The Changing Role of Numbers in 18th-Century Chemistry
By Anders Lundgren
According to a well-established tradition in the history of chemistry, the chemical revolution at the end of the 18th century was a product of the increased use of quantitative methods and of the balance, leavened by the law of the conservation of matter. Despite this emphasis on quantification in accounts of the transformation of 18th-century chemistry, most historians of chemistry have dismissed the topics of quantification, mathematics, and measurements in a few words. To redress the balance, I propose two approaches for studying the changing use of numbers in 18th-century chemistry.
The first approach stresses the influence of economic interests. During the 18th century, chemistry was often classified as an applied science. Mineralogy, metallurgy, and assaying, as aspects of applied chemistry, gave rise to an increased use of the balance and thereby a new role for numbers in chemistry. The second approach assays the influence of experimental physics. Before the beginning of the chemical revolution, important parts of experimental physics had been subjected to quantification. Many chemists during the 18th century
I should like to thank Marco Beretta, Christoph Meinel, and Evan Melhado, as well as the other authors of this volume, for comments and criticism of earlier versions.
invoked physical concepts in explaining their subject matter: witness the assimilation of Newtonian forces and chemical affinities. Lavoisier was only one of a company of chemists who saw themselves as physicists. Consideration of the influence of experimental physics on laboratory instruments and chemical theories will also speak to the use of numbers in chemistry.
The close association of chemistry with practical arts has figured prominently in the history of technology. In the index to the eight-volume History of technology edited by Charles Singer et al., for example, references to chemistry fill almost one page, whereas there is no reference at all to physics. Historians of the chemical revolution have not dwelt on this association with technology, presumably because they have regarded the revolution as a transformation in theory.
In the beginning of the 18th century, traditional chemical theory had little to do with the daily practice of chemistry. Existing theory was antiquated, almost entirely qualitative, and infused with compounds of Aristotelian elements and Paracelsian principles. As chemical descriptions of processes and substances were refined, old theories
lost their empirical foundation. Theories capable of organizing the growing body of empirical observations were in order. Mining practice, one of the most important practical fields and one that contributed to a changing use of numbers in chemistry, will be a focus here. The shape of the resulting systems bore the imprint of mining practice.
By 1700, the balance had long been in use in both metallurgy and assaying. The hydrostatic balance determined density and controlled purity of different substances, especially the noble ones, and assisted in the control of less than noble practices. After introducing the balance in testing gold sand imported from Guinea, the British noticed a marked decrease in "the swindling the natives practiced." But the method could be used only to determine mixtures of metals, never to decide chemical composition. In fact, the hydrostatic balance was afflicted by so many sources of error that it gave results scarcely better than assayer's needles and touchstones. (The needles were gold-silver mixtures of known composition; by matching the color of an unknown sample to that of a needle, the assayer could quickly estimate its makeup.) Despite the fact that the balance played a modest role in practical metallurgy, knowledge of density did not bring the chemist to a better understanding of chemical processes or of the chemical characteristics of a given substance.
The chemical (as opposed to the hydrostatic) balance does not appear in illustrations of laboratories of the 17th and early 18th centuries. Chemists did not use it in their daily work. Only in
commercial mining, which typically involved amounts of material far larger than anything of interest to the chemist or the assayer, was the balance at home. The most sensitive chemical balances were used exclusively for the weighing of noble metals. In De re metallica (1556), Georgius Agricola treats the balance in a section on the assayer's work and the purity of metals, and emphasizes that the most sensitive must be confined to weighing "the bead of gold and silver," since ores and other large weights would injure it. The balance played no part in the production of gold, but only in the measurement of the final, refined result.
Echoes of Agricola's attitude toward sensitive balances can be heard to the end of the 18th century. In 1689 J.J. Becher distinguished three types of balances with respect to sensitivity; his categories recurred in the writings of Johann Cramer in the mid, and of Sven Rinman in the late, 18th century. According to Rinman, the most sensitive balances, which could register changes as small as 1/128 ass (about 0.4 milligram), should be used only to weigh "the smallest bead. . .of the noble metals." Agricola, Becher, Cramer, and Rinman all assigned the same tasks to the balance. They did so independently of any theoretical commitments. Theories in chemistry retained their qualitative character until the end of the 18th century. However, from about 1750 the balance began to take on importance in the shaping of chemical theory.
Synthesis and Analysis Quantified
By synthetic quantification, I mean a recipe that states the amount of each ingredient in numbers and the relative proportion needed to produce a substance with specified properties. In analytical quantification, the chemist determines the proportions among the constituent parts of a given compound substance. Synthetic quantification had a long tradition in pharmaceuticals, mineralogy, and metallurgy, as in the production of different kinds of brass. Analytic quantification scarcely existed before the middle of the 18th century.
Synthetic quantification did not demand exact balances. The figures given in recipes were only approximate: ingredients did not come pure and the recipes were the result of trial and error. A chemist tried one part of A and one of B, then two of A and one of B, and so forth, and then, by comparing the different products, chose the better recipe. A case in point is Johann Heinrich Pott, whose Chymische Untersuchungen of 1754 included extensive "Tabellen von denen Würckungen der verschiedenen Mischungen derer Erden." The tables recorded Pott's attempts to obtain a certain iron product able to withstand high heat. He mixed ingredients in various proportions, but the abundance of expressions like "about 4 parts to 1," "a little more was added," "a greater part," "most of it," in his tables suggest that we should not ascribe much weight to the numbers offered. Pott and his contemporaries resorted to these imprecisions because they lacked standards for measurements and uniform scales and weights. Compounding this lack was a plethora of impurities, which could cause successive weighings to give very different results even if carried out in exactly the same way. This last problem particularly afficted the medical branch of applied chemistry, pharmacy.
In pharmacies balances played much the same roles as in metallurgy and assaying. A pharmacist sold substances by weight, as indicated by the prominent position of the balance on the shop counter, and compounded drugs by weight according to the recipes in the pharmacopoeias. As in the case of the assayer, however, the pharmacist's method of measuring was independent of theory. The role of chemistry in the business was confined to qualitative aspects and in practice influenced the making of very few pharmaceuticals.
As practitioners of an art, apothecaries were trained by apprenticeship. The theory of their business was reserved for philosophical chemists who hobnobbed with savants. Carl Wilhelm Scheele confined his chemical work to his spare time, and Torbern Bergman explicitly reserved the theoretical part of pharmacy for the chemist. In all this pharmacy resembled metallurgy. Many metallurgists in Germany had either practiced as or trained under an apothecary. However, commercial demands on pharmacy did not lead, as in mining, to an increased use of analytical quantification. Synthesis, not analysis, was important for the apothecary; whose financial well-being did not depend on successful analysis. Nor was quantitative analysis important in medicine: with the techniques then available, it would have been impossible to subject the concerns of physicians to quantitative analysis.
The increasing use of analytical quantification in assaying during the 18th century was a consequence of the expanding number of known ores and minerals. In Sweden, for example, where mining
products were the backbone of the economy, discovering new sources of ores took high priority, especially after the failure of the Falun Copper Mine at the beginning of the century. This failure prompted search for substitutes that might have the same economical importance as copper.
In this process chemistry played a central part. Laboratories were built close to the different mines, in places like Ädelfors, Falun, and the little-known Skisshyttan, where Axel Fredrik Cronstedt organized a small chemical research center. The nature and activities of these relatively unknown laboratories need to be examined. They proved equal to their task; they helped solve the problems raised by mining and the iron industry—for example, how carbon influenced the quality of steel.
The significance of chemistry for metallurgy was further emphasized by the fact that the government Board of Mines employed a laboratory worker from the late 1630s, and regularly operated a Laboratorium chymicum from the 1680s. The main task of the laboratory was to prepare pharmaceuticals for the mining industry. From the beginning of the 18th century, it concentrated increasingly on mineral analysis, assaying, and mineralogy. In keeping with its original assignment, the first directors of the laboratory were physicians; but in 1718, when Georg Brandt became its director, the responsibility of running it was transferred to a mineralogist.
In German-speaking central Europe, interest in chemistry likewise grew with the economic importance of mining. Many German chemists, like Becher and Johann Heinrich Gottlob von Justi, also occupy distinguished places in the history of German economic thought.
When the French government wanted information on mining from the rest of Europe, they sent out a mining engineer with a thorough training in chemistry, Gabriel Jars.
Jars' appointment symbolized an important development in our history. Around 1700, empirical knowledge was sufficient to permit miners to decide the value of well-known minerals: a trained eye could sort samples by sight. But traditional knowledge was inadequate to manage the many new minerals discovered during the 18th century. Assayers found themselves in need of chemical knowledge. Only chemists armed with the methods of analytic quantification could answer the question, "How much iron does this ore contain, and how much of it can be made available?" In his Anfangsgründe der Probierkunst (1746), Johann Cramer declared that the assayer must be able to decide the composition of different substances "[i]n order to know what and how much of a [constituent] might be found in the substance under study, or could profitably be obtained from it."
The growing need for analytical quantification raised the balance to a more prominent position in the assayer's workplace, the forerunner of the chemical laboratory. When Cramer described his laboratory, or "Arbeitsstätte," in 1746, ovens and cupels were still the most important instruments, but balances also received attention; Cramer claimed that his newest balance could give "the weight of the smallest body exactly." Still, weight is not everything, and Cramer acknowledged that even the most sensitive balance had to be supplemented by needles and touchstones in order to determine the composition of important substances.
From mineralogy, analytical quantification spread into other fields of chemistry, including wet analysis. Mineral water was a favorite subject of study, even though its constituents were not intended for individual sale. Analysis facilitated subsequent synthetic quantification of naturally occurring mineral waters deemed to be especially valuable. Another sort of wet analysis, titrimetry, also became more important during the second half of the 18th century. Here, too, commercial concerns were at work, since analysis helped in determining the purity of sulfuric acid, among other substances.
Toward the end of the century analytical quantification seeped into ordinary textbooks of chemistry. Bergman's edition of H.T. Scheffer's lectures, Chemiske föreläsningar (Uppsala, 1775), is an instructive example. The material Bergman added almost doubled the size of the original, published thirty years before. In Bergman's additions, the composition of chemical substances is given in the form "100 parts of A contains x parts of B, y parts of C," and so on; no such formulas appeared in Scheffer's original text.
By Bergman's time it was common for chemists to describe composition quantitatively. Formulas did not necessarily afford great precision, however. Analytical quantification was a more difficult assignment than synthetic quantification; the same analytical procedure could yield very different results even when applied twice by the same chemist to the same substance. Insensitivity of the balances and, more importantly, difficulty in procuring and identifying pure substances, contributed to the variations. Early analytical quantifica-
tion thus represented a new use of numbers—whatever their exactness—but did not at first assist in the development of chemical theory.
Constant interaction between chemistry and mineralogy would change the situation. Analytical quantification led to important changes in mineralogical classification systems. Up to the middle of the century, these classifications remained a species of natural history—relatively simple and qualitative, based on the external characteristics of minerals. The more substances mineralogists needed to classify, however, the greater the difficulties they experienced in relying only on external characteristics. Classification by internal characteristics—that is, by chemical composition—offered an attractive alternative or supplement. The shift from classification on the basis of external physical properties to that based on internal, chemical composition began around 1750. Cronstedt's system of 1758 is an early example. He denied the value of description from external factors and plumped for one based on chemical composition. His program gave chemistry a new role as a describer of mineralogical species. Although Cronstedt did not use numbers to express chemical composition in numbers, the growing importance of analytical quantification made it just a matter of time before others did so as a natural part of chemical description.
The determination of composition by weight consequently was not inspired primarily by attempts to formulate a classificatory system, but by the need to describe individual species more accurately. The balance and analytical quantification contributed to improved description, but only within a basically qualitative scheme of classification.
(The order in a system of mineralogical classification, for example, continued to depend on the qualitative properties of the substances to be classified.) It was only with Joseph Louis Proust's system of definite proportions in the 1790s that improved descriptions were incorporated into a quantitative classificatory scheme. Proust arrived at his basic concepts while working as a chemist and mineralogist for the Spanish Government. The research that inspired the theory of definite proportions, which immediately found its way into the mineralogical classification systems and gave a forceful impetus to the change from qualitative to quantitative description, concerned economically important oxides of iron.
Mineralogy thus began to transform itself from a technical art into a science. In it chemistry played important roles, providing both the overall qualitative pattern for classification and exact, quantitative descriptions of mineral species. This dual role finds a parallel in the tension between practice (economic interest) and theory (scientific interest) in the development of mineralogical classifications. Economic interest argued for a system based on monetary value; chemists preferred a system based on composition. Bergman even tried to amalgamate the two systems into one.
Bergman envisioned a scientific mineralogical system whose classes were determined by the most dominant ingredient of the mineral where dominance was defined quantitatively. Value can also be quantified, however, and it was therefore both possible and advantageous in practice to include in one class all minerals containing gold and silver regardless of their content of noble metal. Similarly, any mineral that derived its economic or metallurgical properties from one particular component might be placed in the class of that component, not in the class of the quantitatively dominant component.
Every systematizer used both qualitative and quantitative methods in the laboratory. The blowpipe, for example, never ceased to be an important tool for the mineralogist. According to Berzelius, miner-
alogical classification approached "mathematical certainty" thanks to the theory of chemical proportions. But he also insisted that the constituents of a compound must be found "in their nature as well as in their quantity." He combined the results of his undisputed mastery with the blowpipe with a consistent application of numbers. The game can also be played on pharmaceuticals. Berzelius was the first to do so systematically.
Experimental Physics and Chemistry
During the 18th century, the word "physics" meant both general knowledge of nature and "experimental philosophy." In the second, newer sense, physics was quantified or quantifiable—it attempted to formulate mathematical laws from experimental results obtained by means of specially constructed instruments. It is the influence on chemistry of experimental physics that concerns us here.
Around 1700, there were few connections between physics and chemistry. John Keill's and John Freind's efforts to use Newtonian concepts in calculating chemical affinity did not attract many followers. The influential 31st Query to Newton's Optics referred affinity to atoms interacting by Newtonian forces. No one succeeded in handling the subject mathematically, however, and at the end of the century some chemists working in the Newtonian tradition, like Bergman, declared it to be impossible. Newtonian views of atomism and affinity had little or no effect on concrete chemical work. A common approach to the relations between matter theory and chemical thinking may be seen in the work of Ernst Stahl, who accepted a corpuscular philosophy but based his chemical understanding of
matter on laboratory work. He made a clear distinction between chemistry and physics, saw chemical elements as compounds, and did not consider it the task of the chemist to penetrate deeper. Throughout the 18th century, the atomic theory hovered in the philosophical background against which chemists carried out their work.
The difficulty of applying physical theory to chemical phenomena derived from the fundamental difference between the two disciplines: physics dealt with properties common to all substances, whereas chemistry dealt with their special characteristics. The chemist described unique substances and did not aim at discovering general laws. Even Proust's theory of definite proportions did not explain why these proportions existed. Proust is said to have had little interest in any sort of theory. As late as 1810, when Berzelius considered definite chemical proportions, he could not decide whether the proportions "obey laws, common for all substances, or depend on circumstances unique for each substance."
Working chemists saw physics as very different from chemistry. Johann Friedrich Henckel, in his influential Pyrotologia oder Kiess-Historie (1725), mentioned "gravitas specifica" as a property to be studied with the hydrostatic balance, but the insights thereby gained would be physical rather than chemical. Even though determination of specific weights was still part of metallurgy, Henckel did not own a hydrostatic balance; the remarks about specific weight in his book were instead contributed by his less well-known colleague, Dr. Meuder. To learn chemical properties, Henckel argued, there was no way to avoid tedious laboratory work, despite the opinions of philosophers "who did not like to dirty their hands with coal." Pott liked
to say that physics was the study of the superficial, far from the reality examined in chemistry: "it is an important difference, that superficial physics only describes the external and largely changeable shape of objects, while a reasonable chemistry can discover and bring to light through its experiments the inner forces and characteristics, fundamental composition, and partes constituantes of objects."
In spite of the disparaging rhetoric, ideas borrowed from physics took hold in chemistry, particularly in the English development of pneumatic chemistry with its emphasis on empiricism, instrumentalism, and, ultimately, measurement. Here the influence of Stephen Hales and Joseph Priestley on continental chemistry, and especially on Lavoisier, deserves particular emphasis. Experimental physics made crucial contributions to chemistry—among them, a new attitude toward instruments and the quantitative facts they yielded, and a new methodology, which included an instrumentalist interpretation of theories.
Instruments and Facts
As the use of instruments characteristic of experimental physics spread to chemistry, new sorts of facts seized the chemist's attention. In 1750, Pott could say of fire as a chemical substance: "Although in its subtlety it cannot be investigated by number, measure, or weight, yet chemistry discovers a goodly number of its attributes." The impact of experimental physics changed matters. The means of production of chemical facts in themselves remained much the same—distillation, vaporization, and precipitation, according to the tradi-
tional practices of assayers, pharmacists, and others in chemical trades—but, thanks to the influence of physics, facts yielded up by the balance assumed greater importance.
The major innovation at midcentury was not high accuracy in measurements, but rather numerical measurement per se. Exactness was not essential to the formulation of the theory of definite proportions. Proust derived his ideas about the chemical significance of proportions from his work in ordinary practical metallurgy, the inaccuracy of which left plenty of room for the debates between himself and Berthollet over the nature of chemical combination. The arguments central to Lavoisier's classical investigations on the supposed conversion of water to earth did not depend on great accuracy; they did, however, rest on a numerical base. Nor did his studies of fermentation indicate the importance of exact measurement in the concrete study of chemical processes. To be sure, Lavoisier gave the law of the conservation of mass in mathematical form in order to demonstrate its exactness, but he never came close to exactness in actual experiments. None was needed. The balance merely gave a gravimetric criterion for identifying and describing a unique chemical substance.
Lavoisier thus relied on a rhetoric of numbers. The complication of chemical reality, which could not be idealized, might have compromised the rhetoric. But Lavoisier and others explained away
large numerical discrepancies by invoking unknown chemical processes. In the water conversion experiment, for example, the numerical shortfall was blamed on a chemical reaction between the glass and the water. Still, the precision balance and the law of the conservation of matter conferred upon numbers a rhetorical value similar to what they enjoyed in physics and other fields during the late 18th century. Lavoisier made good use of the eloquence of the balance when arguing for the new chemistry.
Imponderables posed special technical problems. Lavoisier and Joseph Black wished to subject imponderables to quantitative study, but the usual array of chemical instruments offered no help. Other experimental devices were called for, such as the thermometer, which had not been an instrument for the chemist, and above all the calorimeter, recently constructed. These new instruments, introduced into chemistry from physics, became central to the study of chemically important substances. Deliberately constructed to yield quantitative results, they contributed to the introduction of numbers into chemistry. The study of heat stood at the intersection between physics and chemistry. Lavoisier and his physicist colleague Laplace met the problem of measuring the amount of heat participating in a chemical reaction by inventing the ice calorimeter. Bergman, both physicist and chemist, found a way to measure the relative phlogiston content of two metals. He knew that a metal lost its phlogiston in acid solution but could regain it when another metal was added to the solution. He therefore dissolved a certain weight of one metal in acid and then weighed the amount of a second metal necessary to precipitate entirely the first from solution. In his chemistry, the amounts of phlogiston in the two metals were proportional to the weights so determined.
Instruments and Theory
The influence of experimental physics on chemical theory was still negligible in the middle of the century. Although interest in Newtonian ideas about affinity then began to increase, and although affinity was supposed to be a distance force, the tables remained descriptions of empirical facts, in practice irrelevant to any theory of affinity. This generalization holds for the work of the thoroughgoing Newtonian Etienne François Geoffroy and also for Bergman, who brought the affinity tables to their fullest form. Also, the Newtonian concept of the ether did not attract attention during the heyday of phlogiston and affinity studies. It later influenced Lavoisier.
The influence of physics on chemistry was most evident on the continent in France, which, perhaps not coincidentally, had a relatively weak tradition in mining. In Germany and in Sweden, chemists took less interest in physics because of the difficulties of applying it to mineralogy and because of the tendency of Stahl and his followers to keep Newtonian mechanics away from practical chemical work. Chemists who did show interest in physics in the last decades of the 18th century typically had some attachment to the universities. That in any case was true of Sweden.
Relations between mathematics and chemistry were strained by the inability of the one to calculate anything of interest for the other. This was especially true in atomic theory. As Joseph Black put it, the assumption that a certain attractive force existed between certain atoms was void, since "all the mathematicians of Europe are not
qualified to explain a single combination by these means." Macquer, though appreciative of Newtonian methods, thought that some mathematics was needed to formulate a general theory of chemistry; "but that [he said] does not fall into our line of work." As a pharmacist he recognized the complicated reality of the chemist: "Perhaps chemistry is not yet sufficiently advanced to be made the subject of calculation, perhaps it will never be [since] the problems that it will present mathematicians might be so complicated that they would be beyond all human effort." Bergman, who was close to Pierre-Joseph Macquer and Guyton de Morveau, had a thorough knowledge of physics, admired Newtonian methods, and was capable in mathematics. None of these tools seemed useful for the study of the atom. For Bergman, empirical knowledge of atoms was itself impossible; certainly they could not be studied quantitatively.
What chemists did take from experimental physics was an instrumentalist attitude toward theories. William Cullen and Joseph Black, following one methodological approach inherited from Newton, insisted that empirical knowledge and theoretical explanations should be kept separate. The first part of Black's classical treatise, Experiments on magnesia alba (1750), is given over to experiments; the second, to their interpretation and theoretical explanation. The instrumentalist approach fit well with a new view of theories during the late 18th century. English chemists wrote about the caloric
theory of heat in an instrumentalist way and did not commit themselves about its absolute truth.
Instrumentalism triumphed in chemistry with Lavoisier's definition of an element. It was perhaps his most important contribution to a new theoretical role for numbers in chemistry. The definition of a chemical element as the simplest substance available in the laboratory ignored philosophical questions concerning the structure of matter and denied elementary status to the old elements and principles. Lavoisier's definition turned the concept "element" into empirical operations independent of any hypothesis about the structure of matter; at the same time he made the definition the starting point for the construction of theories.
Lavoisier's view of chemical elements represented a break from the earlier concept of chemical facts as so many benchmarks in the search for absolute truth. Lavoisier's definition augmented by the atomic theory, gave numbers a new function in chemistry. This theory can be regarded as a combination of definite chemical proportions, derived from practical chemistry, with the instrumentalist notion of a simple body, derived from experimental physics. Dalton's atomic theory used numbers to express its central concept—atomic weight. With the help of Dalton's rules of simplest combination and the assumption of definite proportions, the different weights could be interrelated. The balance thereby acquired a definitive role in the construction of chemical theory.
