Mécanique Céleste

“ About this period he was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a portion of his time to classics. In the summer of 1822, in his seventeenth year, he began a systematic study of Laplace’s Mécanique Céleste. 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. ”

“ We can here do no more than mention the analysis of the phenomena of the tides, one of the most important and most brilliant of Laplace’s performances. Indeed there is no branch of Physical Astronomy, we might almost say of physics in general, which is not materially indebted to him. Superior to Euler in the power of conquering analytical difficulties, he is almost his equal in the universality of his labours.
The great work of Laplace is the ‘Mécanique Céleste,’ a collection of all that had been done by himself or others, concerning the theory of the universe. ”

“ GRAVITATIONAL ASTRONOMY IN THE 18TH CENTURY.
"Astronomy, considered in the most general way, is a great problem of mechanics, the arbitrary data of which are the elements of the celestial movements; its solution depends both on the accuracy of observations and on the perfection of analysis."
Laplace, Preface to the Mécanique Céleste. 228. The solar system, as it was known at the beginning of the 18th century, contained 18 recognised members: the sun, six planets, ten satellites (one belonging to the earth, four to Jupiter, and five to Saturn) , and Saturn's ring. ”

“ In other respects, the care which Laplace always employed, in pursuing his researches, as far as possible, to their numerical results, will enable those who are disposed to institute a complete comparison between the two rival theories of light, to derive from the Mécanique Céleste the materials of several interesting relations.
Is light an emanation from the sun? Does this body launch out incessantly in every direction a part of its own substance? Is it gradually diminishing in volume and mass? ”

“ In short, the Théorie Analytique des Probabilités is worthy of the author of the Mécanique Céleste.
A philosopher, whose name is associated with immortal discoveries, said to his audience who had allowed themselves to be influenced by ancient and consecrated authorities, "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 depreciating their value, or obliterating their truthful character. ”

“ Though supple and servile in politics, it must be said that in religion and science Laplace never misrepresented or concealed his own convictions however distasteful they might be to others. In mathematics and astronomy his genius shines with a lustre excelled by few. Three great works did he give to the scientific world,—the Mécanique Céleste, the Exposition du système du monde, and the Théorie analytique des probabilities. Besides these he contributed important memoirs to the French Academy. ”

“ The growth of the man of science is represented by his colossal "La Mécanique Céleste," which first appeared in 1799, and was continued by successive volumes until its completion in 1825. Its title, rather than his titles, should be inscribed on his monument.
A little later than these famous confrères, Georges Cuvier appears in Paris—in Hugo's half-truth—"with one eye on the book of Genesis and the other on nature, endeavoring to please bigoted reaction by reconciling fossils with texts, and making the mastodons support Moses." ”

“ The place which it assigned to the disturbing planet was the same, to one degree, as that given by Mr. Adams’ calculations, which I had perused seven months earlier. To this time I had considered that there was still room for doubt of the accuracy of Mr. Adams’ investigations; for I think that the results of algebraic and numerical computations, so long and so complicated as those of an inverse problem of perturbations, are liable to many risks of error in the details of the process: I know that there are important numerical errors in the Mécanique Céleste of Laplace ”

“ At this moment Céleste appeared on the verandah.
"Look, papa, what a lovely orchid I am going to bring you," she called out, with a wealth of love and laughter shining in her eyes.
"No, no, stay where you are," shouted Villebois, "we'll make it a prize." Turning to his companions he added smiling, "Let us race for it; physics, medicine and law running for a prize in botany, and the privilege of having the decoration placed on his breast by Céleste." ”

“ The motive quantity of a centripetal force is the measure of the same, proportional to the motion which it generates in a given time. Thus the Weight is greater in a greater body, less in a less body; it is greater near to the Earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or as I may say, its Weight; and it is ever known by the quantity of a force equal and contrary to it, that is just sufficient to hinder the descent of the body. ”

“ A.--The centrifugal force of bodies revolving in a circle increases as the diameter of the circle, if the number of revolutions remain the same. If there be two fly-wheels of the same weight, and making the same number of revolutions per minute, but the diameter of one be double that of the other, the larger will have double the amount of centrifugal force. The centrifugal force of the same wheel, however, increases as the square of the velocity; so that if the velocity of a fly-wheel be doubled, it will have four times the amount of centrifugal force. ”

“ The potential within the cylinder will then be a function of its length only, and if we suppose the surface of the electrode where the cylinder meets it to be approximately plane, and all its dimensions to be large compared with the diameter of the cylinder, the distribution of potential will be that due to a conductor in the form of a disk placed in an infinite medium. ”

“ To fix our ideas we may conceive the system connected by means of suitable mechanism with a number of moveable pieces, each capable of motion along a straight line, and of no other kind of motion. The imaginary mechanism which connects each of these pieces with the system must be conceived to be free from friction, destitute of inertia, and incapable of being strained by the action of the applied forces. The use of this mechanism is merely to assist the imagination in ascribing position, velocity, and momentum to what appear, in Lagrange's investigation, as pure algebraical quantities. ”

“ Let us be allies in ferreting out this mystery. Will you [Pg 112] help me, Mademoiselle Céleste? Only mind, you must be very discreet."
"Can I depend on you?" asked Céleste, looking up eagerly into his face.
"Like my own soul, mademoiselle," answered Riche solemnly. "We will both keep a watch on Pierre Duval, and on M. Payot as well."
"Oh, thank you, thank you ever so much. It will be just lovely if we can work together. I will do everything you ask me." ”

