C.D. Warner, et al., comp.
The Library of the World’s Best Literature. An Anthology in Thirty Volumes. 1917.
Laplace
By François Arago (1786–1853)
T
Astronomy is the science of which the human mind may justly feel proudest. It owes this pre-eminence to the elevated nature of its object; to the enormous scale of its operations; to the certainty, the utility, and the stupendousness of its results. From the very beginnings of civilization the study of the heavenly bodies and their movements has attracted the attention of governments and peoples. The greatest captains, statesmen, philosophers, and orators of Greece and Rome found it a subject of delight. Yet astronomy worthy of the name is a modern science: it dates from the sixteenth century only. Three great, three brilliant phases have marked its progress. In 1543 the bold and firm hand of Copernicus overthrew the greater part of the venerable scaffolding which had propped the illusions and the pride of many generations. The earth ceased to be the centre, the pivot, of celestial movements. Henceforward it ranged itself modestly among the other planets, its relative importance as one member of the solar system reduced almost to that of a grain of sand.
Twenty-eight years had elapsed from the day when the Canon of Thorn expired while holding in his trembling hands the first copy of the work which was to glorify the name of Poland, when Würtemberg witnessed the birth of a man who was destined to achieve a revolution in science not less fertile in consequences, and still more difficult to accomplish. This man was Kepler. Endowed with two qualities which seem incompatible,—a volcanic imagination, and a dogged pertinacity which the most tedious calculations could not tire,—Kepler conjectured that celestial movements must be connected with each other by simple laws; or, to use his own expression, by harmonic laws. These laws he undertook to discover. A thousand fruitless attempts—the errors of calculation inseparable from a colossal undertaking—did not hinder his resolute advance toward the goal his imagination descried. Twenty-two years he devoted to it, and still he was not weary. What are twenty-two years of labor to him who is about to become the lawgiver of worlds; whose name is to be ineffaceably inscribed on the frontispiece of an immortal code; who can exclaim in dithyrambic language, “The die is cast: I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader since God has waited six thousand years for an interpreter of his works”?
These celebrated laws, known in astronomy as Kepler’s laws, are three in number. The first law is, that the planets describe ellipses around the sun, which is placed in their common focus; the second, that a line joining a planet and the sun sweeps over equal areas in equal times; the third, that the squares of the times of revolution of the planets about the sun are proportional to the cubes of their mean distances from that body. The first two laws were discovered by Kepler in the course of a laborious examination of the theory of the planet Mars. A full account of this inquiry is contained in his famous work, ‘De Stella Martis’ [Of the Planet Mars], published in 1609. The discovery of the third law was announced to the world in his treatise on Harmonics (1628).
To seek a physical cause adequate to retain the planets in their closed orbits; to make the stability of the universe depend on mechanical forces, and not on solid supports like the crystalline spheres imagined by our ancestors; to extend to the heavenly bodies in their courses the laws of earthly mechanics,—such were the problems which remained for solution after Kepler’s discoveries had been announced. Traces of these great problems may be clearly perceived here and there among ancient and modern writers, from Lucretius and Plutarch down to Kepler, Bouillaud, and Borelli. It is to Newton, however, that we must award the merit of their solution. This great man, like several of his predecessors, imagined the celestial bodies to have a tendency to approach each other in virtue of some attractive force, and from the laws of Kepler he deduced the mathematical characteristics of this force. He extended it to all the material molecules of the solar system; and developed his brilliant discovery in a work which, even at the present day, is regarded as the supremest product of the human intellect.
The contributions of France to these revolutions in astronomical science consisted, in 1740, in the determination by experiment of the spheroidal figure of the earth, and in the discovery of the local variations of gravity upon the surface of our planet. These were two great results; but whenever France is not first in science she has lost her place. This rank, lost for a moment, was brilliantly regained by the labors of four geometers. When Newton, giving to his discoveries a generality which the laws of Kepler did not suggest, imagined that the different planets were not only attracted by the sun, but that they also attracted each other, he introduced into the heavens a cause of universal perturbation. Astronomers then saw at a glance that in no part of the universe would the Keplerian laws suffice for the exact representation of the phenomena of motion; that the simple regular movements with which the imaginations of the ancients were pleased to endow the heavenly bodies must experience numerous, considerable, perpetually changing perturbations. To discover a few of these perturbations, and to assign their nature and in a few rare cases their numerical value, was the object which Newton proposed to himself in writing his famous book, the ‘Principia Mathematica Philosophiæ Naturalis’ [Mathematical Principles of Natural Philosophy]. Notwithstanding the incomparable sagacity of its author, the ‘Principia’ contained merely a rough outline of planetary perturbations, though not through any lack of ardor or perseverance. The efforts of the great philosopher were always superhuman, and the questions which he did not solve were simply incapable of solution in his time.
