I

The notion that certain forms of numeracy were characteristic of a civilized state was a distinct outcome of complementary developments in early modern mathematics, print culture, global travel, and notions of civility. From the 1580s, Elizabethan state interest in practical schemes to bolster trade saw the number of London-based mathematical instrument-makers grow significantly, alongside teachers capable of instructing in matters such as cosmography and navigation.Footnote ¹³ The growth of practical mathematics was closely tied to that of print. Mathematicians wrote books as a means to advertise their services as teachers and instrument-makers, while books themselves could play the role of instruments.Footnote ¹⁴ Such tools were important means of self-teaching for aspiring mariners.Footnote ¹⁵ Through the seventeenth century, London emerged as a centre for the publication of mercantile texts, second only to Amsterdam in scale of production.Footnote ¹⁶ By the later Stuart period, as Philip Beeley observes, the city boasted a thriving ‘knowledge community’ which included mathematicians and instrument-makers alongside merchants and booksellers.Footnote ¹⁷ Crucially, these developments not only stimulated the circulation of mathematical knowledge, but also grew the reading public receptive to numerical information in print.

This base of practitioners played an important role in the growth of mathematics as it rose to pre-eminence as a tool of natural philosophical inquiry, state administration, and global trade.Footnote ¹⁸ Maritime commercial ventures necessitated more advanced mercantile and navigational techniques provided by figures such as Thomas Harriot, whose calculations and refinement of instruments were key to early American voyages.Footnote ¹⁹ From the mid-seventeenth century, travellers were equipped with measuring instruments to gather the commensurable numerical data on which scientific inquiry increasingly relied.Footnote ²⁰ The foundation of learned societies in the 1660s, particularly the Royal Society in England and the French Académie des sciences, gave new institutional support to scientific endeavour, while Newton’s cosmology, exemplified in the Principia of 1687, demonstrated the power of mathematics in understanding the universe.

These institutions were instrumental in sponsoring global exploration with the aim of discovering more about foreign peoples, as directed by questionnaires of ‘heads’ and ‘enquiries’ issued to travellers.Footnote ²¹ Though there were no explicit questions regarding numeracy, the early issues of the Royal Society’s Philosophical Transactions attest to some interest in numerical matters across cultures, as in one author’s examination of Chinese characters and counting.Footnote ²² As Daniel Carey shows, John Locke pursued his own ethnographic interests through questionnaires in the Royal Society fashion, which included more explicit enquiries regarding numerical practices in Senegal and India.Footnote ²³

Such developments gave new vigour to learned discourses about the value of numerical knowledge, centring around the term ‘arithmetick’. The word ‘numeracy’ emerged only in the twentieth century, but the early modern use of ‘arithmetick’ was similarly broad.Footnote ²⁴ In one sense, it was the name for a branch of study, a pillar of the classical quadrivium and the basis for further study in geometry and mathematics. The term was also used to describe both the act of counting and different systems of doing so, as in Locke’s question about whether in ‘the east … their arithmatike turns at ten’.Footnote ²⁵ ‘Commercial arithmetic’ was specific to use in trade. The new science of ‘political arithmetick’ involved quantification of people, land, and other economic factors, and proved a valuable tool in colonial ventures.Footnote ²⁶ A crucial facet of this burgeoning discourse was that ‘arithmetick’ came to be understood as a chiefly literate enterprise. Previously common object-based modes of reckoning such as the counting board declined in use, and by the first decades of the seventeenth century Hindu-Arabic figures had largely replaced Roman numerals as the most common number-writing system in England.Footnote ²⁷ Textbooks detailing material and gestural counting such as Robert Recorde’s Ground of artes (c.1540) were replaced by a raft of new titles teaching the subject by only written means.Footnote ²⁸

‘Arithmetick’ was increasingly seen as an indispensable tool of trade, key to national prosperity. In England, Royal Society fellows such as Sir William Petty, father of political arithmetic, recommended its study ‘for all Men in generall’ for its ‘Vast use in all Practical Arts’.Footnote ²⁹ Locke’s assessment was yet more forceful: arithmetic should be taught to students, for it was ‘of so general use in all parts of Life and Business, that scarce anything is to be done without it’.Footnote ³⁰ These appeals to arithmetic’s intrinsic benefits might be regarded as an attempt to legitimize the subject as a form of cultural capital.Footnote ³¹ It was at least implicit in all these assertions that this ‘new arithmetick’ warranted vindication. As Steven Shapin observes, the leading Royal Society figure Robert Boyle had reservations about the proper place of mathematics in natural philosophy.Footnote ³² There was certainly criticism for such methods, as, for example, in the vicious opprobrium of the late Stuart physician Archibald Pitcairne’s ‘mathematical medicine’.Footnote ³³

