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scription of areas at all. Or it may describe it as we may see done, if we suspend a ball by a very long string, and, drawing it a little aside from the perpendicular, throw it round with a gentle impulse. In this case the acting force is directed to the center of the ellipse, about which areas are described equably, and to which a force proportional to the distance (the decomposed result of terrestrial gravity) perpetually urges it. This is at once a very easy experiment, and a very instructive one, and we shall again refer to it. In the case before us, of an ellipse described by the action of a force directed to the focus, the steps of the investigation of the law of force are these: 1st, The law of the areas determines the actual velocity of the revolving body at every point, or the space really run over by it in a given minute portion of time; 2dly, The law of curvature of the ellipse determines the linear amount of deflection from the tangent in the direction of the focus, which corresponds to that space so run over; 3dly, and lastly, The laws of accelerated motion declare that the intensity of the acting force causing such deflection in its own direction, is measured by or proportional to the amount of that deflection, and may therefore be calculated in any particular position, or generally expressed by geometrical or algebraic symbols, as a law independent of particular positions, when that deflection is so calculated or expressed. We have here the spirit of the process by which Newton has resolved this interesting problem. For its geometrical detail, we must refer to the 3d section of his Principia. We know of no artificial mode of imitating this species of elliptic motion; though a rude approximation to it-enough, however, to give a conception of the alternate approach and recess of the revolving body to and from the focus, and the variation of its velocity-may be had by suspending a small steel bead to a fine and very long silk fibre, and setting it to revolve in a small orbit round the pole of a powerful cylindrical magnet, held upright, and vertically under the point of suspension.

(421.) The third law of Kepler, which connects the distances and periods of the planets by a general rule, bears with it, as its theoretical interpretation, this important consequence, viz. that it is one and the same force, modified only by distance from the sun, which retains all the planets in their orbits about it. That the attraction of the sun (if such it be) is exerted upon all the bodies of our system indifferently, without regard to the peculiar materials of which they may consist, in the exact proportion of their inertiæ, or quantities of matter; that it is not, therefore, of the nature of the elective attractions of chemistry, or of magnetic action, which is powerless on other substances than iron and some one or two more, but is of a more universal character, and extends equally to all the material constituents of our system, and (as we shall hereafter see abundant reason to admit) to those of other systems than our own. This law, important and general as it is, results, as the simplest of corollaries, from the relations established by Newton in the section of the Principia referred to (Prop. xv.), from which proposition it results, that if the earth were taken from its actual orbit, and launched anew in space at the place, in the direction, and with the velocity of any of the other planets, it would describe the very same orbit, and in the same period, which that planet actually does, a very minute correction of the period only excepted, arising from the difference between the mass of the earth and that of the planet. Small as the planets are compared to the sun, some of them are not, as the earth is, mere atoms in the comparison. The strict wording of Kepler's law, as Newton has proved in his fifty-ninth proposition, is applicable only to the case of planets whose proportion to the central body is absolutely inappreciable. When this is not the case, the periodic time is shortened in the proportion of the square root of the number expressing the sun's mass or inertia, to that of the sum of the numbers expressing the masses of the sun and planet; and in general, whatever be the masses of two bodies

revolving round each other under the influence of the Newtonian law of gravity, the square of their periodie time will be expressed by a fraction whose numerator is the cube of their mean distance, i. e. the greater semi-axis of their elliptic orbit, and whose denominator is the sum of their masses. When one of the masses is incomparably greater than the other, this resolves itself into Kepler's law; but when this is not the case, the proposition thus generalized stands in lieu of that law. In the system of the sun and planets, however, the numerical correction thus introduced into the results of Kepler's law is too small to be of any importance, the mass of the largest of the planets (Jupiter) being much less than a thousandth part of that of the sun. We shall presently, however, perceive all the importance of this generalization, when we come to speak of the satellites.

