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Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation. William Thomson Lord Kelvin rewrote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity. During the years — he published some memoirs of exceptional power.

In this work, Laplace completely determined the attraction of a spheroid on a particle outside it. This is memorable for the introduction into analysis of spherical harmonics or Laplace's coefficients , and also for the development of the use of what we would now call the gravitational potential in celestial mechanics. Laplace, with scant regard for credit to Legendre, made the non-trivial extension of the result to three dimensions to yield a more general set of functions, the spherical harmonics or Laplace coefficients.

The latter term is not in common use now. This paper is also remarkable for the development of the idea of the scalar potential. A potential function is a scalar function that defines how the vectors will behave. A scalar function is computationally and conceptually easier to deal with than a vector function. Alexis Clairaut had first suggested the idea in while working on a similar problem though he was using Newtonian-type geometric reasoning. Laplace described Clairaut's work as being "in the class of the most beautiful mathematical productions".

However, according to Rouse Ball, the term "potential function" was not actually used to refer to a function V of the coordinates of space in Laplace's sense until George Green 's An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Laplace applied the language of calculus to the potential function and showed that it always satisfies the differential equation : [9].

An analogous result for the velocity potential of a fluid had been obtained some years previously by Leonhard Euler. Laplace's subsequent work on gravitational attraction was based on this result. Laplace's equation , a special case of Poisson's equation , appears ubiquitously in mathematical physics. The concept of a potential occurs in fluid dynamics , electromagnetism and other areas. Rouse Ball speculated that it might be seen as "the outward sign" of one of the a priori forms in Kant's theory of perception.

The spherical harmonics turn out to be critical to practical solutions of Laplace's equation. Laplace's equation in spherical coordinates , such as are used for mapping the sky, can be simplified, using the method of separation of variables into a radial part, depending solely on distance from the centre point, and an angular or spherical part.

The solution to the spherical part of the equation can be expressed as a series of Laplace's spherical harmonics, simplifying practical computation. Laplace presented a memoir on planetary inequalities in three sections, in , , and This dealt mainly with the identification and explanation of the perturbations now known as the "great Jupiter—Saturn inequality". Laplace solved a longstanding problem in the study and prediction of the movements of these planets.

## "astronome" in English

He showed by general considerations, first, that the mutual action of two planets could never cause large changes in the eccentricities and inclinations of their orbits; but then, even more importantly, that peculiarities arose in the Jupiter—Saturn system because of the near approach to commensurability of the mean motions of Jupiter and Saturn. In this context commensurability means that the ratio of the two planets' mean motions is very nearly equal to a ratio between a pair of small whole numbers.

Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's. As a result, the integrated perturbations with this period are disproportionately large, about 0. Further developments of these theorems on planetary motion were given in his two memoirs of and , but with the aid of Laplace's discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate.

It was on the basis of Laplace's theory that Delambre computed his astronomical tables. Laplace now set himself the task to write a work which should "offer a complete solution of the great mechanical problem presented by the Solar System, and bring theory to coincide so closely with observation that empirical equations should no longer find a place in astronomical tables. The former was published in , and gives a general explanation of the phenomena, but omits all details.

It contains a summary of the history of astronomy. This summary procured for its author the honour of admission to the forty of the French Academy and is commonly esteemed one of the masterpieces of French literature, though it is not altogether reliable for the later periods of which it treats. Laplace developed the nebular hypothesis of the formation of the Solar System, first suggested by Emanuel Swedenborg and expanded by Immanuel Kant , a hypothesis that continues to dominate accounts of the origin of planetary systems.

According to Laplace's description of the hypothesis, the Solar System had evolved from a globular mass of incandescent gas rotating around an axis through its centre of mass. As it cooled, this mass contracted, and successive rings broke off from its outer edge. These rings in their turn cooled, and finally condensed into the planets, while the Sun represented the central core which was still left.

On this view, Laplace predicted that the more distant planets would be older than those nearer the Sun. As mentioned, the idea of the nebular hypothesis had been outlined by Immanuel Kant in , [53] and he had also suggested "meteoric aggregations" and tidal friction as causes affecting the formation of the Solar System.

