# Laplace Sentence Examples

But for those who wish to study the molecular constitution of bodies it is necessary to study the effect of forces which are sensible only at insensible distances; and

**Laplace**has furnished us with an example of the method of this study which has never been surpassed.But the Cartesian theory, like the later speculations of Kant and

**Laplace**, proposes to give a hypothetical explanation of the circumstances and motions which in the normal course of things led to the state of things required by the law of attraction.The application of this to telegraphic purposes was suggested by

**Laplace**and taken up by Ampere, and afterwards by Triboaillet and by Schilling, whose work forms the foundation of much of modern telegraphy.In his Naturgeschichte des Himmels, in which he anticipated the nebular theory afterwards more fully developed by

**Laplace**, Kant sought to explain the genesis of the cosmos as a product of physical forces and laws.**Laplace**supposed the existence of a primeval nebula which extended so far out as to fill all the space at present occupied by the planets.As the result of an examination conducted in September 1785 by

**Laplace**, Bonaparte was included among those who entered the army without going through an intermediate stage.After serving for a short time in the artillery, he was appointed in 1797 professor of mathematics at Beauvais, and in 1800 he became professor of physics at the College de France, through the influence of

**Laplace**, from whom he had sought and obtained the favour of reading the proof sheets of the Mecanique celeste.**Laplace**is due the theoretical proof that this function is independent of temperature and pressure, and apparent experimental confirmation was provided by Biot and Arago's, and by Dulong's observations on gases and vapours.The 18th century witnessed a rapid development of analysis, and the period culminated with the genius of Lagrange and

**Laplace**.Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with

**Laplace**, a number of identities between products of determinants of complementary orders.AdvertisementThey remained untold, for he died two days later on the 10th of April, and was buried in the Pantheon, the funeral oration being pronounced by

**Laplace**and Lacepede.This is especially the case between Lagrange and Euler on the one side, and between Lagrange and

**Laplace**on the other.Finally, of the grand series of researches by which the stability of the solar system was ascertained, the glory must be almost equally divided between Lagrange and

**Laplace**.**Laplace**owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way.But

**Laplace**unquestionably surpassed his rival in practical sagacity and the intuition of physical truth.AdvertisementLagrange saw in the problems of nature so many occasions for analytical triumphs;

**Laplace**regarded analytical triumphs as the means of solving the problems of nature.It deserves to be recorded as one of the numerous coincidences of discovery that

**Laplace**, on being made acquainted by Lagrange with his new method, produced analogous expressions, to which his independent researches had led him.The letters remained unnoticed, but

**Laplace**was not crushed by the rebuff.**Laplace**had not yet completed his twenty-fourth year when he entered upon the course of discovery which earned him the title of "the Newton of France."The discordance of their results incited

**Laplace**to a searching examination of the whole subject of planetary perturbations, and his maiden effort was rewarded with a discovery which constituted, when developed and completely demonstrated by his own further labours and those of his illustrious rival Lagrange, the most important advance made in physical astronomy since the time of Newton.AdvertisementIt was followed by a series of profound investigations, in which Lagrange and

**Laplace**alternately surpassed and supplemented each other in assigning limits of variation to the several elements of the planetary orbits.The analytical tournament closed with the communication to the Academy by

**Laplace**, 1 "Recherches sur le calcul integral," Mélanges de la Soc. Roy.The long-sought cause of the "great inequality" of Jupiter and Saturn was found in the near approach to commensurability of their mean motions; it was demonstrated in two elegant theorems, independently of any except the most general considerations as to mass, that the mutual action of the planets could never largely affect the eccentricities and inclinations of their orbits; and the singular peculiarities detected by him in the Jovian system were expressed in the so-called "laws of

**Laplace**."The year 1787 was rendered further memorable by

**Laplace**'s announcement on the 19th of November (Memoirs, 1786), of the dependence of lunar acceleration upon the secular changes in the eccentricity of the earth's orbit.With these brilliant performances the first period of

**Laplace**'s scientific career may be said to have closed.AdvertisementTo this task the second period of

**Laplace**'s activity was devoted.The famous "nebular hypothesis" of

**Laplace**made its appearance in the Systeme du monde.It is curious that

**Laplace**, while bestowing more attention than they deserved on the crude conjectures of Buffon, seems to have been unaware that he had been, to some extent, anticipated by Kant, who had put forward in 1755, in his Allgemeine Naturgeschichte, a true though defective nebular cosmogony.The career of

