Ibn Mun’im: Dust numerals

Marrakesh, in North Africa: between the mountains and the sea. Around 1200 CE.

Take a bundle of silk threads and make a tassel. Take silk of two colours and make a brighter tassel. Take three colours. Take ten.

Or: start with ten different colours and choose only six of them for your tassel. How many ways are there to make the choice? The mind leaps ahead to the abstract questions: how does the number of choices change as you alter the overall number of colours or the number chosen for the tassel? And what other kinds of choice are governed by the same numbers? The number of letters taken from the alphabet to make a word, perhaps, or the number of books selected from a shelf, the arrangement of stresses in a line of verse …

Bhaskara did not know it, but by his day the Brahmi number symbols were already well advanced on the journey that would take them to practically every country in the world. Ways of counting can spread fast in the right circumstances, and widely. Systems of number symbols, perhaps particularly so. They are not tied to any one language, so they can easily cross cultural borders that might be insurmountable for a method of writing words. Number symbols are easier to learn than verbal scripts, more transparent, more self-explanatory. As evidence for this, the numerals in several scripts can be read today even though their words remain undeciphered: the ‘Linear A’ of Minoan Crete, the enigmatic writing of the Indus Valley civilisation.

Buddhist travellers took some information about Indian astronomy to India’s neighbours – China, central Asia, Iran – in the first half of the first millennium. In 662, the Christian bishop Sebokht at Nisibis on the Euphrates remarked on the ‘subtle discoveries’ of the Hindus in astronomy: their methods of calculation and their use of ‘nine signs’ for computation. He may or may not have known about the tenth sign, the dot or circle marking an empty place.

Later, the spread of Islam brought the vast Arabic-speaking world into direct contact with India; Muslim merchants reached the Malabar coast by the eighth century. In the 770s, an Indian mission brought a text about astronomy to the court of al-Mansur in Baghdad. Such texts routinely included expositions both of the Brahmi number symbols and of basic arithmetic using them, and made use of the symbols in their extensive astronomical tables. The symbols were becoming visible to Arabic speakers during the eighth century, even if no exact moment of ‘first contact’ can be identified.

The culture which the Brahmi number symbols now entered already possessed several sophisticated ways of counting. Counters were present in the form of coinage, whether Byzantine gold or the dinars first minted in 697 by the Umayyad caliphs. The Arabic language inherited a decimally based set of number words from its proto-Semitic ancestor, which were therefore ultimately related to those of Akkadian and Hebrew as well as to those of the more distant linguistic cousin Egyptian: wahid, ithnan, thalatha, arba’a, khamsa. A system using the letters of the Arabic alphabet to stand for numbers, structured exactly like that of the similar Egyptian and Greek systems, existed, and indeed continued to exist beside the Brahmi-derived symbols right up to the fourteenth century.

The Roman system of hand-signs for numbers up to 9,999 was also known. They were sometimes described as ‘Rum’, meaning Roman; the term here denoted the eastern half of the Roman Empire, Byzantium. The hand-signs were widely used to store intermediate values when carrying out arithmetic mentally, and down to the tenth century some writers in Arabic continued to recommend their use, a system ‘which calls for no materials and which a man can use without any instrument apart from one of his limbs’.

Despite all this, though, the Indian number symbols had their supporters from early on, and they evidently found much success. A great movement to import and translate texts from Persian, Greek, Syriac and Sanskrit took place in the eighth and ninth centuries under the Abbasid dynasty, centred on Baghdad. Texts were copied, studied, taught from, discussed, improved and commented upon. Learned mathematicians made great strides in astronomy, algebra, trigonometry and other fields. Some experts took the names of Greek scholars as epithets, like al-Uqlidisi, ‘the Euclidean’. Around 825 CE, Muhammad ibn Musa al-Khwarizmi wrote in Arabic a ‘book on Indian calculation’, in which he described the place-value system and its use in calculation, and advocated it for its usefulness and simplicity. Al-Khwarizmi also revised or translated a set of the famously number-heavy Indian astronomical tables, possibly those that had been originally brought to Baghdad fifty years before.

Over the following decades and centuries, there grew up a whole genre of Arabic writings on the Indian system of number symbols and its calculation procedures. All started by describing the nine symbols and noting that a small circle was used to mark a space where there was no symbol. They followed with chapters on the various operations: addition, subtraction, multiplication, division; then on fractions and perhaps the sexagesimal system (inherited via Greek from cuneiform) used in astronomy. Further topics were the finding of square and cube roots, for which procedures existed broadly similar to that for long division.