Laplace was probably aware of this, but, like many writers of his time, he generally did not reference the work of others. The first two volumes, published in , contain methods for calculating the motions of the planets, determining their figures, and resolving tidal problems. The third and fourth volumes, published in and , contain applications of these methods, and several astronomical tables. The fifth volume, published in , is mainly historical, but it gives as appendices the results of Laplace's latest researches.

This is known for the introduction into analysis of the potential, a useful mathematical concept of broad applicability to the physical sciences.

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Laplace also came close to propounding the concept of the black hole. He suggested that there could be massive stars whose gravity is so great that not even light could escape from their surface see escape velocity. In , Laplace bought a house in Arcueil , then a village and not yet absorbed into the Paris conurbation. The chemist Claude Louis Berthollet was a neighbour — their gardens were not separated [57] — and the pair formed the nucleus of an informal scientific circle, latterly known as the Society of Arcueil.

Because of their closeness to Napoleon , Laplace and Berthollet effectively controlled advancement in the scientific establishment and admission to the more prestigious offices. The Society built up a complex pyramid of patronage. The first half of this treatise was concerned with probability methods and problems, the second half with statistical methods and applications.

Laplace's proofs are not always rigorous according to the standards of a later day, and his perspective slides back and forth between the Bayesian and non-Bayesian views with an ease that makes some of his investigations difficult to follow, but his conclusions remain basically sound even in those few situations where his analysis goes astray.

The general relevance for statistics of Laplacian error theory was appreciated only by the end of the 19th century. However, it influenced the further development of a largely analytically oriented probability theory. He begins the text with a series of principles of probability, the first six being:. One well-known formula arising from his system is the rule of succession , given as principle seven.

Suppose that some trial has only two possible outcomes, labelled "success" and "failure". Under the assumption that little or nothing is known a priori about the relative plausibilities of the outcomes, Laplace derived a formula for the probability that the next trial will be a success. It is still used as an estimator for the probability of an event if we know the event space, but have only a small number of samples. The rule of succession has been subject to much criticism, partly due to the example which Laplace chose to illustrate it.

He calculated that the probability that the sun will rise tomorrow, given that it has never failed to in the past, was. This result has been derided as absurd, and some authors have concluded that all applications of the Rule of Succession are absurd by extension.

However, Laplace was fully aware of the absurdity of the result; immediately following the example, he wrote, "But this number [i. The method of estimating the ratio of the number of favourable cases to the whole number of possible cases had been previously indicated by Laplace in a paper written in It consists of treating the successive values of any function as the coefficients in the expansion of another function, with reference to a different variable.

The latter is therefore called the probability-generating function of the former. Laplace then shows how, by means of interpolation , these coefficients may be determined from the generating function. Next he attacks the converse problem, and from the coefficients he finds the generating function; this is effected by the solution of a finite difference equation. The fourth chapter of this treatise includes an exposition of the method of least squares , a remarkable testimony to Laplace's command over the processes of analysis.

In Legendre had published the method of least squares, making no attempt to tie it to the theory of probability. In Gauss had derived the normal distribution from the principle that the arithmetic mean of observations gives the most probable value for the quantity measured; then, turning this argument back upon itself, he showed that, if the errors of observation are normally distributed, the least squares estimates give the most probable values for the coefficients in regression situations. These two works seem to have spurred Laplace to complete work toward a treatise on probability he had contemplated as early as In two important papers in and , Laplace first developed the characteristic function as a tool for large-sample theory and proved the first general central limit theorem.

Then in a supplement to his paper written after he had seen Gauss's work, he showed that the central limit theorem provided a Bayesian justification for least squares: if one were combining observations, each one of which was itself the mean of a large number of independent observations, then the least squares estimates would not only maximise the likelihood function, considered as a posterior distribution, but also minimise the expected posterior error, all this without any assumption as to the error distribution or a circular appeal to the principle of the arithmetic mean.