In 1701 the mathematician John Arbuthnot offered perhaps the era’s most sustained defence of the practice in his Essay on the usefulness of mathematical learning, which argued that the neglect of written Arabic numeral arithmetic ‘would go near to ruine the Trade of the Nation’. Arbuthnot’s treatment of ‘arithmetick’ in the Essay was emblematic of the new social value attached to written mathematical practices, in contrast to the old ‘Roman way of notation by Letters’. Tellingly, he further conflated ignorance of arithmetic with barbarism, drawing on the example of ‘some Americans, who can hardly reckon above twenty’, and ‘those barbarous Nations’ that did not have time measuring devices. Likening the pursuit of numerical knowledge to conquest of land, he described gaps in knowledge of the mathematics of anatomy as a ‘Terra incognita’.Footnote ³⁴ As we shall see, such assimilation of numerical knowledge and imperialist rhetoric was in large part a consequence of travel writing.

The development of such discourses belonged to a recasting of numerical knowledge on socially hierarchical grounds. Once thought necessary only for tradespeople, arithmetic was increasingly regarded as a subject of general mental refinement worthy of gentile study.Footnote ³⁵ Spurred by the writings of Petty, Locke, and Arbuthnot, the subject gradually entered the domain of formal education where social barriers were most rigid. In schools, arithmetic lessons were generally limited to male students and those who could afford to remain in education past the acquisition of basic literacy.Footnote ³⁶ Arithmetic was also central to the educational programmes of the Society for Promoting Christian Knowledge and its Scottish counterpart, founded in 1698 and 1709 respectively, whose growing numbers of schools in Britain and North America sought to impose orthodox religion and, in the case of the Scottish Highlands, eradicate Gaelic language and culture.Footnote ³⁷ New social hierarchies thus emerged around numerical knowledge which in some ways mirrored the colonial power structures spurred by travel writing. It was on such grounds that a 1790 report on education in colonial Quebec observed that the absence of elementary education in reading, writing, and arithmetic would leave ‘the lower classes in all countries … in a state of base barbarism’.Footnote ³⁸

II

Discourses conflating ignorance of written arithmetic with barbarism had roots in sixteenth- and early seventeenth-century notions of civility. Manners, education, deportment, and abstinence from certain forms of violence were all regarded as characteristics of civilized individuals, accompanying societal attributes such as the presence of laws, cities, and advanced modes of land cultivation.Footnote ³⁹ Such notions coloured the views of early travellers to the new world, such as Bartolomé de Las Casas and José de Acosta, who saw European and Chinese cultures as superior to those of the Caribbean.Footnote ⁴⁰ Yet civility was a matter of degree, dependent upon the manner in which it was performed. The ability to write may have differentiated the civil and savage in general, but particular forms of writing and language were telling markers of social distinction.Footnote ⁴¹ Quelling the urge for violence might generally denote a civilized character, but the polite challenge to a duel was another matter.Footnote ⁴² The cultivation of literacy and mathematics too became measures of civility. Bacon, Hobbes, and Descartes were among those for whom the arts of calculation and measurement differentiated more or less civilized places.Footnote ⁴³ Social value was invested not simply in the ability to use numbers, but rather to do so in specific ways familiar to European elites.

In turn, early seventeenth-century English travellers viewed similarities between foreign numerical practices and their own as markers of civility. As Jessica Otis shows, tally sticks were used by the English Exchequer as a means of accounting throughout this period, and writers such as John Smith and Samuel Purchas interpreted the use of similar objects elsewhere as evidence of English manners.Footnote ⁴⁴ In the Generall historie of Virginia (1624), Smith observed that the use of sticks akin to English tallies among some local inhabitants made them ‘the most civill and tractable people’ that his party encountered.Footnote ⁴⁵ Purchas exhibited a keen sense of varying degrees of civility in his 1613 history of travel. Writing on Peru he remarked that ‘They have another kinde of Quippos … with which they will cast up hard acounts which might trouble a good Arithmetician with his pen’, and by the same token denigrated those in Albania who, according to the ancient Greek geographer Strabo, ‘were so simple, that they neither had use of money, nor did they nu[m]ber above an hundred’ and were ‘ignorant of weights, measures, warre, civility, husbandrie’.Footnote ⁴⁶