The processes that were described, promoted and taken up, however, were at first subtly different from their Indian models. Several of the early Arabic texts used in their titles or their descriptions the terms ‘board’ (takht, originally a Persian word) or particularly in the west of the Arabic world ‘dust’ (ghubar). The idea was that a scribe would carry a board; when it was to be used, he would strew sand on it and write the number symbols in the sand with a finger or stylus. This made it easy to erase, replace and move symbols as the calculation required. The whole practice came to be called in Arabic ‘dust arithmetic’ or ‘board arithmetic’, the number symbols themselves ‘dust symbols’. This made arithmetic a wholly temporary process, leaving no permanent record the way the Indian procedures on paper and bark did.

Paper was certainly known in the Islamic world from the mid-eighth century onwards (the story is that Chinese prisoners taught its manufacture around the year 750), and already by the following century some authors were advocating its use for calculation in preference to the dust board. The mathematician al-Uqlidisi argued that the use of the dust board demeaned the serious scholar because of its association with ‘misbehaved’ people ‘who earn their living by astrology in the streets’, and that its difficulties included dirty fingers and sand being blown away by the wind. The key argument, though, was that using paper produced a record of the process of calculation, so that if you needed to check or find an error you could do so: unlike the tahkt, on which at least some steps of the calculation were necessarily erased along the way.

Whether on board or on paper, it is likely that the number symbols quickly began to spread from their initial context of learned astronomical calculation and tabulation, into the arithmetic taught to children and used by merchants and other practitioners. By the later ninth century, and more so through the tenth and eleventh, references to the new notation can be found across the writings of Arabic historians, encyclopedists and religious scholars. Eventually, a mathematical tradition extended over the whole of the Arabic and Islamic world, from northern India to the Atlantic coast of North Africa and Iberia.

The far west of the Islamic world was itself an important centre of culture and learning. One of its exponents was Ahmad ibn Mun’im al-’Abdari, born probably in the second half of the twelfth century, and so roughly a contemporary of Bhaskara 8,000 kilometres to the east. His origin was in the town of Denia in al-Andalus, on the coast near Valencia. At this time the Almohad dynasty controlled both Islamic Iberia and the Maghrib on the North African coast. Theirs was a reforming, purifying impulse and its leaders proclaimed themselves the true Caliphs. Controlling vast resources – mines for sulphur, iron, lead, mercury and cinnabar, textile and ceramic industries – and the whole of the trans-Saharan trade as well as trading connections north of the Mediterranean into Italy, they supported and promoted culture from architecture and poetry to theology, mysticism and philosophy. The Jewish philosophers Averroes and Maimonides both lived under Almohad rule. Medicine, mathematics and astronomy flourished.

Most of ibn Mun’im’s life was spent in the Almohad capital at Marrakesh, where he studied both medicine and mathematics. He was reckoned one of the leading specialists in geometry and number theory, and wrote on both Euclidean geometry and the arrangements of whole numbers known as magic squares.

His book ‘On the Science of Calculation’ was one of those that described the Indian numerals and their use. Aimed presumably at students, it covered the usual topics: the shapes of the ‘dust numerals’ (still so called, though now routinely written on paper); addition, subtraction, multiplication, division, extraction of roots and fractions.

Midway through the book came a section of more advanced discussions. These dealt with topics like the sums of series of odd numbers, of even numbers, of square numbers or of cubes. Ibn Mun’im discussed triangular numbers and other similar arrangements, and the so-called perfect numbers, which are equal to the sum of their factors. And he examined at some length the question of how many words can be made from a given set of letters.

Questions of this general type had been asked for centuries, by linguists and philosophers interested in enumerating the different roots of Arabic words under various constraints. As early as the 790s, scholars performed laborious enumerations of two-, three-, four- and five-letter roots. Up to the twelfth century, such problems were mainly approached simply by listing possibilities and counting them one by one. One writer constructed and described a device consisting of a disc with two rotating wheels, on each of which were written the letters of the Arabic alphabet. By rotating and aligning the wheels it was possible to find all the different alignments of three letters more quickly.

Ibn Mun’im’s approach was mathematical rather than mechanical, and he was keen to calculate rather than simply count. He worked through a series of subsidiary problems on the way to his answer to the general question; in fact he began by considering an ostensibly different problem. Given ten colours of silk, with which you wish to make tassels, how many different choices can be made of, say, five colours? Or three? Or seven? In other words, how many ways are there to choose a certain number of colours out of ten, if the order in which they are chosen does not matter?