Although the atomic theory gave to chemistry numbers invested with theoretical significance, it did not provide generalized laws in mathematical form. The study of discrete atomic weights thus made it possible to combine theory and practice in a quantitative way without commitment to the existence of atoms. The atomic theory
eventually overcame philosophical opposition by its success in explaining experimental facts and by distancing itself from physical atomic theory.
It might seem inappropriate to treat the phlogiston theory here, since it is rightly considered a qualitative system. But as the dominant theory of 18th-century chemistry, and as a theory undermined by Lavoisier's brand of "quantification," the modifications of phlogiston theory in the face of quantitative facts have a claim on our attention.
Phlogiston functioned as a qualitative classifier and offered a rational explanation of the behavior of combustible substances. Like Aristotelian elements and Paracelsian principles, phlogiston could not be isolated in pure form, and it could not easily explain details of chemical processes. Yet there was one important difference: the existence and properties of phlogiston had been inferred from many and repeated empirical observations. It was easy to demonstrate that smoke, heat, fire, and perhaps also matter escaped from burning bodies. Moreover, phlogiston theory was developed by chemists who considered chemistry a practical science and who had a deep knowledge of metallurgical and mineralogical practice.
Swedish and German chemists close to mineralogy generally adhered to the theory and pushed it to its limits. Bergman's attempts to calculate the phlogiston content of different metals is especially interesting in showing how a chemist could combine Newton and Stahl. Adherence to Newtonian method might thus go hand in hand with acceptance of the phlogiston theory. Bergman insisted on strong empirical foundations and on an instrumentalist interpretation
of theories. He distinguished between chemia vulgaris , or traditional descriptive chemistry, and chemia sublimior , or transcendental chemistry; he compared the objects of the latter to the "fluxions and infinitesimals of the more sublime or transcendental geometry." An instrumentalist interpretation of phlogiston theory helped its adherents to make the switch to the oxygen theory. Quantitative facts alone did not kill the phlogiston theory. Increased accuracy of measurements certainly played no important part in its demise. A very crude balance was sufficient to show that weight increased during combustion. The details of the shift deserve careful investigation, and the influence from experimental physics should be taken as a starting point for reinterpretation of the role of phlogiston in the history of chemistry.
During the 18th century an increasing body of chemical facts was expressed in quantitative form. Economic pressures in mineralogy and the general influence of experimental physics inspired the development of new mineralogical systems, in which quantitative descriptions of the units to be classified were an essential feature, and forced a change in the use of the balance for analytic as well as synthetic quantification, which eventuated in the theory of definite proportions.
Experimental physics did not influence chemistry by the direct application of basic physical concepts of affinity or the atom or by the direct use of mathematics. Rather, it encouraged the use of instruments yielding quantitative data: balance, thermometer, and calorimeter. On the epistemological level, it brought an instrumentalist view of theories and so contributed fundamentally to the reinterpretation of known facts that lay at the heart of the chemical
revolution. Instrumentalism was at work, for example, in the phlogistic debates and in the attempts to define an element. Quantification in chemistry did not result in generalized laws until it was augmented by the atomic theory; even then the theory bore the mark of chemistry as a whole—the task of describing the unique. Finally the rhetoric of number played an important role in propagating Lavoisian chemistry.
The change can be summarized in the sorts of questions that were put to the balance. In the beginning of the 18th century, the main question was "How much is needed to produce this substance?"—a question of synthetic quantification. Later in the century, the question shifted to one of analytic quantification: "How many parts of different constituents make up this substance?" With the atomic theory, the question became "What is the atomic weight of this element?" The evolution of questions reflected the growing theoretical importance of the balance. No great increase in the accuracy of the balance prompted or accompanied this change. The exact measurements necessary for a quantitative physics were not necessary for a chemical revolution; what was required was increased awareness within chemistry of the significance of measurement.
The Most Confused Knot in the Doctrine of Reproduction
By James Larson
The systems and methods discussed by Gunnar Broberg and John Lesch in chapters 2 and 3 are among the most characteristic and enduring achievements of the Enlightenment. All of these compendia, from the most circumspect regional flora to the most ambitious inventory of the terraqueous globe, were inspired by the belief that nature's own plan is not only fixed, but intelligible, and known in outline. The physical splendor of these quarto and folio volumes proclaims the scientific and social importance of that belief.
This enormous production, focused entirely upon the stable, observable aspects of natural diversity, has always overshadowed another, smaller but equally important literature written by and for specialists. The transactions and memoirs of academies and universities of the second half of the 18th century are full of studies of degeneration, race, hybridization, metamorphosis, and monstrosity—the irrational back side of the picture devoted to stability and order. Naturalists recognized the existence of these other kinds of diversity with reluctance, and tried to limit the effect of their recognition upon the systems of fixed, natural forms. As knowledge about any one kind of divergence from established order accumulated, specialists segregated the subject and treated it as a separate set of problems. These separate studies did not constitute an integrated body of knowledge, not even at the end of the century, but they did move research in a new direction and they were inspired by a common belief. Naturalists were forced to confront the processes constitutive of living forms and to invent new methods for the study of these processes. If some kinds of diversity seemed to threaten the basic units of natural order with dissolution, it was still possible to believe
that they could be made intelligible, and might even throw light upon the world of stable, established order.
This chapter analyzes one of these specialized studies, the experimental series undertaken by Joseph Gottlieb Koelreuter to discover the reproductive limits of plant species. Koelreuter cannot be seen as a typical or representative figure. The man was unknown outside a small circle of savants in Germany and St. Petersburg. The number of persons who understood his work could be counted on the fingers of one hand. His influence on contemporaries was negligible. In both his professional life and his research, Joseph Gottlieb Koelreuter was, in the exact meaning of the word, unique.
No other experimental sequence in natural history of the late 18th century equaled Koelreuter's in extent. According to Robert Olby, he carried out "more than 500 different hybridizations involving 138 species, and examined the shape, colour, and size of pollen grains from over 1,000 different plant species." It is not just the number of experiments that is unusual: Koelreuter's use of counting and measurement in the analysis of data is unusual. His contemporaries were masters of impersonal, objective observation, and it would be possible to put together an anthology of their work on generation equal in precision to anything ever written. But throughout the 18th century naturalists continued to ignore or resist the use of mathematical methods and symbols in their studies of morphogenesis. The theory of generation, as they understood it, was observational, qualitative, and finalist. The central conceptual tradition was still Aristotelian, and Aristotle's depreciation of mathematics as irreconcilable with the study of final causes was so thoroughly a part of the science that naturalists did not need to justify their resistance to measuring and counting. Almost everyone considered the constitution of living forms so complicated and subject to so many conditions that their complexity was beyond the reach of mathematical analysis.
Koelreuter's concerns were limited, however: the boundaries imposed by nature on crosses between two plant species—"the most confused knot in the entire doctrine of reproduction." He did not
believe he could untie the knot, something for which human wisdom would perhaps prove too weak, but he did intend to set forth the obvious features that lay at its basis. Since in any one cross only two plant species were involved, Koelreuter conceived his problem as analogous to a combinatory. Specific differentia were susceptible of measure and enumeration; combinations would result in the addition of a sum or the subtraction of a difference. Koelreuter's tools, measurement and counting, were the same as those of contemporary systematists. The difference lay in the consistency with which Koelreuter applied these simple tools. His experiments with plant bastards were set up on such a scale and carried out in such a way that he could determine the different forms and sizes under which hybrid offspring appear, arrange these according to their different generations, and ascertain roughly some of their numerical relations. Koelreuter's results were products of the stubborn persistence with which he aimed at the strict numerical determination of carefully limited aspects of plant morphology.
I have mentioned results. Let me admit that I could no more have understood these results than did Koelreuter's contemporaries had it not been for the work of Robert Olby. In two short papers, Olby has made Koelreuter's work accessible as no previous historian has managed to do. However, Olby's insistence upon direct lines between Koelreuter and modern genetics is questionable and different from the approach taken here.
The Problem of Plant Sexuality
The competition on the question of plant sexuality proposed by the St. Petersburg Academy in 1759 provides an obvious point of entry into the study of plant hybrids and hybridization in the late 18th century. By the mid-18th century the question whether plants have sexual organs and reproduce sexually would seem to have been
decided by the many proofs offered by European naturalists. This was by no means the case, and many botanists continued to contest the idea. To end the conflict the Academy of Sciences in St. Petersburg proposed a prize question: "Either to confirm or to deny, by means of new proofs and experiments, as well as those already known, the doctrine of the sex of plants, preceded by a history and an account of all the parts of the plant that play some role in fructification and the formation of seed and fruit." The competition summed up received opinion on the subject of hybridization and initiated new research that marks the beginning of serious investigation of hereditary characters in plants. The Academy received three entries. Two were considered unsatisfactory; the third, however, was judged "praemio omnino dignus"; the prize was awarded to Carl von Linnæus, at the session of September 6, 1760.
Linnæus' paper offered nothing not already familiar to his readers; he simply restated positions on the subject of plant sexuality that he had defended throughout his career. In plants there is no fruit without flower; since the flower is a necessary antecedent to fruit, it follows that sexual organs must exist in the flower. In fact, Linnæus wrote, "the flower consists of nothing but sexual organs." Those parts with the rudiments of fruit, by analogy with the animal kingdom, must be female. Linnæus separated and combined male and female floral parts to show that seed developed only when pollen contacted the stigma, and argued that the possibility of fertilization of a female plant by a male of another species exhibited clearly the sexual duality of plants.
Linnæus considered the production of hybrids the decisive proof of the sexuality of plants. He cited four hybrids, but offered little by way of proof that they were the offspring of two separate species. One (Tragopogon hybridum ) had been hand-pollinated and marked
with a thread. (Koelreuter did not consider this cross above suspicion.) The other three plants (Veronica spuria, Delphinium hybridum, and Hieracium hybridum ), which had been found in the wild or in gardens, exhibited characters intermediate between two known species. Linnæus assumed without demonstration that these plants would reproduce through seed to form "constant varieties." This would constitute a new kind of metamorphosis in plants. "It cannot be doubted that we have here new species brought forth through hybridization." The bastard plant, although resembling the father outwardly, is the image of the mother with respect to the inner medullar substance and fructification. The numerous African Gerania , for example, led Linnæus to conclude that there are as many forms in one genus in the plant kingdom as have emerged from one species through the crossing of flowers. A genus is thus only the epitome of those plant forms that stem from a single mother and various fathers.
These speculations contradicted the few established facts concerning plant crosses. However plausible Linnæus' hypothesis, especially for genera in parts of the world where nature produces a copious variety of species, crosses between species are difficult to achieve. They are, moreover, impossible to perpetuate: because of their absolute or relative infertility, or because of the regressive degeneration to which their issue was subject, the aid of one of the parental stocks is required for fertilization. Insurmountable barriers of sterility, degeneration, and distribution limit the effects of crossing far more stringently than Linnæus imagined. Thus, for each species known to be distinct and constant, most naturalists assumed a common origin and epoch.
The key word is, of course, "assumed." The constancy of species characters and the absolute or relative sterility of crosses between species were subjects about which most naturalists of the 18th century found themselves obliged to make assumptions, just as they assumed that the development of a new individual was simply the
gradual distension of a preformed being. Trained in the identification and analysis of external conformation, they were at a loss to appreciate properties and novelties that breeding experiments alone might reveal. They classified so-called bastards as a special kind of anomaly, outside the rules ordinarily followed in reproduction. Each separate species is designed to function independently, yet to contribute "all those perfections towards the ends for which it had been determined." Intermediate forms lack this teleological justification, and naturalists reckoned among the wiser provisions of nature both the rarity of crosses between species in their free natural state and the infertility of bastards. Crosses between parents with very different organizations were seen as flatly impossible, and conjecture about crosses between genera and orders was idle speculation.
An adjunct of the St. Petersburg Academy, Joseph Gottlieb Koelreuter (1735–1806) had been a proponent of the sexual theory of plant reproduction since his early years as a student at Tübingen. One of his professors, J.G. Gmelin, had been among the first to recognize the significance of Linnæus' work on hybrids, and in 1749 had adopted the subject for his own inaugural lecture. In it, Gmelin called for experiments in hybridization. Koelreuter completed his
degree at Tübingen in 1755 and became an adjunct in natural history at St. Petersburg. When the Academy announced its competition on plant sexuality he recalled Gmelin's recommendation and set to work to produce plant hybrids.
Like Linnæus, Koelreuter considered the production of hybrids a decisive proof of the sexual duality of plants; unlike Linnæus, Koelreuter believed that nature limits this kind of anomaly, thereby preserving the order and harmony that had reigned in Eden. Two different species of animals living in a state of nature do not produce bastards; nature avoids disorder by means of natural instinct. She has equally certain methods for avoiding comparable disorder in plants.
Perhaps it has also been one of her intentions, in order to avoid just such a disturbing disorder, that she disposes one plant in Africa, and gives another its place in America. Perhaps it is partly on this account that she has confined within the limits of a certain region only those plants which in respect to structure have the least likeness with one another, and are consequently least likely to bring about disorder among themselves.
Bastards are products of the artifice of man, as exercised in botanical and zoological gardens.
Here at any rate man gives plants of a certain kind the opportunity that he gives his animals, often assembled from widely separated parts of the world, which he keeps penned in a zoological garden, or in an even narrower space.
Koelreuter was convinced that nature limits the potential for disorder, even under such unnatural conditions, and he set out to discover those limits.
Koelreuter produced his first plant bastards during the fall of 1760, after Linnæus won the Academy competition. The offspring of a cross between two tobacco species, Nicotiana paniculata and N. rustica , flowered the following March, and in the fall of 1761 Koelreuter published a brief account entitled Vorläufige Nachricht von einigen das Geschlecht der Pflanzen betreffenden Versuchen und Beobachtungen . He reported the results of continued experiments in
three Fortsetzungen (1763, 1764, and 1766). The Vorläufige Nachricht and the three Fortsetzungen offer a coherent account, not only of experiments in hybridization, but of the processes of pollination and fertilization. Koelreuter also published a number of individual papers on these subjects in the Commentarii of the St. Petersburg Academy.
The Vorläufige Nachricht continues the discussion occasioned by the prize competition. Indeed, the format of the piece follows closely the order prescribed in the prize question. For Koelreuter the production of plant bastards constitutes a decisive argument for sexual duality. Here, and only here, his analysis parallels that of Linnæus; elsewhere Koelreuter has taken pains to distance himself from Linnæus' wild claims and speculative flights.
The Combinatory of Parental Characters
Koelreuter's conception of hybrid production is inseparable from his account of normal generation. He rejected ovist and spermist theories of preformation; he also rejected Linnæus' theory of cortical and medullar layers—views he considered more clever than correct.
According to Koelreuter, two homogenous fluids of different kinds determined by the Creator for union with one another join to produce organization. The male agent in flowering plants is a product of the pollen grain; the female agent, Koelreuter believed initially, is a sticky secretion of the stigma. These fluids, male and female, differ essentially; that is, "the force of one must be different from the force of the other." From the union of these two fluids results another mean fluid, with a mean force compounded from the two simple forces. This purposive agent which emerges from the organization of the reproductive fluid as a whole is the source of organization for the future plant. For each class of organized beings, a specific compound force produces a determinate structure and specific nature.
Although Koelreuter made many observations and experiments concerning the process of fertilization, the mass of quantitative evidence he accumulated to support theories of epigenesis, equality of parental contributions, and sterility of interspecific offspring constitutes the novelty of his contribution to late 18th-century generational theory.
Koelreuter's work contained an important ambiguity concerning the source of organization in the living body. It is impossible to determine unequivocally whether the source of organization is material or nonmaterial. Koelreuter draws an analogy between the union of two seed materials and the production of salt crystals. When acid and alkaline substances unite, a third, intermediate salt results. In the same way, he argues, the intermediate fluid resulting from the union of male and female seed materials either constitutes the origin or the firm foundation of the vital machine, or produces this vital machine out of itself. Neither the male nor the female seed matter suffices to produce this result by itself, no more than an acid or alkaline substance can in and of itself produce the intermediate salt or form crystals. The formation of the plant requires both the compound of two specific seed materials and the composite active and purposive force resulting from that compound.
Nature works in the same way to produce a cross between two species. Once the male and female seed matters unite, formation proceeds rapidly from the nucleus to the flower, and in the process
the sharpest eye can find no more imperfection than in the natural plant. The resulting bastard, composed from the seed matter of two separate species, reflects twofold nature in both its intermediate form and its absolute or relative infertility.
When Koelreuter first worked to produce a cross between Nicotiana paniculata and N. rustica , he experimented with many flowers. Each time his fertilization succeeded. The result was perfect, somewhat divergent seeds. From 110 seeds he produced 78 plants; of these, he kept 21 over the winter, and in March 1761 they flowered. In the spread of their branches, situation and color of the flower, and individual floral parts, each plant exhibited a mean between the two natural parental species. In repetitions of these experiments, the first-generation bastards consistently exhibited characteristics in "almost geometrical proportion" between differences in the parental species. Koelreuter tabulated his measurements:
Experiments with seven other genera supported the inference of intermediate size in first-generation hybrids. Koelreuter found that even the time of flowering and the odor of hybrid offspring were intermediate between the characteristics of the parents.
The intermediate character of first-generation hybrids, Kolereuter argued, supported the Aristotelian doctrine of reproduction by means of two seed matters. Koelreuter also used it to argue against the contemporary doctrine of generation, that is, "the doctrine of animalcula, or of original embryos and nuclei in the ovaries of animals and plants activated by male seed."
All offspring of the first cross, Nicotiana paniculata × N. rustica , were identical. When Koelreuter reversed the direction of the cross, N. rustica × N. paniculata , the offspring "agreed all together with the plants of the first experiment, and reacted in the same way to the experiments performed upon them." This phenomenon, now called the "identity of reciprocal crosses," also spoke to another contention about the contributions of parents to offspring. The hybrids with which naturalists were best acquainted, plant and animal, show a greater resemblance to the mother than to the father, leading naturalists to assume that this is a universal character of hybrids. Linnæus used the idea in his two-layer theory of generation: "a bastard offspring is with respect to its inner medullar [essential] substance the exact image of the mother, but in leaves and other outer [nonessential] parts [the image of] the father."
Koelreuter's reciprocal crosses and detailed measurements established, however, that parental contributions could not be distinguished so readily. Since Koelreuter did not come across any examples of sex-linked characters in his experimental plants, his inference that each parent contributed equally to an intermediate result was to an extent justifiable.
A second essential character of bastard offspring, fully as important as their intermediate form, is their absolute or relative infertility. Although the formation of the bastard tobacco parallels that of the natural species and its flowers are brilliant, the bastard plant is deficient in the most important character of all: fertility, the final cause of all formation. Koelreuter found the pollen containers of the Nicotiana bastard to be markedly smaller than in the natural species. What pollen they contained was white and dry; the grains did not cohere with one another as in the natural species. When examined under a microscope, the pollen grains proved to be irregular and shrunken. They contained scarcely any fluid, and most were empty husks. These observations led Koelreuter to doubt the fertility of the bastard plant; his experimental results reinforced this suspicion. Of the many flowers on the bastard plants, not one succeeded in bearing a single seed, even after dusting with a large quantity of the plant's own pollen: "instead of 50,000 [they] contained not a single one, and more than a thousand flowers, one after the other, fell without leaving a single capsule behind." In every sense of the word the tobacco proved to be "a true, and as far as I know, the first botanical mule produced by art."
Koelreuter thought infertility only a relative imperfection, which, from the point of view of ultimate consequences, proved to be a positive good. Nature wields sterility to preserve the order established at the Creation. "What an astonishing confusion would the peculiar and unchanged hybrid characters, and consistent fertility of such plants give rise to in nature? What a monstrous swarm of imperfections would they bear, and what evil and inevitable consequences would ensue?"
Koelreuter's pollination experiments, however, quickly convinced him that the infertility of the bastard was only relative. When pollinated with either of the parent species, the Nicotiana hybrid produced some ripe seed; much more was produced when plants of the
original natural species were self-pollinated. For each cross, Koelreuter raised ten plants from the bastard. The results were no longer intermediate between the two parent species, but measurably resembled the pollen parent. In Koelreuter's terminology, the plants were metamorphosed or transformed (his verb is verwandeln ): they again approached the fundamental nature (Grundwesen ) of which the initial crossing had deprived them. Again his measurements told the tale of transformation—in the size of plants; the spread of branches and flowers; and the form, size, and number of flowers.
Koelreuter's evidence for the relative infertility of hybrids and for the progressive reversion of hybrid offspring to the parental species parallels Georges Louis Leclerc Buffon's conjectures concerning animals published three years later. Under certain circumstances, Buffon maintained, a male mule can engender progeny and a female mule can conceive and give birth. Buffon also found a wide range in the productivity of different species of animals. This showed him that fertility is variable. Just as species vary in productivity, so, too, must hybrids—a conjecture that squared with Buffon's transformist views.
Koelreuter was less comfortable with the transformist position. Thus, when his experiments began to produce hybrids with varying degrees of fertility, he attempted to read the results as so many potential reversions to the parental species. He carefully constructed a table of his experimental results with respect to fertility. This was, he emphasized, not just another "useless, hasty, and absurd list of chimeric bastards," but the first systematic catalogue according to a theory of generation certified by experiment. Koelreuter's class of "perfect" bastards includes offspring of two or three natural species of a single genus; normally they are infertile in the highest degree, although some products of only two species prove fertile on the female side when pollinated by either parent, or were fertile to a
diminished degree on both sides when crossed with parental species. "Imperfect" bastards are offspring of two natural species produced when a tincture of pollen from the female supplemented pollen from the male. These offspring are characterized by a diminished degree of fertility on both sides. Finally, what Koelreuter called "varietal" bastards are completely fertile. Koelreuter denied that the parents were different species. He drew a forceful conclusion:
Such a bastard in the true sense is either wholly infertile, or at most in a very limited and unequally diminished degree, by comparison with the true natural species from which it was produced is fertile. On the other hand, a mere bastard variety retains the degree of fertility of its parents, or at least loses nothing observable of this. Thus, I regard the experiment of crossing [species] in every respect as the only true, certain, and infallible touchstone of all separate species and varieties.
