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Laplace

The author of the "Mecanique Celeste" was born at Beaumont-en- Auge, near Honfleur, in 1749, just thirteen years later than his renowned friend Lagrange. His father was a farmer, but appears to have been in a position to provide a good education for a son who seemed promising. Considering the unorthodoxy in religious matters which is generally said to have characterized Laplace in later years, it is interesting to note that when he was a boy the subject which first claimed his attention was theology. He was, however, soon introduced to the study of mathematics, in which he presently became so proficient, that while he was still no more than eighteen years old, he obtained employment as a mathematical teacher in his native town.

Desiring wider opportunities for study and for the acquisition of fame than could be obtained in the narrow associations of provincial life, young Laplace started for Paris, being provided with letters of introduction to D'Alembert, who then occupied the most prominent position as a mathematician in France, if not in the whole of Europe. D'Alembert's fame was indeed so brilliant that Catherine the Great wrote to ask him to undertake the education of her Son, and promised the splendid income of a hundred thousand francs. He preferred, however, a quiet life of research in Paris, although there was but a modest salary attached to his office. The philosopher accordingly declined the alluring offer to go to Russia, even though Catherine wrote again to say: "I know that your refusal arises from your desire to cultivate your studies and your friendships in quiet. But this is of no consequence: bring all your friends with you, and I promise you that both you and they shall have every accommodation in my power." With equal firmness the illustrious mathematician resisted the manifold attractions with which Frederick the Great sought to induce him, to take up his residence at Berlin. In reading of these invitations we cannot but be struck at the extraordinary respect which was then paid to scientific distinction. It must be remembered that the discoveries of such a man as D'Alembert were utterly incapable of being appreciated except by those who possessed a high degree of mathematical culture. We nevertheless find the potentates of Russia and Prussia entreating and, as it happens, vainly entreating, the most distinguished mathematician in France to accept the positions that they were proud to offer him.

It was to D'Alembert, the profound mathematician, that young Laplace, the son of the country farmer, presented his letters of introduction. But those letters seem to have elicited no reply, whereupon Laplace wrote to D'Alembert submitting a discussion on some point in Dynamics. This letter instantly produced the desired effect. D'Alembert thought that such mathematical talent as the young man displayed was in itself the best of introductions to his favour. It could not be overlooked, and accordingly he invited Laplace to come and see him. Laplace, of course, presented himself, and ere long D'Alembert obtained for the rising philosopher a professorship of mathematics in the Military School in Paris. This gave the brilliant young mathematician the opening for which he sought, and he quickly availed himself of it.

Laplace was twenty-three years old when his first memoir on a profound mathematical subject appeared in the Memoirs of the Academy at Turin. From this time onwards we find him publishing one memoir after another in which he attacks, and in many cases successfully vanquishes, profound difficulties in the application of the Newtonian theory of gravitation to the explanation of the solar system. Like his great contemporary Lagrange, he loftily attempted problems which demanded consummate analytical skill for their solution. The attention of the scientific world thus became riveted on the splendid discoveries which emanated from these two men, each gifted with extraordinary genius.

Laplace's most famous work is, of course, the "Mecanique Celeste," in which he essayed a comprehensive attempt to carry out the principles which Newton had laid down, into much greater detail than Newton had found practicable. The fact was that Newton had not only to construct the theory of gravitation, but he had to invent the mathematical tools, so to speak, by which his theory could be applied to the explanation of the movements of the heavenly bodies. In the course of the century which had elapsed between the time of Newton and the time of Laplace, mathematics had been extensively developed. In particular, that potent instrument called the infinitesimal calculus, which Newton had invented for the investigation of nature, had become so far perfected that Laplace, when he attempted to unravel the movements of the heavenly bodies, found himself provided with a calculus far more efficient than that which had been available to Newton. The purely geometrical methods which Newton employed, though they are admirably adapted for demonstrating in a general way the tendencies of forces and for explaining the more obvious phenomena by which the movements of the heavenly bodies are disturbed, are yet quite inadequate for dealing with the more subtle effects of the Law of Gravitation. The disturbances which one planet exercises upon the rest can only be fully ascertained by the aid of long calculation, and for these calculations analytical methods are required.