At an early date, then, civility offered a discursive framework within which European and non-European numeracy were compared. As recent scholarship has emphasized, however, notions of civility which relied increasingly upon global, cross-cultural influences precipitated tensions between elite European codes of behaviour and the violent actualities of imperial conquest.Footnote ⁴⁷ Similar ambiguities lay between the representation of numerical knowledge and the reality of domestic practices. Despite the enthusiasm for new mathematical methods, they remained confined largely to a relatively small group of educated, metropolitan men. Meanwhile, in Britain as elsewhere, non-written practices increasingly condemned as barbarous were declining only slowly, especially among culturally, linguistically, and economically marginal groups.

It was for that reason that the buccaneer and surgeon Lionel Wafer, writing in 1699 of a voyage to the Isthmus of Darien in Central America, recognized similarities between the vigesimal counting he observed among its inhabitants and the Scots and Irish Gaelic with which he had grown up.Footnote ⁴⁸ Non-decimal counting also existed in languages such as Welsh and Manx, while the sixteenth-century physician Andrew Boorde believed that ‘No Cornysheman dothe nomber above .xxx.’.Footnote ⁴⁹ Some authors, including the early eighteenth-century mathematician George Brown, advocated the practice of finger counting for basic numeration and even more complex calculation.Footnote ⁵⁰ In some rural localities, shepherds still counted sheep by the score in regional dialects before notching their crooks and starting again.Footnote ⁵¹ Accounting using rudimentary tally sticks remained common at the turn of the eighteenth century, especially in trades for which it was customary, such as brewing and baking.Footnote ⁵²

Just as some sixteenth-century English commentators had worried that their own vernacular might be barbarous, the endurance of numerical practices in eighteenth-century Britain akin to those observed beyond Europe was an uncomfortable home truth with which few authors grappled.Footnote ⁵³ Writing in 1726, Daniel Defoe – himself a notoriously poor accountant – used the example of an illiterate shopkeeper who counted using spoons to show ‘what an absolute necessity there is for a tradesman to be very diligent and exact in keeping his books’.Footnote ⁵⁴ Lady Mary Wortley Montagu’s remark in a letter of 1753 that ‘the knowledge of numbers is one of the chief distinctions between us and the brutes’ may have been indicative of prevailing sentiment, but it was also quite hypocritical of someone who, according to a Victorian ancestor, ‘knew nothing of common arithmetic’.Footnote ⁵⁵

The way in which even those of limited numerical ability stressed its importance is indicative of the shift in attitudes towards numeracy over the seventeenth century. A new ‘arithmetick’, spearheaded by metropolitan practitioners and learned institutions, perpetuated in the literate media of manuscript and print, and oriented towards global trade, served to antiquate traditional practices. The assimilation of mathematics and civility entrenched new social hierarchies, marginalizing particular groups within the British Isles according to the complexion of their numerical knowledge. As we shall see, these developments also provided a framework within which travellers understood non-European numeracy, bolstering global hierarchies of numerical knowledge as notions of civility crystallized into the concept of civilization.

III

The growing enthusiasm for arithmetic emerged just as new possibilities for colonial expansion saw the production of travel literature increase markedly. The English Short Title Catalogue lists an average of only forty-seven editions of travel books produced per decade in the first half of the seventeenth century, rising to seventy-eight per decade across the 1650s and 1660s, before fully 168 per decade from the 1670s to 1690s. After a slump in the 1710s, production again rose steadily through the mid-eighteenth century before surging to highs of 549 editions produced in the 1780s and 771 in the 1790s.Footnote ⁵⁶ The change was equally noticeable in France, where one commentator of 1680 estimated that as many as 1,300 voyages had been printed.Footnote ⁵⁷ The production of these works was itself symptomatic of the colonial jostling that took place between these northwestern European powers. As the English naturalist and surveyor John Lawson lamented in the preface to his New voyage to Carolina (1709), too few travellers were capable of giving robust accounts, in which matters, he thought, ‘the French outstrip us’.Footnote ⁵⁸ When it came to representing non-Europeans, however, these texts exhibited some cohesion, often referencing one another to broaden accounts of territories beyond the writer’s own observations. Together they fostered a unified, self-reflexive Western European identity based on shared practices and values contrasted with those beyond Europe.Footnote ⁵⁹