Koelreuter also distinguished between first- and second-generation hybrids. At an early stage in the experimental series, for example, he noticed a marked contrast in uniformity and stability. First-generation bastards for any single cross are all alike—intermediate in form between the parental species and either wholly or relatively sterile. Second-generation hybrids, even when produced from a single ovary, tend to be less like the parental bastard and more like the grandparents. After he fertilized perfect bastards of the first generation with pollen from Nicotiana paniculata or from N. rustica , he classified the results of such back crosses as descending (to the natural maternal species) or ascending (to the natural paternal species). Later he succeeded in producing second-generation hybrids from the self-pollination of the tobacco hybrid. Thus, contrary to expectation, pollen from the hybrid tobacco finally proved to have a slight degree of fertility. In Dianthus and Mirabilis bastards he found a greater degree of fertility on both sides.
Koelreuter used a quantitative argument to account for these results. Crosses in which the two seed matters are in equal proportion produce second-generation bastards resembling the first-generation parent; crosses of seed matters in unequal proportions produce second-generation bastards resembling more or less closely one or the other original natural species, depending on which seed matter is dominant and to what degree. These three main groups correspond to the three segregating classes of second-generation hybrids in Mendel's experiments with plants differing in one essential character. In other words, Koelreuter had found roughly the three segregating classes for second-generation hybrids. Koelreuter found puzzling, however, the exception to this neat scheme. Among the characters of second-generation Mirabilis bastards, for instance, he saw reversions, completely fresh characters, and colors too varied to be classified.
The contrasting results between first- and second-generation bastards required only a simple adaptation of the theory of normal reproduction. In any given plant the process of formation liberates the compound matter that gave rise to the form of the plant, and divides this compound once again into the two original matters, concentrated in the ovules and the pollen grains. In any natural fertilization the two seed matters and the simple forces inherent in them unite in equal proportions to form the intermediate product with its corresponding compound force, hypostatized during the process of generation in the specific characters of external form. In some original plant species the outer characters are very different, in others very similar; likewise, the seed matters and simple forces of natural species display differing degrees of affinity. Between species with little affinity—where characters and simple forces differ significantly—no crosses can take place. In cases of close affinity, where there is "no slight resemblance between its parents and a suitable agreement of their natures," crosses can occur, although the Creator had not intended the two ground matters to unite.
Both vital functions in first-generation plant bastards differ from the norm: enhanced vegetative function and completely or partially curtailed reproductive function. Where infertility is not absolute, the next generation of offspring shows an unnaturally wide range of variability. The measurable intermediacy that characterizes all perfect bastards of the first generation characterizes only a small number of individuals in the next generation, the product of seed matters united in equal proportions. More often, however, mixture and union of seed matters do not proceed with the regularity typical of natural products and simple bastards. The principle of equality is broken; the seed matters combine in different proportions, and "All kinds of wrong paths result."
Two routes are open to nature according to Koelreuter. On one, "where she has the laws of close affinity as a guide, she again approaches the high road with something like a straight line; on the other path, where she lacks this guide, she strays . . . ever more from the high road." Here again Koelreuter's speculation parallels the best-informed contemporary opinion in zoology. In a discussion of sheep, goats, dogs, and domestic fowl the St. Petersburg academician Peter Simon Pallas noted that inconsistency of form, once introduced, increased from one generation to the next. Pallas supposed it to result from a vice introduced into the generative faculties of the original species by way of crosses. This vice, he said, parallels in effect the alteration of fluids and solids in a living body under the influence of a miasma.
The presence of essential characters resembling the original natural species was Koelreuter's key to the contrast between first- and second-generation hybrids. He concluded that although new varieties might arise through selfing, plant bastards do not establish the new and constant species Linnæus had predicted. Eventually hybrid races will revert to one or the other of the original natural species.
Offspring of Nicotiana rustica × N. paniculata are intermediate in form, produce sterile pollen, and are only somewhat fertile on
the female side. After pollination with N. paniculata , the plants grown from the seed resemble the pollen parent far more than the first generation. Koelreuter measured the essential likenesses with his usual care: situation, form, and substance of the leaves; number of leafless slender branches; shape and size of the corolla; form, color, and breadth of the flower; and form, size, and external perfection of the capsules.
The next summer (1762) Koelreuter pollinated this second generation with N. paniculata , and in 1763 he sowed 128 apparently fertile seeds, most of which germinated. Of this crop he retained 12 plants; when they bloomed they so resembled paniculata "that one could only have differentiated them with difficulty had they not been labeled with separate numbers." Moreover, the fertility of both pollen and seed had increased noticeably, although this was subject to fluctuations. Nicotiana paniculata was well on the way toward dominating rustica , and Koelreuter foresaw that this series of experiments would end by producing true N. paniculata . "In a word," he wrote, "I no longer have the least doubt of the possibility of transforming one natural species into another."
In the summer of 1765, four years after he had initiated this experimental series, Koelreuter achieved the first complete transformation of N. rustica into Nicotiana paniculata in the fifth generation, or, as he put it, in "the fourth ascending degree." It was as if "he had seen a cat emerge in the shape of a lion." A comparable reversion in the direction of the original mother plant, which he had followed in a parallel series of experiments, would, he predicted, require a different although proportional number of generations. He was also able to announce the successful transformation of Dianthus and Mirabilis . From a cross AB indistinguishable from the natural species B,
Koelreuter had thus produced offspring B. Since the maternal plant of AB was A, he reasoned that he had thereby transformed species A into species B. Moreover, he showed that the number of generations required for a successful transformation varied from species to species. He concluded that the transformation of "one plant into another [occurs according] to the greater or lesser degree of fertility [of] the bastards produced from their equality."
Transmutation of metals served as an analogy for this experimental series. In theory one metal could be transformed into another by taking from it particular properties, conceived as so many independent substances, and substituting other properties for them. Mercury, for example, could be ennobled by removing the two characters upon which its fluidity and volatility rested and substituting other characters for them. In the process two seed matters were needed: the male agent, of a sulfuric nature, possessed the force to make the fluid mercurial female seed matter capable of resisting fire and forming a stable body. Sulfur, in other words, transformed the nature of the mercurial body. The result was a true metamorphosis in which the male, sulfuric agent asserted its superiority (Uebergewicht ) over the mercurial agent.
Koelreuter believed he had achieved the alchemist's dream in botany in the course of a few years. In plants the male seed matter is oily and sulfuric; union with the female seed matter produces a stable organic body, "the initial basis of the future plant." In succeeding generations the male agent gradually takes the upper hand, and in the end the nature of the female has been wholly transformed. The transformation depends on "close affinity, fertility not wholly suppressed in the production of bastards, and dominance in a certain degree."
Koelreuter used his success to argue for the possibility of the transformation of metals. Likewise, one species of animal can probably be transformed into another. From a canary might come a linnet. Experiment had already established that the female retains
fertility in "the second descending degree," and it was therefore probable that she would do so in the "second ascending degree."
Central in Koelreuter's reasoning was the contention that a measurable change of characters in the product of any unnatural union reflects the proportions of the two seed matters. Variability increases in the offspring of bastards since the number of possible combinations of the two seed matters is infinite: that thread of Ariadne, natural affinity, is lacking. Koelreuter did see limits to variation, however. By means of crosses a naturalist can transform one affine species into another, or produce one of the infinite number of possible variations between the forms of the two natural species, but cannot produce wholly new species with entirely new characters. The naturalist thus transforms natural bodies by altering or removing particular characters and substituting others.
Transformations of this kind refuted monoparental heredity, and they showed that crossing was a powerful instrument for change in the world of living forms. But how was change itself interpreted? Koelreuter read his experiments against a conceptual tradition that did not permit development away from the original natural forms. He considered bastard plants artificial products, able to survive only in artificial conditions. "In the orderly arrangement and ordinary situation," he wrote, "established by nature in the plant kingdom, bastard plants would be difficult to produce or even to initiate." Even if we suppose the possibility of a true bastard plant in an open field, "the question would remain whether this chance had not taken place in a region where the natural situation as a whole, either mediately or immediately, had been destroyed or changed: true wilderness as it comes from the hand of nature is one thing; a field, free but in respect to a hundred things often very much altered by the hand of man, is another."
Consider Verbascum , for which the probability of natural crosses is great. Koelreuter did not believe that unnatural products could transform the natural species, even when, as in Verbascum , the
natural species cross regularly. In an open field, where its own and alien pollen reach the stigmata at approximately the same time, a plant will accept the male agent intended for it by nature, excluding the alien matter from fertilization. Neither ancients nor moderns spoke of bastard Verbascum in the field. Linnæus, it is true, mentioned a Verbascum bastard, but this was likely a product of the unnatural conditions in the Uppsala botanical garden. Koelreuter continued, "It is to be wished, nonetheless, that Herr von Linné had given us a more careful description, and more according to nature than according to his fantastic theory of generation, which contradicts nature."
As a naturalist, Koelreuter may have sensed the limits and assumptions of his conceptual tradition, but his was a stubborn defense of the traditional essentialist concept of a species. Koelreuter used his experiments to throw light on the nature of the species as a whole, not the inheritance of individual characters. He set out to prove that a cross between two species does not produce new and constant species. We may imagine that his findings were in this respect a relief. He regarded the essence of species as monolithic; the intermediacy that characterized all first generation bastards confirmed this interpretation. However, second-generation and back-crossed hybrids did not lend themselves to the same neat explanations. Here Koelreuter took comfort from the predominance of forms resembling the original natural species; from this he concluded that hybrid offspring will revert sooner or later to one or the other of the original species. They cannot, he thought, from new and constant species.
Contemporaries were only too happy to accept Koelreuter's conclusions. The rigor and complexity of his experiments were alien to
the mentality of many naturalists of the late 18th century, but his results supported very powerfully the theory of limited variability then being elaborated. Nor was Koelreuter's work so obscure or inaccessible that no one read it. Christian Sprengel took up Koelreuter's work on pollination, arguing that floral structure was an adaptation to secure pollination and showing that cross-fertilization was the rule rather than the exception. C.F. von Gärtner, an erudite and industrious plant hybridizer reaffirmed Koelreuter's entire theory of generation. His evidence consisted of the results of nearly 10,000 separate crossing experiments among 700 species. As Koelreuter's Vorläufige Nachricht and the three Fortsetzungen were published, a reviewer in the Göttingsche Anzeigen astutely picked out the tendency of the argument. Not only had Koelreuter limited severely the possibility of novelty by means of species crosses; he had shown that change was a narrowly defined combinatory of already existing characters.
These summaries, and mémoires published by Koelreuter between 1770 and 1796 in the Commentarii and Acta of St. Petersburg, were read, assimilated, and commented upon by two of the most influential and widely read naturalists of the period, Johann Friedrich Blumenbach and Peter Simon Pallas. In his first important work on the theory of generation, Blumenbach presented Koelreuter's experiments as a refutation of preformation. The very possibility of specific bastards contradicted any conception of preformed germs, while Koelreuter's transformation of one species into another "must undeceive even the most partisan defender of the theory of evolution [i.e., preformationist] from his prejudice."
Peter Simon Pallas claimed that Koelreuter had instituted "experiments on plant crosses the results of which absolutely contradict the opinion of the Chevalier Linné." He continued:
The great difficulty we have in producing crosses, with all our human industry, between two different species, the impossibility, confirmed by fact, of perpetuating these as crosses or as distinct races, either because of their
absolute or relative infertility, or because of the regressive degeneration to which their issue is subject, since these crosses require for fertilization the aid of one of their primitive branches, all this opposes facts to a simple probability and forces us to see that all those species that nature affects to render alike as primitive, projected in the first plan of creation, and destined to form that chain of beings that we admire without being able to account for it, any more than [we can account for] the choice, harmony, and combination of colors and ornaments that the same creative force has used to embellish its works.
The experiments Koelreuter undertook to discover the reproductive limits of plant species may seem commonplace. Older natural history had a modest experimental tradition and breeding experiments utilizing plants and animals were common after 1750. Koelreuter was unusual, however, in his diligence and energy: his experiments involved 138 species and over 500 hybridizations, and required him to examine pollen grains from more than 1,000 different species of plants. No other experimental sequence in natural history equals Koelreuter's in extent.
Koelreuter's use of counting and measurement in the analysis of his data was also unusual in 18th-century natural history. Although his contemporaries were masters of impersonal, objective, precise observation, naturalists throughout the 18th century ignored or resisted the use of mathematical methods and symbols in their studies of morphobiology. Natural history, as they understood it, was observational, qualitative, and finalist. The central conceptual tradition was Aristotelian, with its depreciation of mathematics as irreconciable with the study of final causes. Thus a resistance to measuring and counting came naturally to 18th-century natural history.
Not all naturalists were Aristotelians, however, and even the diehards of the Aristotelian tradition found it impossible to resist the forces toward greater precision that originated in more prestigious spheres of the scientific constellation. Anatomy contributed to the establishment of species characters the procedure of measuring size. Naturalists also counted seeds and pollen grains, observed numerical
ratios, noted numbers of progeny, and proposed mathematical nomenclatures. These efforts toward quantification, however, remained sporadic and uncorrelated; and most naturalists considered the complexity of living forms beyond the power of mathematical analysis.
Koelreuter limited his concern to the boundaries imposed by nature upon crosses between two plant species—"the most confused knot in the entire doctrine of reproduction." He did not intend to untie the knot, but rather to identify its major features. Since only two plant species were involved in any one cross, Koelreuter conceived his problem as analogous to a combinatory. Because specific differentia were susceptible to measure and enumeration, combinations could thus be expressed as a simple sum or difference. Koelreuter applied the instruments of measurement and counting with unusual consistency. The design and scale of his experiments permitted him to determine different forms and sizes of hybrid offspring, arrange them by generation, and estimate their ratios. That his quantitative approach yielded important results was due in large measure to his stubborn concentration on carefully limited problems of plant morphology.
Labs in the Woods: The Quantification of Technology During the Late Enlightenment
By Svante Lindqvist
The concept of quantification can be considered an intrinsic component of technology. Two basic characteristics of technology in modern Western civilization are technical efficiency and economy. Practitioners of technology had labored to increase technical efficiency and economy long before the concept of quantification began to be applied to technology. The latter occurred during the 18th century when technology began to borrow its methods from science, that is, when attempts were made to study technical reality by systematic experimentation and quantification in fixed units using precision instruments. These attempts first met with success after 1770.
The phrase "systematic experimentation and quantification in fixed units using precision instruments" describes a laboratory—a table in a quiet, secluded room with sufficient illumination and heat. Such laboratories became more common in universities, academies of science, and affluent homes in the 18th century. But it was quite a different matter to transfer these ideal conditions from the natural philosopher's laboratory to the technical world outside. Where the
For help in revising an earlier version of this chapter, I am deeply indebted to Eugene S. Ferguson and David A. Hounshell. I also gratefully acknowledge suggestions and comments on this version from R. Angus Buchanan, Arthur L. Donovan, Willem D. Hackmann, Roger Hahn, Charles Haines, Richard L. Hills, Karl Hufbauer, Thomas P. Hughes, Melvin Kranzberg, John Law, Edwin T. Layton, Jr., Otto Mayr, Terry S. Reynolds, Sheldon Rothblatt, Richard Sclove, Bruce Sinclair, Martin Trow, and Wolfhard Weber. The Proceedings of the Royal Swedish Academy of Sciences (Kungl. Vetenskapsakademiens Handlingar ) are abbreviated KVAH .
laboratory was neat, technology was messy. The central component of the experimental method, controlled experiment, was difficult to achieve in technology, with its comparatively large scale. To apply quantitative methods to technology required a degree of control over the material and social world beyond the means of individuals. The late 18th century saw the emergence of institutions with authority and competence to exercise such control.
Three case studies will illustrate general characteristics of this transformation of technology. Our image of 18th-century technology has by and large been shaped by those technologies seen subsequently as spectacular and/or efficient. But the ingenious and complex machines depicted in books of the 18th century—like those depicted by Ramelli, Zonca, Besson, and others in the Renaissance—were often no more than bold, futuristic speculations directed at a restricted audience, irrelevant to the daily reality in which most people lived and worked. We may identify numerous forerunners to 19th-century industrialization, but technological reality in the 18th century lay a long way from the elegant engravings. The average 18th-century man lived by the sweat of his brow, aided by simple wooden implements at a low level of mechanization. Only occasionally was his work made easier by machines powered by his fellow-men or by oxen, horses, water, wind, or steam. As a period in technological history, the 18th century was characterized by the use of wood, water, and work, and this is reflected in the choice of examples for this study.
Industries dependent on mechanical energy in the 18th century included mines, blast furnaces, tilt hammers, other metal works, saw and paper mills, gunpowder factories, brickworks, oil mills, and glassworks. Such industry required high power (work per unit of time) and continuous operation ; the extent to which these requirements could be met by traditional sources of power varied. Muscle power (animal and human) provided continuous operation at limited output. Wind produced high power, but proved inappropriate for industrial
production since continuous operation could never be guaranteed. Only water power fulfilled both requirements. But existing hydraulic resources were limited. In order to utilize fully the available water power, the efficiency of waterwheels had to be increased. For each type of water mill a particular speed gives maximum power. Finding the rules for determining this speed was a problem to which solutions were sought throughout the 18th century. The design of waterwheels was the subject of vigorous debate in learned journals and technical literature in the 18th century, although theoretical considerations had "no direct effect. . . on the construction and installation of wheels," which were governed by a tradition little changed until the end of the century.
Two approaches characterized 18th-century attempts to establish general rules for the most efficient design of waterwheels. In the deductive approach mathematical analysis permitted derivation of general rules from fundamental laws of motion. The inductive approach used systematic experiments with parameter variation and optimization to achieve the same end. Great expectations were matched by great investments of work; the rate of return, at least until the early 19th century, was disappointing. The deductive method resulted in formulas too complicated for ready application. The inductive method yielded unmanageable amounts of unreliable quantitative data. A waterwheel turning peacefully in a stream proved far more complicated than the heavenly clockwork.
Around 1700 French scientists in the Academy of Sciences in Paris made the first analysis of waterwheels in dynamical terms. In his comprehensive history of the waterwheel, Terry Reynolds distinguishes five approximate and overlapping periods following the work in Paris. The first was the establishment of theoretical analysis by
academicians like Parent and his followers between 1700 and 1750. The second was a period of experimental work from 1750 to 1770 by engineers such as De Parcieux, John Smeaton, and Charles Bossut. A third, simultaneous development was the attempt during the same period by Johann Euler and Charles Borda to reconcile the discrepancy between theoretical predictions and experimental findings. This next forty years, 1770–1810, Reynolds labels "the era of theoretical confusion." This confusion abated during the fifth period, 1810–50, when Borda's analysis of 1767 achieved general acceptance.
Facilitating the transition from general theoretical confusion to the reconciliation between theory and experiment were institutions capable of large-scale experiments beyond the means of individuals. The importance of such institutions in water power technology during the late Enlightenment and early 19th century is underscored by comparing a successful Swedish attempt to quantify water power with an earlier, unsuccessful one. From 1701 to 1705 the well-known Swedish inventor Christopher Polhem performed some 25,000 experiments with a hydrodynamic apparatus of his own design. But this heroic experimental effort produced only meager results. A century later, between 1811 and 1815, the Swedish Ironmasters' Association (Jernkontoret ), a private organization of independent ironworks established in 1747, undertook a major investigation of hydrodynamics. Originally intended as a credit agency for the ironworks, the Association assumed an important role in technological development toward the end of the 18th century. Its officers exercised quality control over the various stages in the process of iron manufacturing, and the Association financed a number of large development projects that were
too expensive for any individual ironworks. A large experimental apparatus had been built at the Great Copper Mine in Falun in 1804, which was originally intended for investigating the efficiency of winding gear. However, the metallurgist Eric Thomas Svedenstierna and others supported the idea that the Swedish Ironmasters' Association should finance a lengthy series of experiments in order to establish a general theory of water power. The resulting hydrodynamic investigation of 1811–5 stands in glaring contrast to Polhem's experiments.
Theoretical, instrumental, and, most importantly, institutional factors led to this successful quantification of water power technology during the late Enlightenment. A more profound theoretical framework was available by the end of the 18th century; a heightened appreciation of the applications of mathematics was reinforced by a broader and more critical knowledge of the international literature in the field and an awareness of fundamental principles of experimentation. The institutionalization of science contributed to this conceptual change: technological innovation became verbalized and was documented in journals and monographs produced and distributed under the auspices of new scientific institutions. A higher degree of precision in scientific instruments resulted from a more vigorous market for scientific apparatus and information. In the social organization of technology, responsibility shifted from individuals to institutions. This last was a sine qua non for applying the concept of quantification to the expanded spatial and temporal dimensions of technology. The importance of these three factors in the Swedish case is described in the following subsections.
Although the experimental apparatus used by Lagerhjelm for the investigations of 1811–5 resembled that of Polhem a century earlier, there the resemblance ends. Polhem's experimental apparatus seems to show the influence of the French physicist Edmé Mariotte, who
had carried out experimental studies of water and wind mills. Certainly they adhered to the same empirical tradition. Like Mariotte, Polhem was more concerned with articulating and applying generalizations based on experiments than reducing them to fundamental principles; both relied on common sense to guide their reasoning. Although Polhem's work contains an early example of parameter variation and optimization, he was never able to convert his many experimental results into general rules for the design of waterwheels. Nor did he fully appreciate the merit of mathematical analysis of hydrodynamic phenomena. This is evident in his faint praise for the work of his younger colleague Pehr Elvius, who in 1742 published A mathematical treatise on the effect of water mills . Polhem commented in the Proceedings of the Royal Swedish Academy of Sciences: "Although [Elvius'] book is really written for the learned, who are already familiar with the modern mathematics , which by its discoverer the learned Leibniz is called calculus differentialis and by Newton, fluxio curvarum , so does yet Mr. Elvius show his profound knowledge of such puzzling matters, that he gives hope of becoming a good Mechanicus with time, as well in Practice as now to begin with in Theory." What Polhem had considered "modern mathematics" was a standard tool in the hands of the mine official Pehr Lagerhjelm, whose report on the hydrodynamic experiments of 1811–5 financed by the Swedish Ironmasters' Association was a highly mathematical treatise. Lagerhjelm's report also
included a thorough, critical review of relevant international literature. The first fifty pages of the second volume commented on the works of Smeaton, Euler, Borda, Bossut, Banks, Langsdorf, and others.
