagements with this text that caused the introduction and wide circulation of such words as "algebra" and "algorithm" in the various European languages.

And despite the high regard with which Khuwarizmi's book was received in the Latin West, one can detect that those recipients had their own reasons for disregarding certain parts of it, and for preserving others. The long introduction in which Khuwarizmi explained his own motivation for composing this work, and the circumstances under which he composed it were of no concern to the translators and were dropped completely by Robert of Chester (flourished 1140s), for example. Similarly, and understandably, the second part of the book, which dealt with inheritance problems that were fully contextualized within the Islamic inheritance laws, found no great appeal in the Latin West. The ingenuity of the translators could be demonstrated in the omissions of those problems and their substitution of other commercial problems that made sense to a Western reader. But the theoretical part of Khuwarizmi's book was held in the highest esteem, and that was rendered as closely as possible in the Latin translations and commentaries.

Khuwarizmi himself confessed to a much more modest purpose. In the same introduction to his book, he stated that his primary reason for composing it was "to help people solve the problems they encountered in their daily lives such as their inheritance, their wills and testaments, their share allocations, judgments, commerce and all that they deal with among themselves such as surveying lands, digging of canals, engineering, and such things."

Obviously, the intended readers of this book must have expressed such practical concerns. Issues such as the ones summarily mentioned here, and especially those related to determining exact inheritance portions according to Islamic law, were of paramount importance, and in fact they involved computational techniques that were not trivial. With time, more and more algebraists began to respond to legal questions from the wider public regarding such computations, and even Islamic lawyers themselves at times attempted to learn algebra in order to perform their legal jobs. As a result, a whole new mathematical discipline was developed, called 'ilm al-fara'id (science of inheritance allocations), which was practiced by mathematicians and lawyers alike, the purpose of which was restricted to developing the practical algebraic techniques that touched upon the issues of inheritance.

Despite Khuwarizmi's statement of the mundane motivation for his book, it may also have resulted from the author’s desire to dedicate his book to the ruling caliph of his day, the Abbasid caliph al-Ma'mun (ruled 813‒33). The listing of the types of transactions for which his book could be useful probably should be taken simply as an indication of the actual administrative needs of the government at the time. This should not mask, however, the real intent of the marginal note in the manuscript, just mentioned, which correctly surmised that the author aimed to establish the principles of the new discipline of algebra, in the first book on the subject ever written in Islamic times.

Khuwarizmi did speak in the same introduction of the circumstances that led him to the composition of his book. And while he listed earlier authors as people who composed fresh works to be studied by others, people who simply commented on earlier works and simplified them, or people who found mistakes in existing works and corrected them without waxing proud of their deeds, he wished his own book to be considered of the first type. To designate that, he simply employed the term allaftu (I composed), rather than "I commented" or "I explained" or the like, and thus clearly signaled his awareness of the novelty of his work. As for his expectations from his work, he apparently highlighted its practical applications in the hope that it would attract the attention of the ruling caliph and thus would merit consideration of some monetary compensation. Khuwarizmi may have reasoned that the caliph was in all likelihood much more concerned with such practical matters than with the theoretical considerations that his book would address to fellow scientists, for example.

One of the theoretical considerations to bear in mind is that although the term "algebra" did not exist in any earlier civilization, it does not follow that earlier civilizations were ignorant of algebraic problems and their solutions. Even the ancient Babylonians had their own solutions for algebraic problems, and so did the Indians, the Chinese, the Greeks, and the ancient Egyptians. The Greeks in particular had developed several procedures for solving algebraic problems, some of them involving numbers while others dealt with geometric areas and the like.

What Khuwarizmi aimed to do was to consider this earlier legacy and to use it in order to draw a distinction between what he came to know as arithmetic and what he wished to call algebra, and yet he wished to use the techniques of arithmetic to solve algebraic problems. His purpose can be considered as an attempt to establish a difference between the two disciplines by laying special emphasis on the different strategies that they employed in the solution of problems, and yet highlight the fact that both disciplines used the same procedures of addition, subtraction, and so forth to attain their results. In arithmetic, one started with specific known quantities, like a sum of monetary bills. Those known quantities could then be subjected to a specific set of conditions that constituted the statement of the problem. For example, the problem might be to give away one-fifth of the known sum to charity, one-tenth to a relative, and one-sixth to the local library, and after dispersing these shares to calculate the remainder. The known quantities were then subjected to a specific set of operations, matching the conditions imposed in the statement of the problem. The known sums would then be divided by five, by ten, and by six and the results would be added up and subtracted from the original known sum to find the remainder. Once that was done, and the remainder was found, the problem was considered solved.