Thus when travellers came to describe the numerical practices of peoples they encountered, European ‘arithmetick’ was the standard to which they were compared. As recent analyses have stressed, such Eurocentric epistemologies marginalized indigenous agency and knowledge systems. Jessica O’Leary shows that travel accounts of Brazil appropriated Tupi women’s knowledge into a European classificatory system that obscured its origins.Footnote ⁶⁰ Elisabeth Leake proposes similarly that the European notion of the ‘tribe’ bolstered imperialist rhetoric by imposing an inflexible way of understanding diverse global societies that insisted upon their otherness.Footnote ⁶¹ The imposition of ‘arithmetick’ as a rubric for a heterogeneous range of numerical practices had much the same effect, placing non-European knowledge on an inferior footing, even if the nature and purposes of such numeracy differed greatly from European scientific, commercial, or political arithmetic.

Beginning in the early seventeenth century and intensifying in the decades around 1700, the literate forms of numeracy increasingly advocated by Western Europeans were contrasted with oral, embodied, and object-based modes of reckoning observed elsewhere. These foreign practices were themselves conceptualized with varying levels of approval or reproach. Portrayals ranged from the mysterious, novel, and exotic, to more hostile representations of savagery and backwardness. In all cases, they reflected a sense of otherness which served to reinforce the perceived superiority of European numerical knowledge.

An early text in which the numeracy of different localities was juxtaposed explicitly was Lewes Roberts’ Merchants mappe of commerce (1636). Roberts’ guide to global trade included observations on numerical practices appraised according to European commercial standards. One who was ignorant of ‘the Art of Numbring or Arithmetique’, he believed, ‘may not challenge to himselfe the Title of a Merchant … nor hardly deserve the attribute of a rationall man’.Footnote ⁶² Comparing England and Ireland, Roberts characterized skill in navigation as a capacity which separated civility and barbarism.Footnote ⁶³ Regarding localities in West Africa, he attributed the inhabitants’ lack of trade skills to their ignorance of ‘acompting and reckoning’ and reliance on finger counting. Forms of innumeracy were understood not simply as defects in a particular branch of knowledge, but evidence of more general ignorance.Footnote ⁶⁴ As Morgan shows, such perceptions of inability in trade were used to justify the pursuit of enslavement.Footnote ⁶⁵ Roberts depicted the practice of accounting on wooden tallies in Beijing and its surrounds more favourably, finding it ‘not much unlike the custome of tallies in England’.Footnote ⁶⁶ Roberts’ Mappe was highly influential, enjoying multiple editions to 1700 and proving a mainstay in merchants’ libraries into the eighteenth century.Footnote ⁶⁷ At an early stage, then, Eurocentric standards of commercial arithmetic provided a lens through which global numerical practices were viewed.

Such preoccupations were inherited by later seventeenth- and eighteenth-century writers on China and the Americas. In the former, where Europeans wielded little colonial influence and the purpose of voyages tended towards diplomacy and discovery rather than domination and dominion, accounts were reverent of the country’s apparent economic prosperity, robust social order, and technological and literary sophistication.Footnote ⁶⁸ In turn, travellers depicted Chinese object-based numerical practices sympathetically, while still insisting upon their difference. Such was the case in Johan Nieuhof’s account of a Dutch East India Company trade mission from Canton to Peking from 1655 to 1657, which was translated into English in 1669. Though he was somewhat critical of Chinese mathematics, Nieuhof praised shopkeepers who ‘use Boards to tell upon, which are full of Holes; yet they are so ready at it, that with a Peg they know how to cast up an Accompt with as much Method and Expedition, as the most skilful European with Counters’.Footnote ⁶⁹ A similar appraisal of Chinese methods was visible in the account of French Jesuit traveller and mathematician Louis Le Comte, this time in comparison to written calculation. Le Comte’s journey to Beijing in the mid-1680s was sponsored by the Académie des sciences and guided by a questionnaire inquiring specifically of ‘the perfection and defects of their mathematics, astrology, philosophy, music, medicine, and pulse-taking’.Footnote ⁷⁰ He described the use of the suanpan and remarked that Chinese accountants reckoned ‘with such great dexterity and easiness, that they will keep pace with a Man, let him read a Book of Accompts never so fast’.Footnote ⁷¹ Within broader accounts that exhibited a keen interest in Chinese material culture and exerted significant influence on European sinology, Nieuhof and Le Comte lauded object-based accounting techniques for their sophistication, if only in comparison to European methods.Footnote ⁷²