Lagerhjelm's treatise also evinces a higher level of conceptual awareness. In his preface to the second volume, Lagerhjelm offered an epistemological program to relate theory and experiment for hydrodynamics. His ideas bear a certain resemblance to Kant's theory of knowledge, and the terminology—"phenomenon," "form," and "content"—is similar. For Lagerhjelm, inductive reasoning cannot produce conclusions of universal validity, because "abstractions from a given experience. . .are only valid under the circumstances and within the boundaries essentially associated with the class of phenomena one experienced." The implication was clear: the inductive method followed by Polhem and others, with their thousands of experiments throughout the 18th century, was epistemologically pointless. So, too, was the deductive method, the "speculative root" of knowledge in which Elvius and others had placed their confidence, inadequate in and of itself. The path to truth required a synthesis between "form" and "content," specifically, theory and experiment.
Inaccurate measurements compromised Polhem's data. Because the pendulum in the clock he used was not of the proper length, he arrived at incorrect values for the speed of the waterwheel and hence incorrect values for the output. Polhem tried in 1710 to reduce all these figures to their proper value by means of a correction coefficient, but found the work "so difficult and tedious, that no amount of patience would have sufficed." More seriously, the protractor he used to measure the inclination of the water trough gave different readings as the waterwheel was placed at different levels.
Polhem confided to his assistant: "In fact between ourselves, this work is as useful as a fifth wheel on a carriage."
Lagerhjelm proclaimed explicitly his awareness that calibrated precision instruments were essential if the experiments were to be reproducible and the results of general value. He made linear measurements using "a precisely graduated decimal scale two Swedish feet in length" produced by Johan Gustaf Hasselström, purveyor of mathematical instruments to the Royal Swedish Academy of Sciences. Lagerhjelm also employed "a set of weights calibrated against the Swedish original standard, which is kept in the Archives of the Royal Treasury," and a balance constructed by Gabriel Collin, manufacturer of optical instruments for the Academy, and watched a clock borrowed from the astronomical observatory of the University of Uppsala.
In quality of instruments, Lagerhjelm enjoyed a significant advantage over Polhem. Polhem had used the most accurate instruments he could acquire or construct. Over the course of the 18th century, however, a real market for scientific instruments had developed in Sweden. The market was largely the creation of the Royal Swedish Academy of Sciences, established in 1739. Rivals for this market competed in precision. Instrument-makers like Daniel Ekström, Hasselström, and Collin won the right to call themselves "Purveyors to the Royal Swedish Academy of Sciences"; this distinction implied to prospective customers that every instrument produced in their workshops promised the highest possible degree of precision. Market pressures did thus increase the degree of precision, and the market itself was a result of the establishment of the Academy. Before then, no Swedish instrument-maker could acquire such status; hence there had been little or no competition in degree of precision.
In this way, the institutionalization of Swedish science contributed to the increased degree of precision in scientific instruments during the late 18th century, a development that contributed to the quantification of technology.
The establishment of the Royal Swedish Academy of Sciences contributed, as mentioned above, to a theoretical and instrumental development. But there was also an institutionalization of technology within the largest and most important of Swedish industries, the iron industry: the establishment of the Swedish Ironmasters' Association, which reflected increased interest in general technological problems during the late 18th century. The general importance of this change in the social organization of technology can be described as a shift in responsibility from individuals to institutions . Technological projects were now being undertaken by and with the competence and authority of the institution.
The competence of an institution, with its hierarchical structure based on academic merit, is apparent in the case of Pehr Lagerhjelm, the leader of the project of 1811–5. He had studied at the University of Uppsala, where he passed the mining examination (Bergsexamen ) in 1807. This university degree, established in 1750, had become a prerequisite for officials in the service of the Board of Mines, the governmental department that exercised ultimate authority over the mining industry. The degree required examinations in physics, mechanics, geometry, chemistry, and law. After graduation from the University of Uppsala, Lagerhjelm was duly appointed to the Board of Mines in Stockholm. He became a pupil of the chemist Jöns Jacob Berzelius, and assisted him in calculating the percentage composition of nearly 2,000 chemical compounds. This work was published as a supplement to the third volume of Berzelius' textbook, Lärbok i kemien . In 1808, Lagerhjelm was appointed under-secretary of the
Swedish Ironmasters' Association. He thus reached his position in 1812 as leader of the hydrodynamic experiments by making a career within the formally organized educational system of the Swedish mining bureaucracy. The merit of the system is proven by his many other contributions to Swedish technology during the early years of the 19th century.
By contrast, Christopher Polhem had gained his position as a favorite of Karl XII in the time of the Absolute Monarchy. Polhem had been appointed director of the Laboratorium mechanicum , the section of the Board of Mines with designated responsibility for research and education in mechanical technology. It was here that Polhem's hydrodynamic experiments were undertaken from 1701 to 1705. Although a mechanical genius by any standard, Polhem lacked the broad formal university education Lagerhjelm enjoyed. His investigations were therefore carried out within the context of a more narrow theoretical perspective and were influenced by important personal idiosyncrasies—including his disdain for mathematical analysis.
Comparing these two examples also illustrates the importance of institutional authority . The project in 1811–5 involved several persons, all highly qualified. Although they did not cooperate throughout the many years of the project without personal friction, the authority of the Swedish Ironmasters' Association led to the completion of the project and to the publication of the results in two volumes in 1818 and 1822.
On the other hand, Christopher Polhem submitted to the Board in 1705 what he called an interim report; in fact, it was the only report he ever prepared concerning his hydrodynamic experiments. When he resumed work with the experimental apparatus in 1710, he made the distressing discovery that two crucial quantities had been measured inaccurately throughout the whole series of measurements.
This rendered the results incommensurable and the whole series of experiments nonreproducible. But Polhem, whose individual reputation exceeded his relatively subordinate position as an official in the Board of Mines, had no difficulty in forbidding his assistant to mention to anyone that the data in the report were useless. An institution stronger in terms of hierarchical authority would have insisted that he submit a final report on the experiments for which he had already received his fees, and, on discovering that the data were useless, demanded that he repeat the experiments.
We tend to associate the 18th century with the use of coal and iron, and to look back on the 16th and 17th centuries as, in John U. Nef's phrase, "an age of timber." But the growing importance of coal technology, especially in England, should not obscure the dependence on forests of virtually all aspects of material culture during the 18th century. Not only did mining consume large amounts of wood; so, too, did potash plants, tanneries, glassworks, saltpeter works, train-oil works, lime production, and other industries rely on the forests for fuel and raw materials. Domestic demands included fuel for heating houses and drying grain and malt, and timber for houses, fences, ships, carts, barrels, and agricultural implements. This vital natural resource, however, was perceived to be running out in 18th-century Europe. The fear of imminent shortages spurred both legislative actions and interest in technical improvements aimed at reducing the number of trees felled.
In the 18th century, Sweden—a country devoid of fossil fuel resources for all practical purposes—was gripped by general anxiety about a dearth of timber. It was believed that the forests were laid
waste by excessive felling: "many large areas of the realm are in danger of soon becoming desolate because of the shortage of timber and. . . the mines and towns in many parts of the country are likewise threatened with ruin that cannot long be postponed if an early remedy is not found." In the worst-case scenario, "the fatherland will in the course of time be reduced to a miserable condition." Whether or not the fear of timber shortage was well founded does not concern us here. What matters is that the belief in imminent shortage was widespread and influential.
The production of bar iron accounted for about 70 percent of Sweden's exports in the 18th century. About 50,000 tons of bar iron were produced every year, and every stage of production required much timber. Blast furnaces and forges consumed charcoal equivalent to three million cubic meters of timber a year. Charcoal-burning amounted to half of the total industrial consumption of timber. There were two alternative methods of charcoal-burning: stacking the wood either horizontally in piles called liggmilor or vertically in resmilor . Despite the fear of a forest shortage and the great consumption of wood for charcoal-burning, evaluations of the two methods of extracting charcoal from wood were not easily undertaken; the question was not resolved until the early 19th century.
In 1811 the Swedish Ironmasters' Association financed a series of experiments to assess the relative merits of resmilor and liggmilor and to determine the pile design yielding the maximum amount of charcoal. The report was published as a monograph three years later by the mining official Carl David af Uhr. It was an impressive volume, comparable in scope and depth to the report on the hydrodynamic experiments undertaken at the same time, also with the support of
the Swedish Ironmasters' Association. Uhr's report described the series of experiments in charcoal burning carried out at Furudahls Ironworks in the province of Dalecarlia during the years 1811–3. No less than forty full-scale piles of different types were tested, with twenty parameters recorded for each pile (fig. 10.1). The systematic study was meticulously planned: for example, a specially designed tool was used to measure the diameter of the billets at both ends, in order to compensate for their taper ("frustra of a cone, as they truly are") when calculating the volume of a pile. The volume of the piles was measured in cubic ells to one or two decimal places. The author discusses the effect of various errors in measurement and ways to compensate for these errors. Output, measured in cubic ells of charcoal, was correlated to the total labor, measured in man-days and horse-days, needed from the day the trees were felled to the day the charcoal arrived at the ironworks. Calorimetric experiments helped determine the quality of the charcoal, prompting Uhr to discuss Lavoisier's opinion of the role of oxygen in combustion. He handled this question with the same facility he showed in computing the number of horse-days needed to build a charcoal pile. Results of this study were summarized in a handbook for charcoal-burners that appeared in three editions—a measure of its success.
One looks in vain for quantitative methods in the literature on charcoal-burning at midcentury. In assessing the relative merits of resmilor and liggmilor the quantity of wood was not specified; nor was it clear which units were used in measuring the charcoal. Consider Magnus Wallner, who published at his own expense A brief account of charcoal-burning in Sweden , a Swedish translation of his dissertation under Celsius at the University of Uppsala. The work
contained a brief description of methods and tools, statements by charcoal-burners, a few quotations from foreign literature on the subject, and some of Wallner's own ideas. Entirely lacking was any attempt to give quantities in defined units and to carry out systematic experiments under controlled conditions.
The failure in applying quantitative methods to charcoal-burning may be attributed to the lack of institutions able to recreate a laboratory environment in the forest. No retort on a laboratory bench could reveal the best type of pile for charcoal-burning. "Systematic experimentation" required building many piles of different types and supervising them day and night for several weeks. It was necessary to take into account the species, age, and moisture of the wood; the length and thickness of the billets; the stacking pattern; the total amount of wood; the outer dimensions of the pile; and the ignition method. After the piles were pulled down, the charcoal had to be shoveled into barrels of known size—"quantification in fixed units." All this was far different from laboratory work. It differed first in spatial terms—not only the size of the piles but also the area of the forest required for the tests was large. The temporal requirements were also of a different order. Because building, watching, and pulling down a pile took more than a month, a series of systematic experiments might stretch over several years. Such was the case with the investigations carried out by the Swedish Ironmasters' Association in 1811–3.
The social organization of the work process also argued against a comprehensive study undertaken by an individual and resulting in useful data. Charcoal-burning was a huge, decentralized system of production: peasants and crofters labored under the tenant's obligation to deliver charcoal to the ironworks. Production was the responsibility of individuals, tens of thousands of peasants and crofters, each working independently, deep in the forests. Work in the forest was linked with the changing of the seasons and the tilling of the soil. In autumn, after the harvest, wood was burned to charcoal in the forest
where it had been felled; in winter, when the snow made the pathless forest passable, sledges carried charcoal to the ironworks. The methods of charcoal burning were the product of local conditions and tradition; the variety of circumstances was reflected in the names of different sorts of charcoal piles. To hammer this decentralized variety of methods and measures into a standard form amenable to quantitative comparisons seemed impossible. Hence Wallner contented himself in 1746 with publishing the views he had elicited from charcoal-burners he had met and a few quotations from foreign literature, together with his own individual, casual observations.
That charcoal-burning—like most technologies at the time—was a nonverbal technology complicated the problem of quantification. Witness the failure of the encyclopedists in their attempt to gather and record technical know-how of existing practices like charcoal-burning. John R. Harris has argued that "the difficulties of imparting craft skills by literary and graphic means" limited the "technological gain" possible from the 18th-century encyclopedias. The difficulties of improving a nonverbal technology were akin to the problems of describing it. A process could not easily be formulated as an intellectual problem and reduced to quantitative terms outside the realm of practical experience.
Science first met technology at the edge of the woods, when the sledges loaded with charcoal approached the ironworks to deliver their products. Only here could the individual ironmaster exercise some control over production by making sure that the peasants and crofters delivered the agreed amount of charcoal. Here the ironmaster might enlist the aid of geometry. Two articles in the Proceedings of the Royal Swedish Academy of Sciences addressed his concerns.
They were devoted to the mathematical problem of calculating the volume of the rhombically shaped sledges that carried the charcoal (fig. 10.2). Together, however, the ironmasters could marshal resources enough to launch a concerted, large-scale attack on the technical problem of charcoal production.
Utilization of Manpower
It is only too easy for historians to overlook the importance of muscle power during the 18th century. Horse whims and tread-wheels, despite their widespread use and importance in their day, have been overshadowed by more efficient and spectacular technologies like waterwheels, windmills, and steam engines. But an innovation does not immediately cause the abandonment of earlier, inferior technologies as obsolete. On the contrary, the total use of old technologies declines slowly and asymptotically. Treadwheels, for example, were used in Swedish mines as late as the 1880s, and one is even reported to have been in use as late as 1896. Muscle power was the dominant power technology during the 18th century: the total work produced by men, horses, and oxen in fields, roads, forests, mines, mills, and harbors probably exceeded the combined power of all steam engines, waterwheels, and windmills.
Industry needed sources of mechanical power capable of high output and continuous operation. The muscle power of men and animals
could provide continuous operation, but output was relatively low. A man working a ten-hour day produced approximately 0.1 horsepower, a horse in a good harness roughly six times as much. Horse whims used at the mines, driven by two or four horses, could thus develop approximately 1.2 or 2.4 horsepower. Continuous operation in three shifts required as many as a dozen horses. Perhaps eighty or even forty men could have produced an equivalent amount of work, but animals offered certain advantages when continuous power of this magnitude was required. In other cases, however, the power supplied by a few persons was not only sufficient but preferable—turning a crank or a windlass, pulling at ropes, carrying burdens, tramping in treadwheels. Compared to horses and oxen, men were relatively small and movable; their power output could be regulated with a word or a glance. The factor of control was significant when it came to loading or unloading ships, turning lathes, grinding and polishing, operating textile machines, or building.
Early attempts to determine how much physical labor a man could be expected to do in a day were made around 1700 at the Académie royale des sciences. Inquiries continued through the 18th century. Bernard Forest de Bélidor, Charles Augustin Coulomb, John T. Desaguliers, Johann Euler, and Philippe de La Hire were among those who addressed related questions of a fair day's work and the comparative strength of men and horses. No consensus was reached, but a large body of data was generated.
In these 18th-century studies, Eugene S. Ferguson has written, "the most casual and fragmentary data were being worked up, with the help of algebraic operations, into definite and precise conclusions." A case in point, described in more detail by Ferguson in an earlier paper, is a study published by Coulomb in 1798. On the basis of two single observations of physical labor, Coulomb wrote an equation for the useful work done while carrying one load of firewood upstairs. This equation was then differentiated and set equal to zero. Coulomb claimed to have obtained thereby the optimum load that would lead to the maximum day's work.
In an article published in 1744, the Swedish mathematician Pehr Elvius compared the efficiency of four treadmills then operating in Stockholm: one at the new Royal Palace under construction, one at a glass factory, and two at the construction of Stockholm's lock. For each treadmill, Elvius measured the rate at which a weight was raised and calculated that "the output of every single fellow is so great that 4 2/5 lispounds was hoisted at a rate of one foot every second." After discussing the differences in design and output for the four treadmills, he attempted to draw general conclusions concerning the ideal design for various applications.
Though Elvius recognized that quantitative methods could be used to improve an existing technology, his method suffered from two interdependent weaknesses, one mathematical and the other social. He used a single observation for each type of treadwheel ("a time of 4 minutes exactly or 240 seconds"), not an average value. Furthermore, his study was based on the types of treadwheels that could be seen in action in the course of a leisurely stroll through the
capital. But the large majority of treadwheels were at work in the mines of the countryside.
The opportunity for a systematic study of manpower did not arrive in Sweden until the early 1770s. The place was the naval dockyard of Karlskrona in the southeast, where manual labor was used to operate the pumps. To dry-dock a man-of-war, ninety sailors worked in three shifts of ten to thirteen hours depending on the displacement of the ship. The engineer Johan Eric Norberg studied the efficiency of manual labor during dry-docking on ten different occasions during 1772 and 1773, and published his report in the Proceedings of the Royal Swedish Academy of Sciences. His primary aim was to increase the efficiency of the pumping operation, but he also had in mind to determine in general the output that could be expected from human muscle power. This, Norberg wrote, had not been attempted before, save in a very small number of tests of limited scope. The results had fallen short in both reliability and extent. Norberg thus demonstrated his awareness that many tests under varying conditions were required to obtain results of general validity. Norberg took hundreds of measurements of the work of the ninety men. Figure 10.3, which shows one of his tables, illustrates the systematic nature of his approach.
Norberg's study is the earliest example in Swedish technology of a systematic, full-scale investigation intended to yield a result of general
applicability. It is scarcely surprising that Norberg chose the navy as the locus for his study. The military was the only social institution sufficient in size and authority to assemble and command the large work force needed for so ambitious a study of the efficiency of manual labor.
In his study in the 1740s Elvius had discussed the performance of only a few individuals. Even the physical characteristics of the individual—in particular, the length of his stride—was an important parameter in Elvius' calculation (dust jacket illustration). But in Norberg's study of the dockyard, individuals are reduced to figures in a table, and the combined product of ninety men's work expressed as a single quantity. Norberg was certainly aware of influential external factors: "harmony, the ration of spirits, the gentle persuasion of the officers, fair or foul weather, etc., have the most important influence on the work, and make a great difference as to output, which should otherwise be nearly the same for equal weights of water." These more human aspects of work at the pumps were not reproduced in Norberg's table, however, which expressed output as a product of measurable physical quantities. Norberg distinguished "the outputs of three kinds of people in various corps." In the sixth column of the table the letter V denotes volunteers (Volontairer ), M , marines (Marinierer ) and B , tenement seamen (Rote-Båtsmän ). The attempt to obtain average rather than individual values necessitated suppression of individual characteristics in favor of general traits.
The Need for Control
In their aspiration to apply quantitative methods to technology in the hope of improving its efficiency and economy, the engineers of the 18th century found themselves hampered by their own boun-
daries as individuals. They had the ambition to reproduce the conditions of the laboratory in the field: to carry out systematic experiments under controlled and reproducible conditions. But this demanded a much larger spatial and temporal dimension than the laboratory—with its desktop apparatus and time scale of days or weeks. It was beyond the means of individuals to embrace the technical reality in time and space. In terms of space , their ambition required wooden structures weighing perhaps up to a ton (whether they were treadmills, charcoal piles, or waterwheels). It made even large spatial demands on the immediate surroundings, since it required such assets as large areas of forest, hundreds of men, or a substantial water supply. In terms of time , it required that several people work together on the experiments for several years. In short, considerable resources were needed to build these unwieldy pieces of experimental apparatus and to operate them for a period of years, and these resources were beyond the means of any individual.
This argument can be expressed in another way. To say that the spatial and temporal dimensions of technology demanded considerable resources if the conditions of the laboratory were to be reproduced in the field is the same as to say that one must be able to control the necessary physical and social realm in terms of time and space . This chapter began with the claim that efficiency and economy are the two quantities that characterize modern technology, and it will end by asserting that its qualitative characteristic is control . Technology is an activity that aims at changes of the material world, and this always involves control.
It is therefore not by chance that the first successful attempt to apply quantitative methods to technology discoverable in Swedish history of technology was achieved at the naval dockyard in Karlskrona in the early 1770s. The case of the ninety soldiers, working in shifts at the pumps to dry-dock a frigate while Johan Eric Norberg stood over them with his watch and notebook, symbolizes a turning-point. It marked the beginning of the successful application of quantitative
methods in technology during the late Enlightenment. At that time, only the military possessed the necessary control over the physical and social realm to reproduce laboratory conditions in the field. It would be some time before civilian institutions grew strong enough to be able to exercise the same degree of control, for example, in the federally supported investigation of the Franklin Institute in the 1830s into the causes of steamboat boiler explosions. These institutions were then strong enough in both authority and competence .
Although this study has been limited to a specific aspect of the question of the relationship between science and technology during the 18th century—the application of quantitative methods to technology—it may have some relevance for the more general question. Roger Hahn has argued that one way to grasp the relationship between science and technology during the 18th century would be to examine the institutional development of technology, and in particular the many societies of arts that flourished in Europe on the model of the scientific academies. Science made an impact, Hahn wrote, "not only by lending its ideology, personnel and theories to technology, but also by offering its social organization as a model to be copied." By copying the social organization of the scientific enterprise, these institutions "also accepted the presuppositions of science itself: rationality, objectivity and publicity." Studies of the history of
individual societies, such as Robert E. Schofield's The Lunar Society of Birmingham , have demonstrated and elucidated this institutional development, but the question of their effectiveness in achieving technological change continues to puzzle historians. A. Rupert Hall has, for example, remarked that it requires "a degree of faith" to find a causal relationship between the popularity of science on the one hand and innovation in technology on the other. But if the benefit of a union between science and technology was conceived only on a cultural level during the 18th century, why was it not put into effect on a political and economic level?
This study suggests that we should continue our search for institutional developments, but look for projects that were undertaken by institutions rather than individual efforts. The characteristics to look for in these technological institutions are not only the rationality, objectivity, and publicity of their scientific counterparts but also the power to control the necessary physical and social realm in order to reproduce laboratory conditions in the field.
More specifically, we should look for institutions that exercised authority according to rank in hierarchies based on scientific and technical competence . The quest will lead us to that historical borderland that lies between the inquisitive academies of the 18th century and the efficient industries of the 19th century, to the period between 1790 and 1825, which has been recognized by Cardwell as one of the definite periods of decisive change in the course of technological history. Not only were many of the major technologies of
the 19th and 20th centuries founded then, but "at the same time social changes took place in the organisation of technology and science setting them on the courses that led to modern technological society."
One of the major social changes was the emergence of institutions that could exercise the necessary control for the successful application of quantitative methods to technology; first within the established structure of the military (military academies, arsenals and dockyards) and later in large industries of national-military importance (mining and chemical industry), major civil-engineering projects (canals), and new civilian institutions. These institutions are probably identical with those that Peter Mathias has called "focal points for developing new skills and educational programs" and that were sponsored by the demands of the state for deploying scientific technology for military or official purposes.
