By J.L. Heilbron
I thank my fellow editors for discussions about the structure of this introduction.
In a famous essay on the usefulness of mathematics, the almost perpetual secretary of the Paris Academy of Sciences, Bernard le Bovier de Fontenelle, ascribed the improvement of literature in his time to the spread of the geometrical spirit. To whatever it touched it brought order, clarity, and precision. "A work on ethics, politics, criticism, and, perhaps, even rhetoric will be better, other things being equal, if done by a geometer." When these words were written in 1699, they had little purchase outside their own rhetoric. Sixty years later, according to the Encyclopédie of Diderot and d'Alembert, the idea had caught on: people had come to realize that the mathematical method, the method of exact definition and proof of propositions, was not restricted to mathematics. The big change had come with the writings of Leibniz's intellectual heir Christian Wolff. "Wolff made people see in theory, and especially in practice, and in the composition of all his books, that the mathematical method belongs to all the sciences, is natural to the human mind, and leads to discoveries of truths of all kinds."
From about 1760 on, the spirit Fontenelle and Wolff conjured up did indeed come to invest not only the sciences but also their applications to fields as distant as artificial languages and forest management. Even in the exact sciences, such as astronomy, optics, and
mechanics, we find a newly effective emphasis on precision. We believe that our thesis, that the later 18th century saw a rapid increase in the range and intensity of application of mathematical methods, is an important one. It amounts to specifying the time and surveying the routes by which what may be the quintessential form of modern thought first spread widely through society.
We translate l'esprit géometrique as "the quantifying spirit," which we understand to include the passion to order and systematize as well as to measure and calculate. We recognize that much posturing accompanied the exercise of this passion, that "mathematical" or "geometrical" did not mean the same things to all parties, and that numbers have their rhetoric, a truth we instantiate where appropriate. A novel feature of our work, which we consider to be a strength, is the inclusion of botanists and encyclopedists among the quantifiers. They qualify for inclusion in two ways: by the sheer quantity of the material they handled and by the unimpeachable warrant of the Encyclopédie . There we read that the spirit of a geometer is one of "method and justice." "It is a spirit of computation and of slow and careful arrangement (combinaison ), which examines all parts of an object one after another and compares them among themselves, taking care to omit none." The encyclopedist gives the example of biological classification, with recommendations about the ratio of the numbers of genera to those of species that the naturalist should try to establish.
All systematized or quantified knowledge and information current in the 18th century qualify for our consideration. To cover the topic we would have had to compile an encyclopedia. We have preferred to compose case histories and to concentrate on the second half of the century. In order to point up what we take to be characteristic of that period, however, most essays include material about earlier relevant developments and some extend coverage to the end of the Napoleonic era. In the high and wide realm of philosophical and programmatic discourse, rhetorical continuity runs through the
century, from Fontenelle through d'Alembert and Wolff to Condillac. But we perceive even here an important inflection around 1760. As T. Frängsmyr shows in the first chapter of our book, Wolff's notion of mathematical method, which derived from Greek geometry, proved unequal to his ambitions and fell victim to the principled criticisms of Kant. Thereafter, as R. Rider explains in chapter 4, analysis and algebra, which, in contrast to geometry, had an instrumentalist bias, became the exemplar of the mathematical method.
This instrumentalism was a key ingredient of the quantifying spirit after 1760. Everywhere we see an increased emphasis on the practical uses of number and system. In sciences already quantified, like astronomy and surveying, new instruments and organizations arising from governmental and military needs brought greater precision and larger operations; S. Widmalm examines the case of English geodesy in chapter 6. Likewise, interest in promoting public health and agricultural yields pushed the standardization of meteorological instruments and the study of climatology, as explained by T. Feldman in chapter 5. The bureaucracies of the centralizing nations and principalities developed an enormous appetite for statistics (K. Johannisson, chapter 12) and a compulsion toward rational exploitation of natural resources (H. Lowood, chapter 11). The explosion of information inspired the invention and development of artificial techniques of control from natural history (J. Lesch, chapter 3) to lexicography (G. Broberg, chapter 2). Further examples of instrumentalist approaches are found in chemistry (A. Lundgren, chapter 8) and technology assessment (S. Lindqvist, chapter 10).