The construction of otherness was more ambivalent in an account produced over a century later by the naturalist and Royal Society Fellow Sir George Staunton, who joined the British Macartney Embassy to China from 1792 to 1794. The dismissal of British astronomical instruments gifted to the Qianlong emperor as curiosities rather than examples of scientific progress exposed a divergence between Chinese and European science and contributed to the decline of British reverence for Chinese culture.Footnote ⁷³ Within this context, Staunton’s 1797 account of the embassy framed Chinese arithmetical practices less favourably, rendering arithmetic performed with the suanpan and recorded using Chinese characters as difficult and tedious compared to ‘the concise view of the same quantity in Arabic figures’.Footnote ⁷⁴ Though Staunton’s account differed to those of Nieuhof and Le Comte in its sympathies, each of these works shared a view of the fundamental otherness of Chinese object-based counting.

The assertion of hierarchy was altogether more forceful in depictions of the Americas. Here Europeans exerted far greater colonial power, and numeracy was a focus of more hostile and explicitly value-laden attitudes. Frequently observed oral, gestural, and object-based counting and calculating methods were seen as inadequate and primitive. The perception of their inferiority was especially clear in the Huguenot missionary Charles de Rochefort’s account of the Caribbean, translated into English in 1666, in which the numeracy of Caribbean islanders and both Chinese and European accountants was appraised. Rochefort perceived the islanders’ ‘sottishness and simplicity’ in that ‘they cannot count a number exceeding that of the Fingers of their Hands and the Toes of their Feet, which they shew to express the said number, what exceeds it surpassing with them all Arithmetick’. Echoing a geographical tradition which posited that civilization was spreading gradually across the globe from east to west, Rochefort compared these practices to those of the Chinese, ‘who are such excellent Accomptants, that in a moment they cast up such Sums as it would trouble us much to do’.Footnote ⁷⁵

Such observations of embodied practices and the absence of large number words became characteristic tropes of British and French naturalists’ writings on the Americas. John Josselyn’s 1674 description of New England, which was dedicated to the president and fellows of the Royal Society, detailed people who ‘skill not’ in arithmetic, ‘reckoning to ten upon their fingers, and if more doubling of it by holding their fingers up’.Footnote ⁷⁶ Similarly, in a popular 1722 account of the West Indies, the French botanist Jean-Baptiste Labat described people who could only count to ten, after which they reckoned with peas placed in gourds or knots in cords.Footnote ⁷⁷ Writing in 1745, the mathematician and fellow of the Académie des sciences, Charles-Marie de la Condamine, who became a widely regarded authority on South America, observed Yameo speakers along the Tigre River who had no number words past three. This characteristic he believed they shared with inhabitants of Brazil.Footnote ⁷⁸ Jonathan Carver’s 1778 Travels through the interior parts of North America noted people who had ‘no idea of arithmetic; and though they are able to count any number, figures as well as letters appear mysterious to them, and above their comprehension’. In this late eighteenth-century text, which was published with the assistance of Royal Society president Joseph Banks and saw some sixteen editions to the end of the century, it was simply assumed that ‘arithmetic’ was an inherently written endeavour, although Carver did observe that reckonings of time were ‘very rationally divided by the Indians’.Footnote ⁷⁹

Indeed, non-European temporalities were another aspect of numeracy frequently commented upon by travellers to the Americas. Rochefort encountered Caribbean peoples who counted days using peas, knotted cords, and notched sticks, and further remarked that they had no concept of age, nor how long it had been since their first encounter with Spanish settlers.Footnote ⁸⁰ Josselyn likewise observed inhabitants of New England who reckoned age ‘by Moons, and their actions by sleeps, as, if they go a journie, or are to do any other business they will say, three sleeps me walk, or two or three sleeps me do such a thing’.Footnote ⁸¹ In The history and present state of Virginia (1705), Robert Beverley described locals who had ‘no distinction of the hours of the Day, but divide it only into three parts. … And they keep their accounts by knots on a string, or notches on a Stick, not unlike the Peruvian Quippoes’.Footnote ⁸²

