But the remarkable Sanad ibn Ali had another trick up his sleeve. The great eleventh-century Muslim polymath al-Bīrūni reports in his famous treatise The Determination of the Coordinates of Cities (Kitab Tahdīd al-Amākin) that Sanad had proposed to al-Ma’mūn a far better way of measuring the circumference of the earth that did not involve trudging across hot deserts counting paces. While accompanying al-Ma’mūn on one of his campaigns12 against the Byzantine Emperor Theophilus around 832, Sanad is supposed to have suggested the trick of climbing a mountain that overlooked the sea and measuring the angle of inclination to the horizon. This, along with knowledge of the height of the mountain, could be used, together with some rudimentary geometry, to determine, not the circumference of the earth, but its radius. Of course, it is then a simple matter of multiplying this figure by twice pi to obtain the circumference.
But al-Bīrūni was well known for his modesty in crediting others and we have no other record to suggest that Sanad actually carried out this experiment. Famously, it would take the even greater genius of al-Bīrūni himself to perform this measurement carefully and settle the argument about the size of the earth for good. But that story will have to wait until a later chapter.
The third spectacular project undertaken by al-Ma’mūn’s group of scholars was the most ambitious of all. Another of the great works of Ptolemy was his Geographia. In this book, he included all that was known about the geography of the world at the time, much of it based on the work of an earlier geographer, Marinus of Tyre (70–130 CE), who had come up with the idea of coordinates involving latitudes and longitudes and had defined the line of longitude going down through the Canary Isles as his ‘zero meridian’ and the parallel of Rhodes for his measurements of latitude. Ptolemy’s Geographia was translated into Arabic by a group of scholars in the House of Wisdom with the help of al-Khwārizmi and seems to have been the key text that triggered the early Islamic interest in geography. Here again, we see the audacity and self-belief of al-Ma’mūn. With two new observatories, confirmation of the circumference of the earth and his scholars’ rapidly developing skills in geometry and algebra, he commanded the production of a new map of the world. After all, Ptolemy’s map did not include important Islamic cities such as Mecca or Baghdad; the former had not been important enough and the latter had not existed then. Al-Ma’mūn’s astronomers had worked out the distance between these two cities by taking measurements during a lunar eclipse and found it to be 712 mīl, which is less than 2 per cent off the actual distance.
Al-Ma’mūn’s scholars began to recalculate the coordinates of many of the major landmarks in the known world and soon found discrepancies in Ptolemy’s values. So a new map was produced which contained major improvements on Ptolemy’s. It depicted the Atlantic and Indian oceans as open bodies of water, not landlocked seas as Ptolemy had suggested. While the Greeks had revealed a good knowledge of closed seas like the Mediterranean they showed little understanding of the vast ocean expanses beyond, something Arab traders were able to report back on by the time of al-Ma’mūn. The map also corrected Ptolemy’s gross overestimate for the length of the Mediterranean Sea of 63 degrees of longitude and put it instead at 50 degrees, which was far closer to the correct value. Unfortunately, this Abbāsid map no longer exists, and it has been difficult to reconstruct it with any confidence. What is known about it has come mainly from a treatise produced at the same time entitled Picture of the Earth (Sūrat al-Arth). This treatise has been attributed to none other than al-Khwārizmi himself, who seems to have been central to the map project and is often regarded as the first geographer in Islam. However, it is more likely, from the sheer effort that went into it, that Sūrat al-Arth was, like the Verified Tables, a collective effort. It was modelled closely on the Geographia and completed in 833, the year of al-Ma’mūn’s death. It contained tables of the latitudes and longitudes of more than five hundred cities and grouped locations under five general headings: towns, rivers, mountains, seas and islands, each in tables ordered from south to north, each with its precise coordinates in degrees and minutes of arc.