It may be useful to bring together some of the results relating to practical operations of the geometrical spirit after 1760 and to extend them with a few further examples. The survey proceeds from instruments and instrumentation through applications to statistics, land use, and trades to the information explosion. It concludes with observations on the social support that underlay these developments.
Instruments and Instrumentation
The late Enlightenment rediscovered and fruitfully exploited the complementary character of truth and quantification first made manifest in the epicycles of the ancient astronomers. An example from physics will illustrate both the timing and the content of this turn toward instrumentalism. In the 1740s Alexis Claude Clairaut worked out a theory of capillarity based on a force between particles of water and of glass that acted over distances as great as the radius of capillary tubes. He could not think of a way to discover the precise form of this law, and he considered his failure a blemish. If the unique correct form were inaccessible, how could the theory support inquiry into the fundamental powers of nature? Fifty years later, Laplace broke his head over the subject. Unable to do better than Clairaut, he made a virtue of necessity, redid the physics to require a force that acted only at insensible distances, and rearranged his calculations so that they did not depend on knowing the precise form of the force law. This trick, or insight, which would have been a fundamental flaw in a realist theory, became the basis of Laplace's molecular physics, which became part of what might be called the Standard Model of the physics of the late 18th and early 19th centuries.
The Standard Model invoked imponderable special fluids to carry the various "forces" apparently differentiated in nature. Electrical theory, with its one (or two) repellent and attractive fluids, served as exemplar. The theory was proposed by several people around 1770, demonstrated by Charles-Augustin de Coulomb in 1785, and generally accepted by the end of the century. The matter of heat, caloric, came into existence in the 1770s and 1780s almost in parallel with the electric fluids; the discoveries of latent and specific heats then not only reinforced the old idea of a matter of heat but also provided opportunities for extensive and intricate quantitative experiments.
Two fluids for magnetism and one for light completed the list of fundamental imponderables, to which might be added phlogiston (for as long as it existed), the ethers supposed to mediate gravitation and the interactions of light and matter, radiant heat, and a gravimetric fluid. As the cases of electricity, magnetism, light, and heat evidenced, the invocation of the fluids made possible computations of a sort familiar from the theory of gravitation. "This exchange of lumières ," wrote J.B. Biot in 1803, referring to the then recent increase of the use of mathematics in physics, "is the certain proof of the perfection of the sciences."
Most of the leading proponents of the Standard Model, including Biot, made clear that they understood it in an instrumentalist sense. They conceded that even in the exemplary case, electricity, they could not determine how many, if any, fluids operated; and that every explanation of heat phenomena in terms of caloric had an equivalent account in terms of matter in motion. In return for the convenience of thought and the purchase for calculation afforded by the Standard Model, quantifying physicists or chemists surrendered their claims to truth. In this they found themselves in agreement with the epistemologies of Hume and Kant, and perhaps also with Condillac's teaching that clear and simple language, not intuitions of truth, conduces to the advancement of science.
Calculation involves not only the software of theory but also the hardware with which numbers are garnered from nature. A dramatic increase in the precision of the instruments of physical science occurred during the 18th century: in the apparatus of the oldest exact
sciences, astronomy and geodesy; in the meteorological instruments invented in the 17th century, like the barometer and thermometer; and in devices created more recently, or applied to new purposes, like the calorimeter, electrometer, and chemical balance. This class of instruments might be called "measurers." A second class, which included "explorers" like the air pump and the electrical machine, produced artificial phenomena for demonstration, investigation, and measurement. A third class, "finders," were measurers and explorers with direct application to practical affairs: the telescopes, chronometers, theodolites, and so on that made the increasing trade in the instruments of navigation, surveying, and mensuration. The improvement in accuracy of the measurers and the increase in power of the explorers depended on the growing market for better finders as well as on demand for instruments for instruction and entertainment. The instrument business of the 18th century was driven by raison d'état, that is, by a need for finders of position on land and sea, for the guidance of the army, the navy, and the tax collector, and by raison tout court, that is, by the wish to see and perhaps to study the novelties brought to light by enlightened natural philosophers.