There are obviously ten ways to choose just one colour. By simple counting, you can discover that there are forty-five ways to choose two, a hundred and twenty to choose three … Ibn Mun’im made a table showing the different ways to choose, all the way from choosing one colour from a selection of one, to choosing ten from a selection of ten. By observing the properties of the table, he was able to see ways to calculate its numbers rather than finding them by laborious counting. The table was identical to the arrangement of numbers later called Pascal’s triangle; each number was equal to the sum of the number above it and the number to the left of it.

The ways to choose a set of letters from the twenty-eight available in Arabic are naturally the same as the ways to choose a set of colours from a selection of twenty-eight. To choose three letters (or colours), for instance, there are 3,276 possibilities. This was just the start, though, and ibn Mun’im worked through a dazzling series of more complex questions and examples. Suppose one letter is repeated: how does the number of possible choices change? Suppose more than one letter is repeated. And so on. One of his computations asked, in effect, how many distinct words of nine letters can be formed by rearranging the string ABCCDDEEE. Finally, he introduced restrictions on the alternation of vowels and consonants in order to respect the way in which pronounceable words were really formed in Arabic. For several of these cases, ibn Mun’im constructed separate tables showing the answers – often enormous numbers – arising from a combination of calculation and plain counting.

Empires come and go. Muhammad al-Nasir (1199–1213), at whose court ibn Mun’im worked, is probably best known to history as the leader who lost the battle of Las Navas de Tolosa in 1212 to a coalition of Christian kings, a watershed for the fortunes of the Almohads and the beginning of that empire’s collapse in both Iberia and Africa. He died a year later, and internal divisions hastened the end of the Almohad dynasty and the disintegration of its empire later in the century, to be replaced by several smaller Islamic monarchies.

The Arabic learned tradition continued, of course, and ibn Mun’im’s results about combinations and permutations were passed on to succeeding generations. The subject illustrates a pervasive tendency of advanced exercises in counting to shade into calculation, tabulation and eventually algebra; and, more generally, for the Indian number symbols and calculation on paper to facilitate thinking about numbers in ever more abstract ways.

The question of exactly what the Indian number symbols looked like in the Arabic world is a surprisingly elusive one. The fact is that contemporary manuscripts of the earliest Arabic works on numbers and arithmetic have not survived. For al-Khwarizmi’s work on Indian-style calculation, indeed, even the original text is lost, and the only evidence of what he wrote consists of later translations into Latin. There is absolutely no direct evidence for what the number symbols looked like to him. For ibn Mun’im’s work on arithmetic and combinatorics there is only a single manuscript, made generations after the original author. In fact, the fragility of writing materials has swept away all of the first three centuries of evidence for the Indian number symbols in Arabic contexts.

When the evidence does fade in, during the eleventh century, in the form of manuscripts, astronomical tables and even astronomical instruments such as astrolabes, there is – not surprisingly – a good deal of variation from manuscript to manuscript, reflecting slightly different conventions at different places and times. In fact al-Uqlidisi, as well as advocating calculation on paper using the decimal place-value system, proposed replacing the unfamiliar Indian symbols with the first nine letters of either the Greek or the Arabic alphabet. But that idea did not catch on, and the Indian symbols were retained, albeit modified in detail.

One curious change took place throughout the Arabic-speaking world. In the Brahmi script, which is read from left to right, numbers were written with higher powers on the left. That is, the eye would come to the higher powers first (as it does in modern languages using the Latin alphabet). In Arabic, which is read from right to left, the order of the numerals was not reversed to match the direction of reading: the higher powers still appeared on the left. That meant that the eye would reach the smallest power, the units, first.

As to the shape of the individual number symbols, the tradition eventually split into two branches: the symbols of the eastern and western halves of the Arabic-speaking world, both clearly descended from the original Brahmi forms used in Sanskrit. A Moroccan writer of the late twelfth century remarked on the difference between the two styles, but the divergence had certainly arisen long before that. The Baghdad-centred area, where al-Khwarizmi and al-Uqlidisi had worked, came to use a set of symbols that are the ancestors of those used in modern Arabic: ١, ٢, ٣, ٤, ٥, ٦, ٧, ٨, ٩. The western regions of the Arabic world, however, had a different style of calligraphy and differences in the form of certain letters. All the number symbols were modified to one degree or another there, and the symbols for 6, 7 and 8 became quite different from their eastern counterparts. This western style of writing the Arabic numerals – sometimes called the Maghribi numerals – persisted until the eighteenth or even the nineteenth century before finally being generally replaced by the eastern system.