(422.) It will first, however, be proper to explain by what process of calculation the expression of a planet's elliptic orbit by its elements can be compared with observation, and how we can satisfy ourselves that the numerical data contained in a table of such elements for the whole system does really exhibit a true picture of it, and afford the means of determining its state at every instant of time, by the mere application of Kepler's laws. Now, for each planet, it is necessary for this purpose to know, 1st, the magnitude and form of its ellipse; 2dly, the situation of this ellipse in space, with respect to the ecliptic, and to a fixed line drawn therein; 3dly, the local situation of the planet in its ellipse at some known epoch, and its periodic time or mean angular velocity, or, as it is called, its mean motion.

(423.) The magnitude and form of an ellipse are determined by its greatest length and least breadth, or its two principal axes; but for astronomical uses it is preferable to use the semi-axis major (or half the greatest length), and the excentricity or distance of the focus from the center, which last is usually estimated in parts

of the former. Thus, an ellipse, whose length is 10 and breadth 8 parts of any scale, has for its major semi-axis 5, and for its excentricity 3 such parts; but when estimated in parts of the semi-axis, regarded as a unit, the excentricity is expressed by the fraction .

(424.) The ecliptic is the plane to which an inhabitant of the earth most naturally refers the rest of the solar system, as a sort of ground-plane; and the axis of its orbit might be taken for a line of departure in that plane or origin of angular reckoning. Were the axis fixed, this would be the best possible origin of longitudes; but as it has a motion (though an excessively slow one), there is, in fact, no advantage in reckoning from the axis more than from the line of the equinoxes, and astronomers therefore prefer the latter, taking account of its variation by the effect of precession, and restoring it, by calculation at every instant, to a fixed position. Now, to determine the situation of the ellipse described by a planet with respect to this plane, three elements require to be known: - 1st, the inclination of the plane of the planet's orbit to the plane of the ecliptic; 2dly, the line in which these two planes intersect each other, which of necessity passes through the sun, and whose position with respect to the line of the equinoxes is therefore given by stating its longitude. This line is called the line of the nodes. When the planet is in this line, in the act of passing from the south to the north side of the ecliptic, it is in its ascending node, and its longitude at that moment is the element called the longitude of the node. These two data determine the situation of the plane of the orbit; and there only remains, for the complete determination of the situation of the planet's ellipse, to know how it is placed in that plane, which (since its focus is necessarily in the sun) is ascertained by stating the longitude of its perihelion, or the place which the extremity of the axis nearest the sun occupies, when orthographically projected on the ecliptic.

(425.) The dimensions and situation of the planet's

orbit thus determined, it only remains, for a complete acquaintance with its history, to determine the circumstances of its motion in the orbit so precisely fixed. Now, for this purpose, all that is needed is to know the moment of time when it is either at the perihelion, or at any other precisely determined point of its orbit, and its whole period; for these being known, the law of the areas determines the place at every other instant. This moment is called (when the perihelion is the point chosen) the perihelion passage, or, when some point of the orbit is fixed upon, without special reference to the perihelion, the epoch.

(426.) Thus, then, we have seven particulars or elements, which must be numerically stated, before we can reduce to calculation the state of the system at any given moment. But, these known, it is easy to ascertain the apparent positions of each planet, as it would be seen from the sun, or is seen from the earth at any time. The former is called the heliocentric, the latter the geocentric, place of the planet.

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(427.) To commence with the heliocentric places. Let S represent the sun; A PN the orbit of the planet, being an ellipse, having the sun S in its focus, and A for its peri

helion; and let pa NY represent the projection of the orbit on the plane of the ecliptic, intersecting the line of equinoxes S in y, which, therefore, is the origin of longitudes. Then will SN be the line of nodes; and if we suppose B to lie on the south, and A on the north side of the ecliptic, and the direction of the planet's motion to be from B to A, N will be the ascending node, and the angle SN the longitude of the node. In like manner, if P be the place of the planet at any time, and if it and the perihelion A be projected on the ecliptic, upon the points pa, the angles Sp, v S a, will be the respective heliocentric longitudes of the planet and of

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