In the case of measurers, increase in accuracy can itself be measured. During the century from Tycho Brahe to John Flamsteed, the fineness of graduation improved by a factor of 3, from 1 minute to 20 seconds of arc. (These were of course the best instruments; the ordinary sectors of 1700 were divided to 10 minutes of arc.) During the 18th century graduation improved by a factor of 200, from 20 seconds to a tenth of a second. This transformation built on improvements in metalworking and, above all, in mechanical means of subdividing angles, the so-called ruling engines, among which Jesse Ramsden's model of 1773 perhaps represented the largest advance. The drama of angular division may have been dramatic only to
instrument-makers and their more precise clients; but certain consequences of better measurers could not fail to arouse the interest of a wider public. Astronomers brought to light the aberration of stars (which has a maximum value of 20 seconds of arc), the nutation of the earth (which has a maximum value of 9.2 seconds and a period of 19 years), and the secular change in the inclination of the earth's axis to the plane of its motion (which amounted to around 45 seconds of arc per century ).
A similar story can be told about clocks. Huygens' pendulum clock, which embodied a great leap in chronological performance, was accurate to perhaps 10 seconds a day. During the 18th century, improved escapements and temperature compensation increased accuracy by almost two orders of magnitude. Chronometers available in 1800 could keep time to better than a fifth of a second per day. Among the practical consequences of this accuracy were the determination of longitude at sea to within 2 minutes of arc, or a mile or two in most latitudes, as compared with discrepancies of as much as a hundred miles in 1750; and the specification of the length of a seconds pendulum (which many people proposed as a basis of weights and measures) to within one part in a hundred thousand.
Barometers and thermometers improved in parallel with clocks and sectors. Around 1730 it was deemed unnecessary to correct barometer readings for temperature or to affix anything sturdier than paper scales to thermometers, which in any case were seldom calibrated between fixed points. By 1780, the best barometers could be read to a few thousandths of an inch, an improvement of a factor of 10 in 50 years; and, after compensation for temperature, capillarity, the curve of the miniscus, and so on, these readings gave the true value of the pressure within the accuracy of observation. The best
thermometers of the 1780s were literally incomparably better than those in use in the first third of the century. Careful and uniform procedures for finding the fixed points and meticulous division between them resulted in instruments that could be read to a hundredth of a degree and that, moreover, gave the same readings when immersed in the same temperature bath. Among the consequences of these improvements of wider interest was the perfection of barometric hypsometry—the determination of heights by barometer readings compensated for temperature. By 1790 anyone interested and proficient could obtain the height of a mountain to within 0.5 percent by carrying good meteorological instruments to its top.
The electrometer offers another example of the fact and the consequences of the acceleration in accuracy of instruments during the second half of the 18th century. Around 1750 electrometers came into existence, without standards or standardization, and without much agreement on the part of their makers about what they were measuring. Then the need to standardize measurement—of the settings of machines for trade, of the shocks given in medical treatments, of the degree of atmospheric electricity—produced a strong demand for reliable and sensitive instruments. The demand was met in the 1780s, notably by Alessandro Volta, whose most sensitive straw electrometer registered about 40 volts per degree. The contact of silver and zinc develops about 0.78 volt. Using mechanisms invented during the 1780s for multiplying small charges and driven by galvanism (or by the urge to disprove it), Volta managed to amplify the effect of a single zinc-silver junction until it stimulated his electrometer. His compulsion for the quantitative enabled him to make a
discovery that has made a qualitative difference in the history of the world.
The quantitative information secured by the instruments itself grew in quantity. It is very easy to record the readings of thermometers and barometers and to compute averages to crowds of decimals, but difficult to arrange them usefully. One method increasingly applied during the last third of the century was the tabular display. Johann Carl Fischer's monumental Geschichte der Physik , the longest work on the subject ever published (eight volumes, 7,500 pages, all published between 1801 and 1808), gives even those who only turn its pages a vivid impression of the growing use of numbers and tables in physics during the Enlightenment. Fischer divides the modern period into two parts, from the time of Newton until the discovery of the various types of gases, and from the discovery until the time of his writing. Take the case of heat. There are four tables of data from the first period, sixteen—and many untabulated numbers—from the second. Where the tables cover the same subject in both periods, the detail is much greater for the later: for example, temperatures to a tenth of a degree and expansion coefficients to three or four figures in the first period, temperatures to a hundredth of a degree and expansions to six figures in the second. There are also many new subjects for tabulation in the second period: specific heats, warming by radiant heat, heat of exothermic reactions, expansions of fluids, pressure of superheated steam, and so on. In a word, or rather a number: Fischer needed four times as many tables to set out work done on heat in the last third of the 18th century as he needed to display the measurements of the first two-thirds.