The Calculating Forester: Quantification, Cameral Science, and the Emergence of Scientific Forestry Management in Germany
By Henry E. Lowood
In the second half of the 18th century, few occupational groups rivaled government officials in their attention to numbers. Government officials employed in the duchies, kingdoms, and free cities of German-speaking Central Europe pored over the data on population, imports, and taxes that a growing fiscal apparatus produced in unprecedented volume. Those concerned with the prosperity of the prince and his subjects, from low-level tax assessors to ministers of state, developed an attachment to the quantitative spirit proportionate to the expansion of the state's economic agencies.
Reasons of state and forces of social change brought on the bureaucratization of the state financial apparatus in the 18th century. Rather than dulling their initiative, this bureaucratization created new opportunities for officials, professors, and instructors. In the Age of Englightenment, the improvement of fiscal administration and resource management was seen as requiring a science of state finances, while the proliferation of economic facts and figures raised issues of numeracy and appropriate training for office-holders charged with applying the principles of this new science, which became known as the "cameral sciences" in Germany. The term derived from the Kammer (chamber) in which the prince's advisors tradition-
ally deliberated. The subject matter ranged from economics, finance, and Polizei to mining, agriculture, and trade.
First introduced in Prussia at the universities of Halle and Frankfurt an der Oder in 1727, the Kameral- or Staatswissenschaften were firmly established in the university curriculum throughout Germany by the last third of the century. The call for professional training in cameral science and its gradual emancipation from the faculties of law led to the creation of new professorial chairs and schools for teaching a body of theory and techniques needed for the administration of the state and its domains. It has been argued that these cameral sciences represented a mixed bag of professional training, empirical rules, and warmed-over economic theory. This unflattering characterization overlooks the seminal importance of Kameralwissenschaft in subjecting a variety of economic, administrative, and social practices to rational or "scientific" scrutiny.
Forest management was one aspect of state administration thus scrutinized, in order to fit "scattered pieces of knowledge. . .into systems" and to transform "all sorts of activities previously left to habit. . .into a science." The glue that held these new systems together was economic rationalization. The forest displayed the size of the task of managing the resources from which the prince of the late 18th century ultimately derived his wealth. Discharging the task forged new links between administration and science. The result was quantification and rationalization as applied to both the description of nature and the regulation of economic practice.
German writers on forestry science in the 19th century were struck by the achievements of their compatriots, which "since the middle of the last century can hardly be sufficiently admired." They
touted this example of German cameral science in distinctly national terms: "Compare our literature and the number of our educated foresters to what there is abroad! The beginnings of forestry science are entirely German. " Beginning around 1765, dozens of books and articles published in Germany had established principles and practices of sound forest management; few kindred publications appeared in languages other than German for nearly a century. Theories, practices, and instructional models from Germany provided the starting point for every other national effort in forestry science and management until the end of the nineteenth century.
The work of the classical writers of German forestry science, such as George Hartig, Johann Heinrich Cotta, Johann Hundeshagen, and Friedrich Pfeil, built upon an established tradition of quantitative approaches to the measurement and regulation of the forest. This debt has generally been neglected in the writings of historians of forestry. The origins of rational forest management in the quantitative "forest mathematics" of the last half of the 18th century constitute the subject of this chapter. It will demonstrate that the first advocates of forestry science quantified in spirit in order to bring profits in practice; in the process, they established a tradition of quantitative resource management.
As a substantial portion of the prince's domain, forests constituted one of the largest sectors of the state economy in central Europe. Other forested lands in Germany belonged to the cities and the landed nobility and provided indispensable products for the local and regional economies under their control. Wood in one form or another was essential for home heating and construction, iron manufacture, glassmaking, shipbuilding, and other crafts and trades, while secondary products of the forest found applications in myriad occupations, such as tanning and agriculture. Before the age of coal, which would not begin in many parts of Germany until the middle of the nineteenth century, wood was king.
After the acute and widespread devastation and neglect that resulted from the Seven Years' War (1756–63), the state fixed its gaze on economic recovery. The specter of shortages of wood fuel caught the attention of a small group of conscientious foresters and enlightened bureaucrats, who saw evidence that the deterioration of the woodlands, reported here and there since the Middle Ages, had dramatically accelerated. In the Palatinate, for example, a survey of the forests carried out between 1767 and 1776 spoke of "woods in places so ruined that. . .hardly a single bird can fly from tree to tree." The state of Germany's forests reached its nadir just when rulers like Frederick the Great sought to encourage population growth and force the expansion of industry and trade, measures bound to increase the pressure of demand for wood and other forest products. The fear of impending crisis in the supply of wood lodged in the minds of government officials throughout the remainder of the century, and was periodically intensified by reports of rapidly rising prices.
Officials vigorously pursued economy in the use of wood. But redesigning fireplaces, door-frames, and spoons offered help only on a limited scale; to expand that scale would be a tedious undertaking. Better understanding of the nature of combustion and material properties of wood offered some hope for greater efficiency in wood burning, and scattered experimental reports on these matters of forest physics appeared before 1800. The alternative of expanding the wood supply promised larger gains. Here a bold innovation might succeed in increasing the amount of firewood and lumber available to an entire town, city, or region. Almost in proportion to the potential payoff, however, the complex problem of proper forest management exceeded the meager qualifications of the vast majority of foresters. As a rule their primary appointments as caretakers, game wardens, and master of the hunt required neither practical nor theoretical training in forestry. In Prussia, for example—even under Frederick the Great—posts in the forest administration, which carried the revealing title of Jäger , served as sinecures for military retirees. In the absence of qualified personnel, how could a new approach to forest management arise?
After the middle of the century, the establishment of private forestry schools and publication of books and even journals devoted to forestry began to raise expectations for the training and competence of future foresters and forestry officials. The last year of the Seven Years' War saw the foundation of the first forestry school (by H.D. van Zanthier, in the Harz Forest), the appearance of the first book to use "forestry science" in its title (Johann Beckmann's Beyträge zur Verbesserung der Forstwissenschaft ), and the first journal devoted exclusively to forestry (J.F. Stahl's Allgemeines oekonomisches
Forstmagazin ). One of the first points to settle was the very definition of forests. Traditional privileges and the continued use of the forest for such agricultural purposes as grazing or mast (windfall nuts) had long discouraged a conceptually precise demarcation of the forest. Beginning in the 1760s, however, better-trained officials, equipped with publications for the exchange of ideas, promoted the notion that the forest could be defined precisely and studied objectively.
The first writers on forestry science were led by men trained in the cameral sciences—financial officials and chief foresters who expected economic disaster if the condition of the forests continued its downward slide. As these officers of the local prince consolidated their control over state-managed economies throughout Germany, they attended to the forests in their jurisdiction. Where bureaucratization and centralization of political authority extended the official's sphere of action, as in Prussia, forestry science flourished. The year 1757 marked the appearance of the first of many books on forestry geared specifically to cameral officials: Wilhelm Gottfried von Moser's Principles of forest-economy . Like other cameral officials, the head forester came to his post after considerable study. Every cameralist learned about forest administration, a subject of acknowledged importance: "First, because they are a considerable source of revenue for the state, and second, because they constitute a vital necessity for the sustenance of its citizens, without which these lands—especially in the north—would hardly be habitable." Cameralist writers such as
George Ludwig Hartig placed the new forestry alongside the "state sciences," since the two "make up a complete whole."
The new breed of officials trained in cameral science described the living forest quantitatively before subjecting it to economic reason. They brought to the task a familiarity with mathematics. Mathematics figured prominently among the required subjects, especially in the first year or two of coursework, in the university curriculum in the cameral sciences and also in special forestry schools. Published curricula and schedules of lectures consistently featured mathematics as a Hilfswissenschaft , both for the work of the future government official and as exercise for his mind. At the Cameral College in Kaiserslautern, for example, mathematics was one of the subjects required of every student, and "empiricists" wishing to proceed straight to practical studies without this preparation were not welcome. Heinrich Cotta's Forest Institute at Zillbach, which originated as a site for private instruction in mathematics during the idle Saxon winters, featured the same progression from theoretical to practical. Forestry had become a "complicated science," and it fell to "patriotic men" to ensure that foresters entrusted with the resources of the state were adequately prepared in this new science.
The program won over skeptics. An anonymous reviewer of one book on mathematics for the forester had questioned whether forestry required its own mathematical literature. Careful reading removed his doubts: forest management presented a set of problems worthy of special attention, which they surely would not receive within the body of mathematical literature. Moreover, the reviewer pointed out,
new sciences need to stand on their own feet, and specialized textbooks help to disseminate new rules and procedures and to establish new sciences as independent disciplines.
Writers on forestry presented problems and applications of special techniques, not elementary mathematical instruction. Their goal was to demonstrate how the forester should proceed mathematically, not to produce a new mathematics. With the exception of solutions to a few obscure problems of stereometry and xylometry (measurement of volume and specific gravity of wood), mathematical virtuosity was not necessary. Cotta argued that the "practiced algebraist," to whom calculating the value of a forest was a trivial exercise, would not be the least bit interested in applying his art to it. Cotta also knew that most foresters, unencumbered by such mathematical sophistication, were likely to faint at the slightest scent of a mathematical problem. A reviewer of another early book on the mathematics of forestry concurred: "[the author] demands from the forester planimetry, stereometry, trigonometry, levelling, transformation of figures, third-order and second-order equations. Terrible demands for most foresters!" A prominent advocate of forestry schools argued that one cannot make "great scholars out of uneducated people." But one could turn trees into thalers by replacing the time-worn "routines" of the old Jäger with Forstwissenschaft , it was generally agreed.
This approach was decidedly German. Reforms under Louis XIV had resulted in plans de forêts for state-owned forests and promoted the concept of dividing the forest into annual cutting areas. Jean-Baptiste Colbert's ambitious plan for improving France's forests in 1669 had prompted new statutes, administrative reorganizations, and inventories throughout the 18th century. But a scientific forest management did not take root in France until it was imported from
Germany in the 1820s. English authorities, ignoring such expressions of concern as John Evelyn's Sylva (1664), did not even inventory the remaining forests until the founding of the Board of Agriculture in 1793. As late as 1885, select committees in Parliament debated the merits of emulating the German model of forestry schools and forest science. In Switzerland and Austria, government officials exerted control over a lesser proportion of the forests than did their counterparts in Prussia and Saxony. Moreover, the physiocratic doctrine fashionable in late 18th-century Vienna and Bern offered a rationale for avoiding the problem by selling off woodland and converting it to farmland.
Doing the Work
In central Germany, particularly in Hesse and Saxony, a few foresters had applied the same enthusiasm to managing the forest as to directing the hunt. These conscientious holzgerechte Jäger of the mid-century set annual cuttings according to easy rules based on areal divisions of the forest. After demarcating and measuring the acreage covered by the woods under their supervision, foresters estimated the number of years that the dominant types of trees should be allowed to grow between clearings or cuttings. They then partitioned the forest into a number of divisions equal to the number of years in this growth cycle, from which they proposed to derive equal annual yields, assuming that equal areas yield equal amounts of wood for harvest each year. This straightforward method worked reasonably
well for relatively short growth periods typical of coppice farming and the periodic clearing of underwood. It permitted limited variations, such as shelterwood (Schirmschlag ) or relative cutting (Proportionalschlag ), in which the harvest from a given section of the forest or the size of individual sections could be adjusted according to soil quality and other contingencies.
These methods may have sufficed for a minimally trained huntsman, but not for the fiscal or forest official imbued with Wissenschaft . The crude assumptions underlying the traditional areal division of the forest proved wholly unsatisfactory for the cash crop of forestry—the long-lived high timber, or Hochwald ; the older the trees, the greater the variation in the timber produced by each of the divisions of the forest. Furthermore, the irregular topography and uneven distribution of German woodlands confounded ocular estimation of area without the aid of instruments. Only in the 1780s did Johann Peter Kling, chief administrator of forests in the Palatinate and Bavaria under Elector Karl Theodor, systematize forest mensuration and cartography into instructions for making forest maps of unprecedented detail.
Other fundamental problems also plagued area-based forest management. First, a division of the forest into equal cutting areas did not provide the most useful information to those responsible for fiscal planning and management. They needed to know the amount of firewood or lumber. Correlation of acreage with actual distribution of lumber and firewood required principles not formulated and measurements not routinely executed under the old forestry. Second, the prudent forester could not easily respond to inevitable quirks of nature over the many decades in a single forest cycle, because the area-based system did not provide a flexible method for directly adjusting the harvest from year to year, let alone predicting annual yields over the long cycle from the outset. The most meticulous forest management under these methods, while an improvement over neglect, fell short of the high principles of Kameralwissenschaft .
After midcentury, an approach to forest economy based on the mass or volume of wood gradually displaced area-based systems. The first prominent advocate of wood-mass as the quantitative basis for sound forestry emerged from the holzgerechte Jäger . Johann Gottlieb Beckmann, a forest inspector in Saxony, gave the forest priority over the hunt; his knowledge of forestry derived from experience, not education. Beckmann's deep concern for preserving the wood supply led him to construct a system of forest economy that rested on a practical technique for measuring the quantity of standing wood in the forest. Beckmann instructed his team of assistants, whom he supplied with birch nails of various colors, to walk side by side through the forest at intervals of a few yards. Each member of the formation fixed his gaze to the same side and noted every tree he passed. He made a quick estimate of the size category in which the tree fell and marked it with a nail of the appropriate color. At the end of the day, unused nails were counted and subtracted from the original supply to indicate the number of trees in each category. The forester and his assistants knew from experience the approximate yield of wood from trees in each size category; with multipliers thus assigned, the number of nails used could be converted through a simple calculation into the quantity of standing wood in the forest. Beckmann's case suggests that the clever quantifier need not be a calculator or mathe-
matician nor carry out detailed measurements or stereometric calculations in order to determine the mass of wood. A vigorous and productive author, Beckmann began around 1760 to campaign for the method of forest economy based on wood mass. Soon Beckmannianer sprang up throughout Germany to propagate his ideas.
Within a few years, a group of mathematically adept foresters followed along the trail cleared by Beckmann. Carl Christoph Oettelt, Johann Vierenklee, and Johann Hossfeld assigned the task of measuring the area of the forest to the Forstgeometer , a surveyor hired to demarcate the borders of the forest, prepare maps, and carry out other prescribed tasks for a set fee. The geometer, along with the army of marching assistants, gathered the data. Forsttaxation , or forestry assessment—a mix of calculation, analysis, and planning—fell to the chief forester and his superiors. Forest mathematicians like Oettelt and Vierenklee were moved by a new confidence in the power of mathematics to solve problems associated with the conversion of the forest into an equivalent quantity of wood mass. Assessment, the scientific component in Forstwissenschaft , required general principles and techniques based on them. Without them the unrelated numbers and observations reported by foresters and surveyors would overwhelm planners and administrators. Forestry science supplied the necessary organizing principle: "evaluation, or the ascertaining of the mass of wood, which is to be found for a given place at a given time." Identifying wood mass as the crucial variable of forestry set the stage for quantitative forest management.
Counts to Calculation
Theoretical computations of tree volume began to appear in the 1760s. In the first definitive work of scientific tree measurement (Holzmesskunde ), Carl Christoph Oettelt's Practial proof that mathematics performs indispensable services for forestry , the problem of estimating the quantity of wood on a tree without felling it figured prominently. Oettelt was an experienced surveyor and had held the title of "Forest-Geometer" in the civil service of Saxony-Gotha before taking over the forest department in Ilmenau, where he would later serve under Goethe. In the Practical proof , Oettelt criticized the crude techniques commonly used to estimate the quantity of wood. Most foresters used the so-called Bruststärke , or a stack of wood piled to chest height, to veil their wild guesses as to how many boards a tree had delivered. Estimating in this way, they commonly made the value of a tree proportional to its diameter. Heinrich Wilhelm Döbel, one of the most conscientious writers on forestry around 1750, exemplified the problem. In his influential Gamekeeper's practicum , Döbel struggled to find a simple computation for the problem—in fact, relatively easy—of estimating the volume of a felled trunk. Oettelt invoked geometry: "A tree is the same as a cone with a circular base." With the appropriate formula for the volume of a cone, calculating the volume and mass of trees was not so troublesome.
Oettelt's treatment of wood mass as a mathematical quantity was a radical departure. The holzgerechte Jäger had shown little potential for forest geometry. Döbel argued vehemently that exact calculations of wood mass were unnecessary, "since you don't measure wood like
you do gold." He preferred the simple "farmer's calculation" to disputations and proofs. The mathematically oriented foresters, among them Johann Vierenklee and Carl Wilhelm Hennert, joined Oettelt's cause. They corrected and improved his geometric calculations in a series of books that culminated in 1812 in the definitive work on forest stereometry by Johann Hossfeld. As abstract, mathematics-based forestry gained sway during the 1780s and 1790s, compilations of tables based on controlled measurements replaced the older crude techniques described by Oettelt.
Those who compiled such tables had to bridge the gap between tree conics and precise measurement. Consider the problem of converting from cubic measures of wood mass to Klafter , the unit of stacked cordwood familiar to the forester, and back again. The interstices and warping of real wood might defeat the most exact geometrical analysis of its volume. Since mass or volume constituted the central quantity of the new forestry science, small errors due to branches, warped stocks, and imperfections of nature multiplied rapidly as one reasoned from the tree to the forest. Equating the economic measure—volume of stacks of hardwood—and the computed volume did not work out.
The quantifiers, beginning with Oettelt and Hennert, searched for scientific sandpaper to achieve a greater semblance of precision. Oettelt measured as accurately as possible the volume of the cord, then ordered the wood chopped into small pieces. The volume of each piece could be measured with greater accuracy. He summed these individual measurements, and compared the sum to the original cord. After repeated tests he determined that a typical span of cordwood measuring approximately 110 cubic feet contained 14 to 18 cubic feet of empty space, about 15 percent of its volume. Hennert borrowed Diogenes' barrel: he poured water into a box filled with wood; the volume of the box less the volume of the water yielded the solid content of wood (Derbgehalt ). By 1812 Hossfeld, in his Lower and higher practical stereometry (1812), had replaced Hennert's water with sand and contrived even more accurate xylometers. Such innovations made feasible "measurement and calculation of all regular and irregular bodies, and especially trees in the forest."
In the German tradition, the mathematician's forest was populated not by the creations of undisciplined nature, but by the Normalbaum . Forest scientists planted, grew, and harvested this construct of tables, geometry, and measurements in their treatises and on it based their calculations of inventory, growth, and yield. Writers and instructors gave foresters in the field the tools for reckoning the dimensions of the standard tree. Most treatises contained instructions for averaging measurements made on a test plot, but foresters were happier to use the Normalbaum . Tables of numbers representing measurements and calculations, or Erfahrungstabellen , provided data organized by classes of trees under specified conditions. A small number of variables governed the forester's choice of one or another of these tables. For example, the wood mass of the typical sixty-year-old pine on good soil was given as a function of its height and circumference. These tables, which appeared in every complete manual of rational forestry
practice, generally did not bother with regional variation, the bugaboo of 18th-century agricultural treatises.
By the end of the 18th century, German writers on forest management had worked out steps for determining, predicting, and controlling wood mass. Heinrich Cotta presented the clearest and most widely read exposition of these steps in his Systematic instruction for the assessment of woods , published in 1804; they were elaborated in his Directions for the organization and assessment of the forest , which appeared sixteen years later. Cotta's first book, which consisted of lectures originally prepared for students attending the forestry school under his direction, was an example of systematization induced by the necessity of teaching. In his method, the "geometric survey" of the woods supplied the Taxator with information about the extent of his forests. The next step required calculations of wood mass of individual trees, then of stands, and finally of the forest as a whole; growth rates were computed for each level of organization. Finally, Cotta's forester qua cameralist linked the forest balance sheet to the monetary budget by determining the value of the yield.
If the standing forest is capital and its yield is interest, the forester can complete the chain of conversions from wood to numbers to units of currency: an estimate for the worth of the forest can thus be used to predict income, calculate taxes, assess the worth of the forest, or determine damage to it resulting from a natural disaster. For Cotta, the fundamental problem of forestry management was determining the "standing value" of a forest, given uninterrupted maintenance costs and full harvest some 100 to 150 years hence. Cotta's forestry science thus consisted of sound methods for inventory and prediction: "From summary investigations based entirely on verified
judgment, we go through various stages to more exact investigations, first of individual trees, then of the supply, growth, and yield-determination of individual stands, and finally of whole forests." Similar procedures, from the forest to the tree and back again, also appeared in practical manuals such as Georg Hartig's New instructions for the Royal Prussian forest-geometers and forest-assessors .
In one respect, Cotta differed from Oettelt's line. He preferred careful ocular estimates based on tables to geometrical deduction, which he not only considered impractical in the field, but also inaccurate, since branches and other irregularities confound the comparison of trees to cylinders and cones. For Cotta, the only absolutely sure method was to chop up a tree and measure its volume (or mass) in the same unit of measure to be adopted in the taxation itself. This view did not weaken his allegiance to mathematical forestry. He was skeptical only of geometrical estimates, not of quantification.
The Forstwissenschaftler , and particularly Cotta, championed use of "experience tables." Their use reinforced the notion of a forest filled with standard trees. The forester was to instruct his assistants in the use of these tables so that a mental picture of a tree encountered in a forest corresponded to an entry in the tables. With sufficient repetition, a good forester could make an instant association from the mental picture triggered by the tree to the value of the wood mass contained in the table. The next step was to generalize: every tree of the same height has the same mass (or volume). The standard forester was trained to find the standard tree. For Cotta, the "eyeball measure" could displace the "measuring hand" if every forester learned to see the archetypical. The practiced eye could indeed attain this mechanical perfection, "as subsequent measurements and calculations prove[d]."
The head forester thus trained his assistants to internalize Erfahrungstabellen and become computers of wood mass. He remained at his desk manipulating the Normalbaum and numerical data based on local measurements. He could produce his own tables if necessary; according to Hartig, the Taxator was responsible for all "mathematical preliminaries" of forest assessment—determining growth rates, preparing maps and calculating tables—before delegating to his staff routine measurements and the mechanical application of tables. The assistants marched, tallied, catalogued, and marked under the watchful gaze of their supervisor, who—according to Hartig's directions—never counted with them. Instead, his duty was to "dictate principles, record the results in the Assessment-Register, and make sure that there are no mistakes."