“ Cyclic Superposition.—An example may be given in two-dimensional motion in the case of the cyclic superposition (Fig. 48) . We know that the energy contained in a case of cyclic motion around a cylinder or cylindrical filament in space is infinite, for the linear size of the , squares forms a geometrical progression, and any finite number of such squares, however great, may be circumscribed by a circle of finite diameter; that is to say, no finite quantity of energy, however great, will cover the whole field. ”

“ Newton was indebted to Gilbert for his discovery of the so-called law of gravitation. Keely defines gravity as transmittive inter-etheric force under immense etheric vibration, and electricity as a certain form of atomic vibration. When Gilbert, court-physician to Queen Elizabeth, announced his discovery of electricity, he was asked by his compeers of what use it was. No one dreamed then of it as a motive power. He replied, “Of what use is a baby? It may develop into a man or a woman, and, although we cannot make any use of electricity now, the world may in time find out uses for it.” ”

“ If the body L, by a radius drawn to the other body T, describes areas, which compared with the times, are very unequal; and that other body T be either at rest or moves uniformly forward in a right line: the action of the centripetal force tending to that other body T, is either none at all, or it is mix'd and compounded with very powerful actions of other forces: and the whole force compounded of them all, if they are many, is directed to another (immovable or moveable) centre. The same thing obtains, when the other body is moved by any motion whatsoever ”

“ If a cylinder move uniformly forward in the direction of its length, the resistance made thereto is not at all changed by augmenting or diminishing that length; and is therefore the same with the resistance of a circle, described with the same diameter, and moving forward with the same velocity in the direction, of a right line perpendicular to its plane. For the sides are not at all opposed to the motion; and a cylinder becomes a circle when its length is diminished in infinitum. ”

“ We have observed that the same causes which produce pressure on a body restrained, will produce motion if the body be free. Accordingly, if a body be moved by any efficient cause in a curve, it will, by reason of the centrifugal force, fly off, and the moving force with which it will thus retreat from the centre round which it is whirled will be a measure of the centrifugal force. Upon this principle an apparatus called a whirling table has been constructed, for the purpose of exhibiting experimental illustrations of the laws of centrifugal force. ”

“ Therefore a homogeneous and perfect globe will not retain several distinct motions, but will unite all those that are irnpressed on it, and reduce them into one; revolving, as far as in it lies, always with a simple and uniform motion about one single given axis with an inclination perpetually invariable. And the inclination of the axis, or the velocity of the rotation will not be changed by centripetal force. ”

“ There is an immense difference between vibratory physics, in which field Keely is researching, and mechanical physics. The consumption of coal to expand water for the production of steam power, in the operation of engines, cannot be compared to a force which is yielded in sympathetic vibration or by sympathetic flows. In mechanical physics, no matter what the nature of the force may be, its production must necessarily be accompanied by a corresponding expenditure of force in some form or other. ”

“ The negative charge has no desire to flow away through our body to earth, for it is attracted by the charge of the sphere, but the positive charge of the cylinder is pushed away by the action of the tubes of force and will flow as far as it can. Before the cylinder was touched it went to the farthest point, namely, the left-hand end of the cylinder, but the moment we touch it we give it a path to flow much farther, namely, through our body to earth, that is, right away to zero potential. ”

“ The foregoing illustration of the idiosyncrasies of mechanical construction offer a good example of why perpetual motion is not possible, viz., that no mechanism was ever made that would not consume a certain per cent of the power delivered to it, through friction and inertia. Of course, expert knowledge of the physical laws of mechanics allow of the application of force along the lines of least resistance, and it is with this feature that the new improvements in the Comptometer have to do. ”

“ If any part of the kinetic energy depends on the product of an ordinary velocity and the strength of a current, it will probably be most easily observed when the velocity and the current are in the same or in opposite directions. We therefore take a circular coil of a great many windings, and suspend it by a fine vertical wire, so that its windings are horizontal, and the coil is capable of rotating about a vertical axis, either in the same direction as the current in the coil, or in the opposite direction. ”

“ Since , the forces acting on the particle at a point M can be grouped into two forces, a force that acts along the direction and another force that acts in the direction. These forces can themselves be resolved into a tangential force along the trajectory , and a normal force perpendicular to the trajectory . The normal force provides the local centripetal acceleration , where is the local radius of curvature of the trajectory. Thus, our principle of deriving mechanics from a principle of maxima and minima conforms with experience. ”

“ Measurement of Speed of Rotation.
To measure the speed of rotation, a tuning-fork, bearing on one prong a steel mirror, was used. This was kept in vibration by a current of electricity from five "gravity" cells. The fork was so placed that the light from the revolving mirror was reflected to a piece of plane glass, in front of the lens of the eye-piece of the micrometer, inclined at an angle of 45°, and thence to the eye. When fork and revolving mirror are both at rest, an image of the revolving mirror is seen. When the fork vibrates, this image is drawn out into a band of light. ”

“ Again, the kinetic energy of a dynamical system can be expressed in terms of the squares and products of the velocities of its several parts. It can also be expressed by multiplying the velocity of each driving-point by the momentum corresponding to that driving-point, and taking half the sum of the products. Suppose, now, we are dealing with a system consisting of a number of wire circuits in which currents are running, and let us suppose that we may represent the current in each wire as the velocity of a driving-point in our dynamical system. ”