Meteorology offers a luxurious example of rampaging numbers. Before the middle of the 18th century, people who liked to measure the weather tended to do so desultorily, with imperfect instruments, and with little regard for the ways and means used by others. Efforts to establish more regular observing with standard equipment always failed. The Royal Society of London collected reports from all over
Britain and from parts of the continent in the early 18th century; the miscellaneous and mismatched returns allowed only qualitative generalizations, such as "Pisa's prodigious rains make it the 'Piss-pot of Italy'." In a second try, begun in 1723, the Society distributed thermometers, barometers, and rain boxes, all by the same maker, and detailed instructions for their use. It received in return ill-digested, capricious, inaccurate, and worthless reports and registers from people who had soon felt the inconvenience of reading several instruments several times a day, rain or shine. Much changed after mid-century. Tabular displays improved in quality and increased in number; climatologists obtained intercomparable instruments; the Royal Society's earlier objective, surveying the weather by network rather than by occasional reliable informant, was realized on the continent, where the resources of centralized and enlightened government had been brought to inspire observers to stay close to their barometers. (These matters are considered in the detail they inspire in the essay by T. Feldman.) The Societas meteorologica palatina, which took the lead on the continent, also distinguished itself by publishing some of the results of its observers in graphical form.
The Play of the Spirit
In 1778 there appeared a pseudonymous work entitled Recherches et considérations sur la population de la France . The author rhapsodized as follows: "Experiment, research, calculation are the probe of the sciences. What problems could not be so treated in administration! What sublime questions could not be submitted to the law of calculation!" The probe or key to intelligent administration, the
most important number in the Kingdom—a number customarily kept secret by governments up to the middle of the 18th century—was the number of the subjects of the King. This strategic number provided an index to the strength of the state. The population could be considered an instrument, and its reading an indicator of health or decline. Lavoisier and the intendant des Pommelles likened it to a thermometer, "the thermometer of public prosperity," a pleasant image since, in French usage of the time, "température" was to the air what "tempérament" was to the bodily humors, that is, an indicator of condition or temper, and climate was a recognized factor in public health. Other protostatisticians saw an analogy to the barometer. The Paris Academy of Sciences recognized a connection with science, and opened its pages during the last years of the Ancien Régime to tables of births, deaths, and marriages recorded for various districts marked out on the great map of France drawn up under the direction of its members, the Cassinis.
There is a further analogy between population statistics and meteorology. Coordinating a national census presented many of the same problems as organizing an international survey of the weather. The first effort at a thorough census in France was the instruction sent in 1697 to the thirty-two intendants of the administration of Louis XIV to report the number of towns, villages, hamlets, and inhabitants within their jurisdictions. The intendants had neither the means nor the inclination for steady observation of the demographic barometer. They returned a medley of bad numbers taken by hearth rather than head counts, or from old tax rolls; nine of the intendants did not bother to estimate total populations; one omitted all towns,
another all servants; and the whole suffered from errors in multiplication and transcription.
The physical difficulties with which census takers of 1700 had to contend, like poor communications, ameliorated during the course of the century. Not so the psychological. The people did not see the beauty in acting as a barometer of public prosperity; they saw only the certainty that their taxes would increase when their prosperity was better known. Des Pommelles, writing in 1789, concluded that the opposition of the people made a head count impossible. For this reason, and because of the cost, the central administration did not push for a general census. Still they wanted a demographic index. They hit upon a clever method of approximation. In 1772, just before Turgot came to power as controller-general of finance, his predecessor ordered that the intendants forward every year, retrospective to 1770, the numbers of births, deaths, and marriages that took place in their jurisdictions. Turgot then demanded that they make a head count in the leading towns and a few neighboring country parishes. Division of this head count by the average number of births in the same regions over the preceding decade or quinquennium would give a coefficient—let us call it k —by which all the data about births for all France could be converted into a figure that might be defined as the French population.