By 1800, the forest assessor trained in the cameral sciences specialized in theoretical principles, mathematical preliminaries, and the cumulation and analysis of data, a far cry from Beckmann with his colored nails and squad of assistants. An array of numbers stood for the quantity of wood in the forest. The forester or cameralist trained in forestry science felt no need to step off every acre with the exactness given to the test plot, the geometrical abstraction, or exact measurements of the volume of cordwood. Instead, he could sample and generalize. The work of the assessment and management of the forest thus required only standard trees and Erfahrungstabellen . As Cotta argued, the crucial quantities of his science were "determined mathematically" from the "premises" of forestry science, not through "direct real measurement ." The scientific forester had abandoned Beckmann's empiricism in favor of "sure mathematical deductions, experiments and experiences in the given and understood units of measure." Under the banner of Wissenschaft , the new breed of qualified forester breathed the quantitative spirit into administrative practice.
By the end of the 18th century, the new breed of foresters in Germany, those with diplomas from forestry schools or degrees from the university, adopted the methods popularized in the clear prose of writers like Hartig and Cotta. Their appetite for a rational synthesis of calculation and cameralism was whetted by identification of mass and yield as suitable quantities to measure. As in Lavoisier's chemistry, new fundamental measures required new terms of analysis. By 1800, the ideal of the "regulated forest" proclaimed the preservation of the forest's maximum yield under a sound system of forest economy. Three regulae silvarum found throughout the writings of the Forstwissenschaftler linked the desideratum of the regulated forest and the methodological focus on measurement and calculation: "minimum diversity," "the balance sheet," and "sustained yield."
Direct measurements of wood mass or volume would have provided the forester with the data he needed for determining fellings or predicting monetary yield, but such numbers were hidden in the diversity and complexity of vegetation in the forest. New units of analysis gave categories better suited to forest computation than the vast, green sea of individually appreciated trees: the "standard tree" (Normalbaum ), the "size class" (Stärkeklasse ), the "sample plot" (Probemorgen ), and the "age class" (Periode, Altersklasse ), as used in textbooks and instructions.
Johann Wilhelm Hossfeld typifies the Forstmathematiker as leveler. His precocious fondness for mathematics, combined with an argumentative temperament, made him unpopular with his teachers; he turned the tables and became an instructor of mathematics at schools specializing in commerce and forestry at Eisenach, Zillbach (under Cotta), and Dreissigacker, where he finally settled in 1801 with the title of Forstkommissar . Here he moved his mathematical skills from the lectern to the forest. Hossfeld made his name among foresters as a leading proponent of stereometrical and geometrical methods in the determination of wood volume and as the inventor of methods to calculate the value of the forest. His writings are a train of
mathematical exercises, with solutions. Hossfeld worked his way from the volumes of cubic forms representing ideal tree trunks through growth, yield, expected demand, costs, and the budget to an all-encompassing "integral of all results pertaining to the value of a forest." He defended his mathematical approach on the basis of economy of effort: a purely empirical assessment counting every tree in a forest might take one observer several years, whereas a mathematician could produce a useful formula after a dozen or so careful observations. Nature "makes no leaps," he claimed, so that a series of multiplied averages based on one or two easily observed characteristics, such as the height of a stand of trees, is as good as an exact and painstaking summation of all the individual cases. The mathematician need not fear hidden pockets of diversity.
Minimizing nature's diversity and reconstructing the forest to make life easier for foresters and assessors were typical of the authoritative writings of the Forstklassiker . Hartig advocated strict adherence to results drawn from a few sample plots. He recommended that the forester keep things simple by following a small number of general rules and reliable methods. With characteristic dogmatism, Hartig ruled that one should always cut out "arbitrary" details of nature that might distract from the systematic Taxation . Cotta agreed with Hartig on the need to ingnore disparate details and concentrate on useful numbers derived from a sample plot. Cotta argued that selective measurements generate acceptable values for quantities like typical yield or growth, which then become the characteristics of ideal types presented in tables and other summations and
multiplications of data from test plots: "the assumed quantum of growth is really abstracted from many trees of the same kind; the sum of the whole is always the basic measure." These sums cannot be directly measured by any practical method; they can only be determined "mathematically, that is, with the aid of an inference based on single values that are known." The source of the values did not really matter; measurements in sample plots, geometrical deductions, or experience-tables were equally acceptable if the method produced a standard—the "single value." One need not worry about the cumulation of errors; individual differences cancel out in the aggregate. This assumption brought freedom from the need to poll every tree, without increasing the risk of error. The new science rewarded the forester who did not see the trees for the forest.
The Balance Sheet
Although cameralists had in common with forest scientists a faith in numbers as worth a thousand words of old forestry, their underlying assumptions differed. Oettelt, Hossfeld, and Cotta saw management as dependent on mathematics, not the reverse: "the workings of nature and mathematical truths do not subjugate themselves to words of authority." Even kings and ministers had to bow to this ruler of the kingdom of reason.
Officials in the fiscal bureaucracy with broader responsibilities than the forester's showed less enthusiasm for the ultimate rule of mathematics in forestry science. They clearly appreciated numbers as the rudimentary facts of accurate inventory and accounting. Sophisticated forest management provided efficient tools for monitoring the quantities that the state bureaucracy sought to control from year to year. If expressed coherently in numbers, represented clearly in charts
and tables, and placed in the hands of the cognizant minister at court, these vital signs eased the task of keeping the body economic healthy, much as a thermometer aids the physician. To the cameralist, the role of quantification in forestry science was descriptive, not prescriptive.
A common denominator nonetheless related the disparate values that scientists and cameralists attached to quantitative information. The annual accounting of the bureaucrat had to be linked with a long-term plan of resource management based on scientific principles. One prominent Forstwissenschaftler , Friedrich von Burgsdorf, called the common problem "keeping the forest's books," and defined procedures to follow in terms of the quantities of interest to forestry science. The bond between forestry science and cameralism was the conversion from an amount of wood to its value. From that point, the practitioners could go their separate ways, the cameral official to the preparation of the Geld-Etat , or monetary budget, and the forestry scientist to the Forst-Etat , the budget that compared the yield to what the forest could bear over time.
Hartig described the task of creating the Forst-Etat as seeking an equilibrium, as opposed to the bottom line in a fiscal budget. "Where a sure balance sheet of forest use, based on mathematics and natural philosophy, is lacking, wood will always be over- or underutilized." In the former case, balance would have to be restored through conservation, raising more land for the forests, or abandoning a less vital productive arm of the economy; in the latter, by exporting lumber or founding new industries. Hartig used terms like "forest use budget" and "natural forest budget" to describe the related components of planning and biological growth that concerned the forester in his effort to balance supply and demand. Hossfeld likewise spoke of budgets and balances. He explicitly identified forestry assessment with the process of evaluating disturbances to the equilibrium of the forest, whether natural (fires and pests) or artificial (management). After calculating the magnitude of these disturbances, the forester
could prescribe means for restoring the equilibrium of growth and yield over time. The image of the budget, whether of nature or gold, linked forestry, cameralism, and quantification, as foresters learned to manage both the Forst-Etat and the Geld-Etat according to the books.
As we have seen, the books themselves consisted largely of numbers. Hartig wrote hundreds of pages on the gathering of data, calculations, and organization of charts and tables necessary for the production of ledgers; the charts mimicked the columnar arrangement of the accountant's books. Hartig and Cotta both offered book-length examples of their methods of forest bookkeeping, complete with templates for the tables they had used. In general, journals and records kept by low-level foresters were to be turned in quarterly to the supervising forester in each district, who compiled and summarized. A Forst-Rentmeister would calculate the monetary budget from these and parallel records according to prescribed forms, while the Forest Commission, consisting of higher financial officials, would review, analyze, and summarize. According to Ernst Friedrich Hartig, Georg's younger brother and colleague, the results concerning consumption, production, and distribution of wood could thereby be arranged so that "the balance in every forest, district, administrative region, and province can be easily reviewed at a glance." The recurring themes of equilibrium and the balance sheet harmonized with those of administrative convenience and scientific resource management.
The third quantitative principle in German forestry science was sustained yield (Nachhaltigkeit ). Chopping down enough trees to meet immediate needs satisfies the balance sheet. The bureaucratic
annual cycle and associated methods in forestry management deal with immediate and short-term record-keeping and assessment. Year after year, cuttings reduce the wood mass according to ephemeral prices, needs, and the conditions of nature. All can be precisely measured and monitored. But the life of individual trees, let alone the forest as a whole, contains dozens and dozens of annual cycles. Long after the incompetent forester is gone, his mismanagement and irresponsibility survive. As Johann Matthäus Bechstein proclaimed in 1801 to students entering his forestry school, the forester must be capable of calculating "more than one or two generations into the future." Planning the growth, cutting, and replenishment of a forest over the longue durée requires an idea more powerful than the balance sheet. Foresters found it in sustained yield.
The rudimentary concept of sustained yield appeared in one of the earliest texts on forestry mathematics, Johann Ehrenfried Vierenklee's Mathematical first principles of arithmetic and geometry, to the extent they are needed by those who wish to devote themselves to the most necessary subject of forestry , which appeared in the first of three editions in 1767. Vierenklee judged that the forester must know "how to divide up a forest into a definite number of annual cutting areas, from which he obtains a definite amount of wood each year." Vierenklee relied on mathematics for the formulas to achieve this division, and based his work on growth calculations for high timber.
A full generation of Forstwissenschaftler later, sustained yield figured as the cornerstone of Hartig's dogmatic system of forestry management: "always deliver the greatest possible constant volume of wood." The grail of sustained yield has guided the quest for rational forest economy ever since. With this concept, time entered forestry science. How much wood can the forest deliver over a century or two? How should this yield be harvested in one year so as to ensure that the same yield will still be available 100 years hence?
Questions like these redefined the forester's task as curator of the forest, not simply its measurer. As Hartig put it, "no lasting forest economy is conceivable if the output of wood from the forests is not calculated according to sustained yield."
The proper way to ensure the "permanence, certainty, and relative equality of the yield" is not immediately obvious. Yield, unlike wood mass or forest area, is not a "quantity determined by nature"; it cannot be measured, save for the year at hand. A system of forestry based on sustained yield requires prediction and planning. Some relevant factors, like the present mass of wood in the forest, can be measured; others, such as growth rates, must be extrapolated from the performance of sample plots, and the assumption of "good," "average," and "bad" soil. From this blend of quantities and qualifiers, the scientific forester can determine a schedule of cuttings for the forest of standard-trees under the "particular aspects of each system of culture," such as timber forest, coppice, or a mixed form. Conditions such as the present state of the forest and expected growth rates must then be factored in; these, as Cotta pointed out, cannot be calculated according to "algebraic formulae." Inconsistencies in soil, weather, and natural devastations complicate the application of the method. Moreover, equating annual yield to the expected biological growth is a risky proposition. "How can man presume to determine such events of the future in advance, when they are dependent on a thousand accidental events?"
Undaunted by the obstacles to accurate prediction, the Forstklassiker specified procedures for "forest regulation" (Forsteinrichtung ); before long, many foresters throughout Germany adopted these methods. Unlike descriptive assessment, forest regulation was predictive and prescriptive. It offered a framework of long-term seeding and cutting based on the mathematical forest and standard practices
for application in the wooded forest. Scientific forest regulation also exercised many aspects of the forester's art, from cartography, description, and techniques for regeneration to silviculture and assessment. The role and authority of the vigilant chief forester who oversaw and adjusted the plan to circumstances, were reinforced by the scientific principles of forestry management.
Approaches to forest regulation multiplied quickly and differed considerably. Hossfeld and Cotta used geometry and arithmetic to construct flexible systems based primarily on wood mass and areal divisions of the forest; plans derived from their methods could be adjusted as local conditions dictated. Heinrich Christoph Moser, Commissar of Forests in Bayreuth, published a method of determining the "periodic yield" based on "proportion constants" and sample plots. Johann Leonhard Späth, Professor of Mathematics and Physics at Altdorf, proposed a detailed algebraic method. The result of these investigations in almost all cases was a visual arrangement of age-classes and plots, linked with the quantities of wood and cuttings over time. Fold-out tables were common; Hartig used one to extend his plan into the 21st century (fig. 11.1). Like the business plans of a later day, attention to graphic clarity propped even the most chimerical of schemes against the firm oak of faith in numbers.
During the 19th century, the tradition of German forestry science persisted as cameralism gave way to economic liberalism. It produced the monocultural, even-age forests that eventually transformed the Normalbaum from abstraction to reality. The German forest became an archetype for imposing on disorderly nature the neatly arranged
constructs of science. Witness the forest Cotta chose as an example of his new science: over the decades, his plan transformed a ragged patchwork into a neat chessboard (fig. 11.2). Practical goals had encouraged mathematical utilitarianism, which seemed, in turn, to promote geometric perfection as the outward sign of the well-managed forest; in turn, the rationally ordered arrangement of trees offered new possibilities for controlling nature. For example, the technique of periodic area allotment (Flächenfachwerk ) favored by Cotta generated the now familiar checkerboard scheme of growth periods. The mathematical exercise that generated the pattern could be modified to order the sequence of cutting so that older stands protected younger trees against prevailing winds. In the hands of a suitably trained forester, mathematical order and practical utility became one enterprise.
During the 19th century, Forstwissenschaft advanced along the lines established by the early forest mathematicians: sustained yield, regulation according to age-classes and wood mass, and construction of the "normal forest" as an artifact of mathematical reasoning applied to quantitative data. By the end of the 19th century, reformers of forestry in other natiøns—France, England (via the Indian Forest Service under Sir Dietrich Brandis), and the United States—had also discovered the need for conservation and forest management based on professional training and scientific principles. In each country, beginning with France during the 1820s and culminating with the American conservation movement, the inspiration and example was German Forstwissenschaft .
In Germany, however, resistance to the mathematically formulated forest economy began to grow, spurred by natural devastations caused largely by strict reliance on monocultures. By the end of the 19th century, foresters such as Karl Gayer, inspired by new-found loyalty to the natural diversity of species, called for turning "back to nature." Careful consideration of the forest as a multi-faceted biological ecosystem came into vogue. Even in the face of this opposition, quantitative techniques elaborated by the Forstklassiker survived in practice. Above all, the doctrine of sustained yield remained sacrosanct. Franz Heske, writing in 1938 for American foresters, reaffirmed the legacy of 19th-century Forstwissenschaft based on the work of the classical writers:
For all time, this century [the 19th] of systematic forest management in Germany, during which the depleted, abused woods were transformed into well-managed forests with steadily increasing yields, will be a shining example for forestry in all the world. German experience over a century makes it considerably easier for the rest of the world to pursue a similar course, because the attainable goal is now known, at least in principle. The sponsors of sustained-yield management in countries where forestry is still new can find in the results of this large-scale German experiment a strong support in their battle with those who know nothing, who believe nothing, and who wish to do nothing. This experiment and its outcome have rendered inestimable service in the cause of a regulated, planned development and use of the earth's raw materials, which will be an essential feature of the coming organic world economy.
Society in Numbers: The Debate over Quantification in 18th-Century Political Economy
By Karin Johannisson
In his Modest proposal of 1729, Jonathan Swift took aim at political arithmetic—at its number mania, its intimate links with state power, its impudence in measuring human worth in money. A half century later, Adam Smith, himself no stranger to issues of quantification, joined in Swift's attack: "I have no great faith in political arithmetic." In a few more years Robert Malthus would express similar sentiments.
Karl Marx, on the other hand, would accept political arithmetic as just the all-embracing social science its founders had envisioned, and would cite William Petty as one of the most inspired and original of economists. In modern surveys of the history of the social sciences, political arithmetic has likewise been assessed in conspicuously favorable terms. Quantitative methods hold prestige in the context of social science; numerical data are seen as a mark of scientific neutrality toward the phenomena under study. Only in recent years has this objectivity come under scrutiny, with the recognition that statistics offer no special guarantee of freedom from ideological influences. Numerical data are not collected, they are selected and sorted according to criteria shaped by ideology and politics.
The 18th-century debates over the merits of political arithmetic reveal sober recognition of the ideological content of the subject, as well as shining faith in order, systematics, and reason. The optimism that attended the application of political arithmetic was countered by a reluctance to claim much for its methods and utility. What emerged was not the vision of a measurable society, but rather a new view of social statistics as a useful instrument for describing aspects of society and economy.
Staatenkunde and Statistics
By the middle of the 18th century, "statistics" or Staatenkunde had been a subject of study in German-speaking countries of over fifty years. Its object was comparative description of the resources of different states; its aim was to assess their political strength. The purely verbal descriptions neither employed numbers nor aspired to generalization or to the formulation of general laws. This early Staatenkunde , which lacked both a quantitative method and a connection with the natural sciences, grew into a university discipline of great prestige equipped with an increasingly refined methodology. The descriptions it generated served as a bank of knowledge from which facts could be drawn by government officials as they drafted domestic and foreign policy. University statistics thus came to be known as "the right eye of the politician," whose duty it was to watch out for the nation's resources and prosperity.
To the German statisticians, schooled in Aristotelian philosophy, the welfare of the state was not merely a question of quantities and materials; their concerns also encompassed intangibles like national character, satisfaction of the citizenry, and realization of the aims of the state. From an enormous mass of information, the statistician had to extract the facts pertaining to the happiness of the masses. In his Theorie der Statistik , August von Schlözer identified the task of his discipline: "to measure the happiness of peoples, and whether this is increasing or declining." This meant "power and strength, to be sure! But these are only part of the happiness of a people. And not always these: for are there no states which are outwardly all-powerful but whose citizens live in wretchedness?" Von Schlözer's words capture a basic ideology. A country's strength is not to be measured only in the superficial and the visible. Such assessments miss crucial factors like character, quality, and depth, which serve to distinguish the nations of the earth.
The statistical net hauled in unmanageable quantities of information. Toward the end of the 18th century some statisticians began systematically to use the table as a means of organizing this information. The tabular form, with its columns of countries and rows of categories, facilitated comparative analysis and offered new perspectives. At first, the tables mixed verbal and numerical information, but the use of columns soon favored facts in the form of figures. Numerical language, uniform and efficient, produced compact tables. The instrument of the table in turn created a corps of advocates, with clear preference for just those categories that could be most easily quantified. Such "tabular statistics," as its detractors dubbed it, differed markedly from the qualitative discipline of university statistics.
Although they wielded a numerical sword, the advocates of tabular statistics did not fight under the banner of quantitative social
analysis. The purpose of their tables remained description, albeit with the aid of numbers. Such descriptive quantification should be distinguished from analysis of a society believed to be the real product of individual, quantifiable constituents. Nonetheless, tabular statistics was seen as a real threat by the proponents of the older, university statistics. As supposed materialists and social mechanics bent on dismantling a beautiful and intricate reality, the tabular statisticians came under steadily heavier fire.
To the university statisticians, the word was the medium of statistics. Numbers might occasionally prove useful as an auxiliary instrument to give concrete form to particular descriptions or to express relationships. But numbers can never pierce the surface, they argued, or explain more than material circumstance. The influential review close to the professoriate of the University of Göttingen dismissed the new "table hacks" and "table fabricators" as common journeymen, whose reliance on the instrument rendered their work both shallow and coarse and reduced a beautiful art to a soulless technology. Only the nobility of university statistics grasped the idealistic factors inherent in the state. "The tabular method [seeks to] reduce everything to figures. . . . If one has a few columns giving the figures for square miles, revenue, population and our dear livestock, one has a summary of the strength of the state; for national spirit, love of freedom, genius and character. . .there are no columns. . .and yet it is much less the body than the spirit that determines the strength of the state." The metaphor of the body recurred: "Has not. . .the whole science of statistics—one of the noblest—been debased to a skeleton, to a veritable corpse, on which one cannot look without loathing?. . . . The state is something nobler than a machine. . .it forms a moral body."
The note of desperation in these words reflects a profound transformation in the nature of the dispute. A quarrel that appeared to concern the form in which statistical data should appear came to represent rival philosophies. The quantitative approach, regarded as
reducing reality to the material and excising the spiritual, stood as a challenge to the basic ideology of Romanticism, with its idealism, organic concept of the state, and emphasis on individuality. How naive to see the state as machine! How could anything so multifaceted as a state aspiring to fulfillment be expressed in mere numbers? The university statisticians instead saw the state as a "being" (Wesen ), people as Volk , imbued with Volkgeist . The collective presupposed a social code based on spiritual and traditional values. People should not, could not be reduced to a factor of production; they were not the means to prosperity but its purpose.
As this colorful rhetoric may suggest, the university statisticians felt their position slipping out from under them. Deprived of more and more content as new specialties (political economy, geography, ethnology, and so on) broke away, university statistics began to wither away. As a political science it would be ruthlessly discredited by events during and after the French Revolution. University statistics, in failing to identify the popular discontent that found its voice and program in the Revolution, or to foresee that mighty Prussia would be trampled like a sand castle under the feet of Napoleon's troops, sounded its own death knell: "Nothing, nothing at all was achieved by the higher [university] statisticians."
The tables to which the German university statisticians objected were, or could be, instruments of political arithmetic. In England and Sweden, debate about the use of statistics centered explicitly on the program of quantitative social and economic analysis that descended from the work of William Petty and his contemporaries at the end of the 17th century.
William Petty was held in high esteem in circles populated by the likes of Robert Boyle, John Wallis, Samuel Hartlib, and Thomas Hobbes. Considered by Samuel Pepys "the most rational man that I ever heard speak with a tongue," Petty fit in well with the successful men of science who met in the Invisible College and the Royal Society. He dreamed of a perfectly rational society, a social edifice as stable and unassailable as a mathematical theorem. Each element had to be weighed, measured, and evaluated before it could be incorporated into this rational system. Petty made the point explicitly: "The Method I take. . .is not very usual; for instead of using only comparative and superlative Words, and intellectual Arguments, I have taken the course. . .to express myself in Terms of Number, Weight or Measure ; to use only Arguments of Sense, and to consider only such Causes, as have visible Foundations in Nature; leaving those that depend upon the mutable Minds, Opinions, Appetites, and Passions of particular Men."
Petty aimed at a science that used quantitative methods (counting and measuring) to isolate, describe, and analyze the elements that created a society's prosperity. Available observations of population, land, and resources—all in numerical form—were used in the calculation of values of other, as yet unknown, resources.
To reduce Petty himself to arithmetic, or rather geometry, his program lay at the intersection of inspiration of Baconian empiricism and Newtonian natural philosophy. He called for the assembly of individual measurements (of, for example, birth and mortality rates, work capacity, output, consumption, and fertility) to form a valid picture of a general reality. In practice, however, he often relied on estimates and averages; and his columns of numbers necessarily remained isolated from one another since correcting principles analogous to the law of gravity eluded him. Petty and his successors Gregory King and Charles D'Avenant framed a general program of social analysis based on computations and systematic collection of facts: "He who will pretend to Compute, must draw his Conclusions from many Premises; he must not argue from single Instances, but from a thorough view of many Particulars; and that Body of Political Arithmetick, which is to frame Schemes reduceable to Practise, must be compos'd of a great variety of Members."