Five geometers—Clairaut, Euler, D’Alembert, Lagrange, and Laplace—shared between them the world whose existence Newton had disclosed. They explored it in all directions, penetrated into regions hitherto inaccessible, and pointed out phenomena hitherto undetected. Finally—and it is this which constitutes their imperishable glory—they brought under the domain of a single principle, a single law, everything that seemed most occult and mysterious in the celestial movements. Geometry had thus the hardihood to dispose of the future, while the centuries as they unroll scrupulously ratify the decisions of science.
If Newton gave a complete solution of celestial movements where but two bodies attract each other, he did not even attempt the infinitely more difficult problem of three. The “problem of three bodies” (this is the name by which it has become celebrated)—the problem of determining the movement of a body subjected to the attractive influence of two others—was solved for the first time by our countryman, Clairaut. Though he enumerated the various forces which must result from the mutual action of the planets and satellites of our system, even the great Newton did not venture to investigate the general nature of their effects. In the midst of the labyrinth formed by increments and diminutions of velocity, variations in the forms of orbits, changes in distances and inclinations, which these forces must evidently produce, the most learned geometer would fail to discover a trustworthy guide. Forces so numerous, so variable in direction, so different in intensity, seemed to be incapable of maintaining a condition of equilibrium except by a sort of miracle. Newton even suggested that the planetary system did not contain within itself the elements of indefinite stability. He was of opinion that a powerful hand must intervene from time to time to repair the derangements occasioned by the mutual action of the various bodies. Euler, better instructed than Newton in a knowledge of these perturbations, also refused to admit that the solar system was constituted so as to endure forever.
Never did a greater philosophical question offer itself to the inquiries of mankind. Laplace attacked it with boldness, perseverance, and success. The profound and long-continued researches of the illustrious geometer completely established the perpetual variability of the planetary ellipses. He demonstrated that the extremities of their major axes make the circuit of the heavens; that independent of oscillation, the planes of their orbits undergo displacements by which their intersections with the plane of the terrestrial orbit are each year directed toward different stars. But in the midst of this apparent chaos, there is one element which remains constant, or is merely subject to small and periodic changes; namely, the major axis of each orbit, and consequently the time of revolution of each planet. This is the element which ought to have varied most, on the principles held by Newton and Euler. Gravitation, then, suffices to preserve the stability of the solar system. It maintains the forms and inclinations of the orbits in an average position, subject to slight oscillations only; variety does not entail disorder; the universe offers an example of harmonious relations, of a state of perfection which Newton himself doubted.
This condition of harmony depends on circumstances disclosed to Laplace by analysis; circumstances which on the surface do not seem capable of exercising so great an influence. If instead of planets all revolving in the same direction, in orbits but slightly eccentric and in planes inclined at but small angles toward each other, we should substitute different conditions, the stability of the universe would be jeopardized, and a frightful chaos would pretty certainly result. The discovery of the actual conditions excluded the idea, at least so far as the solar system was concerned, that the Newtonian attraction might be a cause of disorder. But might not other forces, combined with the attraction of gravitation, produce gradually increasing perturbations such as Newton and Euler feared? Known facts seemed to justify the apprehension. A comparison of ancient with modern observations revealed a continual acceleration in the mean motions of the moon and of Jupiter, and an equally striking diminution of the mean motion of Saturn. These variations led to a very important conclusion. In accordance with their presumed cause, to say that the velocity of a body increased from century to century was equivalent to asserting that the body continually approached the centre of motion; on the other hand, when the velocity diminished, the body must be receding from the centre. Thus, by a strange ordering of nature, our planetary system seemed destined to lose Saturn, its most mysterious ornament; to see the planet with its ring and seven satellites plunge gradually into those unknown regions where the eye armed with the most powerful telescope has never penetrated. Jupiter, on the other hand, the planet compared with which the earth is so insignificant, appeared to be moving in the opposite direction, so that it would ultimately be absorbed into the incandescent matter of the sun. Finally, it seemed that the moon would one day precipitate itself upon the earth.