Indian number symbols in the western Arabic form, from a manuscript made in fifteenth-century Granada.

University of Pennsylvania Library, LJS 464, fol. 2v. Public domain.

The interest of the western-style Arabic number symbols lies in their onward journey: for it was this form that would travel to Europe and become the ancestor of the ‘Arabic numerals’ known around the world today. Indeed, during the lifetime of Bhaskara and ibn Mun’im, that process was already under way.

Hugo of Lerchenfeld: Toledan numerals

Similarly to the admiring reference by the Bishop Sebokht in Syria in the seventh century, the Arabic numerals received a first glimmering mention in Europe in the form of a passing reference in a manuscript about something else. The monk Vigila, of the monastery of Albelda in the Rioja, was responsible, adding to his copy of a Latin encyclopedia the following words:

the Indians have a most subtle understanding and all other peoples yield to them in arithmetic and geometry and the other liberal arts. And this is clear in the nine figures with which they are able to designate each and every degree of each order [of numbers]; of which these are the shapes …

The shapes given were essentially those of the western-style Arabic numerals. This was in 976, the period of the Umayyad caliphate based at Cordoba: evidently some of the vast learning of that splendid city had made its way north into Catalonia. And thus, paradoxically, the earliest surviving evidence for the western forms of the Arabic number symbols comes from manuscripts written in Latin.

During much the same period around 970, a young man named Gerbert of Aurillac travelled to Catalonia, there to study mathematics for three years. Once back in France, he would become archbishop of Rheims, and subsequently Pope, as Sylvester II. At Rheims he devised and promoted a new – some would say bizarre – kind of counting board, whose key feature was that the counters were marked with number symbols. Instead of putting, say, six counters in the tens column, you would put a single counter there: one marked with a ‘6’.

For two centuries, Gerbert’s board was the preferred tool for learning number lore in the monastic schools, with around three dozen separate treatises devoted to it. One or two actual ‘boards’ in this style – made of paper – survive to this day, but sadly none of Gerbert’s specially marked counters. In the later treatises, the symbols on the counters were Arabic numerals, and it is likely that that was Gerbert’s own choice, deriving from his sojourn in Catalonia and what he had learned there.

The procedures for adding, subtracting, multiplying and dividing on Gerbert’s counting board were essentially identical to those for written number symbols on a dust board, the counters being successively replaced by others as the computation progressed. This, then, was to all intents and purposes a European version of the Indian – and now Arabic – practice of calculation using written symbols. Instead of writing the symbols on a surface, you placed them, moved them and removed them in the form of the counters. A minor difference was that on a board there was no compelling need to mark an empty column with a ‘zero’ counter: you could just leave it empty. In fact, ‘zero’ counters do seem to have existed, redundant though they arguably were. In Latin the zero was at first sometimes called a ‘wheel’ and sometimes a theta, after the Greek letter whose shape it resembled. But fairly soon, Latin writers settled on terms derived from the Arabic sifr, meaning ‘null’ or ‘void’, itself borrowed from the Sanskrit sunya meaning ‘empty’. In Latin the word became cifra or chifra, the ancestor of words in modern European languages such as English ‘cipher’ and French chiffre. In these languages, the term sometimes came to designate not just the zero but the whole set of number symbols, or indeed their use (‘ciphering’). Sometimes the term’s meaning was extended to symbolic characters in general, especially secret ones. Thus in Italian something ‘cifrato’ is in a secret code, in French a ‘chiffre’ may be a monogram initial letter, in German ‘Ziffer’ means the system of Arabic numerals, while one of the meanings of ‘cipher’ in English is simply a ‘zero’.

Gerbert’s counting board was cumbersome to use compared with more traditional models, and there is little evidence that it was used outside the teaching context of monasteries and their schools; little sign that it was used for real-world calculation rather than for learning about numbers and their manipulation in the abstract. After the later twelfth century, there is no further evidence for it even in the monasteries, and it seems to have fallen out of use and out of memory.

Meanwhile, and more widely in Europe, Arabic numerals had begun to appear among the manuscripts entering the continent through Antioch, through Sicily, and increasingly through Spain. Copies went as far north and west as France and England by the mid-twelfth century. They prompted a period of experiments, perhaps broadly similar to the Indian experiments in representing numbers in writing: most short-lived, and some very strange.