13
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.^[1]

[1] A different dimension of these issues, with special reference to the Traité de mécanique céleste , is treated by Michel Serres in his "Histoire des sciences et histoire de la philosophie au XIXe siècle," Revue de l'Université de Bruxelles , 1973, 433-44.

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.^[2] There were three important stages in his life when the issue of mathematization was prominent.

[2] For details, see Roger Hahn, Laplace as a Newtonian scientist (Los Angeles: William Andrews Clark Library, 1967); Charles C. Gillispie et al., "Laplace, Pierre-Simon," Dictionary of scientific biography (DSB), 15 , 273-403.

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.^[3] 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

[3] Robin E. Rider, Mathematics in the Enlightenment: A study of algebra, 1685–1800 (Ph.D. dissertation, University of California, Berkeley, 1980; DAI 42/01A, 351), 70–2.

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 .^[4] 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.^[5] 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.

[4] M.E.M.B.C.D.S.M. [Etienne Mauger], Nostradamus, ou le Physicien plaideur , (Leyde [Caen], 1779).

[5] Jean Adam, Philosophia ad usum scholarum accomodata , 3 vols. (Caen: J.C. Pyron, 1771–5).

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."^[6]

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

[6] Archives nationales, Paris, M196.

parental expectations or initial career paths.^[7] 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.^[8]

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.^[9] 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-

[7] Charles B. Paul, Science and immortality. The eloges of the Paris Academy of Sciences (1699–1791) (Berkeley: University of California Press, 1980), 75–9.

[8] Roger Hahn, "Scientific research as an occupation in eighteenth-century Paris," Minerva , 13 (1975), 506-8.

[9] Stephen M. Stigler, "Laplace's early work: Chronology and citations," Isis, 69 (1978), 234–54.

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.^[10] 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.

[10] Denis I. Duveen and Roger Hahn, "Laplace's succession to Bézout's post of Examinateur des élèves de l'artillerie," Isis, 48 (1957), 416–27.

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.^[11] 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.^[12] His public service included calling the first international congress of weights and measures in 1798 and laboring at the implementation of its decisions.^[13] 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

[11] Henry Guerlac, "Chemistry as a branch of physics: Laplace's collaboration with Lavoisier," Historical studies in the physical sciences, 7 (1976), 183–276, on 224.

[12] For the connection between the calculations surrounding heat experiments and the metric system, see Guerlac, ibid., 253.

[13] Laplace to Delambre, 10 pluviôse an VI, published in Revue d'histoire des sciences et de leurs applications, 14 (1961), 287–90; Maurice P. Crosland, "The Congress on definitive metric standards, 1798-1799: The first international scientific conference?" Isis, 60 (1969), 226–31.

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.^[14] 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.^[15]

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.^[16] 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

[14] Jean André Deluc, Idées sur la météorologie , 2 vols. (London: Spilsbury, 1786–7), 1 , 5.

[15] Theodore S. Feldman, "Applied mathematics and the quantification of experimental physics: The example of barometric hypsometry," Historical studies in the physical sciences, 15 (1985), 127–95.

[16] Académie des Sciences, Registre des procès-verbaux, 26 Nov 1783.

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.^[17]

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.^[18] 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:^[19]

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

[17] John G. Burke, Origins of the science of crystals (Berkeley and Los Angeles: University of California Press, 1966); Marika Blondel-Mégrelis, "Le modèle et la théorie: Analyse d'un exemple de la cristallographie de Haüy," Revue philosophique de la France et de l'Etranger, 171 (1981), 283–302.

[18] J.L. Heilbron, Electricity in the 17th and 18th centuries: A study of early modern physics (Berkeley: University of California Press, 1979), 426.

[19] René Just Haüy, "Exposition abrégée de la théorie sur la structure des cristaux," Journal d'histoire naturelle, 1 (1792), 159 and Journal de physique , Aug 1793, 1.

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.^[20] 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

[20] Jean Dhombres, "L'enseignement des mathématiques par la méthode révolutionnaire: Les leçons de Laplace à l'Ecole Normale de l'an III," Revue d'histoire des sciences et de leurs applications, 33 (1980), 315–48.

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."^[21] 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.

[21] Gillispie, "Laplace, Pierre-Simon," 343.

[22] Pierre Simon Laplace, Exposition du système du monde (Paris: Cercle Social, an IV [1796]), 2 , 310–11.

Here is his sermon-like peroration:^[22]

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."^[23] 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."^[24] 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

[23] Ibid., 6th ed. (Brussels: De Vroom, 1827), 531.

[24] Ibid., 442, note; ibid., 1st ed., 2 , 292.

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.^[25]

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."^[26] Nonetheless, the further advancement of

[25] Ibid., 6th ed., 457, 503.

[26] Ibid., 1st ed., 2 , 285.

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:^[27]

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."^[28]

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

[27] Ibid., 1st ed., 2, 289–290. For the section given in brackets, 2d ed. (Paris: Crapelet, 1799), 339.

[28] Ibid., 6th ed., 506, n. 1.

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.^[29] For him, mathematization truly embodied the spirit of rationalism, the greatest virtue offered by the Age of Enlightenment.

[29] Roger Hahn, "Laplace and the mechanistic universe," in David C. Lindberg and Ronald L. Numbers, eds., God and Nature (Berkeley: University of California Press, 1986), 269.