In works such as Beverley’s, explicit appraisal of numerical knowledge made way for less prejudiced ethnographic description. Similarly, in the wake of the disastrous Scottish attempt to colonize the Isthmus of Darien in the 1690s, Lionel Wafer published a perceptive account of non-written numeracy among its inhabitants. He noted that they counted ‘by Unites and Tens, and Scores, to an Hundred; beyond which I have not heard them reckon’, and also counted with grains and expressed quantity with locks of hair. Wafer took seriously the task of delineating these practices precisely. He was careful to record the names of number words and, as we have seen, even compared the practice of counting in scores of twenty to the Gaelic of his childhood.Footnote ⁸³

Native American numerical games were a particular focus of travellers’ admiration. John Lawson observed one involving ‘a sort of Arithmetick’ in which players guessed the number of sticks in a bunch thrown by their opponent. ‘Some are so expert at their Numbers’, he remarked, ‘that they will tell ten times together, what they throw out of their Hands’.Footnote ⁸⁴ This observation was repeated almost verbatim in the Irish physician John Brickell’s Natural history of North Carolina (1737).Footnote ⁸⁵ Beverley observed a similar game in Virginia, involving ‘handfuls of Sticks or hard Straws, which they know how to count as fast, as they can cast their Eyes upon them, and can handle with a surprizing dexterity’.Footnote ⁸⁶

Yet even these less hostile portrayals of numeracy contributed to the construction of otherness through the framework of the travel account. The very existence of these books, themselves products of European numerical civilization, was a point of juxtaposition to the cultures delineated in their pages. Many texts established from the outset their authors’ affinity for mathematical methods. In the accounts of La Condamine, Labat, and Lawson, intricately printed and mathematically scaled charts, followed by discussion of navigation by precise timing, measurement, and bearing, asserted the numerical competence of their makers.Footnote ⁸⁷ La Condamine’s work began with a map of the Amazon basin drawn from astronomical measurements, while continual recourse to numerical language served as a rhetorical strategy to legitimate an account which sometimes relied upon suspect second-hand sources.Footnote ⁸⁸ By using these books as vehicles for communicating European numerical knowledge, travel accounts further assimilated numeracy and literacy through the medium of print. In Carver’s account this relationship was illustrated quite literally by an anecdote in which Naudowessie people (later known as Sioux or Dakota) took interest in a plate in Carver’s astronomy book. Carver showed that he could tell the number of leaves in the book without counting them simply by reading the folio number, while the supposedly uncomprehending Naudowessie instead concluded that the book was a spirit communicating with its owner.Footnote ⁸⁹

Whether or not the accounts of numerical practices in these works were entirely true, and however explicit were the value judgements imbued therein, numeracy in non-European localities was most often portrayed as fundamentally alien. Descriptions which arbitrarily bound together different counting, calculating, and measuring practices under the rubric of European ‘arithmetick’ served to construct co-dependent hierarchies of numeracy and otherness that rested upon boarder colonial power structures.

IV

The reception and subsequent incorporation of travel writing into both learned and lay culture in Britain saw numerical practices encompassed within broader conceptions of human rationality and civilization. John Locke’s highly influential philosophy of cognition and language was inspired to some degree by accounts of foreign lands, including those supplied in travel literature.Footnote ⁹⁰ As his sustained epistolary campaign of inquiries attests, Locke actively sought information on global numerical matters. Such insights informed discussion of counting in his Essay concerning humane understanding (1690), which argued that numbering was a fundamental cerebral capacity. While numeration itself was a straightforward task, Locke reasoned, without words for each number, counting to large sums was difficult. For demonstration, he cited the example of Native Americans who could not count to 1,000 because they had no word for that number and were ‘unacquainted either with Trade or Mathematicks’ even though they were ‘otherwise of quick and rational parts enough’. Instead they conveyed quantities using gatherings of hair, rather like people on the Isthmus of Darien described by Wafer. Locke cited Jean de Léry’s Histoire d’un voyage faict en la terre du Bresil (1578) in observing that the Tupinambá had no names for numbers above five.Footnote ⁹¹ Subsequently, in Some thoughts concerning education (1693), notions of numeracy’s value underlay Locke’s case for the necessity of teaching arithmetic to students.Footnote ⁹²