Values of k founded on various samples congregated around 26, the number used in the estimates published by the Academy of Sciences. The question naturally arose whether the method had any merit. Turgot's mathematical advisor, the marquis de Condorcet, sought an answer, but without much success. Laplace then took it up, and managed to solve it by reducing it to operations with the standard apparatus of the game theorist, that is, an urn filled with very large numbers of black balls and white balls. Let the black balls stand for the existing population, the white balls for the average number of births. The records returned by the intendants gave the
head counts for a sample s , supposed representative, of the total population t , and the number of births, bs and bt , for the sample and for the total. The numbers s and bs can be likened to the number of black balls and of white balls, respectively, taken from the urn in s + bs separate draws. Imagine now a new play at the urn, giving bt white balls and an unknown number t of black balls. What is the probability that t lies within any arbitrary number of (bt /bs )s = ks ? The answer depends on the size of the multiplier k . To be on the safe side, Laplace recommended taking samples of a million or more. When this was done just after the turn of the century, k came out to be 28.3.
The introduction of annual reporting of births and deaths to the central administration around 1770 marked an epoch in French statistics—the advance from occasional, imperfect, one-time, static surveys to continuous data collection and useful time series. The later 18th century was in fact a watershed in the gathering of all sorts of quantitative statistics, industrial (particularly in the low countries) as well as demographic. The first national statistical bureau, appropriately called the "department of tables" (Tabellverket ), came into existence in Sweden in 1749; it owed its precocity to the availability of a corps of inexpensive and experienced workers, the parish pastors, who had been keeping registers of their parishioners for fifty years or more. (K. Johannisson sets forth in chapter 12 the Swedish case and its connection with English political arithmetic.) Norway and Denmark took their first national censuses in 1769 and set up their own bureaus of tables in 1797. The central German states, Mecklenburg, Hesse-Darmstadt, and Bavaria, and also Austria, counted themselves in 1776 and 1777. In 1791, a colonel of militia mobilized the clergy of Scotland by threatening to quarter his troops
on them if they did not send him inventories of their parishes. "The Ministers have it in their choice [said he], either to write to the Colonel, or to treat his soldiers." This maneuver brought the soldier-statistician the wherewithal to compile twenty-one volumes in eight years.
Back in France, the spirit of the Revolution puffed new vigor into the statisticians, who had it decreed, in 1791, that every administrative unit should furnish each year, during November and December, the name, age, birthplace, residence, profession, and other means of subsistence of all citizens living in its territory. Three out of 36,000 communes replied. No one succeeded in enforcing the decree until 1801, when a number was obtained about a million less than Laplace had calculated. Another count, in 1806, came out almost 800,000 larger than he had allowed. Napoleon liked these readings of the thermometer of public prosperity, but did not care to try again, lest the results reveal a decline in population and an argument for critics of his stewardship. The census of 1806 was followed by that of 1821. Meanwhile the British faithfully fulfilled the requirement of a decennial census, which they took upon themselves for the first time in 1801.
The land is easier to measure than the people. Our quantifying spirit came to ground in the late 18th century not only in exact cartography and geography (as in the cases of the Ordnance and metric surveys) but also in the exploitation of old forests and virgin lands. Two examples are worth our attention, one from the Old World, where the quantifying landscaper faced constraints imposed by centuries of unregulated growth, and one from the New World, where the United States felt itself free to cut up the unsettled western territories as it pleased.
The European mathematical landscapers made their greatest conquests in the princely forests of Germany. The products of these forests provided an essential element in the budgets of German states and principalities and an equally important item in the well-being of its inhabitants. By the end of the Seven Years' War in 1763, enlightened princelings had recognized the need for prudent management of the remaining forest lands. (The following is derived from H. Lowood's account in chapter 11.) Forstwissenschaft required an inventory of the number and sorts of trees in the prince's domain, a plan for harvesting and reforestation so as to maximize yield while guaranteeing constant productivity, and a method for estimating the value of the cut timber. Foresters developed methods of calculating wood mass from estimates of tree height; the more daring among them had recourse to the integral calculus to handle the irregular shapes of standing trees; others idealized the shapes to truncated cones; and all recognized the utility of the fiction of a Normalbaum , or standard tree, to which the natural specimens could be assimilated.