The oldest surviving world maps from the Islamic Empire are copies of earlier ones dating back to the early eleventh century and containing many references to al-Khwārizmi’s tables. Several years ago, considerable excitement greeted what appeared to be a remarkable discovery. A renowned historian by the name of Fuat Sezgin based in Frankfurt claimed to have discovered a fourteenth-century representation of al-Ma’mūn’s original map in the Topkapi Museum in Istanbul. But not everyone is convinced of its authenticity just yet.
Muslim cartography rapidly built on the work of al-Khwārizmi and his circle and the subject evolved into two different schools of cartography: those maps that followed a Baghdadi mapmaker by the name of al-Balkhi (850–934), which tended to be more like stylized diagrams than literal maps as we understand them (rather like the map of the London Underground), and those following the later style of a twelfth-century Andalusian cartographer by the name of al-Idrīsi. Other Muslim scholars who wrote on geography included Ibn Sīna (Avicenna) and al-Bīrūni in the eleventh century, along with the historian Ibn Khaldūn and the famous traveller Ibn Battūta in the fourteenth. But the important contribution of al-Ma’mūn’s cartographers to the development of the field of mathematical geography cannot be overestimated.
*
There is one final area of scholarship that became something of an obsession for al-Ma’mūn, one that might sound surprising to us today. Indeed, it was something of a revelation to me when I first learnt about it: Egyptology.
The Pyramids of Gīza just outside Cairo date back to the middle of the third millennium BCE, making them already well over three thousand years old by the time of al-Ma’mūn. The ancient Egypt of the pharaohs was thus already a civilization lost in the mists of time by the time Islam arrived. Here is what the tenth-century historian al-Mas’ūdi has to say about the pyramids:
The temples of Egypt are very curious structures … then there are the pyramids, which are very high and built in a remarkable way. Their sides are covered in all kinds of inscriptions, written in the scripts of ancient nations and of kingdoms that no longer exist. No one can read this writing or know what was intended by it. These inscriptions relate to the sciences, to the properties of things, to magic and to the secrets of nature.
Later he recounts:
I have questioned the most learned Copts of Upper Egypt and other provinces on the meaning of the word ‘pharaoh’, but no one has been able to tell me anything about it, for this name does not exist in their language. Perhaps originally it was the general title of all their kings.
You will find strange tales of the treasures and monuments of Egypt and the wealth which both its kings and other nations who ruled this land buried in the earth and which are sought even today.13
Al-Ma’mūn travelled to Egypt in 816 to quell an uprising and while there became fascinated with the Pyramids. It is said that he searched in vain for someone to explain their purpose to him. Medieval Arab fascination with accounts of ancient Egypt comes from several references made to it in the Qur’an, particularly in the story of Moses and the pharaohs, as well as what was gleaned from the translations of Greek classical writers such as Homer and Herodotus. But we can imagine the impact that actually seeing the Pyramids must have had on al-Ma’mūn.
His obsession with the translation of ancient texts extended naturally to his desire to decipher the hieroglyphic symbols on the walls of the tombs of Egypt. It was believed by the early Arab scholars that these symbols held ancient secrets associated with astrology and alchemy, as indeed many of them did. The alchemists closely associated with mystic Sufism were particularly fascinated with Egyptian scripts, the most prominent of these being none other than Jābir ibn Hayyān.
While in Egypt, al-Ma’mūn enlisted the services of a sage named Ayyūb ibn Maslama, who he hoped could translate hieroglyphs for him. After all, the Coptic language still spoken by many of the indigenous
population there was itself a descendant of the old Egyptian language. Unfortunately, and to the disappointment of the caliph, Ayyūb was unable to make much sense of any of the inscriptions.