The strong political context of political arithmetic may be discovered in a fight in Parliament over a proposal to provide it with one of its basic instruments. In 1753 advocates of a coherent program of social statistics, among them the mathematician James Dodson, called for a general census. But the census bill fell victim to strong opposition fed by fears of an expanding government. Five years later Parliament rejected a similar proposal for mandatory registration of births, marriages, and deaths. Prosperity appeared better served by capital, industry, and steam power than by the calculations of a power-hungry central government. Parliament thereby denied to proponents of political arithmetic both the means of collecting
data—the census and the registration of vital statistics—and the function intended by Petty and his contemporaries—that of a political instrument of scientific legitimacy.
That, of course, did not prevent private groups from compiling vital statistics for their political purposes. Republicans and religious dissenters picked up political arithmetic and wielded it to combat the "faithful guardians of the state" and reduce their authority. These advocates of local power counted local conditions as more significant than national aggregations of class, rank, and occupation. A good example is Richard Price's statistical studies of national debt and local prosperity based on data privately collected. Another is development of the work of John Graunt and Edmund Halley on the mathematical analysis of life expectancy. Graunt had demonstrated the utility of vital statistics for establishing the laws of demography; Halley's studies (published in 1693) of mortality lists from the city of Breslau had shown that generally valid calculations of life expectancy could be based on mathematical analysis of available, incomplete data on births and deaths. As a basis for calculating insurance premiums and annuities, quantitative data thus offered real practical value. By the second half of the 18th century, both governments and private investors recognized insurance ventures as a promising prospect. Price saw annuity societies, if guided by mathematicians, as a solution to England's economic ills.
While republicans and dissenters in England reserved political arithmetic for their own purposes and rejected its use as a tool of central government policy, Swedish officials sought a socioeconomic
strategy based on social measurement that might replace the capital and manpower the country had lost during long years of war. To remedy these ills, members of the Swedish parliament followed orthodox mercantilist lines based on the conception that population is the best measure of a nation's real riches. "A plenitude of poor people is a country's greatest wealth." In the mercantilist analysis a surplus of people meant a surplus of need, which would inspire economic initiative, industry, and cultivation of land. A Swedish economist of the time wrote, borrowing a line from D'Avenant, "It may be better for the people to suffer a shortage of land than for the land to suffer a shortage of people." The line symbolized a tie: in 18th-century Sweden, the old English idea of political arithmetic would enjoy political support, the notice of the Royal Academy of Sciences, an institutional platform in the parliament, and enthusiastic public backing. The key role of quantitative analysis in the Swedish debate on ways and means to national prosperity can be traced in pamphlets and programmes; in the Transactions of the Royal Academy of Sciences; in parish surveys, state memoranda, and confidential parliamentary reports.
In accord with the proposition that people attracted industry, rather than the reverse, colonization projects of all kinds, resettlement projects, and measures to stimulate population growth abounded in Sweden beginning in the 1730s. So, too, did optimism about the rate at which the population of Sweden might increase. Some fortune tellers saw a doubling in the space of twenty years. In the long run, the population of Sweden (which then included Finland) might reach ten, twenty, even thirty million inhabitants. (The actual population figures in the mid-18th century, not disclosed at the time, were closer to two million.) The mainstream of public
discussion overflowed with optimistic calculations from leading social theorists, and from reports submitted to the Riksdag by the Office of Tables.
How could these calculators arrive at such preposterous figures? In part, the optimism derived from conceptions of Sweden's great natural resources inherited from the patriotic historiography of the preceding century. In the years when Sweden was emerging as a great power, the country of the north was often depicted as the vagina gentium , the land chosen by the sons of Noah for its natural riches.
Mercantilism reinforced this patriotic tradition in a land of unexploited natural resources. "No man dares keep any land unusable and unfruitful, now and in the future, once he has seen the new ideas that can spring from ingenuity when spurred by need in a populous country," proclaimed the director of the Land Survey Board in 1758. In Sweden, all agreed, the good Lord had "let abundance drip from His footsteps." He had also arranged for an advantageous climate. The cold protected Sweden's populace from infectious diseases and made them "merry, lively, and manly." Snow on the ground prevented the evaporation of nutritious substances; once it melted, rotting leaves and needles gave way to rich humus soil. The woods teemed with useful game ("if anyone would seriously try to domesticate our elk, they might well become our camels"); lakes and rivers overflowed with salmon and other splendid fish, pearl-filled mollusks, oysters, and lobsters. Nonetheless, God's handiwork could be improved upon. "If wildernesses and wastes are cultivated, a whole new land can be created, even more fruitful, milder in climate, more pleasant in every way, rich and able to support and feed millions more people than today."
The language and the imagery were as rich as Sweden herself. But how to measure and master these natural resources? Political arithmetic offered a key to calculating Sweden's potential and devising a new strategy for growth and prosperity. For three decades, from 1740 to 1770, intense public debate would focus on the interpretation and application of political arithmetic. Three varieties of Swedish statistics emerged in these years: utopian, practical, and descriptive.
Swedish Statisticians at Work
Utopian statisticians regarded political arithmetic chiefly as a means of forecasting and calculating the prosperity they took for granted. Their method went back to Petty: a mixture of exact measures and approximations, the equation of human worth and capital value, and a willingness to mask imprecision with strings of decimal places.
The visionary statisticians put their program to work in quantitative parish surveys. Their Description of Lajhela Parish in Österbotten was explicitly intended as a model for similar surveys of all Sweden's parishes. The director of the Land Survey Board, E.O. Runeberg, drew up the plans and published the results in the Transactions of the Academy of Sciences for 1758–9. The report breaks down the area of the parish into exact figures for cultivated, cultivable, and uncultivable land; refines the analysis with a series of subdivisions; and specifies watercourses (numbers of lakes, rivers, streams, and springs) and roads (classed by degree of passability). It analyzes woodland ("450 trees on each tunnland [approximately 1.2 acres], yielding 36,733,120 trees in the parish") and animals ("there are 590 horses in the parish, 2,124 cows, 236 oxen, 944 young steers, 4,720 sheep, and 474 calves, [which] together total 9,086 head of livestock, fed over 7 months with 6,443 bales of hay at 7/10 bale a head").
The report includes details on the number of dwellings, barns, water mills, windmills, and so on, but excludes churches from the count as unproductive resources.
The descriptions had the higher purpose: making it possible to calculate the potential resources of the parish. Most striking in this regard was Runeberg's quantitative analysis of population. Averaging yielded a peculiar figure of 1,800 1/4 for the total population of the parish. Runeberg then analyzed the parish mortality and fertility rates. "In Lajhela 3.83 marriages, or for each marriage 3.83 years, are required to produce a child, but since the majority of children die, 9.15 marriages, or 9.15 years of each marriage, are required to increase the populace by one child." Runeberg found these figures all the more dispiriting when he computed from the parish's potential natural resources it could support 28,000 inhabitants.
Runeberg's most significant calculations addressed the parish's capacity for work. Like the political arithmeticians of the preceding century, Runeberg drew a clear distinction between a person and a worker . He set the highest value on a married workman (2,390.99 dalers), somewhat less on an unmarried workman, and even less on "a person in general," by which he meant an average value over the total population. A woman was assigned a capital value three-fourths that of a man. Then came children, divided by age group. Runeberg judged infants of negligible worth, since they required on average "one-fifth of a person in care and tending." They thus had to be reckoned as a debit equal to one-fifth of the full value of an adult. The reasoning went as follows: "If we assume that by the eighteenth year a youth is equivalent to a full adult workman, and that a child of common people does not begin to be of use until his ninth year, and that not until the eighteenth year has he atoned for all the inconvenience and damage he caused before his ninth year, then a youth can be seen as non-withdrawable capital, increased by the accrual of compound interest, which only begins to yield an annual return through simple interest after the eighteenth year. Thus if a youth is assigned a political value of 1,195 dalers at the eighteenth
year, he must be valued at 998.8 at the fifteenth, 746.3 at the tenth, 557.6 at the fifth, and 416.7 dalers in the cradle." On this basis, Runeberg reduced the value of the almost 2,000 inhabitants of the parish to that of 800 workers.
The report then presents the calculations in tabular form and compares actual and potential resources. Note the exactitude of the entries for what Runeberg called "political evaluation:"
Instead of gazing at distant vistas of great wealth, practical statisticians focused on the foreground of poverty and wretchedness, a country in crisis. To some observers the cause was obvious: serious imbalance between different sectors of the economy. The solution seemed equally clear: restore the balance, guided by measurement, counting, and calculation. Here quantification acquired an instrumental function. Quantitative data served as the foundation for functional social models, which could then be translated into immediate political action. Like utopian statistics, practical statistics aimed at improving society, not merely describing it.
The basis of practical statistics lay in the doctrine of proportions. As prescribed in the biblical text, all things are ordered by measure and number and weight. A social order called for well-defined harmony and balance among population, land, industry, and so on; and the definition of harmony rested on number. The task at hand was to measure social and economic components, compare them to the ideal, and shift and alter the components to achieve the desired harmony.
The leading advocate of practical statistics was Anders Berch, a professor of economics at Uppsala University. As a representative of the ruling party in the Riksdag (the party known as "the Hats"), Berch was strongly influenced by mercantilism. The title of his Politisk arithmetica (1746) shows where he stood. His ambition was to establish political arithmetic as a science; then and only then could a solid, exact economic policy be constructed. The transformation of social analysis into social strategy required four clearly defined stages.
The first stage was to uncover and understand the omniscient Creator's plan for a world in perfect balance. In the ideal state, everything would stand in harmonious proportion: population to area, economic activity to natural resources, production to consumption, men to women, and the duration of human life to "the supply of all human need." Next came measurement and collection of data—quantitative analysis of the present state of society. Anything and everything was to be measured: people, land, natural resources, productivity, efficiency, and consumption. A crucial factor was human productive capacity, which required careful assessment of the results of work in terms of time expended. A sufficiently broad base of calculations and data could overcome individual variations and yield an accurate value for the country's work force as a whole.
The data gathered would inevitably reveal imbalance in respect to the ideal proportions. The third stage was thus to calculate and balance every conceivable factor affecting the nation's capital strength against every other: people, agriculture, industry, and trade. The practical statistician used computations and estimates to settle on the suitable production of offspring or consumption of aquavit for a stated number of cities or of farmhands per farm. He also needed to
balance the costs of war against the value of war booty, and the expense of ambassadorial travel against diplomatic advantage.
In his insistence on the fourth and final stage of implementation, Berch showed his perception of the gulf between theory and practice, or calculation and political action. He criticized political arithmetic in England, which had been allowed to remain the preserve of scientific circles and never approached implementation. Graunt, Petty, and D'Avenant also came in for rebuke for their narrowness of vision; in their hands measurement broke down into fragments without consistency or system. The grandeur of the Swedish program lay in the intent to make of quantitative social analysis an instrument for regulating the whole economy.
Berch's dream of a fully planned economy was founded on a faith in the state and its officials and a presumption of the loyalty of individuals to a powerful state. Once the data were put in the hands of the authorities and the balances struck, laws and ordinances would oblige individuals to distribute themselves and their resources to conform with the proposed model for prosperity. Slowly but surely a new social edifice would emerge. But what if all attempts to force the data into tabular form failed? How then to derive mathematical formulas for prosperity? Realization of the difficulties inherent in practical statistics prompted some to retreat to a less ambitious enterprise. For them, "statistics" meant the art of compiling and processing numerical information, with no purpose beyond the figures themselves.
The leading representative of descriptive statistics was the astronomer Pehr Wilhelm Wargentin, long-time secretary of the Royal Swedish Academy of Sciences, who played an instrumental role in securing an institutional base for Swedish statistics. Descriptive statistics had as its objective to reveal, describe, and interpret data, but not to prescribe how the data might be used. In parallel with the limitation of its aims, descriptive statistics came to be confined to a subject where data might be gathered without insuperable difficulty—to population studies. Slowly, sound and methodologically conscious
population statistics began to squeeze out the extravagant attempts at precision and the lofty social aspirations of utopian and practical statistics. By the 1770s, population statistics would become the only type of statistical work undertaken in Sweden.
The general outlines of the growth of Swedish population statistics are well known. In 1749 influential Swedish mercantilists and the Academy of Sciences succeeded in their campaign to establish an Office of Tables (which would become the Central Bureau of Statistics in 1858). Parish priests were required annually to complete printed forms reporting the numbers of births (classed by sex), deaths (by sex, age, and cause), and marriages, as well as the total population (by age, sex, estate, and occupation) for the parish. The tables were forwarded through a series of governmental agencies to the Commission of Tables, whose task was to summarize the results and transmit them to the Riksdag and the king. The Office of Tables thereby compiled the first set of population statistics in the world based on regular counts of total population. An efficient parish registration system that did not miss a single soul, a permanent institutional base, and a population unusual for its ethnic and religious homogeneity, disciplined by an established church with ample opportunity to exercise formal and informal control, contributed to the success of the venture.
But the very success of the Office of Tables represented a retreat from larger ambitions. Its reason for existence derived from the central importance of population in the mercantilist program, but population studies alone were only part of the social analysis urged by the utopian and practical statisticians.
Initially, the staff of the Office of Tables shared the optimism of other statisticians. In particular, the Commission of Tables (dominated by statisticians and civil servants) dreamed of a gigantic survey of all components of the economy. Collated, combined, and
compared, the numbers and tables would constitute a map of Sweden's resources, strengths, and weaknesses, and provide the political authorities with an effective instrument for governing the country. This ambitious program is evident in the highly secret reports delivered by the Commission to the Riksdag and the king in 1755, 1761, and 1765. Mind-boggling arrays of figures classified Sweden's population under a total of sixty-one headings; virtually all individuals were linked to their work and capital-producing capacities.
In their aim, these reports appear to be a faithful application of political arithmetic. Individuals were assigned categories according to their economic and hence political value to the state. First came providers, then consumers, and finally a category of wholly "superfluous members" (notably tavern staff and servants) numbering 10,336. The figure for emigration—8,059 in 1761—is just as precise; when converted into value using the methods of utopian statistics, it represented an annual capital loss to Sweden of 9 1/2 million dalers. When the potential of the emigrants to produce offspring was figured in, the loss amounted to no less than 19 million dalers.
At first the Swedish parliament showed much interest in the data and their implications. It appointed commissions and ordered certain reforms, especially in the medical field. Soon, however, the initiatives were tabled or defeated. Decisions disappeared mysteriously en route to the king for implementation. The important table of estates and occupations was originally required annually. But already in the 1750s the requirement was changed to reporting once every three years; later this was reduced to once every five years. Quantitative analysis of natural data like births and deaths remained noncontroversial, but attempts to derive social diagnosis or prescribe social therapy from the figures excited objections. Political arithmetic fell out of favor as a political instrument. With the constitution of 1772, parliamentary reponsibility for the Office of Tables formally ceased.
As officials lost enthusiasm for statistics, so too did advocates of descriptive statistics rebel against the use of their subject as an
instrument of state power. For fifteen years, population information, broken down by estate, occupation, and age group, had been kept under wraps by the Office on Tables. During the early 1760s this suppression of population figures as a state secret occasioned heated debate. Only in 1764 was the official population of Sweden (2,383,113) first disclosed. Before long the detailed information underlying the estimate was available for study by anyone who wanted it. As the gap between statistics and state widened, statistics had the opportunity to develop independently of power interests or practical applications. A gradual drop in the number of references in the Academy's Transactions to practical political aspects of statistics reflects this shift.
A quantitative social science was born of high hopes that social phenomena could be studied with the same precision as natural phenomena, yielding exact knowledge applicable in practical and political contexts. Yet the 18th-century conviction that the methods of natural science could be made to apply to all fields of knowledge hesitated at the crossroads of theoretical and practical goals in political economy.
The practical obstacles were daunting. Efficient collection and utilization of data required not only a firm institutional base (such as political arithmetic enjoyed in Sweden), but also workable methods for reducing masses of information to manageable and functional tables. How were consumption, efficiency, or the utility of diplomacy to be assessed and expressed in numerical terms? How were soil quality, popular morale, or unexploited natural resources to be set down in tables? The ambition to embrace society in its entirety overreached the practical limitations of 18th-century quantitative analysis.
The relationship between state and statistics in the 18th century was a complicated one. The more closely the quantitative method was linked to the interests of the state and the more obviously its political function was defined, the greater the danger that the method itself would be undermined. If the practical application of a science normally strengthens its empirical character, here the opposite seems to prevail. In England the quantitative method strayed from empiricism as it became more closely identified with a national strategy for prosperity. In the state-directed, accelerated program for progress in Sweden, quantitative social analysis slipped into a rut of utopianism that led nowhere. Only by reducing its field of operation to vital statistics did quantitative social analysis meet with success. In Germany, the development of a quantitative approach was contingent upon freeing statistics from the ideology of the state.
In England and Sweden, mercantilism had fostered a mechanistic view of society that favored quantitative social analysis. As society was broken down into its material components of population, resources, industry, and so on, so the populace was composed of faceless, voiceless atoms. The quantitative program further reduced the individual to an equivalence in work or capital value. In Germany, where cameralism, not mercantilism, held sway, more complex concepts of Land and Leute argued against the reduction of social well-being to a set of material components or the reduction of human beings to interchangeable particles.
It is noteworthy that social statistics on the quantitative, English model reached a zenith in Sweden around 1750, just when Swedish natural science was flourishing. Thus in 18th-century Sweden, as in 17th-century England, quantitative social science grew in the same soil as a vigorous and prestigious natural science.
The Laplacean View of Calculation
By Roger Hahn
Laplace's adherence to the geometrical spirit of the late Enlightenment appears at first glance so conventional that its discussion might serve best as an example or summary of the new attitudes of the age. One is therefore tempted to look at his work as reflecting each aspect of mathematization prevalent in the century. Many of the individual features that mark his predecessors are in evidence: his approval of the concision afforded by the language of equations; his adherence to a rigorous and logical scheme of presentation that echoed the Euclidean model; his insistence on accurate and precise measurements necessary to concretize explanatory theories and to provide them with unambiguous empirical tests; and his repeated efforts to transform the vague uncertainties attached to empirical laws into measurable degrees of certitude by applying the calculus of probabilities wherever appropriate. Laplace is perhaps the most consistent consumer (and certainly the most influential one) of a philosophical attitude that considers mathematization as the key feature of modern science's success and the guarantor of its continued prosperity in all its branches. He epitomizes the movement we are describing.
Every scientific article Laplace wrote reflects this belief. Our task will be to try to understand how he reached this position and to indicate how it fit with his general philosophy. The difficulty lies in Laplace's peculiar aversion to self-reflective discussions. He clearly preferred monographic treatment of scientific issues to general treatises of philosophy. In his vocabulary, the term "philosophy"—used in the sense of systematic statements referring to epistemological or ontological issues—is rarely in evidence, featured prominently only once in the title adopted for the introductory section to his Traité analytique des probabilités , the "Essai philosophique" of 1813.
Laplace is no outspoken philosopher; nor does he engage in debate with the prominent philosophers whose works he must have read and studied. Yet he was taken by his peers as an exemplar and commonly recommended in France as the spokesman of the scientific approach. He wrote an immensely popular Exposition du système du monde ; published the lectures on mathematics he gave to huge audiences at the short-lived Ecole normale in 1795; and these and his other treatises were often awarded as prize-books for school valedictorians. Most of the noted teachers of science of the early 19th century in France—Prony, Lacroix, Biot, Arago, Poisson—and many other practicing scientists, including his colleague Berthollet and his disciples Gay-Lussac, Malus, Alexander von Humboldt, and Quetelet (to name but a few), adopted his techniques and their implied philosophy. Even though the principles were not set down explicitly, Laplace's approach was taught at the Ecole polytechnique, and his philosophy was tested in the examinations given at the Faculty of Sciences and adopted as a general yardstick for measuring accomplishments at the Bureau des longitudes and the Academy of Sciences. His attitudes permeated all of the physical sciences in France in the early 19th centruy, and may even have been influential in the newly emerging life sciences.
Since Laplace is so reluctant to speak for himself in a systematic fashion, we will attempt to squeeze out his message by a select review of his life's activities. The biographical approach promises to offer insights into the development of his "geometrical spirit" that cannot easily be extracted from textual analysis. There were three important stages in his life when the issue of mathematization was prominent.
Mobility through Mathematics
The most pivotal of these turns on Laplace's decision to become a scientist. His first appearance on the scientific stage is in September 1769 as a teacher of elementary mathematics at the Ecole royale militaire in Paris, where for seven years he dispensed his knowledge to teenage sons of impoverished nobles. Since he received neither special praise nor blame in this occupation, one presumes he discharged his duties competently. The lectures he presented on elementary mathematics twenty-five years later at the Ecole normale—the only other teaching stint he accepted in his long career—show a solid command of the main features of mathematics and, in at least one instance, creative abilities as a mathematician probing the foundations of calculus. But one must not look to Laplace's career for major contributions to the mathematical discipline, or cast him in the role of a rival to his creative contemporaries Lagrange, Legendre, or Gauss. Lecturing on mathematics was a job he initially accepted as a means to escape the provinces where family plans would undoubtedly have taken him in a totally different direction. For Laplace, the move to Paris was a cathartic declaration of independence from his peasant origins and their social implications. Mathematics was the liberating agent that offered him a chance to begin a new life. The experience was also traumatic because mathematics constituted the philosophical antipode of the career his family had chosen for him in theology.
Laplace came from a modest family in Normandy who sent him to a local school from which he was recruited by the University of Caen for the Church. The standard career path for an intelligent youth born in a rural district in the middle of the 18th century led to administration or the clergy. His uncle, who taught him the "three R's," held a sinecure at a nearby parish that left him in a state of secure indolence. Young Pierre Simon could aspire to a similar position after completing his Master's in Theology, which he was awarded only three months before his move to Paris. In his student
days, however, Laplace had already succumbed to the lure of mathematics.