In contrast, algebra, according to Khuwarizmi's coined term, proceeded in the reverse order, so to speak. The algebraist started with an unknown quantity, like a sum of money, but knew that one fifth of that sum was given to charity, one tenth to a relative, and finally one sixth to the local library, without knowing the exact values of those shares of fifth, tenth, and sixth. He or she also knew that after all those shares were disposed of, the remainder was a specific known number, say 5,000. In Khuwarizmi's terms the strategy required that one began by calling the original unknown sum a "thing," or in modern terms " x. " One would then divide that "thing" by five, ten, and six respectively to get x /5, x /10, and x /6. After adding up all the results and subtracting them from the original "thing," to get x - (x /5 + x/10 + x /6), that would be the known remainder of 5,000 that was originally given in the statement of the problem. The problem then was to force the original "thing" to assume a known value, in this case the value of 9,375. In Khuwarizmi's terms, this process of al-jabr (forcing) the original unknown to become known is the essence of the algebraic strategy that distinguishes the discipline of algebra from arithmetic.

But not all problems were so simple. Some involved much more complicated procedures, and in some instances they did not even yield a final acceptable solution. In Khuwarizmi's time negative integers, for example, were not yet accepted as being of the same status as the positive integers. Thus if a problem led to a negative sum, that result was not considered acceptable. So an algebraic problem, like x 2 = 25, which would yield x = ± 5, would have the one acceptable solution x = 5 and the solution x = -5 would be discarded. With time, and during the long duration of Islamic civilization, negative numbers, and even fractions, that is, real numbers, were finally incorporated into the general number system and were treated like any other number. Even the concept of decimal fractions began to appear as early as the middle of the tenth century. It was theoretically formalized and defined a century or so later and further developed until the 16th century to cover all types of fractions, decimal as well as sexagesimal.

More importantly, the algebraic strategy also allowed the "thing" to designate any numerical quantity or geometric unit, or in fact any entity in general. It could in principle resemble a black box in which a person can carry out all the regular arithmetic operations and treat the black box like any other regular number. The result could then be oranges, apples, currency units, or fields, triangles, or any other geometric spaces.

Although Khuwarizmi's introduction laid emphasis on the practical applications of algebra, the first part of that book was dedicated to such abstract theoretical considerations as how to manipulate unknown quantities by applying arithmetical procedure to them. Through Khuwarizmi one could then simply say that the discipline itself was moved from answering specific questions, like finding the value of the "thing," to the study of the relationships that may exist among entities, irrespective of the actual value of those entities. Thus, strategies for solving equations in general became the object of algebraic studies.

It was Khuwarizmi who initiated this theoretical abstract trend. In his book, he both solved specific problems that involved inheritance, surveying, commercial dealings, and the like, as he stipulated in the introduction of the book, and he also raised the question as to the number of ways in which all second-degree equations could be solved. He developed a new term for the second-degree concept,  which he called mal (estate), and the equations were written as, for example: A specific quantity of mal were added to a specific quantity of  "things," and from the sum so many "units," which were called dirhams (currency units), were subtracted to produce a specific number. The question would then be to find the value of the "thing", or rather to "force" the thing to assume a value.

So in this theoretical part of the book, Khuwarizmi managed to solve at least one example of every conceivable type of second-degree equation. By treating equations so exhaustively, and by developing strategies for their solution in a systematic manner, he transformed the algebra from a field that concentrated on the abstract study of relationships between numbers and concepts. In a sense, through this process of abstraction, the field of algebra became theoretically divorced from the very practical solutions involved in commercial transactions and the like. But it did not lose its applicability to such concerns, and left the door wide open for those who wished to exploit the field in that direction.

It was this general theoretical trend of solving all conceivable equations of a particular degree that challenged the  mathematician Omar Khayyam (1048‒1131) to continue this line of research and to raise the general question of systematically solving all third-degree equations. In that process, he found out that some of those problems could only be solved by applying the more sophisticated intersections of the conic curves, the circle, parabola, ellipse, and hyperbola. At those boundaries the discipline of algebra began to merge with what is now called the field of analysis and the borders between arithmetic, geometry, and algebra began to blur.

The importance of the field to modern science is undeniable. One can hardly think of any aspect of modern life that does not involve the solution of one algebraic equation or another, and in terms very reminiscent of those that were developed almost 1,500 years ago by Khuwarizmi and his contemporaries.