Al-Ma’mūn next ordered an excavation of the Great Pyramid of Khūfū. A team, accompanied by the caliph himself, managed to break their way in and found behind the opening a jar of gold, which al-Ma’mūn took with him back to Baghdad. Once inside, they discovered ascending and descending corridors. At the top, they came across a small chamber, in the middle of which was a sealed marble sarcophagus with the pharaoh’s mummified remains still inside. At this point, al-Ma’mūn, not wishing to continue the desecration any further, ordered the excavation to stop.14
As a postscript to this account I cannot pass up the opportunity to tell how one Arab scholar who lived in Kūfa not long after al-Ma’mūn’s reign did in fact succeed in cracking about half of all hieroglyphic symbols. I recently visited Saqqara, the ancient necropolis in Egypt dating back to the twenty-seventh century BCE, before the Pyramids of Gīza were built. I was shown around several tombs by London-based Egyptologist Okasha El Daly, who has made a comprehensive study of ancient Egypt in medieval Arabic writing. He makes a convincing case that a man by the name of Ibn Wahshiyya, who flourished in the ninth/tenth century, can rightly be regarded as the world’s first real Egyptologist. So, whereas it is commonly assumed in the West that the hieroglyphic code was not cracked until 1822 when the Englishman Thomas Young and the Frenchman Jean-François Champollion deciphered the writings on the Rosetta Stone, I suddenly found myself marvelling at yet another little facet of the translation movement. Ibn Wahshiyya’s text on various ancient alphabets, Kitab Shawq al-Mustaham, gives a list of hieroglyphic symbols and their meaning, either as words or sounds, together with their Arabic equivalent; this nearly a millennium before Young and Champollion.
Let us return, though, to the House of Wisdom in Baghdad and al-Khwārizmi. For, as I have mentioned already, his greatest legacy to science was not in the field of geography, but in mathematics. The extent of the debt we owe to this man will become clear over the coming two chapters, when I explore the development of mathematics during the time of al-Ma’mūn.
7
Numbers
I will not say anything now of the sciences of the Hindus, who are not even Syrians, of their subtle discoveries in this science of astronomy, which are even more ingenious than those of the Greeks and Babylonians, and of the fluent methods of their calculations, which surpasses words. I want to say only that it is done with nine signs.
Severus Sebokht, Bishop of Syria
The ‘five-bar gate’ tally system, familiar in movies when scratched onto prison cell walls to mark off the days, is one way of counting that allows a running tally to be kept for a constantly updated number. This and other forms of tallies are also our most ancient way of counting, and go back many thousands of years. Cavemen during the Upper Paleolithic Age (40,000–10,000 years ago) first started using animal bones as tally sticks for counting. The earliest surviving example is probably the Lebombo bone, dating back 35,000 years, which was a small piece of the fibula of a baboon, marked with twenty-nine clearly defined notches, discovered within the Border Cave in the Lebombo Mountains of Swaziland.
Even before the invention of a sensible number system there would still have been a need to keep a tally – of the sheep in a flock, for instance. As each sheep passes through the farmer’s gate out to graze in the morning, he cuts a notch in a stick. At nightfall, when the flock returns, he checks his sheep against the tally by running his finger along the notches, moving from notch to notch as each sheep passes. In this way he knows if any are missing without having to know the number of sheep in his flock. The tally stick served its purpose just as well as counting the flock serves the farmer today, and could be passed from one person to another just as easily as numbers are passed by word of mouth, or written down.
A simple modification of the primitive tally stick extended its scope to a remarkable degree. All that is needed is a second stick, which does more than simply double the space available for notches. This is how it works: one stick, called the standard, already has a number of notches on it, say twenty, while the other – the tally stick – is uncut apart from a dividing line splitting it into two sections. The farmer counts the sheep by moving his finger along the notches on the standard. As soon as he reaches the end of the standard he marks a notch on the bottom half of the tally stick, indicating one unit of twenty. He then starts again on the standard. Each time he reaches the twentieth notch he cuts a new one on the tally. When the last sheep has been recorded on the standard, he cuts that number of notches on the upper half of the tally stick – the units section. Now, if he has, say, four notches on the bottom half and seven on the top, his tally is 4 × 20 + 7 = 87. But of course he does not need to know this total as an actual number as he can reverse the process by marking off on both sticks when the sheep return.