Laplace's teacher of metaphysics and natural philosophy was an unreconstructed Aristotelian who bore the distinctively primitive name of Jean Adam. Judged by the size of his classes, Adam was an enticing popular lecturer who performed divertingly on demonstration equipment he bought from the outlawed Jesuits. But he became a pathetic figure whose exploits were ridiculed in a scathing anonymous play (written by a student) ingeniously entitled Nostradamus . Whereas contemporaries were arguing about the merits of Newton over Descartes, our abbé Adam was still championing Aristotle. His printed lectures could have been written centuries earlier. They display the worst features of scholasticism, filled with hairsplitting verbal distinctions, elaborate and useless Latin terminology, and questionable principles. His attempt to modernize his manual with examples from current studies on hydraulics and electricity were so inept that his students found in it cause for mockery. The abbé Adam was also a fierce defender of the faith, particularly those values championed by Jesuits, who had been a dominant force earlier in the century at Caen. Adam took special delight in bringing before the bar of justice a fellow priest and professor at Caen named Christophe Gadbled on the grounds that he had failed in his duties as a teacher and clergyman. In his lectures, Gadbled had criticized Adam for his philosophical ineptness, had raised metaphysical questions about God's omnipotence, and had even ventured to hypothesize about a physical world operating without His immanence. The abbé Gadbled was also accused of offering flimsy excuses for failing to attend Mass, thereby setting an intolerable example for youth.
As an impressionable and curious youth, Laplace naturally took notice of Gadbled. A totally new world opened up to him. His new teacher was thoroughly versed in Newtonian physics. He substituted the succinct language of calculus for the tangled verbiage of Aristotle.
He wove a beautiful tapestry of the system of the world, using mathematical threads of gold that dazzled the youth and challenged his mind. To appreciate the new picture, Laplace quickly absorbed the calculus, probably using the new texts of Euler recommended by Gadbled. There was in the manipulation of mathematics a clarity and sharpness that could not be extracted from Adam's lectures. Gadbled was also in touch with practitioners of modern natural philosophy, incorporating the latest advances in the scientific literature, particularly from the work of Clairaut and d'Alembert. Moreover, Gadbled applied mathematics in a useful fashion to navigation in his lectures on hydrography. As one contemporary document puts it, Gadbled had "made calculation and geometry fashionable in Caen."
The critical moment in Laplace's conversion probably came when the hapless Adam, not content to squabble with Gadbled, decided to take on d'Alembert as well. The only known copy of his combative pamphlet was destroyed during the D-Day invasion. It may be assumed, however, that it accused the Newtonian coauthor of the Encyclopédie of criminal disrespect for the teachings of the Church and of Aristotle. While Laplace's Adam may have dazzled some of his students, he did not draw as much as a rejoinder from d'Alembert, who was most likely informed about the attack by his correspondent Le Canu, Gadbled's major assistant. It was this same Le Canu who apprised the high priest of the mathematical sciences about a promising twenty-year-old abbé named Laplace, and provided the young man with a letter of introduction. Pierre Simon set off for Paris armed with this letter and an audacious essay criticizing one of d'Alembert's minor writings on the law of inertia. D'Alembert had missed a mathematical point. Laplace's conversion from theology to science was completed when a properly chastened d'Alembert found him a well-paid position in mathematics at the Ecole militaire.
D'Alembert had additional reasons to empathize with Laplace, since he had in his youth experienced a similar transition from theological studies to the sciences. Recruitment into a mathematical occupation for most of the century often involved painful shifts from
parental expectations or initial career paths. La Caille, for example, renounced a life as a priest once he "discovered" Euclid for himself. Condorcet, Laplace's elder by six years, and tired of his traditional Jesuit education, turned his back on his uncle the bishop of Auxerre and a military life, embracing instead the uncertainties of a mathematical career, supported at first only by a gifted teacher at the Collège de Navarre. Others fortunate to be in Paris—like Coulomb, Legendre, and Lacroix—took to mathematics because of inspiring lecturers, while still others including Borda, Monge, and Carnot were drawn to the subject through military schooling. The individual paths may each have been different, but they all signaled a break from the security of established patterns for bright youths. In Paris especially, mathematics was an exciting and burgeoning activity that captivated many risk-takers.
As the central feature of Laplace's new life, mathematics came to be quite naturally the symbol of emancipation from the errors of the past and the agent of personal success. In his first three years in Paris, Laplace wrote sixteen original papers on a variety of topics that immediately established him as a leading contributor to the mathematical sciences of his era. He was twenty-three when he took his seat in the Academy of Sciences on a bench behind d'Alembert, following the same path as his elders Condorcet, Vandermonde, and Cousin.
Once in Paris, Laplace's professional life until the French Revolution was entirely centered on the Academy of Sciences. As an academician, he was called upon to review a host of papers and projects submitted for approval to the learned body. On the average, he sat on over a dozen review committees annually and wrote many of the reports, the texts of which still survive in good numbers. His membership in the Academy's section on mechanics and his mathe-
matical abilities brought him the tedious task of reporting on eccentric projects for the making of mechanical devices to douse fires, to float on water, or to erect perpetual-motion wheels. Most of them were rejected, often with scathing comments that reveal the mathematical illiteracy of the projectors. The task consumed so much time that in 1775 the Academy accepted d'Alembert's proposal to refuse outright to consider papers on squaring the circle, trisecting an angle, or perpetual motion schemes. Several years later, Laplace sought to use elementary mathematics tests to screen out artisan crackpots who did not deserve serious attention, much to the chagrin of amateurs like Marat and Brissot. A mathematics test was already in use by examiners of military and naval schools to rank aspiring officers. In effect, Laplace had already been practicing what he preached for artisans since 1783 in his capacity as entrance examiner of artillery and naval engineering schools. Next to proof of noble origin, mathematical competence was the only means adopted for separating the wheat from the chaff in the officer corps of the Old Regime. The extension of this practice to artisans seemed natural to Laplace; it also reveals in a pointed way the great importance he attached to this kind of literacy. There was a threshold of learning beneath which one could not be considered as a serious contributor to science.
Equally instructive were his positive reactions as a referee. Laplace was asked to comment on several projects on demography, life insurance schemes, and tables of amortization, all involving mathematical counts and statistics. In these instances, he was either an enthusiastic supporter or found ways to criticize the projects constructively. Laplace was constantly called upon to examine new scientific devices, particularly scales to record numerical data, techniques of interpolation, or other means to improve thermometers, barometers, microscopes, pyrometers, and other apparatus. The reports invariably focused on the improvements the new instruments provided for measuring. He hailed each device leading to more precise and reliable data as a progressive step.
Laplace's voice was heard and respected even before he made a name for himself in this line of work. In 1781 he devised a telescopic pyrometer to assist Lavoisier in experiments on thermal expansion of solids. A year later, he invented the ice calorimeter, whose sole purpose was to turn qualitative arguments about exothermic reactions into measurable data that could be used to test hypotheses. Lavoisier had enlisted Laplace in his experimental work as a keen critic of instrument-making and as a scientist endowed with a sharp mind trained to uncover errors and to devise methods to turn them. Despite its design limitations, the calorimeter was evidence that they appreciated the critical role instruments must play in a developing science still encumbered with vague principles. We know that later during the Revolution, Laplace participated with singular attachment in every phase of the establishment of the meter and the gram as national standards, and that he continued to promote the diffusion of the metric system in France and its conquered territories as Minister of Interior and later, as Senator. His public service included calling the first international congress of weights and measures in 1798 and laboring at the implementation of its decisions. His correspondence is replete with details about techniques of measurement, sources of error, and the consistency yielded by repeated measurement.
Laplace was not the first scientist to be so concerned. No doubt he developed an appreciation for instruments from close association with observational astronomers on whom he constantly relied for his work in celestial mechanics. Though not an expert observer himself, Laplace was as keenly concerned with sighting and timing devices as Tycho Brahe had been two centuries earlier. What is particularly
noteworthy is Laplace's desire to extend this attitude to all the sciences, particularly newly emerging ones. Two examples are in order here.
The Genevan physicist Deluc was one of Laplace's favorite scientific correspondents, despite Deluc's admittedly weak grasp of mathematical theories. In his mature years, Deluc turned into an irrepressible type quite distasteful to Laplace. Nevertheless, at an earlier time, Laplace vigorously encouraged him to reorganize the field of atmospheric research by elaborating his critical history of the thermometer and barometer, published in 1772. Laplace's stated hope was that Deluc could give meteorology its proper place among the more legitimate and established physical sciences. All that meteorology needed, so he imagined, was a carefully developed theory linking altitude to heat and pressure supported by accurate data taken from Alpine climes. So encouraging did Laplace's support prove to be that Deluc originally wrote his revised treatise on modifications of gases as a series of letters to Laplace. Even though the results did not fully meet Laplace's expectations, except in hypsometry, the goal pursued for almost two decades is testimony to Laplace's commitment to an increasingly accurate quantitative approach.
A more successful sponsorship following the same lines was Laplace's advocacy of the abbé Haüy's research into crystallography. In late 1783, Laplace was on the committee assigned to examine the manuscript of an Essay on a theory of the structure of crystals . He wrote a glowing report. Haüy had discovered a theory that had all the makings of a legitimate organizing principle linked directly to observation. Moreover, the theory of lattice structure he proposed was acknowledged to stem from a conscious effort to apply mathematics to natural history, as recommended by the mathematician Bézout. Unlike Bergman and Romé de l'Isle, who had tried to
organize crystals into classes or to relate their formation to the supposedly analogous processes of the formation of the earth, Haüy sought to link the geometry of standard crystals to the geometry of elementary particles that constituted their fundamental units. He succeeded in explaining various simple, quantitative phenomena.
Laplace considered this approach so promising that he developed a close personal attachment to the abbé, seeking to appropriate his talents for the physical sciences, symbolically capturing a portion of natural history (the mineral realm) for mathematics. The strategy worked so well that Haüy's next project brought him squarely into a newly quantified portion of physics. In 1787 Haüy offered the Academy a theory of electricity and magnetism that won him overwhelming praise for combining the theories of Aepinus and the empirical laws of Coulomb. During the Revolution, Haüy was selected (probably on the recommendation of Laplace) to lecture on physics at the Ecole normale; and in 1803 he was commissioned by the government to prepare the standard textbook on physics to be used in secondary schools. In all of these works, he expressed a philosophy totally in accord with and probably inspired by Laplace. Haüy demonstrated what a well-articulated theory intimately linked to verifiable, quantified data could be. If we take the writings of the abbé as a mirror of Laplace's views, we realize that their devotion to mathematics was not an end in itself, but a powerful tool for taming nature to human understanding. In describing the significance of his work in 1792, Haüy stated:
The theory of the structure of crystals can only be furthered with the aid of calculation (calcul analytique ). Analysis has the merit of encompassing in a single formula the solution to a large number of varied problems, and it can
alone impress upon theory the mark of rigorous certitude by arriving at results completely in accord with those of observation.
Variations on this theme were repeatedly asserted in scientific literature of the early 19th century, sometimes with direct reference to the inspiration provided by Laplace. The chemist Berthollet (Laplace's neighbor in Arcueil), who shared the same attitudes in his Essai de statique chimique (1803), acknowledged that his views were shaped by prolonged exchanges with Laplace. The latter never seems to have wavered from the belief that the advancement of scientific learning depended centrally upon the intelligent and appropriate use of mathematization. The view seemed fully sanctioned by many examples of progress in the physical sciences during the late Enlightenment.
Mathematics and Revelation
Laplace experienced a second major change in his life that affected his career and provided an opportunity to develop his view on the calculating spirit. He suffered through the trauma of the French Revolution. Until the eve of the Revolution, his life had been singularly focused on his career. His social circle was limited to professional colleagues who served to establish or reinforce his academic stature. But marriage at the ripe age of thirty-nine and becoming the father of two children imposed a reorientation of his habits. He took time to be more accessible, mixing with the educated public from both the aristocracy and the urban middle class. Political events eventually forced him to find a refuge for his family outside of the capital. With the surge of populist power came the inevitable criticism of elitist science and its practitioners. Old patterns at the Academy were disrupted and ultimately destroyed; many friends went their separate ways to exile, the provinces, prison, or the guillotine; and his sources of income were seriously threatened. While remaining true to the professional ideals he had espoused earlier, Laplace was led by the new circumstances to add a public dimension to his life.
He cleansed himself of political suspicion by serving on several governmental committees connected with science. He became friendly with men of influence, but not so openly that he was
compromised when they happened to fall out of political favor. He wisely adopted the role of the technocrat, an expert on science and its applications, loyal to the nation rather than to any particular political party. This strategy paid off handsomely. As France emerged from the Terror, he was recognized as a leading spokesman of science and consulted on all the important phases of the reorganization of the cultural life of France. Good political instincts led him into Bonaparte's circle, and he wound up temporarily as Minister of Interior after the 1799 coup that brought the General to power. Kicked upstairs to the Senate, he became one of the favorite courtiers of Napoleon. When the Emperor fell, Laplace was so highly respected that the new Bourbon regime had to hold him in esteem despite his earlier close association with Napoleon. By then, he was a fixture in public circles and an integral part of France's cultural elite.
In late 1794, Laplace was selected with the aging Lagrange to lecture on mathematics at the Ecole normale before 1,200 auditors, the flower of French educators, sent to the capital to absorb the quintessence of learning from the foremost scientists of the day. The lectures were recorded by stenographers and immediately published in the media, which represented the event as evidence that the French could continue to absorb themselves in cultural activities despite their new-won reputation as blood-thirsty barbarians. Everything, even the childing remarks about Leibniz' silly belief in a deity and Newton's aberrant interest in the Apocalypse, was reported in the press and spread abroad. It was the first time Laplace had faced the world so openly.
He and Lagrange did a remarkable job, considering that neither had tried such a thing before. Setting aside the grand manner of philosophizing reminiscent of Condillac, d'Alembert, and the Académie française, Laplace chose a direct style of exposition on a fairly sophisticated level with little moralizing. He employed a minimum of mathematical symbols, mixing common-sense platitudes with profound conclusions about the nature of numbers, lengths, and their
manipulation. The so-called elementary lectures were, in fact, a superior condensation and synthesis of the best thought of the age about mathematics and its uses in everyday life and in the scientific enterprise. He included a lecture on the metric system and announced the composition of a new work meant for the same audience to cover astronomy, the Exposition du système du monde .
It is in this Exposition , published first in 1796, rather than in the ten lessons on mathematics, that one sees how Laplace's new role as a popularizer expanded his thoughts on mathematization. Not only was it presented as the central means to effect the progress of science, but that progress itself was offered as an example of the noblest aspects of humanity. In the midst of the Revolution, Laplace became a grand spokesman for his profession, justifying the activity he had chosen to pursue as the most glorious of all secular pursuits.
Challenged to expound his life's work to an educated audience presumed to be mathematically untutored, he eschewed all algebraic equations in the body of the work. That did not make it, however, "a handbook of cosmology." It was in fact a nonmathematical version of his planned Celestial mechanics , written for literate readers following the format of standard popularizations. It presents astronomy in a simulated inductive fashion, dealing first with direct observations of apparent motions in the heavens, followed by a description of the real (Copernican) movements of the planets and their satellites. These sections lead to a purely verbal discussion of the laws of motion and the theory of universal gravitation. The final section is a review and summary of the entire work offered in the form of an elementary history of astronomical discoveries. It is this last section that provides the key to its author's central purpose. Laplace wants to present the evolution of astronomy as a model for the finest features of modern science and the current state of celestial science as a reflection of the most elevated characteristics of liberated humanity.
Here is his sermon-like peroration:
Taken as a whole, astronomy is the most beautiful monument of the human mind—the noblest voucher of its intelligence. Seduced by the illusion of the senses and of vanity, man considered himself for a long time as the center about which the celestial bodies revolved, and his pride was punished by the vain fears they inspired. The labor of many ages has at length withdrawn the screen that concealed the system of the world. And man now appears [to dwell] upon a small planet, almost imperceptible in the vast extent of the solar system, itself only an insensible point in the immensity of space. The sublime results to which this discovery has led may console him for the [inferior] rank assigned to him in the universe.
In a later edition, he added that thinking beings should especially take pride in their ability to have measured this universe, given the tiny base from which they were operating. "Let us carefully preserve, and even augment the number of these sublime discoveries, which constitute the delight of thinking beings." The loss of anthropomorphic centrality is compensated by the power of the human mind, which despite its translocation, is able to contemplate and provide a proper assessment of the heavens.
The originality of Laplace's treatment lies in his characterization of the historic path to progress by the systematic movement from observation to induced laws of nature, and from laws to their causes. Within this framework, he identifies improved observation and mathematics as the two propellants responsible for progress, while unsupported speculation and religious obsession are its most notorious obstacles. Time and again he proclaims "observation and calculation as the only [solid] grounds for human knowledge." His historical analysis does not rest alone, however, on this trite generalization. By "calculation" he means several things, each of which denotes the spirit of the géomètre philosophe .
Most obvious is the way increased precision has led to the discovery of celestial regularities (and hence to the possibility of establishing laws of nature). Laplace points to Hipparchus, Ulugh Beg, Tycho Brahe, Galileo, and Kepler. Improvement in precision, he
observes, may come about in a variety of ways: through the construction of better observatories and instruments; with the introduction of simplifying techniques of calculation, such as logarithms; by the more systematic tabular collection of data; or simply through the greater conscious attention to detail by observers.
A second dimension is the critical evaluation of data that comes from juxtaposing expected positions calculated from elements of planetary orbits alongside the results of observation. As the degree of conviction about the validity of the calculations increases over time (especially since Newton), observation is subjected to ever greater criticism. Laplace repeatedly reminds his readers that it is "this analytical connection of particular with general facts that constitutes theory," and that this theoretical outlook distinguishes modern Western astronomy from its predecessors.
A third and more complex notion advanced by Laplace refers to the new potentialities of algebra and the calculus for astronomy. It was an issue close to his heart because in the Traité de mécanique céleste , he replaced all of Newton's geometrical demonstrations with what was then called analyse . The same term was used to denote a particular epistemological method, and Laplace at times shifts from one usage to another without realizing he is dealing with distinct issues. Thus when praising Newton's methodological approach, he adopts the distinction between induction (analyse ) and deduction (synthèse ). In the Principia , Newton argued the truth of his system by means of synthesis, using a geometrical form. While admiring Newton's use of induction to discover his principles, however, Laplace criticizes him for having chosen a geometrical form of exposition to establish their truth. To be sure, he recognized that there were extenuating circumstances: "The state of imperfection in which the infinitesimal calculus was in the hands of its inventor did not permit him to resolve completely the difficulties pertaining to the theory of the system of the world; and he was often given to positing uncertain conjectures, until such time as they were to be verified by rigorous analysis." Nonetheless, the further advancement of
astronomy depended upon abandoning the geometrical approach, of which Newton was the last grand master. Laplace credits calculus not only with bettering the inductive or analytical processes that led from evidence to general principles, but also with advancing the deductive or synthetic phase that allows one to derive particulars from the new theory.
Whatever confusion exists in Laplace's use of terminology, he emerges as a strong advocate of the value of calculus for further progress:
Geometrical synthesis has the advantage of never allowing us to lose sight of its goal. . .; whereas algebraic analysis quickly allows us to forget the principal goal in the form of abstract combinations, and it is only at the end [of the operation] that it brings us back to it. But in isolating itself from this goal after having taken what is needed to arrive at the required result; and then by giving ourselves over to the operations of analysis. . . one is led by the power and generality of this method [and by the inestimable advantage of transfering reasoning into a mechanical process to arrive at] results often inaccessible to synthesis.
No other [mathematical] language lends itself so elegantly. . .to the long train of interconnected expressions, all flowing from one fundamental equation. Analysis also offers the advantage of always leading us to the simplest methods. One need only make a judicious selection of unknowns using the proper methods and give the results the form most easily reducible to. . .numerical calcualtion.
He offers as an illustration of the truth of his observation the solution of lunar inequalities, which "would be impossible to arrive at by synthesis."
Mathematics against Uncertainty
To this inventory of virtues in quantification for astronomy, Laplace was later to add the merits of the calculus of probability. Chronologically, his Philosophical essay on the theory of probability falls in a dark period of his life, near what he thought was the end of his career in 1813. This third crisis brought him face to face with the
limits of mortality and the uncertainties of life. His bachelor son, an artillery officer and aide-de-camp of Napoleon in the Russian campaign, nearly lost his life; his only daughter died in childbirth, causing him to sink into a profound depression; and Napoleon, who had come to power as a force of stability, was leading the Empire to dissolution and France to surrender. One might easily expect in this essay a Pascalian cry of despair about the weakness of the human mind or a Stoic sense of fatalistic resignation. But Laplace's intellectual resilience and self-confidence prevailed to turn his essay into a remarkably optimistic piece worthy of comparison with his departed friend Condorcet's Sketch for an historical picture of the human mind .
In a tour de force, Laplace stands traditional worries about fortuity on their head and asserts his famous view of determinism at the outset. According to his philosophy, the world operates by immutable laws that we can begin to know by applying the calculus of probabilities to the phenomena we observe. His concern with this calculus came directly from a desire dating back to his youth to find a systematic way of moving from the gathering of observations to the statement of true laws of nature. He had invented (or perfected) a calculus of statistical inference that allowed him to estimate the likelihood that a particular configuration of events would lead to a subsequent arrangement known through observation. This a posteriori calculus, presented in a pioneering paper of 1774 he subsequently developed, promised to become a powerful tool for his natural philosophy. Philosophically, its power stemmed from capturing game theory from the domain of conjecture (chance), and turning it into a method for calculating likelihood (probability). The uncertainties of chance were replaced by the manipulable concept of degrees of likelihood, bringing the operation within the reach of the mathematician. As a technical tool, statistical inference was prized for helping natural philosophers to distinguish between likely and spurious causes, thus preventing them from lapsing into unfounded speculation. Systematic errors of observation attributable to instruments could thus be distinguished from those dependent upon human failings. Statistical inference also offered the possibility for treating problems of civic life—such as the differential birth rates for the two sexes, annuities,
sampling techniques, voting methods, and judicial decisions—in a rigorous and rational manner, bringing mathematical light to obscure social problems.
For Laplace, calculation was no mere scientific tool of limited significance. Its scope went far beyond the technical advantages it had already supplied for the progress of natural philosophy. Mathematization offered a central path to an enhanced epistemology that would progressively reduce the errors of human ways and permit the assertion of intellectual powers. Given enough evidence, and using the proper analytic tools provided by mathematicians, humans could conceivably attain the skills necessary to rival the Supreme Being. It is no accident that Laplace proclaimed the possibility of an ideal, supreme, but human calculator (an absolute Intelligence) in the only philosophical essay he devoted to a mathematical subject. For him, mathematization truly embodied the spirit of rationalism, the greatest virtue offered by the Age of Enlightenment.