There was nothing doubtful or speculative in these sinister forebodings. The precise dates of the approaching catastrophes were alone uncertain. It was known, however, that they were very distant. Accordingly, neither the learned dissertations of men of science nor the animated descriptions of certain poets produced any impression upon the public mind. The members of our scientific societies, however, believed with regret the approaching destruction of the planetary system. The Academy of Sciences called the attention of geometers of all countries to these menacing perturbations. Euler and Lagrange descended into the arena. Never did their mathematical genius shine with a brighter lustre. Still the question remained undecided, when from two obscure corners of the theories of analysis, Laplace, the author of the ‘Mécanique Céleste,’ brought the laws of these great phenomena clearly to light. The variations in velocity of Jupiter, Saturn, and the moon, were proved to flow from evident physical causes, and to belong in the category of ordinary periodic perturbations depending solely on gravitation. These dreaded variations in orbital dimensions resolved themselves into simple oscillations included within narrow limits. In a word, by the powerful instrumentality of mathematical analysis, the physical universe was again established on a demonstrably firm foundation.
Having demonstrated the smallness of these periodic oscillations, Laplace next succeeded in determining the absolute dimensions of the orbits. What is the distance of the sun from the earth? No scientific question has occupied the attention of mankind in a greater degree. Mathematically speaking, nothing is more simple: it suffices, as in ordinary surveying, to draw visual lines from the two extremities of a known base line to an inaccessible object; the remainder of the process is an elementary calculation. Unfortunately, in the case of the sun, the distance is very great and the base lines which can be measured upon the earth are comparatively very small. In such a case, the slightest errors in the direction of visual lines exercise an enormous influence upon the results. In the beginning of the last century, Halley had remarked that certain interpositions of Venus between the earth and the sun—or to use the common term, the transits of the planet across the sun’s disk—would furnish at each observing station an indirect means of fixing the position of the visual ray much superior in accuracy to the most perfect direct measures. Such was the object of the many scientific expeditions undertaken in 1761 and 1769, years in which the transits of Venus occurred. A comparison of observations made in the Southern Hemisphere with those of Europe gave for the distance of the sun the result which has since figured in all treatises on astronomy and navigation. No government hesitated to furnish scientific academies with the means, however expensive, of establishing their observers in the most distant regions. We have already remarked that this determination seemed imperiously to demand an extensive base, for small bases would have been totally inadequate. Well, Laplace has solved the problem without a base of any kind whatever; he has deduced the distance of the sun from observations of the moon made in one and the same place.
The sun is, with respect to our satellite the moon, the cause of perturbations which evidently depend on the distance of the immense luminous globe from the earth. Who does not see that these perturbations must diminish if the distance increases, and increase if the distance diminishes, so that the distance determines the amount of the perturbations? Observation assigns the numerical value of these perturbations; theory, on the other hand, unfolds the general mathematical relation which connects them with the solar distance and with other known elements. The determination of the mean radius of the terrestrial orbit—of the distance of the sun—then becomes one of the most simple operations of algebra. Such is the happy combination by the aid of which Laplace has solved the great, the celebrated problem of parallax. It is thus that the illustrious geometer found for the mean distance of the sun from the earth, expressed in radii of the terrestrial orbit, a value differing but slightly from that which was the fruit of so many troublesome and expensive voyages.
The movements of the moon proved a fertile mine of research to our great geometer. His penetrating intellect discovered in them unknown treasures. With an ability and a perseverance equally worthy of admiration, he separated these treasures from the coverings which had hitherto concealed them from vulgar eyes. For example, the earth governs the movements of the moon. The earth is flattened; in other words, its figure is spheroidal. A spheroidal body does not attract as does a sphere. There should then exist in the movement—I had almost said in the countenance—of the moon a sort of impress of the spheroidal figure of the earth. Such was the idea as it originally occurred to Laplace. By means of a minutely careful investigation, he discovered in its motion two well-defined perturbations, each depending on the spheroidal figure of the earth. When these were submitted to calculation, each led to the same value of the ellipticity. It must be recollected that the ellipticity thus derived from the motions of the moon is not the one corresponding to such or such a country, to the ellipticity observed in France, in England, in Italy, in Lapland, in North America, in India, or in the region of the Cape of Good Hope; for, the earth’s crust having undergone considerable upheavals at different times and places, the primitive regularity of its curvature has been sensibly disturbed thereby. The moon (and it is this which renders the result of such inestimable value) ought to assign, and has in reality assigned, the general ellipticity of the earth; in other words, it has indicated a sort of average value of the various determinations obtained at enormous expense, and with infinite labor, as the result of long voyages undertaken by astronomers of all the countries of Europe.