Considering a linear regression problem, he restricted his attention to linear unbiased estimators of the linear coefficients. After showing that members of this class were approximately normally distributed if the number of observations was large, he argued that least squares provided the "best" linear estimators.

Here it is "best" in the sense that it minimised the asymptotic variance and thus both minimised the expected absolute value of the error, and maximised the probability that the estimate would lie in any symmetric interval about the unknown coefficient, no matter what the error distribution. His derivation included the joint limiting distribution of the least squares estimators of two parameters.

In , Laplace published what is usually known as the first articulation of causal or scientific determinism : [62]. We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

This intellect is often referred to as Laplace's demon in the same vein as Maxwell's demon and sometimes Laplace's Superman after Hans Reichenbach. Laplace, himself, did not use the word "demon", which was a later embellishment. As translated into English above, he simply referred to: "Une intelligence Even though Laplace is known as the first to express such ideas about causal determinism, his view is very similar to the one proposed by Boscovich as early as in his book Theoria philosophiae naturalis.

As early as , Euler , followed by Lagrange , had started looking for solutions of differential equations in the form: [65].

This integral operator transforms a function of time t into a function of position or space s. In , Laplace took the key forward step in using integrals of this form to transform a whole differential equation from a function of time into a lower order function of space; The transformed equation was easier to solve than the original because algebra could be used to manipulate the differential equation into a simpler form. The inverse Laplace transform was then taken to convert the simplified function of space back into a function of time.

Laplace built upon the qualitative work of Thomas Young to develop the theory of capillary action and the Young—Laplace equation. Laplace in was the first to point out that the speed of sound in air depends on the heat capacity ratio. Newton's original theory gave too low a value, because it does not take account of the adiabatic compression of the air which results in a local rise in temperature and pressure.

Laplace's investigations in practical physics were confined to those carried on by him jointly with Lavoisier in the years to on the specific heat of various bodies. He prudently withdrew from Paris during the most violent part of the Revolution. In November , immediately after seizing power in the coup of 18 Brumaire , Napoleon appointed Laplace to the post of Minister of the Interior. The appointment, however, lasted only six weeks, after which Lucien Bonaparte , Napoleon's brother, was given the post.

Evidently, once Napoleon's grip on power was secure, there was no need for a prestigious but inexperienced scientist in the government.

Geometrician of the first rank, Laplace was not long in showing himself a worse than average administrator; from his first actions in office we recognized our mistake. Laplace did not consider any question from the right angle: he sought subtleties everywhere, conceived only problems, and finally carried the spirit of "infinitesimals" into the administration.

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Grattan-Guinness, however, describes these remarks as "tendentious", since there seems to be no doubt that Laplace "was only appointed as a short-term figurehead, a place-holder while Napoleon consolidated power". Although Laplace was removed from office, it was desirable to retain his allegiance. In copies sold after the Bourbon Restoration this was struck out. Pearson points out that the censor would not have allowed it anyway.

In it was evident that the empire was falling; Laplace hastened to tender his services to the Bourbons , and in during the Restoration he was rewarded with the title of marquis. According to Rouse Ball, the contempt that his more honest colleagues felt for his conduct in the matter may be read in the pages of Paul Louis Courier. His knowledge was useful on the numerous scientific commissions on which he served, and, says Rouse Ball, probably accounts for the manner in which his political insincerity was overlooked.

Roger Hahn in his biography disputes this portrayal of Laplace as an opportunist and turncoat, pointing out that, like many in France, he had followed the debacle of Napoleon's Russian campaign with serious misgivings. Napoleon had originally come to power promising stability, but it was clear that he had overextended himself, putting the nation at peril. It was at this point that Laplace's loyalty began to weaken. Although he still had easy access to Napoleon, his personal relations with the emperor cooled considerably.

As a grieving father, he was particularly cut to the quick by Napoleon's insensitivity in an exchange related by Jean-Antoine Chaptal : "On his return from the rout in Leipzig , he [Napoleon] accosted Mr Laplace: 'Oh! I see that you have grown thin—Sire, I have lost my daughter—Oh!

