665. There is no object within the scope of astronomical observation which affords greater variety of interesting investigation to the inhabitant of the earth, than the various motions of the moon : from these we ascertain the form of the earth, the vicissitudes of the tides, the distance of the sun, and consequently the magnitude of the solar system. These motions which are so obvious, served as a measure of time to all nations, until the advancement of science taught them the advantages of solar time; to these motions the navigator owes that precision of knowledge which guides him with well-grounded confidence through the deep.

Phases of the Moon.

666. The phases of the moon depend upon her synodic motion, that is to say, on the excess of her motion above that of the sun. The moon moves round the earth from west to east; in conjunction she is between the sun and the earth ; but as her motion is more rapid than that of the sun, she soon separates from him, and is first seen in the evening like a faint crescent, which increases with her distance till in quadrature, or 90° from him, when half of her disc is enlightened : as her longation increases, her enlightened disc augments till she is in opposition, when it is full moon, the earth being between her and the sun.

169. If a solid body receives an impulse in a direction passing through its centre of gravity, all its parts will move with an equal velocity; but if the direction of the impulse passes on one side of that centre, the different parts of the body will have unequal velocities, and from this inequality results a motion of rotation in the body round its centre of gravity, at the same time that the centre is moved forward, or translated with the same velocity it would have taken, had the impulse passed through it. Thus the double motions of rotation and translation are produced by one impulse.

170. If a body rotates about its centre of gravity, or about an axis, and is at the same time carried forward in space ; and if an equal and contrary impulse be given to the centre of gravity, so as to stop its progressive motion, the rotation will go on as before it received the impulse.

171. If a body revolves about a fixed axis, each of its particles will describe a circle, whose plane is perpendicular to that axis, and its radius is the distance of the particle from the axis. It is evident, that every point of the solid will describe an arc of the same number of degrees in the same time ; hence, if the velocity of each particle be divided by its radius or distance from the axis, the quotient will be the same for every particle of the body. This is called the angular velocity of the solid.

The infinite varieties of motion in the heavens, and on the earth, obey a few laws, so universal in their application, that they regulate the curve traced by an atom which seems to be the sport of the winds, with as much certainty as the orbits of the planets. These laws, on which the order of nature depends, remained unknown till the sixteenth century, when Galileo, by investigating the circumstances of falling bodies, laid the foundation of the science of mechanics, which Newton, by the discovery of gravitation, afterwards extended from the earth to the farthest limits of our system.

This original property of matter, by means of which we ascertain the past and anticipate the future, is the link which connects our planet with remote worlds, and enables us to determine distances, and estimate magnitudes, that might seem to be placed beyond the reach of human faculties. To discern and deduce from ordinary and apparently trivial occurrences the universal laws of nature, as Galileo and Newton have done, is a mark of the highest intellectual power.

Simple as the law of gravitation is, its application to the motions of the bodies of the solar system is a problem of great difficulty, but so important and interesting, that the solution of it has engaged the attention and exercised the talents of the most distinguished mathematicians ; among whom La Place holds a distinguished place by the brilliancy of his discoveries, as well as from having been the first to trace the influence of this property of matter from the elliptical motions of the planets, to its most remote effects on their mutual perturbations.

309. The three laws of Kepler furnish the data from which the principle of gravitation is established, namely :—

1. i. That the radii vectores of the planets and comets describe areas proportional to the time.

2. ii. That the orbits of the planets and comets are conic sections, having the sun in one of their foci.

3. iii. That the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun.

310. It has been shown, that if the law of the force which acts on a moving body be known, the curve in which it moves may be found; or, if the curve in which the body moves be given, the law of the force may be ascertained. In the general equation of the motion of a body in article 144, both the force and the path of the body are indeterminate; therefore in applying that equation to the motion of the planets and comets, it is necessary to know the orbits in which they move, in order to ascertain the nature of the force that acts on them.

311. In the general equation of the motion of a body, the forces acting on it are resolved into three component forces, in the direction of three rectangular axes ; but as the paths of the planets, satellites, and comets, are proved by the observations of Kepler to be conic sections, they always move in the same plane: therefore the component force in the direction perpendicular to that plane is zero, and the other two component forces are in the plane of the orbit.

1. The activity of matter seems to be a law of the universe, as we know of no particle that is at rest. Were a body absolutely at rest, we could not prove it to be so, because there are no fixed points to which it could be referred; consequently, if only one particle of matter were in existence, it would be impossible to ascertain whether it were at rest or in motion. Thus, being totally ignorant of absolute motion, relative motion alone forms the subject of investigation: a body is, therefore, said to be in motion, when it changes its position with regard to other bodies which are assumed to be at rest.

2. The cause of motion is unknown, force being only a name given to a certain set of phenomena preceding the motion of a body, known by the experience of its effects alone. Even after experience, we cannot prove that the same consequents will invariably follow certain antecedents; we only believe that they will, and experience tends to confirm this belief.

3. No idea of force can be formed independent of matter; all the forces of which we have any experience are exerted by matter; as gravity, muscular force, electricity, chemical attractions and repulsions, &c. &c, in all which cases, one portion of matter acts upon another.

Definitions, &c.

225. A fluid is a mass of particles which yield to the slightest pressure, and transmit that pressure in every direction.

226. Mobility of the particles constitutes the difference between fluids and solids.

227. There are, indeed, fluids in nature whose particles adhere more or less to each other, called viscous fluids; but those only whose particles do not adhere in any degree, but possess perfect mobility, are the subject of this investigation.

228. Strictly speaking, all fluids are compressible, for even liquids under very great pressure change their volume; but as the compression is insensible in ordinary circumstances, fluids of perfect mobility, are divided into compressible or elastic fluids, and incompressible.

229. The elastic and compressible fluids are atmospheric air, the gases, and steam. When compressed, these fluids change both form and volume, and regain their primitive state as soon as the pressure is removed. Some of the gases are found to differ from atmospheric air in losing their elastic form, and becoming liquid when compressed to a certain degree, as lately proved by Mr. Faraday, and steam is reduced to water when its temperature is diminished; but atmospheric air, and others of the gases, always retain their gaseous form, whatever the degree of pressure may be.

230. It is impossible to ascertain the forms of the particles of fluids, but as all of them, considered in mass, afford the same phenomena, it can have no influence on the laws of their motions.