In the second half of the eighteenth century conceptions of non-European numeracy became more deeply entrenched, understood not simply as products of ignorance but a more ‘primitive’ state of being. Long-standing notions of civility were incorporated into a teleological view of human development as a civilizing process.Footnote ⁹³ The emergence of stadial theories of history, which posited that human populations each progressed from common hunter-gather origins through stages of development from pastoral to agricultural and finally commercial society, provided an intellectual framework within which travellers’ descriptions were interpreted. The othering of foreign peoples well suited stadial historians’ cosmopolitan emphasis on a common European civilization which had reached a state of maturity.Footnote ⁹⁴ Tracing the emergence of civil society meant investigating the character of localities considered further behind on the developmental timeline, and the rich source material travel writing appeared to offer led to the incorporation of numeracy into civilizing narratives.Footnote ⁹⁵

The most explicit conflation of numeracy and civilization within a stadial model appeared in William Robertson’s popular History of America (1777). As a leading minister, principal of Edinburgh University, and one of the best-selling historians of the age, Robertson was a key player in the Scottish Enlightenment. Despite his initial reluctance to adopt stadial theory, he became one of its chief exponents and offered particularly stark accounts of the superiority of European civilization, searching for evidence of social development in the behaviour, character, culture, and habits of American peoples.Footnote ⁹⁶

Within a broader examination of these matters, Robertson argued that:

The analysis rested upon an uncritical reading of several of the travel accounts discussed above. Like Locke, Robertson drew on Léry’s Historie d’un voyage, as well as later accounts such as those of Labat and La Condamine, in stressing the limitations of counting words among some peoples, ‘who cannot reckon farther than three; and have no denomination to distinguish any number above it’.Footnote ⁹⁸ By associating arithmetical ability with possessions and property, Robertson employed similar tropes as Morgan has identified regarding descriptions of African peoples.Footnote ⁹⁹ The notion that arithmetic had a long history in Europe would become a salient feature of Eurocentric histories of mathematics in more recent times.Footnote ¹⁰⁰

Reinforcing the universality of stadial theory, Robertson noted that these characteristics were not unique to Americans, ‘but all nations, which extremely rude, seem to be unacquainted with the art of computation’. Degrees of numeracy were taken as a yardstick for civilization. The Iroquois, who were said to have numbers up to 1,000, and the Cherokee up to 100, were considered ‘much more civilized than the rude inhabitants of Brasil, Paraguay, or Guiana’. As the references to travel writing in Robertson’s footnotes attested, it was by no means new to see arithmetic as a capacity that separated Europeans from others, but in the History of America, a stadial view of human development placed numerical ability at the heart of what made modern, commercial society. So integral was numerical knowledge to civilizations, thought Robertson, it ‘may be considered as one standard, by which to estimate the degree of their improvement’.Footnote ¹⁰¹

Yet such analysis was not accepted universally. As Stewart Brown notes, soon after the publication of Robertson’s History, he received criticism for an overly negative portrayal of American peoples, rendering them savages too readily, failing to condemn strongly acts of Spanish barbarity, and for reliance on European sources.Footnote ¹⁰² Accordingly, between the second and third editions of the Encyclopaedia Britannica (1778–88), the entry on America which drew heavily on Robertson’s account was revised.Footnote ¹⁰³ In turn, when Robertson came to produce his final work, a history of India entitled An historical disquisition (1791), he gained plaudits for his positive portrayal of Indian culture, including its development of mathematics and invention of the numerals and place value system which underpinned European arithmetic.Footnote ¹⁰⁴

Even if works such as Robertson’s were not always read uncritically, they were widely received and remained intellectually potent within the context of Enlightenment historiography. Mark Towsey has calculated that fully 50 per cent of Scottish private libraries held copies of the History of America, making it accessible to a wide range of middle-ranking borrowers.Footnote ¹⁰⁵ Nor was Robertson the only historian to adopt these ideas as their cultural purchase grew. The equation of European modes of numeracy with modern civilization was already present in French historian Antoine-Yves Goguet’s Origin of laws, arts, and sciences (1758), which saw fifteen editions in five European languages by the early nineteenth century, including an English translation in 1761. Goguet diverted from the Scottish four-stage theory of historical development to a simpler two-phase model dividing primitive and civilized nations. Again, arithmetic was said to have emerged first in those societies where it was needed for commerce and navigation, citing examples of ancient Egypt, Babylon, and China, as well as Peru and Mexico, ‘the only two great monarchies in America’. If these examples were not proof enough that ‘There was never any civilized and regularly governed nation, which had not some tincture of arithmetic’, Goguet further argued that in ‘certain nations lately discovered’ in America, number words were largely absent and quantity was shown with heaps of sand or handfuls of hair. Once again his sources were a familiar cast of travel accounts, including those of La Condamine and Wafer.Footnote ¹⁰⁶

