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.
Some European scholars tried to make a hybrid of Roman and Arabic number symbols, by using a selection of the Arabic symbols as abbreviations within a system structured like the Roman one. The results were monsters like MC87 for 1,187 or 52M,CC20 for 52,220. Others took up the place-value system from the Arabic numerals, but not the Arabic symbols. Abraham ibn Ezra, in Pisa, used the first nine letters of the Hebrew alphabet in a system that was structurally identical to the Brahmi numerals; a colleague in Flanders used the first nine letters of the Latin alphabet in the same way. Evidently the new number symbols were stimulating and provocative, but equally evidently they created confusion and anxiety just as often as clarity.
It was in the context of astronomy and astronomical tables that European writers first learned to use the Arabic numerals properly. The Spanish route was becoming the most important one for Arabic material entering Latin Europe, and the city of Toledo was established as a major centre of translation and transmission. The astronomical tables of al-Khwarizmi received adaptations for use first in Cordoba and later in Toledo. The Toledan version acquired a wide reach and importance in Latin; more than two hundred manuscript copies survive today, and these ‘Toledan tables’ were in turn adapted for use in various other European cities, becoming a model for other astronomical works in Latin.
This material was of interest in Europe because it could potentially solve urgent problems with the Latin calendar. The underlying difficulties were to do with the fact that there are not a whole number of days in a year: that is, for each orbit it makes around the sun, the Earth spins on its axis not 365 or 366 times but a fractional number between the two. The problems would not receive a lasting solution until the Gregorian calendar reform of the late sixteenth century, which produced the rules about leap years still in use today.
Absorbing Arabic calendar lore, these scholars also learned how to use the Arabic numerals. The Toledan tables were accompanied in some manuscripts by Latin versions of al-Khwarizmi’s book about the numerals, which explained their use for arithmetic. The association of the numerals with Toledan material was so strong that some Latin commentators called them the ‘Toledan’ – rather than the Arabic or the Indian – numerals.
When Arabic (or Toledan) numerals began to appear in Latin manuscripts, they once again underwent changes in their exact shapes, though they remained clearly derived from ancestors of the western Arabic type. It is possible they were influenced by Visigothic conventions for writing and abbreviating Roman numerals, but the differences may just be the result of the usual vicissitudes of handwriting style over several centuries. At least two Latin writers actually remarked on the difference between the ‘Toledan figures’ and the original Arabic ones, giving examples of each, although it was understood that the place-value system in which they worked was the same; that the differences were superficial.
One unlikely centre of expertise and experiment in both astronomy and the Toledan numerals was the German city of Regensburg. There, a small circle of interested scholars collected together a clutch of mathematical manuscripts, including copies of the Toledan tables and of a Latin version of al-Khwarizmi’s arithmetic book. They tried out different ways of adapting the thirty-year Arabic calendar cycle, or the nineteen-year Jewish one, to Christian use. They even indulged in distinctive local variations on the shapes of the number symbols, giving ‘2’ and ‘3’ long vertical tails and occasionally writing ‘3’ to look like a sort of cleft stick.
It was here that, so far as the evidence goes, Arabic numerals first began to leak out of the contexts of astronomy and the study of calendars, into more everyday uses of numbers and counting.
Hugo, citizen of Regensburg, became a canon at the cathedral there after the death of his wife, probably in 1178, and remained so until his own death in or after 1216. He is an obscure figure, his life recorded as the merest outline: except that he spent a proportion of his time compiling an idiosyncratic notebook. Now in the Bavarian State Library in Munich, it contains seventy-eight sheets of parchment, kept together not by binding but by wrapping them in a larger piece of parchment.
It was a messy compilation, constructed in part by copying – with variations – a volume of annals from the neighbouring monastery, and by supplementing it after its end in 1167. As Hugo grew older, his handwriting became larger and messier in the later entries. Other parts of the book consisted of inserted passages and pages with notes on all kinds of subjects: family history, letters, the Hebrew alphabet, spiritual reflections, solar and lunar eclipses … The loosely inserted sheets must have been in constant danger of ending up dropped, lost or disordered.
Altogether, the annals ran from 35 to 1201 CE by the time Hugo ceased to write. He evidently also had astronomical or arithmetical material from the monastery in front of him, because instead of using Roman numerals for the dates in his annals he used the Arabic numerals: not MCLXVII but 1167; not MCCI but 1201.
It is hard to convey just how eccentric Hugo was to date his annals in a way that now seems like the natural choice. No more than a handful of his contemporaries in the city would even have recognised the Indian/Arabic/Toledan symbols as numbers, never mind been able to read them. Perhaps in a private notebook, it didn’t matter; perhaps Hugo relished the novelty, the cleverness, the hint of secrecy about the new numerical code. Thus he became in his small way a pioneer, the first in Europe to use the Arabic numerals outside the realm of astronomical tables or arithmetical treatises.
A page from Hugo’s notebook, showing the Toledan number symbols.
München, Bayerische Staatsbibliothek Clm 14733, fol 32v. CC BY-NC-SA 4.0.