**Laplace**was one of scarcely interrupted prosperity.Biot relates that, when he himself was beginning his career,

**Laplace**introduced him at the Institute for the purpose of explaining his supposed discovery of equations of mixed differences, and afterwards showed him, under a strict pledge of secrecy, the papers, then yellow with age, in which he had long before obtained the same results.This instance of abnegation is the more worthy of record that it formed a marked exception to

**Laplace**'s usual course.With Lagrange, on the other hand, he always remained on the best of terms.

**Laplace**left a son, Charles Emile Pierre Joseph**Laplace**(1789-1874), who succeeded to his title, and rose to the rank of general in the artillery.It might be said that

**Laplace**was a great mathematician by the original structure of his mind, and became a great discoverer through the sentiment which animated it.**Laplace**was, moreover, the first to offer a complete analysis of capillary action based upon a definite hypothesis - that of forces "sensible only at insensible distances"; and he made strenuous but unsuccessful efforts to explain the phenomena of light on an identical principle.**Laplace**treated the subject from the point of view of the gradual aggregation and cooling of a mass of matter, and demonstrated that the form which such a mass would ultimately assume must be an ellipsoid of revolution whose equator was determined by the primitive plane of maximum areas.Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and

**Laplace**, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids.Finally, in a celebrated memoir, Theorie des attractions des spheroides et de la figure des planetes, published in 1785 among the Paris Memoirs for the year 1782, although written after the treatise of 1784,

**Laplace**treated exhaustively the general problem of the attraction of any spheroid upon a particle situated outside or upon its surface.These researches derive additional importance from having introduced two powerful engines of analysis for the treatment of physical problems,

**Laplace**'s coefficients and the potential function.By his discovery that the attracting force in any direction of a mass upon a particle could be obtained by the direct process of differentiating a single function,

**Laplace**laid the foundations of the mathematical sciences of heat, electricity and magnetism.**Laplace**nowhere displayed the massiveness of his genius more conspicuously than in the theory of probabilities.The theory of probabilities, which

**Laplace**described as common sense expressed in mathematical language, engaged his attention from its importance in physics and astronomy; and he applied his theory, not only to the ordinary problems of chances, but also to the inquiry into the causes of phenomena, vital statistics and future events.The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but

**Laplace**first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.**Laplace**published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable.The first formal proof of Lagrange's theorem for the development in a series of an implicit function was furnished by

**Laplace**, who gave to it an extended generality.In 1842, the works of

**Laplace**being nearly out of print, his widow was about to sell a farm to procure funds for a new impression, when the government of Louis Philippe took the matter in hand.A grant of 40,000 francs having been obtained from the chamber, a national edition was issued in seven 4to vols., bearing the title Ouvres de

**Laplace**(1843-1847).An enumeration of

**Laplace**'s memoirs and papers (about one hundred in number) is rendered superfluous by their embodiment in his principal works.**Laplace**'s first separate work, Theorie du mouvement et de la figure elliptique des planetes (1784), was published at the expense of President Bochard de Saron.An edition entitled Les Ouvres completes de

**Laplace**(1878), &c., which is to include all his memoirs as well as his separate works, is in course of publication under the auspices of the Academy of Sciences.**Laplace**'s treatise on specific heat was published in German in 1892 as No.Diophantine problems were revived by Gaspar Bachet, Pierre Fermat and Euler; the modern theory of numbers was founded by Fermat and developed by Euler, Lagrange and others; and the theory of probability was attacked by Blaise Pascal and Fermat, their work being subsequently expanded by James Bernoulli, Abraham de Moivre, Pierre Simon

**Laplace**and others.Meanwhile the astronomical theories of development of the solar system from a gaseous condition to its present form, put forward by Kant and by

**Laplace**, had impressed men's minds with the conception of a general movement of spontaneous progress or development in all nature.Thus he carried on the narrative of orderly development from the point at which it was left by Kant and

**Laplace**- explaining by reference to the ascertained laws of physics and chemistry the configuration of the earth, its mountains and seas, its igneous and its stratified rocks, just as the astronomers had explained by those same laws the evolution of the sun and planets from diffused gaseous matter of high temperature.In 1826 Fourier became a member of the French Academy, and in 1827 succeeded