With an armament of mathematical methods which had been perfected since the days of Newton by the labours of two or three generations of consummate mathematical inventors, Laplace essayed in the "Mecanique Celeste" to unravel the mysteries of the heavens. It will hardly be disputed that the book which he has produced is one of the most difficult books to understand that has ever been written. In great part, of course, this difficulty arises from the very nature of the subject, and is so far unavoidable. No one need attempt to read the "Mecanique Celeste" who has not been naturally endowed with considerable mathematical aptitude which he has cultivated by years of assiduous study. The critic will also note that there are grave defects in Laplace's method of treatment. The style is often extremely obscure, and the author frequently leaves great gaps in his argument, to the sad discomfiture of his reader. Nor does it mend matters to say, as Laplace often does say, that it is "easy to see" how one step follows from another. Such inferences often present great difficulties even to excellent mathematicians. Tradition indeed tells us that when Laplace had occasion to refer to his own book, it sometimes happened that an argument which he had dismissed with his usual formula, "Il est facile a voir," cost the illustrious author himself an hour or two of hard thinking before he could recover the train of reasoning which had been omitted. But there are certain parts of this great work which have always received the enthusiastic admiration of mathematicians. Laplace has, in fact, created whole tracts of science, some of which have been subsequently developed with much advantage in the prosecution of the study of Nature.

Judged by a modern code the gravest defect of Laplace's great work is rather of a moral than of a mathematical nature. Lagrange and he advanced together in their study of the mechanics of the heavens, at one time perhaps along parallel lines, while at other times they pursued the same problem by almost identical methods. Sometimes the important result was first reached by Lagrange, sometimes it was Laplace who had the good fortune to make the discovery. It would doubtless be a difficult matter to draw the line which should exactly separate the contributions to astronomy made by one of these illustrious mathematicians, and the contributions made by the other. But in his great work Laplace in the loftiest manner disdained to accord more than the very barest recognition to Lagrange, or to any of the other mathematicians, Newton alone excepted, who had advanced our knowledge of the mechanism of the heavens. It would be quite impossible for a student who confined his reading to the "Mecanique Celeste" to gather from any indications that it contains whether the discoveries about which he was reading had been really made by Laplace himself or whether they had not been made by Lagrange, or by Euler, or by Clairaut. With our present standard of morality in such matters, any scientific man who now brought forth a work in which he presumed to ignore in this wholesale fashion the contributions of others to the subject on which he was writing, would be justly censured and bitter controversies would undoubtedly arise. Perhaps we ought not to judge Laplace by the standard of our own time, and in any case I do not doubt that Laplace might have made a plausible defence. It is well known that when two investigators are working at the same subjects, and constantly publishing their results, it sometimes becomes difficult for each investigator himself to distinguish exactly between what he has accomplished and that which must be credited to his rival. Laplace may probably have said to himself that he was going to devote his energies to a great work on the interpretation of Nature, that it would take all his time and all his faculties, and all the resources of knowledge that he could command, to deal justly with the mighty problems before him. He would not allow himself to be distracted by any side issue. He could not tolerate that pages should be wasted in merely discussing to whom we owe each formula, and to whom each deduction from such formula is due. He would rather endeavour to produce as complete a picture as he possibly could of the celestial mechanics, and whether it were by means of his mathematics alone, or whether the discoveries of others may have contributed in any degree to the result, is a matter so infinitesimally insignificant in comparison with the grandeur of his subject that he would altogether neglect it. "If Lagrange should think," Laplace might say, "that his discoveries had been unduly appropriated, the proper course would be for him to do exactly what I have done. Let him also write a "Mecanique Celeste," let him employ those consummate talents which he possesses in developing his noble subject to the utmost. Let him utilise every result that I or any other mathematician have arrived at, but not trouble himself unduly with unimportant historical details as to who discovered this, and who discovered that; let him produce such a work as he could write, and I shall heartily welcome it as a splendid contribution to our science." Certain it is that Laplace and Lagrange continued the best of friends, and on the death of the latter it was Laplace who was summoned to deliver the funeral oration at the grave of his great rival.