Bhaskara, ibn Mun’im and Hugo of Lerchenfeld were alive at the same time, in the late twelfth century. They wrote down numerals in their Sanskrit, Arabic and Latin texts within a decade or so of one another. Their stories tell of the enormous reach the Indian number symbols had attained by this date. In India they already had a history of hundreds of years; of four centuries in the Arabic-speaking world. In Latin contexts, their use on parchment was only just beginning – though they had been manipulated on counters since about 1000 – and only hindsight makes it clear that they had a future in the Latin world, and that that future would involve the Toledan version of the numerals rather than one of the other experiments or versions of the period. Chance played a role in this, as did the high status of Toledo as a centre of translation from Arabic, and the prestige and visibility of the texts issuing from that city.
The Indian numerals had, of course, a vast subsequent history in all three locations. In the Latin world, at least four different versions of al-Khwarizmi’s treatise on Indian-style arithmetic had been made by the end of the twelfth century, representing various degrees of reworking and refinement as translators attempted to make the material more comprehensible to their readers. If the place-value system itself was familiar from the counting board – whether traditional or Gerbertian – the idea of calculation on parchment seems to have required laboured explanation, demonstration and discussion. So widespread did these texts become that the name ‘al-Khwarizmi’, which in Arabic meant the man from Khwarezm, came to mean the process of calculation with the Arabic numerals, or a treatise describing it. The Latin word, inevitably somewhat garbled, was alchorismus or later algorismus. Eventually it came to mean arithmetic in general, whatever the method, and in modern languages an ‘algorithm’ can be any step-by-step procedure. It was quickly forgotten that the word had its origin in a personal name, and some later medieval authors even attempted to reason back from algorismus to ‘Algus’ or ‘Argus’, its supposed inventor. Chaucer once referred without explanation to ‘Argus, the noble counter’.
The numerals in the Toledan form gradually increased in visibility and importance; in country after country a critical mass of the literate switched to using them for calculating, and for recording the results of counts and calculations. Their proponents effectively captured the narrative and succeeded in associating with the Arabic numerals a sense of prestige that went rather beyond their practical advantages. The older system – counters for counting and calculation, and Roman numerals or words for communicating results – had worked for a thousand years and had enabled the Romans to administer their vast empire: it was not in any clear sense broken.
Italy took up the new way first, with both merchants’ and public accounts adopting the Arabic numerals alongside the techniques of double-entry bookkeeping between the thirteenth century and the fifteenth. Specialist schools taught the techniques; by the early 1500s Florence, for instance, had six schools teaching up to twelve hundred pupils ‘algorism’. Northern and Western Europe followed, with the proportion of Arabic numerals in documents Europe-wide hitting 50 per cent by around 1500. (There appears to be no evidence for the claim, still occasionally repeated, that the new numerals – or zero in particular – were resisted, feared or blanket-banned because of their foreign, Hindu or Muslim associations or some supposed association with magic.)
Hindsight has given this rise a feeling of inevitability, as though the claims of late medieval boosters in favour of the numerals were the unvarnished truth. They were indeed efficient for calculation, if you were prepared to invest the initial effort in memorising addition and multiplication tables, and learning algorithms for addition, subtraction and so on. And, as practised on paper rather than dust boards, slates or blackboards, they provided a record of the process of calculation, making it relatively easy to check and spot errors. But boards and counters continued to be important well into the era of print, tally sticks were used at the English exchequer until late in the eighteenth century, and Roman numerals linger to this day in contexts where prestige or stability need to be communicated (the numbering of monarchs and popes, the dates of films and public foundations) or ambiguity between two series of numbers needs to be avoided (page numbers in books, month and day numbers in certain styles of date).
What did counting in Europe look like, after the Arabic numerals had completed their conquest?
The account keeper: Counting on paper
A Dutch interior in the 1650s. A wooden counter: inkwells, a ledger. A few coins. A middle-aged woman sits at her books, the window open to catch the light. She is working on her accounts: perhaps for her household, perhaps for her business. Beside her, keys, another reminder of her responsibilities. A barrel, some jugs, a bowl; a basket with the hint of food or drink in it. It is a cluttered room; it could even be the corner of a shop or a kitchen. From the wall behind her, a bust of Juno peers down: patroness of commerce, she is a sort of guardian of the woman’s goods and wealth. Beside Juno, a huge map larger than the counter and the accountant herself: Africa, East Asia, the Americas; dragons and serpents beyond.
Sadly, the account keeper has fallen asleep. Though her pen is still poised over the ledger – the top one of a stack of three ledgers in fact – her eyes are closed, her head slumped into her hand. The perils of overwork; the weight and tedium of calculation with number symbols on paper.
Nicolaes Maes, who painted The Account Keeper in the Netherlands in 1656, was one of Rembrandt’s most talented pupils, and had a long career as a portraitist as well as producing a few dozen works like this one depicting Dutch interiors. Most were closely observed studies of women at work: making lace, peeling vegetables, praying, reading – or doing the accounts. Maes’s own craftsmanship was exquisite, reminiscent of Vermeer in its control of colour and light. With The Account Keeper, he created a rare record of what the practice of accounting looked and felt like in the Dutch golden age.
Nicolaes Maes, ‘The Account Keeper’.