**Laplace**as president of the council of the Ecole Polytechnique.He was also a great physicist and had arrived at the nebular hypothesis theory of the formation of the planets and the sun long before Kant and

**Laplace**.That is to say, instead of using Boyle's law, which supposes that the pressure changes so exceedingly slowly that conduction keeps the temperature constant, we must use the adiabatic relation p = kpy, whence d p /d p = y k p Y 1= yp/p, and U = (yp/p) [

**Laplace**'s formula].In the first of these, entitled " Recherches sur l'attraction des spheroides homogenes," published in the Memoires of the Academy for 1785, but communicated to it at an earlier period, Legendre introduces the celebrated expressions which, though frequently called

**Laplace**'s coefficients, are more correctly named after Legendre.In this memoir also the function which is now called the potential was, at the suggestion of

**Laplace**, first introduced.During forty years the resources of analysis, even in the hands of d'Alembert, Lagrange and

**Laplace**, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached.The introduction of the coefficients now called

**Laplace**'s, and their application, commence a new era in mathematical physics.The third memoir relates to

**Laplace**'s theorem respecting confocal ellipsoids.For the first time we have a correct and convenient expression for

**Laplace**'s nth coefficient."**Laplace**also justified the method by means of the principles of the theory of probability; and this led Legendre to republish the part of his Nouvelles Methodes which related to it in the Memoires de l'Academie for 1810.Late in 1793, Bailly quitted Nantes to join his friend Pierre Simon

**Laplace**at Melun; but was there recognized, arrested and brought (November 10) before the Revolutionary Tribunal at Paris.In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessel's functions are particular cases of

**Laplace**'s functions.He not only agrees with

**Laplace**and Lyell about the evolution of the solar system, but also supposes that the affinities, pointed out by Lothar Meyer and Mendeleeff, between groups of chemical elements prove an evolution of these elements from a primitive matter (prothyl) consisting of homogeneous atoms. These, however, are not ultimate enough for him; he thinks that everything, ponderable and imponderable or ether, is evolved from a primitive substance, which condenses first into centres of condensation (pyknatoms), and then into masses, which when they exceed the mean consistency become ponderables, and when they fall below it become imponderables.A life of Bowditch was written by his son Nathaniel Ingersoll Bowditch (1805-1861), and was prefixed to the fourth volume (1839) of the translation of

**Laplace**.Pursuing the investigations of

**Laplace**, he demonstrated with greater rigour the stability of the solar system, and calculated the limits within which the eccentricities and inclinations of the planetary orbits vary.**Laplace**in the Mecanique celeste was its larger aim, for the accomplishment of which forty years of unremitting industry barely sufficed.It was not till the 25th of June 1783 that in conjunction with

**Laplace**he announced to the Academy that water was the product formed by the combination of hydrogen and oxygen, but by that time he had been anticipated by Cavendish, to whose prior work, however, as to that of several other investigators in other matters, it is to be regretted that he did not render due acknowledgment.In addition to his purely chemical work, Lavoisier, mostly in conjunction with

**Laplace**, devoted considerable attention to physical problems, especially those connected with heat.He was faithful to the Bourbons during the Hundred Days; in fact, was 1 This prediction is sometimes attributed to

**Laplace**.His well-known correction of

**Laplace**'s partial differential equation for the potential was first published in the Bulletin de la societe philomatique (1813).The earlier forms of ice-calorimeter, those of Black, and of

**Laplace**and Lavoisier, were useless for work of precision, on account of the impossibility of accurately estimating the quantity of water left adhering to the ice in each case.In the theory of surfaces, in hydrokinetics, heat-conduction, potentials, &c., we constantly meet with what is called "

**Laplace**'s operator," viz.And we now see that the square of V is the negative of

**Laplace**'s operator; while V itself, when applied to any numerical quantity conceived as having a definite value at each point of space, gives the direction and the rate of most rapid change of that quantity.This explanation of the action of the solid is equivalent to that by which Gauss afterwards supplied the defect of the theory of

**Laplace**, except that, not being expressed in terms of mathematical symbols, it does not indicate the mathematical relation between the attraction of individual particles and the final result.Leslie's theory was afterwards treated according to