The investigations of Laplace are, generally speaking, of too technical a character to make it possible to set forth any account of them in such a work as the present. He did publish, however, one treatise, called the " Systeme du Monde," in which, without introducing mathematical symbols, he was able to give a general account of the theories of the celestial movements, and of the discoveries to which he and others had been led. In this work the great French astronomer sketched for the first time that remarkable doctrine by which his name is probably most generally known to those readers of astronomical books who are not specially mathematicians. It is in the "Systeme du Monde" that Laplace laid down the principles of the Nebular Theory which, in modern days, has been generally accepted by those philosophers who are competent to judge, as substantially a correct expression of a great historical fact.

The Nebular Theory gives a physical account of the origin of the solar system, consisting of the sun in the centre, with the planets and their attendant satellites. Laplace perceived the significance of the fact that all the planets revolved in the same direction around the sun; he noticed also that the movements of rotation of the planets on their axes were performed in the same direction as that in which a planet revolves around the sun; he saw that the orbits of the satellites, so far at least as he knew them, revolved around their primaries also in the same direction. Nor did it escape his attention that the sun itself rotated on its axis in the same sense. His philosophical mind was led to reflect that such a remarkable unanimity in the direction of the movements in the solar system demanded some special explanation. It would have been in the highest degree improbable that there should have been this unanimity unless there had been some physical reason to account for it. To appreciate the argument let us first concentrate our attention on three particular bodies, namely the earth, the sun, and the moon. First the earth revolves around the sun in a certain direction, and the earth also rotates on its axis. The direction in which the earth turns in accordance with this latter movement might have been that in which it revolves around the sun, or it might of course have been opposite thereto. As a matter of fact the two agree. The moon in its monthly revolution around the earth follows also the same direction, and our satellite rotates on its axis in the same period as its monthly revolution, but in doing so is again observing this same law. We have therefore in the earth and moon four movements, all taking place in the same direction, and this is also identical with that in which the sun rotates once every twenty-five days. Such a coincidence would be very unlikely unless there were some physical reason for it. Just as unlikely would it be that in tossing a coin five heads or five tails should follow each other consecutively. If we toss a coin five times the chances that it will turn up all heads or all tails is but a small one. The probability of such an event is only one-sixteenth.

There are, however, in the solar system many other bodies besides the three just mentioned which are animated by this common movement. Among them are, of course, the great planets, Jupiter, Saturn, Mars, Venus, and Mercury, and the satellites which attend on these planets. All these planets rotate on their axes in the same direction as they revolve around the sun, and all their satellites revolve also in the same way. Confining our attention merely to the earth, the sun, and the five great planets with which Laplace was acquainted, we have no fewer than six motions of revolution and seven motions of rotation, for in the latter we include the rotation of the sun. We have also sixteen satellites of the planets mentioned whose revolutions round their primaries are in the same direction. The rotation of the moon on its axis may also be reckoned, but as to the rotations of the satellites of the other planets we cannot speak with any confidence, as they are too far off to be observed with the necessary accuracy. We have thus thirty circular movements in the solar system connected with the sun and moon and those great planets than which no others were known in the days of Laplace. The significant fact is that all these thirty movements take place in the same direction. That this should be the case without some physical reason would be just as unlikely as that in tossing a coin thirty times it should turn up all heads or all tails every time without exception.