Two aspects of this management are relevant to the quantifying spirit. One is the amount of quantitative dog work Forstwissenschaft demanded. The manuals set out exemplary tables—paradigms in the true sense—to be filled in with details about the location, type, and yield of every commercial species grown in the forest. A representative example concerns the value of pine stands of a particular domain in the Jägerthal. It has space for 400 numbers; and it is only one of many tables in the manual. The second aspect of interest is the compulsion to remake the forest to place the different species where they grew best so as to maximize not only the harvestable wood mass but also its quality and hence its value. Nature and custom prevented the full realization of this plan, but its design suggests what the mathematical landscaper might do when given a free hand.
That happened in the newly liberated British colonies in America. The several states had to honor the grants of land they had made to the officers and men of the Revolutionary army and the debts they owed to foreign governments. By 1785 most of the states had ceded most of their claims to western territories to the federal government. That year Congress passed its first ordinance directing the method
for surveying and cutting up its new lands for disposal to soldiers, creditors, and settlers. The ordinance represented a great victory for the quantifiers, led by Thomas Jefferson, who insisted that the lands be surveyed into equal spans before being offered, and subdivided into parts affordable by small farmers; their opponents, primarily southern aristocrats and plantation owners, favored large grants to companies or wealthy individuals who would undertake to divide it up into such shapes and by such boundaries as suited their interests. A similar compact between geometers and democrats developed during the metric reform.
The Ordinance of 1785 stipulated that a Geographer's Line be run due west beginning at the intersection of the Ohio River with the western boundary of Pennsylvania, and that every 6 miles lines be run due south until they met the Ohio. These north-south strips, known as ranges, were to be cut into squares 6 miles on a side by parallels to the Geographer's Line. The squares, known as townships, might be further divided into 36 lots of one square mile each, later called sections. In each such township the federal government reserved for parks or other purposes four lots symmetrically placed, and, for a school to be run by the settlers, an additional lot at the center.
It is not possible to make the boundaries of the townships parallels of latitude and meridians of longitude and also to have all townships 6 miles square irrespective of their distance north or south of any east-west baseline. Surveyors adjusted to this impertinence of the earth's curvature by incorporating correction lines parallel to the baseline at 24-mile intervals. The middle townships are slightly larger, the northerly ones slightly smaller. At the correction line, the shorter-than-average sides terminating the townships below were expanded into the longer-than-average sides terminating the townships above.
The earth still exacts a penalty for the presumption of those who would cut it into equal pieces. Motorists in the Great Plains, who can speed from east to west along roads that go straight as an arrow forever, must stop every 24 miles where north-south boundary roads jog suddenly at the correction lines.
Jefferson's original proposal would have divided the western territories into what he called "hundreds"—squares with sides of 10 nautical miles—rather than the townships of 36 square statute miles Congress eventually authorized. Each hundred was to contain 1,000 Jeffersonian acres and so on, for Jefferson, who invented the American system of pennies, dimes, and dollars, championed decimal division almost as strongly as democracy. He and the professor of mathematics who helped him work out his system of rigid squares recommended it for its order and clarity, and as an obstacle to cheating. They argued that irregular lots inspired fraud, and could point to the experience of Massachusetts, which discovered that holdings in the country typically held 10 percent more land, and often 100 percent more land, than had been granted. Friendly surveyors set the boundaries where their clients wished. The square grid made the practice much more difficult and allowed purchasers to get more or less what they paid for.
Industry did not escape attempts at rationalization induced by the quantifying spirit. Its ingredients of inventorying and systematization expressed themselves in the Dictionnaire des arts et métiers of the Paris Academy and in the articles and plates of the Encyclopédie . The first volumes of both sets came into print in the early 1760s, obedient to our periodization and consequent to a rush to secure priority. The inventory was to be the first step toward a rationalization that would be accomplished by artisans and philosophers in unlikely collaborations. The grandest vision in this direction was possessed by a
French general, Jean-Baptiste de Gribeauval, who beginning around 1765 plumped for standardized armaments with standardized parts. His notion of uniformity or interchangeability came to the United States via its champion of numeracy, Thomas Jefferson. To realize interchangeability required great advances in precision in machine tools and their application. The enlightened French general saw truly if distantly: the exactness that his quantifying and military spirit knew to be achievable was indeed accomplished by force of arms, by gun-makers to the U.S. Army.