You are a mathematician; put this event in an equation, and you will find that it adds up to zero. In the second edition of the Essai philosophique , Laplace added some revealing comments on politics and governance. Since it is, he says, "the practice of the eternal principles of reason, justice and humanity that produce and preserve societies, there is a great advantage to adhere to these principles, and a great inadvisability to deviate from them".

States that transgress these limits cannot avoid being "reverted" to them, "just as is the case when the waters of the seas whose floor has been lifted by violent tempests sink back to their level by the action of gravity". About the political upheavals he had witnessed, Laplace formulated a set of principles derived from physics to favour evolutionary over revolutionary change:.

Let us apply to the political and moral sciences the method founded upon observation and calculation, which has served us so well in the natural sciences.

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Let us not offer fruitless and often injurious resistance to the inevitable benefits derived from the progress of enlightenment; but let us change our institutions and the usages that we have for a long time adopted only with extreme caution. We know from past experience the drawbacks they can cause, but we are unaware of the extent of ills that change may produce. In the face of this ignorance, the theory of probability instructs us to avoid all change, especially to avoid sudden changes which in the moral as well as the physical world never occur without a considerable loss of vital force.

In these lines, Laplace expressed the views he had arrived at after experiencing the Revolution and the Empire. He believed that the stability of nature, as revealed through scientific findings, provided the model that best helped to preserve the human species. Laplace died in Paris in It was reportedly smaller than the average brain.

A frequently cited but potentially apocryphal interaction between Laplace and Napoleon purportedly concerns the existence of God. Although the conversation in question did occur, the exact words Laplace used and his intended meaning are not known. A typical version is provided by Rouse Ball: [9]. Laplace went in state to Napoleon to present a copy of his work, and the following account of the interview is well authenticated, and so characteristic of all the parties concerned that I quote it in full.

Someone had told Napoleon that the book contained no mention of the name of God; Napoleon, who was fond of putting embarrassing questions, received it with the remark, 'M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.

An earlier report, although without the mention of Laplace's name, is found in Antommarchi's The Last Moments of Napoleon : [77]. Je m'entretenais avec L I congratulated him on a work which he had just published and asked him how the name of God, which appeared endlessly in the works of Lagrange, didn't occur even once in his. He replied that he had no need of that hypothesis. It was not God that Laplace had treated as a hypothesis, but merely his intervention at a determinate point:.

In fact Laplace never said that. Here, I believe, is what truly happened. Newton, believing that the secular perturbations which he had sketched out in his theory would in the long run end up destroying the Solar System, says somewhere that God was obliged to intervene from time to time to remedy the evil and somehow keep the system working properly. This, however, was a pure supposition suggested to Newton by an incomplete view of the conditions of the stability of our little world. Science was not yet advanced enough at that time to bring these conditions into full view. But Laplace, who had discovered them by a deep analysis, would have replied to the First Consul that Newton had wrongly invoked the intervention of God to adjust from time to time the machine of the world la machine du monde and that he, Laplace, had no need of such an assumption.

It was not God, therefore, that Laplace treated as a hypothesis, but his intervention in a certain place. Faye writes: [78] [79]. I have it on the authority of M. Arago that Laplace, warned shortly before his death that that anecdote was about to be published in a biographical collection, had requested him [Arago] to demand its deletion by the publisher. It was necessary to either explain or delete it, and the second way was the easiest. But, unfortunately, it was neither deleted nor explained. The Swiss-American historian of mathematics Florian Cajori appears to have been unaware of Faye's research, but in he came to a similar conclusion.

It's just that he doesn't intervene, to break the laws of Science. The only eyewitness account of Laplace's interaction with Napoleon is from the entry for 8 August in the diary of the British astronomer Sir William Herschel : [82]. The first Consul then asked a few questions relating to Astronomy and the construction of the heavens to which I made such answers as seemed to give him great satisfaction.

