No state, no age, no man, nor child, but here may wisdome win
For numbers teach the parts of speech, wher children first begin.
And number beares so great a sway, even from the most to least
That who in numbring hath no skill, is numbred for a beast:
For what more beastly can be thought? nay what more blockish than
Then man to want the onely art, which proper is to man,
For many creatures farre excell mankind in many things,
But never none could number yet, save man, in whom it springs.
If numbring then be (almost) al, betweene a man and beast,
Come learne o men to number then, which arte is here profest.
—THOMAS HILL
The Arte of Vulgar Arithmetick, both in Integers and Fractions (London, 1600) [Original spellings slightly modified.]
In 1600, Elizabethan England had good reason to sing the praises of the art of arithmetic. The century just brought to a close had been characterized by continual expansion. Marked population growth, monetary inflation, and overseas discoveries stretched the boundaries of thought and custom in ways that called to mind size, number, and measure. Certainly it was an indulgence in hyperbole for the author of the paean to arithmetic, Thomas Hill, to single out counting as the essence of humanity, the skill that distinguished man from the beasts. But the exaggeration served to emphasize the paramount importance of arithmetic in a country rapidly becoming a center of commercial capitalism.
Arithmetic skills had in fact been only rather recently and selectively developed in England when Hill published his book. Although his poem went on to list nearly a score of occupations and stations in life for which knowledge of arithmetic was allegedly requisite, including shepherds, philosophers, and soldiers, Hill was not describing reality so much as he was attempting to widen the market for his book. In the early seventeenth century, arithmetic was a commercial subject, studied chiefly by men who followed the life of commerce. Arabic numerals did not come into general use until the middle of the sixteenth century, and the earliest English textbook of the rules of arithmetic appeared in the 1540s, the first of a trickle that became a flood around 1600. The expansion of the commercial sector of the population generated a demand for reckoning skills that books like Hill’s were attempting to meet.
As knowledge of arithmetic became slowly diffused in the seventeenth century, the activities of counting and measuring soon moved somewhat beyond the confines of commerce and found expression not only in improved methods of navigation and surveying, but in political thought as well. A new word, “pantometry,” appeared in seventeenth-century dictionaries, indicating that the measuring of all things could be considered to be one activity subsumed in one word. Arguments erupted over the true heights of mountains and the exact number of years since God’s creation of the world. Dozens of men acquired thermometers and barometers and consulted them thrice daily to measure the weather. The idea of commercial bookkeeping, transposed to the realm of government, led to “political arithmetic,” an accounting of the total resources of the country for the purpose of formulating rational policy.
To the twentieth-century mind, all these varieties of counting and measuring may seem to be only remotely related. Today arithmetic is applied to so many different phenomena that it scarcely seems instructive to draw connections between sports statistics and the consumer price index. They have numbers in common, but, then, number is the new literacy of the twentieth century ; sports and prices are only two of a very large set of items easily described in quantitative terms. In the seventeenth century, applications of number were still limited, and many obstacles stood in the way of an easy diffusion of numeracy. Measuring things was in fact a kind of sport, relatively few people were engaged in it, and the set of items thought to be susceptible of quantitative expression was rather small. It is therefore legitimate to ask what bounded that set. What did these instances of quantification have in common, and what limited the further application of numerical reasoning?
A key unifying element is that a mere handful of people produced most of the instances of quantification. An urge to measure had caught the fancy of a number of men, many of whom knew each other through the Royal Society of London or through local mathematics clubs. They became intent on measuring, counting, and weighing as a method of acquiring knowledge, and they moved easily from one sort of measurement to another. Their oddity was that they were trying to affix numerical certainty to what most people were content to leave to supposition, estimate, or qualitative description. A single author would write books on arithmetic, navigation, bookkeeping, gauging (measuring the capacity of containers), and interest rates. The man who coined the term “political arithmetic” had earlier in his career been a navigator and surveyor. A minister fascinated by the demography of his parish readily turned his hand to the problem of the heights of lunar mountains. The mathematical similarity in all these endeavors was precisely what attracted them. Prefaces to gauging and arithmetic books often specified ingenious persons of mathematical bent as part of the intended audience, in addition to bookkeepers or artisans with practical need of specific arithmetic techniques. That was how they saw themselves: as men disposed to delight in numbers.1
While the common feature of the new and diverse instances of quantification was their origin amongst a few men, the limiting factor, the constraint that kept quantification confined to certain areas, is less easy to discern. It might appear sufficient to observe that people measure only what they need to measure, and, as the need for precise knowledge expands, the practice of measurement expands with it. Thus it might be argued that the rise of commercial capitalism required that merchants have ways to figure their exact profits and losses on ventures. Trading in distant markets turned the art of gauging into a necessity, for one had to master several intricate systems of weights and measures to be able to compare products from one place with products from another. The mercantilist balance-of-trade theory suggested the utility of an office of import and export statistics. However, to assume so direct a link between a need for knowledge and an urge to measure and number is far too simplistic. In fact, most seventeenth-century capitalists, traders, and mercantilists got along quite well without using the quantitative techniques—double-entry bookkeeping, geometrical gauging, national account books—that the quantifiers praised. Something more than plain need, then, produced the urge to measure among the handful of quantifying men; clearly, something quite apart from need was at work in their efforts to measure the heights of mountains or the heat of the day. Why did they develop curiosity about the dimensions of some things but not others? One clue may lie in their frequent declarations that numbers brought satisfaction because they signified certainty, quite apart from any practical application. What was measured in the seventeenth century, then, was not only what was thought to be necessary but also what most urgently needed to be made certain.