Even in the rudimentary, gross industry of the 18th century a program to rationalize through experiment and measurement may be discerned. The interrelations of physics, chemistry, mathematics, self-help, nonconforming education, and invention in industrializing Britain, especially the Midlands, during the later part of the 18th century have been examined at length. Clock-makers, the very model of the exact machinist, made the textile industry tick; instrument-makers created the precision tools that made possible the realization of Watt's engines. Here the quantifying spirit expressed itself in the balance sheet: not only in the counting house, but also in the engineering shop, where, as in the practice of John Smeaton, designs and processes were optimized by systematic variation of pertinent parameters that changed output in measurable ways. This approach contrasted with that of French engineers, like Coulomb or Charles Borda, who liked to work from the principles of analytic mechanics. Watt perhaps occupies an intermediate position. From our point of view, the old dispute over the quantity of science that figured in the industrial revolution derives its smoke as well as its fire from a false
and poorly defined dichotomy. One did not need to proceed from the axioms of the Principia to a new machine, or wish to do so, to share a primary trait of the men of science of the late 18th century: an instrumentalist use of mathematics.
In chapter 10, S. Lindqvist schematizes the advance of the quantifying-instrumentalist approach to technological improvement by examining episodes in Swedish experience. To dramatize the difference between academic science and engineering imperative, the episodes take place in "labs in the woods"—or, anyway, on location, where facilities for large-scale experiment existed. In the case of water power, the very considerable advance in effective quantification between the early 18th and the early 19th centuries is made clear by a comparison between the 25,000 disjointed, inaccurate, useless experiments undertaken by Christopher Polhem on his own initiative and the sustained, precise, theoretically motivated and mathematically analyzed measurements made under the direction of Pehr Lagerhjelm on a commission from the Swedish Ironmasters' Association. Studies of charcoal-burning show the same pattern: some rough, qualitative assessments around 1750 and a thorough, careful, quantitative investigation by the Ironmasters around 1810. The Ironmasters were by no means the first in Sweden to explore productivity carefully and quantitatively. Lindqvist gives as the earliest example an effort to apply brain to brawn: a detailed study of the efficiency of human muscle power, carried out in the naval yards at Karlskrona in the early 1770s.
We touch upon quantification in technology in two additional contexts. A. Lundgren (chapter 8) points to the increased use of the balance in mineralogy and pharmacy after the middle of the 18th century and its more frequent presence in chemical laboratories. Here an instrument applied primarily to improve technique entered an environment where it was to effect—or to help effect—a revolution in physical principles. R. Rider's example (chapter 4) goes the other way. She moves from the elevated reaches of the theory of universal
languages to the very practical problem of devising an efficient, long-range optical telegraph. The signaling codes, which found employment during the Napoleonic wars, disclose evidence of the operation of the quantifying spirit.
The Encyclopédie required 28 folio volumes, 71,818 articles, and 2,885 plates to do justice (or, as many thought, injustice) to the knowledge of the time. That is no doubt impressive; but to say the truth, Diderot and d'Alembert were but indolent encyclopedists. Johann Heinrich Zedler's Grosses vollständiges Universal-Lexikon , completed in 64 volumes just as the Encyclopédie began to appear, could not stop with completion, and began to issue "necessary" supplements, which reached the letter "C" in four volumes, and died there. And, as G. Broberg demonstrates beyond peradventure in chapter 2, these summae were made possible only by suppressing most of what was known. Take the notorious case of insects. John Ray, writing in the 1690s, guessed that there might be as many as 20,000 species of them. Fifty years later, Pieter van Musschenbrock made it 130,000; another forty years, and it had reached 875,000. The total number of species of plants, animals, polyps, and microscopical creatures evidently exceeded the number of insects. How survey it all?
Encyclopedists and lexicographers had an easy and obvious method: they could list their information alphabetically and leave synthesis to the reader. The chemists, perplexed with some hundreds of mixed salts and the prospect of many more, found their salvation in the new gases, the composition of water, and the linguistics of Condillac. Their nomenclature of 1787 and its purely instrumentalist definition of "element" were so well constructed that they survived the discovery that their most controversial ingredient, oxygen, is not, as its carefully chosen name implied, the acidic principle. Natural
historians handled their multitudes by arithmetic: classification by numbers of flowers, teeth, nipples, and toes—anything superficial and denumerable. The system of Linnæus was a godsend to naturalists at sea in the quantity of their own discoveries. His way was not only arithmetical but also geometrical. The placement of species identified by arithmetical criteria under genera and higher orders permitted a two-dimensional layout of God's plan. It fit perfectly the instrumentalist character of the quantifying spirit. The physicist Biot made it the first great achievement of 18th-century science, "a universal systematic method," capable of arraying a vast amount of information "down to the smallest detail." As J. Lesch shows with a wealth of examples in chapter 3, the Linnæan method became the instrument of classifiers from the medical through the mechanical sciences. The binomial approach to knowledge was as characteristic of the Age of Reason as silk stockings.