Certain remarks of Laplace himself bring into strong relief the profound, the unexpected, the almost paradoxical character of the methods I have attempted to sketch. What are the elements it has been found necessary to confront with each other in order to arrive at results expressed with such extreme precision? On the one hand, mathematical formulæ deduced from the principle of universal gravitation; on the other, certain irregularities observed in the returns of the moon to the meridian. An observing geometer, who from his infancy had never quitted his study, and who had never viewed the heavens except through a narrow aperture directed north and south,—to whom nothing had ever been revealed respecting the bodies revolving above his head, except that they attract each other according to the Newtonian law of gravitation,—would still perceive that his narrow abode was situated upon the surface of a spheroidal body, whose equatorial axis was greater than its polar by a three hundred and sixth part. In his isolated, fixed position he could still deduce his true distance from the sun!
Laplace’s improvement of the lunar tables not only promoted maritime intercourse between distant countries, but preserved the lives of mariners. Thanks to an unparalleled sagacity, to a limitless perseverance, to an ever youthful and communicable ardor, Laplace solved the celebrated problem of the longitude with a precision even greater than the utmost needs of the art of navigation demanded. The ship, the sport of the winds and tempests, no longer fears to lose its way in the immensity of the ocean. In every place and at every time the pilot reads in the starry heavens his distance from the meridian of Paris. The extreme perfection of these tables of the moon places Laplace in the ranks of the world’s benefactors.
In the beginning of the year 1611, Galileo supposed that he found in the eclipses of Jupiter’s satellites a simple and rigorous solution of the famous problem of the longitude, and attempts to introduce the new method on board the numerous vessels of Spain and Holland at once began. They failed because the necessary observations required powerful telescopes, which could not be employed on a tossing ship. Even the expectations of the serviceability of Galileo’s methods for land calculations proved premature. The movements of the satellites of Jupiter are far less simple than the immortal Italian supposed them to be. The labors of three more generations of astronomers and mathematicians were needed to determine them, and the mathematical genius of Laplace was needed to complete their labors. At the present day the nautical ephemerides contain, several years in advance, the indications of the times of the eclipses and reappearances of Jupiter’s satellites. Calculation is as precise as direct observation.
Influenced by an exaggerated deference, modesty, timidity, France in the eighteenth century surrendered to England the exclusive privilege of constructing her astronomical instruments. Thus, when Herschel was prosecuting his beautiful observations on the other side of the Channel, we had not even the means of verifying them. Fortunately for the scientific honor of our country, mathematical analysis also is a powerful instrument. The great Laplace, from the retirement of his study, foresaw, and accurately predicted in advance, what the excellent astronomer of Windsor would soon behold with the largest telescopes existing. When, in 1610, Galileo directed toward Saturn a lens of very low power which he had just constructed with his own hands, although he perceived that the planet was not a globe, he could not ascertain its real form. The expression “tri-corporate,” by which the illustrious Florentine designated the appearance of the planet, even implied a totally erroneous idea of its structure. At the present day every one knows that Saturn consists of a globe about nine hundred times greater than the earth, and of a ring. This ring does not touch the ball of the planet, being everywhere removed from it to a distance of twenty thousand (English) miles. Observation indicates the breadth of the ring to be fifty-four thousand miles. The thickness certainly does not exceed two hundred and fifty miles. With the exception of a black streak which divides the ring throughout its whole contour into two parts of unequal breadth and of different brightness, this strange colossal bridge without foundations had never offered to the most experienced or skillful observers either spot or protuberance adapted for deciding whether it was immovable or endowed with a motion of rotation. Laplace considered it to be very improbable, if the ring was stationary, that its constituent parts should be capable of resisting by mere cohesion the continual attraction of the planet. A movement of rotation occurred to his mind as constituting the principle of stability, and he deduced the necessary velocity from this consideration. The velocity thus found was exactly equal to that which Herschel subsequently derived from a series of extremely delicate observations. The two parts of the ring, being at different distances from the planet, could not fail to be given different movements of precession by the action of the sun. Hence it would seem that the planes of both rings ought in general to be inclined toward each other, whereas they appear from observation always to coincide. It was necessary then that some physical cause capable of neutralizing the action of the sun should exist. In a memoir published in February, 1789, Laplace found that this cause depended on the ellipticity of Saturn produced by a rapid movement of rotation of the planet, a movement whose discovery Herschel announced in November of the same year.
