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’.

Saint Louis Art Museum, object number 72:1950. Public domain.

The painting only shows a tantalising glimpse of the accounts book itself: but accounts books survive in huge numbers from the sixteenth century to the nineteenth, deriving from households, businesses and individuals. Each page could be as simple as a list of items and their prices, with a total at the foot of the page (and some working by the side, providing a record of how that total had been arrived at). Other books were elaborately ruled in multiple colours, with separate columns for profits and losses, and the ‘total’ column updated after each entry: much like a modern bank statement, and manifesting the same awkward combination of dullness and importance. If something is not right here, it can mean that something is terribly wrong – fraud, theft, embezzlement – in the real world of counters and goods. Maes’s accounts book looks like the more complex kind, and the fact that there are two more volumes stacked closed on the counter suggests a business or household of some complexity, in which different categories of income and expenditure have to be tracked separately, or in which a single run of accounts has to be compared over a long period of time.

How did you learn to do such things? For this, too, a great deal of evidence has survived. Now that counting was well established as part of reading, and calculating as part of writing, you learned it wherever you learned your reading and writing. At home or at school, you made a ‘cyphering book’, in which you wrote out definitions and exercises starting with ‘numeration’ and passing through addition and subtraction, multiplication and division until you – possibly – moved on to more advanced procedures like reasoning about proportions or keeping accounts. The word ‘ciphering’ came from ‘cipher’: that is ultimately from Sanksrit sunya and Arabic sifr: a little reminder of the long heritage of this way of recording numbers, and this curriculum of study.

Hundreds of the books survive, from all over Europe and North America. They typically consisted of a quire of paper sewn together, in a generous size: often 30cm by 20cm. Children from well-to-do families might buy one ready-made, but most students would have sewn, roughly bound and decorated the book themselves. Some ruled lines to help them write: these were meant to be fair-copy books, but the students using them began as young as nine or ten and often had quite uncertain handwriting. Many, all the same, turned out careful and neat, with different inks elaborately used, headings, borders and careful management of the space on the page to make a truly personal textbook. Illustrations were common, ranging from the accomplished watercolour to the childish doodle.

The basic technique of instruction was always the same. The teacher would provide a ‘rule’ for some arithmetical process – numeration, addition or subtraction, say – which the student would copy into the book. The teacher would also provide some examples of the use of that rule, to try out as practice on slate or waste paper. The students, called forward individually, would show their exercises. If they were wrong, they had to be done again; once they were right they could be copied into the ciphering book.

Estimates varied, even at the time, as to how well this worked. Sometimes wrong sums found their way into the ciphering book if the teacher was careless; sometimes students got the answers right merely by copying from one another. And sometimes work that was correct but used a slightly different method from the teacher’s was judged wrong unfairly. It could be a frustrating system, and it was certainly a repetitious and a laborious one. A student who stayed in school for several years might produce several hundred pages of ciphering work; some students filled more than a dozen separate books. The ciphering system assumed that counting was part of reading and writing, and that the way to learn arithmetical operations was by repetitiously writing them down: that the route to the mind was through the pen and the hand. There was no mental arithmetic, and precious little speaking at all.

The curriculum started with ‘numeration’: learning to read and write the Arabic numerals fluently. Most students wrote out a table showing how to read Arabic numerals up to perhaps eight or ten digits long: hundred millions, ten millions, millions … all the way down to hundreds, tens and units. After that, practice: write eighty-five in numbers. Write one hundred and eight. Write one million, eleven hundred and one. You would do these on scrap paper until you or your teacher thought they were right, then copy them into the book.

Often there was an emphasis on large numbers. Up to nine digits was typical, but some textbooks went up to twenty or more; one went to seventy-eight digits, the perhaps ludicrous ‘duodecillion’. Large numbers provided more arithmetical work per exercise, of course; they also gestured none-too-subtly at prosperity, motivating students – or trying to – by promising that if they learned this material well they too could one day deal in hundreds of cows, thousands of barrels or millions of silver coins.