The assimilation of numeracy and civilization was visible not only in these histories but in a wider variety of entertaining and didactic texts aimed at sizeable audiences. Daniel Defoe’s 1719 novel Robinson Crusoe depicted numerical practices in a manner clearly resembling more ostensibly ‘factual’ travel accounts. An avid consumer of such books, Defoe created an eponymous protagonist whose European numeracy appears uncalibrated with the realities of life on the Caribbean island where he is shipwrecked. Worried that he might lose his ‘reckoning of time for want of books, and pen and ink’, Crusoe resorts to carving notches in wood.Footnote ¹⁰⁷

Travel accounts themselves were increasingly recommended as educational tools to broaden children’s horizons, while their depictions of foreign peoples entered the pages of geography textbooks.Footnote ¹⁰⁸ The compleat geographer (1709) discussed the use of quipus in Peru and lauded its inhabitants’ effectiveness in counting with grains.Footnote ¹⁰⁹ Another favourable account was offered in A new system of geography (1764) by the widely read educational writers Daniel Fenning and Joseph Collyer, which praised highly trained Indian Brahmins, whom ‘Few people excel … in the practical part of arithmetic … for, from their infancy, they are taught to cast up sums by their fingers, without the help of a pen’.Footnote ¹¹⁰

Equally, more hierarchical portrayals of numeracy found repetition in educational texts. The Edinburgh schoolmaster Alexander Adam’s Summary of geography and history (1794) offered a view of Native American culture which drew heavily on ‘the admired works of Dr Robertson’, echoing the observation that some could not count past three, while the ‘more civilised’ Iroquois had words to 1,000.Footnote ¹¹¹ One late eighteenth-century encyclopaedia talked of the ‘barbarous manner’ of ‘Kamtschatkans, Koreki, and Kuriles’ people in the far east of Russia, who were ‘so totally ignorant of arithmetic, that it is said they cannot reckon above 20, and even that only by the help of their fingers and toes’. Regarding time, the passage continued, ‘They reckon ten months in the year, some of which are longer and some shorter; for they do not divide them by the changes of the moon, but by the order of particular occurrences that happen in those regions’.Footnote ¹¹²

V

In a much broader array of texts, therefore, observations that originated in travel accounts and gained intellectual grounding in the work of authors such as Locke and Robertson were expounded to a wider reading public. Ideas about the primacy of certain forms of numeracy had roots in transformations from the late sixteenth century, which saw the mathematics of trade and travel, practitioners and print, gain elite institutional backing. Simultaneously emerging discourses of global civility saw the assimilation of this ‘new arithmetick’ and elite European codes of behaviour in the early decades of the seventeenth century. For travellers in the late seventeenth and eighteenth centuries, therefore, these distinctly European forms of numeracy became an important lens through which to appraise the knowledge of foreign peoples. In turn, the broader dissemination of these ideas within studies of European civilization reinforced this hierarchy of numerical knowledge within Britain.

As such, the evolution of numeracy in this period constitutes an important means of understanding the relationship between social stratification in national context and global power structures. The literate and mathematical modes of numeracy lauded in travel accounts were certainly proliferating in the British Isles, but the process was gradual, and for all its benefits served to marginalize those unable to acquire these skills. For colonized peoples, moreover, the hierarchy of numeracy which travel writing helped to construct was not merely discursive. Such literature was ‘instrumental in the economy and machinery of Empire’, helping to justify colonial expansion.Footnote ¹¹³ As Morgan shows, numeracy was not only a discursive tool in travel writing on Africa for legitimating colonial projects, but its practical manifestation in logbooks, ledgers, and tables evaluating human lives was key to the administration of the transatlantic slave trade.Footnote ¹¹⁴ In ways that demand still further examination, numeracy was integral to broader knowledge hierarchies that bolstered power structures both domestically and abroad. In an important sense, then, notions of the pre-eminence of mathematics in today’s world are an inheritance of the way in which early modern global encounters were understood.