We can express the argument numerically. Calculation proves that such an event would not generally happen oftener than once out of five hundred millions of trials. To a philosopher of Laplace's penetration, who had made a special study of the theory of probabilities, it seemed well-nigh inconceivable that there should have been such unanimity in the celestial movements, unless there had been some adequate reason to account for it. We might, indeed, add that if we were to include all the objects which are now known to belong to the solar system, the argument from probability might be enormously increased in strength. To Laplace the argument appeared so conclusive that he sought for some physical cause of the remarkable phenomenon which the solar system presented. Thus it was that the famous Nebular Hypothesis took its rise. Laplace devised a scheme for the origin of the sun and the planetary system, in which it would be a necessary consequence that all the movements should take place in the same direction as they are actually observed to do.

Let us suppose that in the beginning there was a gigantic mass of nebulous material, so highly heated that the iron and other substances which now enter into the composition of the earth and planets were then suspended in a state of vapour. There is nothing unreasonable in such a supposition indeed, we know as a matter of fact that there are thousands of such nebulae to be discerned at present through our telescopes. It would be extremely unlikely that any object could exist without possessing some motion of rotation; we may in fact assert that for rotation to be entirety absent from the great primeval nebula would be almost infinitely improbable. As ages rolled on, the nebula gradually dispersed away by radiation its original stores of heat, and, in accordance with well-known physical principles, the materials of which it was formed would tend to coalesce. The greater part of those materials would become concentrated in a mighty mass surrounded by outlying uncondensed vapours. There would, however, also be regions throughout the extent of the nebula, in which subsidiary centres of condensation would be found. In its long course of cooling, the nebula would, therefore, tend ultimately to form a mighty central body with a number of smaller bodies disposed around it. As the nebula was initially endowed with a movement of rotation, the central mass into which it had chiefly condensed would also revolve, and the subsidiary bodies would be animated by movements of revolution around the central body. These movements would be all pursued in one common direction, and it follows, from well-known mechanical principles, that each of the subsidiary masses, besides participating in the general revolution around the central body, would also possess a rotation around its axis, which must likewise be performed in the same direction. Around the subsidiary bodies other objects still smaller would be formed, just as they themselves were formed relatively to the great central mass.

As the ages sped by, and the heat of these bodies became gradually dissipated, the various objects would coalesce, first into molten liquid masses, and thence, at a further stage of cooling, they would assume the appearance of solid masses, thus producing the planetary bodies such as we now know them. The great central mass, on account of its preponderating dimensions, would still retain, for further uncounted ages, a large quantity of its primeval heat, and would thus display the splendours of a glowing sun. In this way Laplace was able to account for the remarkable phenomena presented in the movements of the bodies of the solar system. There are many other points also in which the nebular theory is known to tally with the facts of observation. In fact, each advance in science only seems to make it more certain that the Nebular Hypothesis substantially represents the way in which our solar system has grown to its present form.

Not satisfied with a career which should be merely scientific, Laplace sought to connect himself with public affairs. Napoleon appreciated his genius, and desired to enlist him in the service of the State. Accordingly he appointed Laplace to be Minister of the Interior. The experiment was not successful, for he was not by nature a statesman. Napoleon was much disappointed at the ineptitude which the great mathematician showed for official life, and, in despair of Laplace's capacity as an administrator, declared that he carried the spirit of his infinitesimal calculus into the management of business. Indeed, Laplace's political conduct hardly admits of much defence. While he accepted the honours which Napoleon showered on him in the time of his prosperity, he seems to have forgotten all this when Napoleon could no longer render him service. Laplace was made a Marquis by Louis XVIII., a rank which he transmitted to his son, who was born in 1789. During the latter part of his life the philosopher lived in a retired country place at Arcueile. Here he pursued his studies, and by strict abstemiousness, preserved himself from many of the infirmities of old age. He died on March the 5th, 1827, in his seventy-eighth year, his last words being, "What we know is but little, what we do not know is immense."

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