**Laplace**'s mathematical methods by James Ivory in the article on capillary action, under "Fluids, Elevation of," in the supplement to the fourth edition of the Encyclopaedia Britannica, published in 1819.**Laplace**investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal.In the Supplement to the Theory of Capillary Action,

**Laplace**deduced the equation of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface.In this supplement

**Laplace**gave many important applications of the theory, and compared the results with the experiments of Louis Joseph Gay Lussac.This condition when worked out gives not only the equation of the free surface in the form already established by

**Laplace**, but the conditions of the angle of contact of this surface with the surface of a solid.Gauss thus supplied the principal defect in the great work of

**Laplace**.He did not, however, enter into the explanation of particular phenomena, as this had been done already by

**Laplace**, but he pointed out to physicists the advantages of the method of Segner and Gay Lussac, afterwards carried out by Quincke, of measuring the dimensions of large drops of mercury on a horizontal or slightly concave surface, and those of large bubbles of air in transparent liquids resting against the under side of a horizontal plate of a substance wetted by the liquid.He proceeded to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and arrived at the conclusion that on this hypothesis none of the observed capillary phenomena would take place, and that, therefore,

**Laplace**'s theory, in which the density is supposed uniform, is not only insufficient but erroneous.In particular he maintained that the constant pressure K, which occurs in

**Laplace**'s theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concluded this is itself defective.**Laplace**assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance.The result, however, of Poisson's investigation is practically equivalent to that already obtained by

**Laplace**.In the form of the theory given by

**Laplace**, the density of the liquid was supposed to be uniform.Integrating the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of

**Laplace**.The expression for the intrinsic pressure is thus simply K= 2 iro 2 f 1,G(z)dz (28) In

**Laplace**'s investigation o- is supposed to be unity.As

**Laplace**has shown, the values for K and T may also be expressed in terms of the function cs, with which we started.For further calculations on

**Laplace**'s principles, see Rayleigh, Phil.**Laplace**does not treat systematically the question of interfacial tension, but he gives incidentally in terms of his quantity H a relation analogous to (47).This hypothesis was suggested by

**Laplace**, and may conveniently be named after him.So far the results of

**Laplace**'s hypothesis are in marked accordance with experiment; but if we follow it out further, discordances begin to manifestthemselves.The fact that a pair of plates which repel one another at a certain distance may attract one another at a smaller distance was deduced by

**Laplace**from theory, and verified by the observations of the abbe Haiiy.Attempts have been made by

**Laplace**and his successors to fix certain limits within which the obliquity of the ecliptic shall always be confined.In the summer of 1822, in his seventeenth year, he began a systematic study of

**Laplace**'s Mecanique Celeste.Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; for

**Laplace**'s great work, rich to profusion in analytical processes alike novel and powerful, demands from the most gifted student careful and often laborious study.Having detected an important defect in one of

**Laplace**'s demonstrations, he was induced by a friend to write out his remarks, that they might be shown to Dr John Brinkley (1763-1835), afterwards bishop of Cloyne, but who was then the first royal astronomer for Ireland, and an accomplished mathematician.In the following year his memoir on the secular acceleration of the moon's mean motion partially invalidated

**Laplace**'s famous explanation, which had held its place unchallenged for sixty years.After her marriage she made the acquaintance of the most eminent scientific men of the time, among whom her talents had attracted attention before she had acquired general fame,

**Laplace**paying her the compliment of stating that she was the only woman who understood his works.**Laplace**developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant.The development of the science by the successors of Newton, especially

**Laplace**and Lagrange, may be classed among the most striking achievements of the human intellect.The practical methods of computing perturbations of the planets and satellites were first exhaustively developed by Pierre Simon

**Laplace**in his Mecanique celeste.These successes paved the way for the higher triumphs of Joseph Louis Lagrange and of Pierre Simon

**Laplace**.It was especially adapted to the tracing out of " secular inequalities," or those depending upon changes in the orbital elements of the bodies affected by them, and hence progressing indefinitely with time; and by its means, accordingly, the mechanical stability of the solar system was splendidly demonstrated through the successive efforts of Lagrange and

**Laplace**.The proper share of each in bringing about this memorable result is not easy to apportion, since they freely imparted and profited by one another's advances and improvements; it need only be said that the fundamental proposition of the invariability of the planetary major axes laid down with restrictions by