The number of notches on the standard is, of course, quite arbitrary, and a standard of twenty has no special advantage. Nevertheless, counting sheep by way of making a score, or notch, on a tally stick for each unit of twenty is itself the origin of the word ‘score’ in Old English. In the Bible, the number seventy is written as ‘threescore and ten’, the word ‘threescore’ for sixty going back to the fourteenth century – as in ‘Thre scoor and sixe daies’ from John Wyclif’s Bible, the very first translation from Latin into English.1
Such a base-20 counting system is known as vigesimal. Modern-day French numbering is still partially vigesimal: twenty (vingt) is used as a base in the names of numbers from sixty to ninety-nine. The French for eighty, for example, is quatre-vingts, which literally means ‘four twenties’, and soixante-quinze (literally ‘sixty-fifteen’) means seventy-five. This convention was introduced after the French Revolution in order to unify the various counting systems in existence at the time around France.
Other numeral systems have been used with different bases. A base of twelve, known as the duodecimal system, is one of the earliest of these. Its use began perhaps because there are approximately twelve cycles of the moon (lunar months) in one year, and because the multiples and divisors of the number 12 are convenient: 12 = 2 × 2 × 3 = 3 × 4 = 2 × 6, while 60 = 12 × 5 and 360 = 12 × 30, and so on. The use of twelve as a base number was widespread in Europe and the word ‘dozen’ comes from the old form of the French word douzaine, meaning ‘a group of twelve’. The word ‘gross’, from the Latin grossus meaning ‘large’, was taken to represent the number 144, meaning a ‘large dozen’, or a dozen dozens.
Ultimately, though, the fact that humans have ten digits on their hands provided a standard so readily accessible and convenient that the base-10 (decimal) system has been adopted almost universally.
Pythagoras (c. 580–500 BCE) is rightly regarded as the first great mathematician in history and the school of thought that carried his name made huge advances, despite being more of a religious movement than a mathematical one. His philosophy was based on the notion that numbers were intimately connected with the reality of the universe; he regarded them as the abstract yet fundamental building blocks of physical matter. However, it should be noted that his life is shrouded in mystery and it has even been suggested by some historians that he never actually existed.
Even earlier than Pythagoras, around the eighteenth century BCE, the Babylonians used what is called a ‘sexagesimal’ numbering system, which, unlike the decimal system that changes units every ten, is based on sixtieths of the next highest unit. It is therefore from the Babylonians that we inherit the division of an hour into sixty minutes and a minute into sixty seconds. Similarly, the split of angles into degrees, minutes and seconds of arc is sexagesimal. The Babylonians had symbols for numbers up to 59, beyond which the next unit starts off as 1 again. Thus, to write our numbers in sexagesimal notation, we can divide the units by a comma. The number 61 would then be written as (1,1). Likewise, the number 123 is written
as (2,3) since it is made up from 60 × 2 + 3. It follows that, for instance, the number 4321 would be written as (1,12,1), since 4321 = 3600 × 1 + 60 × 12 + 1, and so on. This sexagesimal notation is continued into the fractions, where a semicolon can be used to separate integers from fractions. Thus, while (1,30) means the number 90 (1 × 60 + 30), (1;30) means the number 1.5, as 30/60 is the same as ½. Likewise, (2;45) means 2.75, since 45/60 is the same as ¾.
There is a small Babylonian tablet (now at Yale University in the USA) showing a remarkably good approximation for the square root of 2. It is written in sexagesimal form involving fractions as (1;24,51,10). We can write this number in its full fraction form and add up all the fractions:
And given that the exact value is …, we see that this is a startlingly good approximation. But this in itself is not what is really impressive. The Yale tablet, and many other known Babylonian tablets, prove (according to Otto Neugebauer, who, despite championing Babylonian science, was one of the most cautiously conservative scholars of the history of ancient science) that the Babylonians were well aware of the ‘Pythagorean theorem’: of determining the length of the diagonal of the square from the length of its side – this, one thousand years before Pythagoras!2 Thus, a square with sides equal to one unit has a diagonal that is the square root of the sum of the squares of the two sides. In a right-angled triangle it forms the hypotenuse:
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