Arabic numerals filtered into Europe in the tenth century through North Africa and Spain, but they were hardly known before the early thirteenth century. Leonardo of Pisa learned of them in his travels in Egypt and Syria, and on his return to Italy in 1202 he wrote a treatise describing an amazing number system that allowed basic computations to be carried out with the written numbers themselves. Roman numerals, the traditional notation system then in use in Europe, allowed only for the expression of quantity. There was no way to add, subtract, multiply, divide, or write fractions by manipulating the written symbols. Nevertheless, medieval and early modern Italian merchants found this traditional system entirely adequate to their purposes. In the twelfth and thirteenth centuries Italian city-states had developed rather complicated systems of trade, exchange, banking, and credit, and they managed to maintain all records and transactions with roman numerals. Basic operations were carried out with an abacus or on a counting board, a table top with lines representing an abacus and with coins or small stones used as counting tokens. Business transactions could be concluded right on the counter in this mechanical fashion, without the use of paper and without the necessity of doing simple sums in the head.2
Leonardo’s treatise on arabic numerals had little immediate impact on this traditional system of figuring. Indeed, in 1299 one Italian city, Florence, passed a law against the use of arabic numerals because the numbers were more easily falsified than the roman. It was argued that a zero could be altered into a six or a nine too easily, or a one to a seven. Even more grievous falsification was permitted by the addition of extra digits to the beginning or end of a number; 428 could grow to 4281 with the stroke of a pen. By contrast, roman numerals expressed a number with finality: usually the last letter in a group was given an extra mark to identify it as the last, for example xij instead of xii for 12.3 These reasons for rejecting arabic numerals show that the basic function of written numbers in medieval and early modern commerce was to record transactions, not to create manipulable bodies of data. Further, the limited availability of paper argued in favor of retaining the old notation, where calculations were carried out on counting boards and only the final results were committed to writing.
In the century between 1450 and 1550 arabic numerals gradually supplanted roman numerals in European commerce. The changeover in England came in the reign of Henry VIII. As that king’s title suggests, roman numerals did not entirely fade away. They continued to be used in written documents where specification of quantity or number was essential and where no calculation with the number was intended. Legal documents, wills, and court records retained the roman, as did the days of the month in almanacs, chapter numbers in books, and numerical designations of successive monarchs. But where calculation with numbers was desired, arabic numerals were adopted. Commerce, bookkeeping, navigation, and surveying all were altered under the impact of arabic numerals.
The first English books of arabic arithmetic presented their subject as a systematic set of rules governing the elementary manipulation of the new numbers. Robert Recorders Grounde of Arts appeared in 1543, followed by a second text, The Whetstone of Witte, in 1557. The titles squarely placed mathematics on the ground floor of intellectual development by suggesting that arithmetic was the foundation of knowledge as well as an activity that sharpened the mind. Both books were reprinted in many editions over the next century, and they established the model followed by other pure arithmetic texts.4 They began with numeration and explained the power of arabic place notation, where nine symbols plus zero could express any quantity. The basic operations with numbers were set forth, examples were worked, and a systematic development was sustained from each stage to the next, through the first four “rules” (addition, subtraction, multiplication, division), to fractions and decimals, up to surde (irrational) numbers and the “cossicke practise” (algebra).5 The reader of such texts learned mathematics as a coherent body of thought about numbers in the abstract.
At the end of the sixteenth century, books of applied arithmetic came on the scene. Most of them presupposed a thorough knowledge of basic arithmetic on the part of the reader. The authors were typically mathematical men whose interests spanned several fields. For example, in the middle years of Elizabeth’s reign the mathematician Leonard Digges wrote about navigation, gunnery (the flight of projectiles), architecture, and the all-encompassing “pantometria,” the art of measuring all lines, surfaces, and solids. Seventy-five years later one William Leybourn churned out at least twenty books on surveying, navigation, gunnery, “dialing” (constructing sundials), accounting, and “arithmetical recreations.” Since they assumed a preexisting mathematical competence, their readership could not have been wide; but the practical applications addressed problems of growing importance in Tudor-Stuart England.
Navigation and surveying, for example, had taken on new importance in this era of overseas expansion, settlement of new lands, and a reorganization of existing land-tenure arrangements at home. To some extent, navigation and surveying have always concerned men of property and of the sea. But there was no premium on mathematical precision when ships clung to coastwise routes or when property lines stayed in their traditional places, coinciding with natural features of the land, such as river bottoms. In the sixteenth and seventeenth centuries, vessels on the high seas required more exact methods of reckoning position. Disruptions of land boundaries, such as those caused by the confiscation of...