Measuring Angles and Triangles

In his study of the Arabic heritage in the exact sciences the most famous historian of Islamic science, the emeritus professor of the American University of Beirut, Edward Kennedy, had this to say about the developments in trigonometry during Islamic times: "This subject, the study of the plane and spherical triangle, was essentially a creation of Arabic-writing scientists...." As for its importance, the most obvious manifestation of it is in the ubiquitous use of trigonometry in our modern life. Islamic studies of trigonometry were only surpassed in modern times, and only after the center of such studies had moved to Europe during the 18th century.

Why such studies flourished so well in Islamic civilization may have something to do with the ritual requirements of the religion of Islam itself. For those requirements included, among other things, the command that during the five daily prayers every Muslim was religiously obligated to perform, the individual had to face the Kaaba, a sacred building in the city of Mecca in Arabia, no matter from which spot on the earth the prayer was being performed. Historical sources reveal that the command was taken seriously, and Muslims, no matter where they happened to live on the terrestrial globe, always tried to ascertain the direction of the Kaaba, a direction for prayer that is usually called the qibla (the direction one faces), for their own locality. That seemingly simple requirement proved to be an immense mathematical challenge when it was realized that the terrestrial globe of the earth was in fact a spherical body. Thus, facing any point on the globe from any other point on the same globe could lead to multiple solutions if one did not determine a unique circle that passed through these two points. The situation is quite different from plane surfaces, for which one would simply draw a straight line between any two points on the surface and that line would be unique and thus would determine the direction of one point from the other. On the surface of a sphere, one can draw an infinite number of circles that would pass through two points on the spherical surface. The unique circle that was agreed upon for the purposes of determining the qibla direction was the great circle that passed through the zenith of the place where an individual lived, the zenith of the city of Mecca in Arabia, and the center of the earth. And with that definition and the determination of great circles and the like, the problem was immediately transformed to a problem of spherical trigonometry where the laws governing trigonometric functions were mostly analogous but not identical to those that prevailed on plane surfaces.

On the spherical surface, the calculation of the qibla required the development of both the sine and cosine laws, and also required very sophisticated knowledge of determining accurate longitudes and latitudes for any point on the face of the globe. The actual calculation of this qibla function and its reduction to the determination of an angle on the local horizon that the praying person had to actually observe required knowledge of the longitude and latitude of that person’s own city, the longitude and latitude of Mecca itself, and that the individual know trigonometric functions enough to perform the multiplication and division of products of such functions, as well as extract the actual values of angles once their trigonometric values were determined. These developments in qibla calculations, which were already performed in the ninth century, proved to be quite sophisticated, or at least sophisticated enough that they still constitute a problem of major proportions even to modern-day followers of Islam.

Another religious ritual required of every able Muslim was a pilgrimage to the city of Mecca at least once in a lifetime. To do that, people needed to know how far they were from Mecca. And that knowledge quickly led to the consideration that we are not living on a plane surface, where such distances and directions could be much more easily determined. In order to solve such problems, the mid-ninth-century mathematician and astronomer called Habash al-Hasib felt that he could create a quick tool that Muslims could use, not only to determine the direction of Mecca but also the distance between their cities and Mecca.

The tool that he invented was designed as an astrolabe, which was an instrument that was already known to the ancient Greeks. The ancient Greek astrolabes were theoretically designed to render the spherical projections onto the plane of the equator, with the projection center being either at the south or the north poles for northern or southern astrolabes respectively. However, Habash's projection took the city of Mecca itself to be the center of the projection, and determined all the circles that passed through Mecca and the other cities on the globe. And unlike the Greek astrolabes that only preserved the angles between the circles on the surface of the sphere, Habash's projection preserved the directions to the city of Mecca and allowed for an actual reading of the distance to that city.

This novel projection allowed a Muslim believer to rotate a specially designed ruler over an equally specially designed brass disk in order to determine very quickly the directions of, and distances to, the city of Mecca from any other city on the globe. No actual tool of this design has survived. But we do know that the instrument was further sophisticated in 17th-century Persia to produce a jewel-like brass map of the layout of the major cities of the known world at the time in relation to their direction and distances from Mecca.