If we descend from the heavens to the earth, the discoveries of Laplace will appear not less worthy of his genius. He reduced the phenomena of the tides, which an ancient philosopher termed in despair “the tomb of human curiosity,” to an analytical theory in which the physical conditions of the question figure for the first time. Consequently, to the immense advantage of coast navigation, calculators now venture to predict in detail the time and height of the tides several years in advance. Between the phenomena of the ebb and flow, and the attractive forces of the sun and moon upon the fluid sheet which covers three fourths of the globe, an intimate and necessary connection exists; a connection from which Laplace deduced the value of the mass of our satellite the moon. Yet so late as the year 1631 the illustrious Galileo, as appears from his ‘Dialogues,’ was so far from perceiving the mathematical relations from which Laplace deduced results so beautiful, so unequivocal, and so useful, that he taxed with frivolousness the vague idea which Kepler entertained of attributing to the moon’s attraction a certain share in the production of the diurnal and periodical movements of the waters of the ocean.
Laplace did not confine his genius to the extension and improvement of the mathematical theory of the tide. He considered the phenomenon from an entirely new point of view, and it was he who first treated of the stability of the ocean. He has established its equilibrium, but upon the express condition (which, however, has been amply proved to exist) that the mean density of the fluid mass is less than the mean density of the earth. Everything else remaining the same, if we substituted an ocean of quicksilver for the actual ocean, this stability would disappear. The fluid would frequently overflow its boundaries, to ravage continents even to the height of the snowy peaks which lose themselves in the clouds.
No one was more sagacious than Laplace in discovering intimate relations between phenomena apparently unrelated, or more skillful in deducing important conclusions from such unexpected affinities. For example, toward the close of his days, with the aid of certain lunar observations, with a stroke of his pen he overthrew the cosmogonic theories of Buffon and Bailly, which were so long in favor. According to these theories, the earth was hastening to a state of congelation which was close at hand. Laplace, never contented with vague statements, sought to determine in numbers the rate of the rapid cooling of our globe which Buffon had so eloquently but so gratuitously announced. Nothing could be more simple, better connected, or more conclusive than the chain of deductions of the celebrated geometer. A body diminishes in volume when it cools. According to the most elementary principles of mechanics, a rotating body which contracts in dimensions must inevitably turn upon its axis with greater and greater rapidity. The length of the day has been determined in all ages by the time of the earth’s rotation; if the earth is cooling, the length of the day must be continually shortening. Now, there exists a means of ascertaining whether the length of the day has undergone any variation; this consists in examining, for each century, the arc of the celestial sphere described by the moon during the interval of time which the astronomers of the existing epoch call a day; in other words, the time required by the earth to effect a complete rotation on its axis, the velocity of the moon being in fact independent of the time of the earth’s rotation. Let us now, following Laplace, take from the standard tables the smallest values, if you choose, of the expansions or contractions which solid bodies experience from changes of temperature; let us search the annals of Grecian, Arabian, and modern astronomy for the purpose of finding in them the angular velocity of the moon: and the great geometer will prove, by incontrovertible evidence founded upon these data, that during a period of two thousand years the mean temperature of the earth has not varied to the extent of the hundredth part of a degree of the centigrade thermometer. Eloquence cannot resist such a process of reasoning, or withstand the force of such figures. Mathematics has ever been the implacable foe of scientific romances. The constant object of Laplace was the explanation of the great phenomena of nature according to inflexible principles of mathematical analysis. No philosopher, no mathematician, could have guarded himself more cautiously against a propensity to hasty speculation. No person dreaded more the scientific errors which cajole the imagination when it passes the boundary of fact, calculation, and analogy.
Once, and once only, did Laplace launch forward, like Kepler, like Descartes, like Leibnitz, like Buffon, into the region of conjectures. But then his conception was nothing less than a complete cosmogony. All the planets revolve around the sun, from west to east, and in planes only slightly inclined to each other. The satellites revolve around their respective primaries in the same direction. Both planets and satellites, having a rotary motion, turn also upon their axes from west to east. Finally, the rotation of the sun also is directed from west to east. Here, then, is an assemblage of forty-three movements, all operating alike. By the calculus of probabilities, the odds are four thousand millions to one that this coincidence in direction is not the effect of accident.
It was Buffon, I think, who first attempted to explain this singular feature of our solar system. “Wishing, in the explanation of phenomena, to avoid recourse to causes which are not to be found in nature,” the celebrated academician sought for a physical cause for what is common to the movements of so many bodies differing as they do in magnitude, in form, and in their distances from the centre of attraction. He imagined that he had discovered such a physical cause by making this triple supposition: a comet fell obliquely upon the sun; it pushed before it a torrent of fluid matter; this substance, transported to a greater or less distance from the sun according to its density, formed by condensation all the known planets. The bold hypothesis is subject to insurmountable difficulties. I proceed to indicate, in a few words, the cosmogonic system which Laplace substituted for it.