Natural species did not exhaust the species of nature. With human help, plants and animals disinclined to breed in the wild could be made to father and mother hybrids. In chapter 9, in our closest approach to sex, J. Larson describes J.G. Koelreuter's crossing of two species of tobacco plant and analyses of the resulting bastards. More interesting for our purposes than Koelreuter's deductions about the transformability of plants was his quantitative method. His compulsion to measure equaled that of his contemporary Ezra Stiles, president of Yale University, who took the temperature of the weather twice a day for thirty years, weighed his children regularly before breakfast, and counted 888 houses, 439 warehouses, 16 stills, 77 oxen, 35 cows, and 1,601 sheep on supernumerary walks in Rhode Island one autumn. Koelreuter followed the progress of hybridization by minute measurement of flower parts.
The rapid progress of the quantifying spirit during the last three decades of the Ancien Régime indicates wide social support. It is scarcely a challenge to discover in the cultural and political history of the 18th century developments that created or strengthened this support. The more obvious include the rise of the benevolent despots of central Europe, with their cameralist bureaucracies and programs of economic rationalization; the multiplication in France of enlightened philosophers and their fellow travelers, intent on transforming the arts, sciences, education, and government; the acceleration of industrial innovation in Britain owing to a happy conjunction of capital, skilled labor, natural resources, and expanding markets; and, throughout the Atlantic countries, that insistence on guiding society by the lamp of human reason, however dim it might be, rather than by the light of revelation or the radiance of sun kings, that marked the "age of democratic revolutions."
The tempo of these developments picked up around the middle of the 18th century. The exemplary despots of Prussia and Austria then established their positions. In 1751 the philosophes published the first volume of their mightiest weapon, the Encyclopédie . The precise origins of great social revolutions do not lend themselves to exact dating. Nonetheless, the quantitative instrument of the historian, the time scale, demands the effort. The balance of contemporary learned opinion seems to incline toward the same date for the quickening of the industrial revolution in Britain that Arnold Toynbee proposed a century ago: 1760. That is over precisely the identical date that the historian of democratic revolutions settled on for the beginning of his saga, and that we have identified as the watershed in the activities of our quantifying spirit.
Links between these wider movements and ours may be made at several levels. The need of the increasingly bureaucratic state to organize itself and control its resources gave an impulse to the
collection and analysis of vital and other statistics; to forestry and rational agriculture; to surveying and exact cartography; and to public hygiene and climatology. Industrial innovation encouraged quantitative experiment. The political philosophy of the Enlightenment and its application in revolution spread notions of reason and nature compatible with the rationalization realizable by numbers. The interconnection of the themes of revolutionary rhetoric, numeracy, bureaucratic imperative, and precision measurement is evident in the course and nature of the reform of weights and measures effected during the French Revolution, which is the subject of chapter 7.
The institutions of benevolent despotism and enlightened bureaucracy provided many niches for mathematicians. One set of niches harbored salaried academicians. Their opportunities increased rapidly shortly after 1750, when the rate of founding of academies doubled over what it had been in the previous half-century. This enlargement continued until the French Revolution and affected the quality as well as the quantity of academicians. The insistence upon expertise in a science as a condition for admission, although not always imposed, did create an environment more favorable than earlier more lenient times to the advancement of quantifying spirits. Other important niches opened in armies, navies, and state industries.
The upward mobility of the mathematically talented is nowhere better illustrated than in the career of Laplace, who has already appeared several times in these few pages, as a contributor to the Standard Model, to the theory of population statistics and (tacitly) to the metric reform. We end our book with R. Hahn on Laplace's career as we began with T. Frängsmyr on Wolff's. We thus frame our accounts of the quantifying spirit with scientific biographies that, between them, encapsulate the transformation documented in the balance of the book.