Following numeration came the four arithmetical operations: addition, subtraction, multiplication and division. The syllabus, its order, and even some of the details of the algorithms for calculation were the same as they had been in medieval Europe, in the Maghrib and Baghdad, in India itself. Next, the ‘rule of three’, which taught how to find the missing number from a set of four linked quantities. If it takes three men two days to build a certain wall, how long will it take five men to build the same wall? The same subject, with some strikingly similar examples, was covered in Bhaskara’s textbook.

Once a student had done all that, she unfortunately had to start all over again with so-called ‘denominate’ numbers: that is, numbers denoting money or weight or measure. The systems in use were numerous; just for weight there were troy weight, avoirdupois weight and apothecaries’ weight, and there were many more for volumes, lengths, areas and of course money. All of this taught the arithmetic that would enable you to keep domestic or small-business accounts in adulthood, dealing with the costs of food, clothing and equipment as well as bargains, barter and loans:

Divide twelve pounds, 17 shillings and six pence equally among four men.

Bought a hogshead of molasses for twenty-one pounds and five shillings. How many gallons did it contain, valuing the gallon at two shillings and six pence?

Two merchants A and B agree to trade together. A puts in £332 for four months, and B £475 for seven months; they gained £520. What was each merchant’s part of the profits?

Many ciphering books tailed off rather than finishing neatly: students left school, and the series of definitions and examples came to a more or less abrupt halt. Nevertheless, adults frequently retained their ciphering books as personal textbooks, using them to display their competence at arithmetic and handwriting to schools, colleges or potential employers, or referring to them for the techniques of calculation that were useful in daily life. Some added to them, modified or corrected the original exercises. Some books were passed down among siblings or from parent to child; some became dense palimpsests recording the arithmetical education of a whole family. However orderly the sequence of definitions and exercises, chaos had a tendency to creep in around the edges.

Maes’s Account Keeper is a tranquil scene, invested in the details of wood and plaster, crockery and curling paper. It says something about the dignity of household work, and of accounting in particular. It speaks of the dedication it required, the time and devotion to be given to the task. Of an activity worth pursuing to the point of exhaustion. While the woman’s clothing is that of a solid, not a wealthy, householder, three separate ledgers would be a lot for a merely domestic set of accounts. The map on the wall behind her seems a clear hint that her ledgers dealt with trade beyond the borders of the Netherlands: perhaps far beyond, to the ends of the world she knew.

Did the artist intend to moralise about the woman and her labours? The picture has points in common with traditional depictions of avarice or worldliness: the prominent scatter of coins; the prominent image of, literally, the world. It would not take much to turn it into a depiction of a miser in a counting house: the resulting accounts book might look almost identical, but it would be designed to provoke a rather different response and of course tell a rather different story about what counting and the practices derived from it do to people.

Yet this woman is not a miser, and the fact that she has fallen asleep makes her look just as much like an emblem of melancholy, perhaps even of sloth. The ledgers are about to fall to the floor, the orderly scene to collapse (the woman to wake, disturbed by the crash). It was Maes’s style to look on such things with a gently humorous eye: his other pictures of people asleep invite a laugh or at least a wry smile, as dreamers have their pockets picked or risk a scolding for neglecting their chores. Here, Juno herself – the guardian of goods and wealth – also seems to have nodded off, echoing the sleepiness of the account keeper, and hinting that matters are not as well stewarded as they seem. And the world map, the sign of the woman’s ambitions and international connections, is half a reproduction of contemporary maps, half a fantasy, embellished with fantastic beasts copied from maps of the stars.

Keeping accounts – writing numbers down – always comes with a tinge of anxiety that the stability and solvency they record might one day collapse. It always has the character of an island of order and rule, threatened on every side by chaos. It was the character of the Renaissance and the Enlightenment to cultivate and expand those islands of order, bringing more and more aspects of the world under the purview of numbers, tables and facts. But a glimpse of chaos often remains at the edges.