According to Laplace, the sun was, at a remote epoch, the central nucleus of an immense nebula, which possessed a very high temperature, and extended far beyond the region in which Uranus now revolves. No planet was then in existence. The solar nebula was endowed with a general movement of rotation in the direction west to east. As it cooled it could not fail to experience a gradual condensation, and in consequence to rotate with greater and greater rapidity. If the nebulous matter extended originally in the plane of its equator, as far as the limit where the centrifugal force exactly counterbalanced the attraction of the nucleus, the molecules situate at this limit ought, during the process of condensation, to separate from the rest of the atmospheric matter and to form an equatorial zone, a ring, revolving separately and with its primitive velocity. We may conceive that analogous separations were effected in the remoter strata of the nebula at different epochs and at different distances from the nucleus, and that they gave rise to a succession of distinct rings, all lying in nearly the same plane, and all endowed with different velocities.
This being once admitted, it is easy to see that the permanent stability of the rings would have required a regularity of structure throughout their whole contour, which is very improbable. Each of them, accordingly, broke in its turn into several masses, which were obviously endowed with a movement of rotation coinciding in direction with the common movement of revolution, and which, in consequence of their fluidity, assumed spheroidal forms. In order, next, that one of those spheroids may absorb all the others belonging to the same ring, it is sufficient to suppose it to have a mass greater than that of any other spheroid of its group.
Each of the planets, while in this vaporous condition to which we have just alluded, would manifestly have a central nucleus, gradually increasing in magnitude and mass, and an atmosphere offering, at its successive limits, phenomena entirely similar to those which the solar atmosphere, properly so called, had exhibited. We are here contemplating the birth of satellites and the birth of the ring of Saturn.
The Nebular Hypothesis, of which I have just given an imperfect sketch, has for its object to show how a nebula endowed with a general movement of rotation must eventually transform itself into a very luminous central nucleus (a sun), and into a series of distinct spheroidal planets, situate at considerable distances from one another, all revolving around the central sun, in the direction of the original movement of the nebula; how these planets ought also to have movements of rotation in similar directions; how, finally, the satellites, when any such are formed, must revolve upon their axes and around their respective primaries, in the direction of rotation of the planets and of their movement of revolution around the sun.
In all that precedes, attention has been concentrated upon the ‘Mécanique Céleste.’ The ‘Système du Monde’ and the ‘Théorie Analytique des Probabilités’ also deserve description.
The Exposition of the System of the World is the ‘Mécanique Céleste’ divested of that great apparatus of analytical formulæ which must be attentively perused by every astronomer who, to use an expression of Plato, wishes to know the numbers which govern the physical universe. It is from this work that persons ignorant of mathematics may obtain competent knowledge of the methods to which physical astronomy owes its astonishing progress. Written with a noble simplicity of style, an exquisite exactness of expression, and a scrupulous accuracy, it is universally conceded to stand among the noblest monuments of French literature…. The labors of all ages to persuade truth from the heavens are there justly, clearly, and profoundly analyzed. Genius presides as the impartial judge of genius. Throughout his work Laplace remained at the height of his great mission. It will be read with respect so long as the torch of science illuminates the world.
The calculus of probabilities, when confined within just limits, concerns the mathematician, the experimenter, and the statesman. From the time when Pascal and Fermat established its first principles, it has rendered most important daily services. This it is which, after suggesting the best form for statistical tables of population and mortality, teaches us to deduce from those numbers, so often misinterpreted, the most precise and useful conclusions. This it is which alone regulates with equity insurance premiums, pension funds, annuities, discounts, etc. This it is that has gradually suppressed lotteries, and other shameful snares cunningly laid for avarice and ignorance. Laplace has treated these questions with his accustomed superiority: the ‘Analytical Theory of Probabilities’ is worthy of the author of the ‘Mécanique Céleste.’
A philosopher whose name is associated with immortal discoveries said to his too conservative audience, “Bear in mind, gentlemen, that in questions of science the authority of a thousand is not worth the humble reasoning of a single individual.” Two centuries have passed over these words of Galileo without lessening their value or impugning their truth. For this reason, it has been thought better rather to glance briefly at the work of Laplace than to repeat the eulogies of his admirers.