There were other tools for determining the direction of Mecca from any locality in the then-known world. Astronomers, and later on muwaqqits (timekeepers), would solve the lengthy trigonometric problems for every locality, that is, for every longitude and latitude of the then-known world, and would record the answer in tabulated form. In order to solve this problem comprehensively, it would be necessary to find the trigonometric formula for 360 degrees of longitude and another 360 degree of latitude. That would mean a total sum of 360 x 360 = 129,600 problems to solve and tabulate. Luckily neither the whole world was then known, nor was it all inhabited, and much of it was covered with water anyway. And yet, there were astronomers who attempted to produce such comprehensive solutions in tables that contained more than 20,000 entries, which they calculated to within a high level of accuracy, so much so that when they were checked by modern computers, the maximum error value did not exceed a few minutes of arc, and never reached as much as half a degree of arc. And if qibla values could be calculated, then they could be geographically plotted, as was often done on the backs of astrolabes.

It is not difficult to see how two simple religious requirements could indeed lead to such trigonometric developments, which in turn led to new map projections and laid the foundations for a whole new discipline of mathematical geography. In fact, much of that field had already been developed by the middle of the 11th century, as can be attested by a treatise entitled The Determination of Coordinates of Cities, which was written by the 11th-century polymath Muhammad ibn Ahmad al-Biruni (circa 973‒1048).

Other astronomers and mathematicians who worked in the Islamic civilization used those newly found techniques to develop other theorems of their own, and at times they explicitly gave their own theorems names such as al-shakl al-mughni (the substitute theorem), as a signal that they were going beyond what was inherited from earlier civilizations and a statement that they had no further need for those earlier productions. Al-Biruni’s new theorem is now known as the sine law.

And like all other scientific disciplines, once a discipline is developed for a specific purpose, it immediately becomes a tool that could be used for other purposes, which may not have any relation to the original purpose. In this case the developments in the field of trigonometry were widely used in astronomy proper to develop novel theorems for planetary motions. And they were also used in the field of astrology, which was frowned upon by religious authorities, in order to cast more sophisticated horoscopes and the like.

The field that profited the most from trigonometric developments, however, was undoubtedly a religious field, called the 'ilm al-miqat (science of timekeeping), which was developed specifically to answer to the requirements of some of the religious rituals.

The person who practiced that field was commonly a muwaqqit and normally functioned as one of the employees of a major mosque. Much sophisticated astronomy was developed by those gifted timekeepers. But what was the religious interest in such subjects, and why would the community pay a functionary of the mosque to perform trigonometric calculations?

The answer to such questions has to do with the requirements that Muslim prayers be performed in the specific direction of Mecca, and also that they were all supposed to be performed at times that were defined with reference to atmospheric and mathematical geographic phenomena. And both of those disciplines require an intimate knowledge of trigonometry. Take for example for requirement of the first daily prayer, which is called salat al-fajr (the dawn prayer). This prayer has to be performed after the end of the night, and before sunrise, i.e., at dawn. But when does dawn start, and when does it end? The same question could be asked about twilight in the evening, when the evening prayer has to be performed. Both dawn and twilight are atmospheric phenomena (a fact that was very well known to Muslim astronomers), and both depended on the position of the sun with respect to the local horizon. And as the apparent path of the sun with respect to the celestial sphere is called the ecliptic circle, that ecliptic circle does not always form a fixed angle with the other celestial circle called the horizon. Note that even in the introduction of the discussion of dawn and twilight such major spherical circles as the ecliptic and the horizon are already involved and trigonometry is already incorporated. And in order to calculate the duration of such phenomena as dawn and twilight, one needed to know how long it would take the apparent sun to sink by a specific degree below the horizon to form the twilight, and how long would it take to rise over the horizon at the time of dawn. Both calculations depended of course on the mean motion of the sun, or in modern language the mean motion of the earth, and that in turn depended heavily on trigonometric functions to be determined.

But the most challenging calculations were reserved for the time for salat al-'asr (afternoon prayer), which was defined in terms of shadow lengths. The technical definition of the duration for the afternoon prayer stipulated that it could begin from the time when the shadow of a straight gnomon reached the shadow it had at noon on that day plus the length of the gnomon itself. And it would end when the shadow reached the same length it had at noon plus twice the length of the gnomon itself. So both ends of the period of prayer are defined in terms of shadow length. And anyone could see that shadows are never the same from minute to minute, or from day to day. And sometimes they could not even be observed, as on a rainy or cloudy day, for example. So how was one to determine when to perform the afternoon prayers?

Such questions were first answered by astronomers and mathematical geographers and with time fell within the professional preview of the muwaqqits. Those muwaqqits could have solved those problems one at a time, or when the need arose. What they did instead, was to solve the problems, which involved very sophisticated spherical trigonometric functions, once and for all, and tabulated the results in such a manner that a practicing Muslim could read off the table the time when the afternoon prayer started for any locality on the surface of the earth and for any day of the year.