He also addressed himself to Mr Laplace on the same subject, and held a considerable argument with him in which he differed from that eminent mathematician. The difference was occasioned by an exclamation of the first Consul, who asked in a tone of exclamation or admiration when we were speaking of the extent of the sidereal heavens : 'And who is the author of all this! De la Place wished to shew that a chain of natural causes would account for the construction and preservation of the wonderful system. This the first Consul rather opposed. Much may be said on the subject; by joining the arguments of both we shall be led to 'Nature and nature's God'.

Since this makes no mention of Laplace saying, "I had no need of that hypothesis," Daniel Johnson [83] argues that "Laplace never used the words attributed to him. Born a Catholic, Laplace appears in adult life to have inclined to deism presumably his considered position, since it is the only one found in his writings. However, some of his contemporaries thought he was an atheist , while a number of recent scholars have described him as agnostic. He owned that he was an atheist.

It appeared to Guettard that Laplace's atheism "was supported by a thoroughgoing materialism ". Hahn states: "Nowhere in his writings, either public or private, does Laplace deny God's existence. Let Him be always present to your mind, as also your father and your mother]. Glass, quoting Herschel's account of the celebrated exchange with Napoleon, writes that Laplace was "evidently a deist like Herschel".

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It is, he writes, the "first and most infallible of principles It is the sheerest absurdity to suppose that "the sovereign lawgiver of the universe would suspend the laws that he has established, and which he seems to have maintained invariably". Maurice thought that the basis of Laplace's beliefs was, little by little, being modified, but that he held fast to his conviction that the invariability of the laws of nature did not permit of supernatural events.

However, according to his biographer, Roger Hahn, it is "not credible" that Laplace "had a proper Catholic end", and he "remained a skeptic" to the very end of his life. Platina's account does not accord with Church records, which do not mention the comet. Laplace is alleged to have embellished the story by claiming the Pope had " excommunicated " Halley's comet. From Wikipedia, the free encyclopedia. For other uses, see Laplace disambiguation. Beaumont-en-Auge , Normandy, Kingdom of France. Paris , Kingdom of France. Main article: Theory of tides. Spherical harmonics.

Second law of motion. As they age, stars also fuse those heavier elements to make more complex ones. This process of stellar nucleosynthesis is what populates the universe with many of the elements heavier than hydrogen and helium. It's also an important part of the evolution of stars, which Cecelia sought to understand.

The idea that stars are made mostly of hydrogen seems like a very obvious thing to astronomers today, but for its time, Dr. Payne's idea was startling. One of her advisors — Henry Norris Russell — disagreed with it and demanded she take it out of her thesis defense. Later, he decided it was a great idea, published it on his own, and got the credit for the discovery. She continued to work at Harvard, but for time, because she was a woman, she received very low pay and the classes she taught weren't even recognized in the course catalogs at the time.

In recent decades, the credit for her discovery and subsequent work has been restored to Dr. She is also credited with establishing that stars can be classified by their temperatures, and published more than papers on stellar atmospheres, stellar spectra. She also worked with her husband, Serge I. Gaposchkin, on variable stars. She published five books, and won a number of awards. She spent her entire research career at Harvard College Observatory, eventually becoming the first woman to chair a department at Harvard. Despite successes that would have gained male astronomers at the time incredible praise and honors, she faced gender discrimination throughout much of her life.

Nonetheless, she is now celebrated as a brilliant and original thinker for her contributions that changed our understanding of how stars work. As one of the first of a group of female astronomers at Harvard, Cecelia Payne-Gaposchkin blazed a trail for women in astronomy that many cite as their own inspiration to study the stars.

In , a special centenary celebration of her life and science at Harvard drew astronomers from around the world to discuss her life and findings and how they changed the face of astronomy. Largely due to her work and example, as well as the example of women who were inspired by her courage and intellect, the role of women in astronomy is slowly improving, as more select it as a profession. She got interested in astronomy after hearing Sir Arthur Eddington describe his experiences on an eclipse expedition in She then studied astronomy, but because she was female, she was refused a degree from Cambridge.

After she received her doctorate, Dr. Payne went on to study a number of different types of stars, particularly the very brightest "high luminosity " stars.