**Laplace**in 1773, was finally established by Lagrange in 1776; while**Laplace**in 1784 proved the subsistence of such a relation between the eccentricities of the planetary orbits on the one hand, and their inclinations on the other, that an increase of either element could, in any single case, proceed only to a very small extent.The crowning trophies of gravitational astronomy in the r8th century were

**Laplace**'s explanations of the " great inequality ".A periodic character was thus indicated for the disturbance; and

**Laplace**assigned its true cause in the near approach to commensurability in the periods of the two planets, the cycle of disturbance completing itself in about goo (more accurately 929) years.**Laplace**'s calculations, it is true, were inexact.The Mecanique celeste, in which

**Laplace**welded into a whole the items of knowledge accumulated by the labours of a century, has been termed the " Almagest of the 18th century " (Fourier).The first step in constructing this theory was taken by

**Laplace**, who showed that the secular acceleration was produced by the secular diminution of the earth's orbit.**Laplace**'s immediate successors, among whom were Hansen, Plana and Pontecoulant, found a larger value, Hansen increasing it to 12.5", which he introduced into his tables.**Laplace**first showed that modern observations of the rpoon indicated that its mean motion was really less during the second half of the 18th century than during the first half, and hence inferred the existence of an inequality having a period of more than a century.The existence of one or more such inequalities has been fully confirmed by all the observations, both early and recent, that have become available since the time of

**Laplace**.It derives its celebrity ffom the demonstration by

**Laplace**that to whatever mutual actions all the bodies of a system may be subjected, the position of this plane remains invariable.Having received his early education from his father Louis Francois Cauchy (1760-1848), who held several minor public appointments and counted Lagrange and

**Laplace**among his friends, Cauchy entered Ecole Centrale du Pantheon in 1802, and proceeded to the Ecole Polytechnique in 1805, and to the Ecole des Ponts et Chaussees in 1807.Having adopted the profession of an engineer, he left Paris for Cherbourg in 1810, but returned in 1813 on account of his health, whereupon Lagrange and

**Laplace**persuaded him to renounce engineering and to devote himself to mathematics.Considering that our sun is but a star, or but one of the millions of stars, it is of interest to see whether any other systems present indication of a nebulous origin analogous to that which

**Laplace**proposed for the solar system.Their quantitative experiments were, however, too rough to permit of accurate generalization; and although Lavoisier and

**Laplace**stated the principle that the same amount of heat must be supplied to decompose a compound as would be produced on its formation, the statement was not based on exact experiment, and only received experimental confirmation much later.As notable instances may be mentioned

**Laplace**'s discoveries relating to the velocity of sound and the secular acceleration of the moon, both of which were led close up to by Lagrange's analytical demonstrations.The investigation of the figure of equilibrium of a rotating fluid mass engaged the persistent attention of

**Laplace**.The expressions designated by Dr Whewell,

**Laplace**'s coefficients (see Spherical Harmonics) were definitely introduced in the memoir of 1785 on attractions above referred to.Poisson, although previously demonstrated by

**Laplace**for the case when p=0.In science the state can boast of John Winthrop, the most eminent of colonial scientists; Benjamin Thompson (Count Rumford); Nathaniel Bowditch, the translator of

**Laplace**; Benjamin Peirce and Morse the electrician; not to include an adopted citizen in Louis Agassiz.The introduction of the coefficients now called

**Laplace**'s, and their application, commence a new era in mathematical physics."Thus, although the method of least squares was first formally proposed by Legendre, the theory and algorithm and mathematical foundation of the process are due to Gauss and

**Laplace**.**Laplace**, in whose footsteps Poisson followed, regarded him almost as his son.Fresnel's arguments in favour of that theory found little favour with

**Laplace**, Poisson and Biot, the champions of the emission theory; but they were ardently espoused by Humboldt and by Arago, who had been appointed by the Academy to report on the paper.In the first part of our own investigation we shall adhere to the symbols used by

**Laplace**, as we shall find that an accurate knowledge of the physical interpretation of these symbols is necessary for the further investigation of the subject.Having been requested by Lord Brougham to translate for the Society for the Diffusion of Useful Knowledge the Mecanique Celeste of

**Laplace**, she greatly popularized its form, and its publication in 1831, under the title of The Mechanism of the Heavens, at once made her famous.