relations of the principal axes of the ellipses they describe may be found. Consequently, if the relative dimensions of each planetary orbit be determined by observation, and the chief axis of any one of them be ascertained, the respective major axes of the others may be found by the above analogy: and indeed they are generally determined in this manner; though other methods are in some cases had recourse to, by which the accuracy of this simple method is clearly proved.

287. When the mutual actions of the sun and primary planets upon each other are taken into consideration, the process is thereby rendered more tedious; but it would then appear that the orbit of each planet is nearly an ellipsis, whose focus is in the con.mon centre of gravity of the sun and all the planets inferiour to it; and this centre of gravity (Art. 219) is always either in the body of the sun, or very near it.

288. The celebrated M. John Dominic Cassini, in a Treatise on the Origin and Progress of Astronomy, proposed a curve for the orbit of a planet, somewhat different from the common ellipsis: in the common ellipsis, the sum of two lines drawn from the foci to any point of the curve is equal to the transverse diameter; but in the Cassinean ellipsis, the product of two lines from the foci to the curve is equal to a constant or given number. In this figure, if the less axis exceeds the distance of the foci, the curve is every-where concave towards the centre: if, while the principal axis remains the same, the distance of the foci is lessened, the minor axis will be increased; and when the foci meet in the centre, the ellipsis will become a circle: but if, on the contrary, the distance of the foci be increased, the less axis will be lessened, and the curve will at length have a point of contrary flexure, and will, at the ends of the minor axis, be convex towards the

centre; and when the distance of the foci is so far increased, as to be in the same proportion to the major axis as the side of a square to the diagonal, that is, as I to√2, the less axis will become nothing, and the curve extend to the centre on each side: if the distance of the foci be greater in proportion to the major axis, than in the above ratio, the figure turns into two conjugate ones at a distance from each other; and as the distance of the foci is farther in. creased, the two conjugate figures will at last become merely points. It must be evident, after thus tracing a few of the properties of this curve, that it cannot possibly be the orbit of a planet: for it is certain, that in all those cases where it passes into two conju gate figures, it deviates from what is essential to the nature of an orbit, namely, continuity; and in all those cases where it is, at the end of the minor axis, convex towards the centre, the planet would need a centrifugal force to describe such parts of its orbit; that is, it would require at equal distances from the sun sometimes a centrifugal, and sometimes a cen tripetal force to retain it in its orbit, which is totally incompatible with all the laws of nature. It may fairly be concluded, that when all the species of a figure beyond a certain limit are unfit for discharging any office of nature, the remaining species on the other side of the limit should be rejected also: and when, in addition to this, it is considered that the ce lestial observations are not consistent with this curve, it can by no means be admitted into astronomy*.

* A very beautiful elementary treatise of Physical Astronomy may be seen in the learned Dr. Stewart's Tracts, Physical and Mathema tical. The doctrine of centripetal forces is there laid down in a series of twenty-eight propositions, demonstrated (if the quadrature of eurves be admitted) with the utmost rigour, and requiring no previous knowledge of any part of the mathematics, besides the ele ments of plane geometry, and of conic sections. But the most comprehensive, elaborate, and profound work, on Physical Astronomy, in all its branches, is that published by M. La Place, under the title of Traité de Mecánique Celeste.

CHAPTER XI.

On the Magnitude and Situation of the Earth's Orbit.

ART. 289. IT is related of the ancient philoso pher Archimedes, that when Hiero was admiring the skill and judgment displayed in his machines, he replied to the monarch, "These are nothing; but

give me another place to stand upon, and I will "move the earth :" with much more truth it may be said by a modern astronomer, Give me a proper basis for my geometrical operations, and I will scale the heavens. This basis Nature has furnished us with, in the elliptical orbit of the earth; the dimensions of which we, therefore, now proceed to determine, as a necessary preliminary in asceṛtaining the magnitudes and distances of the planets, and the dimensions of their orbits. But we must first premise a few definitions of terms used in this part of astronomy.

290. The path in which the sun appears to move, is in the same plane as that in which the earth really moves; therefore (Art. 40.) the earth's orbit is in the plane of the ecliptic: the points in which the orbits of the planets cut the orbit of the earth, or the ecliptic, are called the nodes; as are also the points where the orbits of the secondary planets, or satellites, intersect their respective primaries. That node is called ascending where the planet passes from the south to

the north side of the ecliptic, and is marked thus : the other node is called descending, and is denoted by this mark 8. An imaginary line drawn from one node to the other is called the line of the nodes.

291. That angle which the plane of a planet's orbit makes with the plane of the earth's orbit, is called the inclination of that planet's orbit.

292. if a perpendicular be let fall from a planet to the ecliptic, the angle at the sun between two lines, one drawn from it to that point where the perpendicular falls, and another to the earth, is called the angle of commutation.

293. The curtate distance of a planet from the sun or the earth, is the distance from the sun or the earth to the planet, reduced to the ecliptic; or the line between the sun or the earth, and that point where the perpendicular let fall from the planet meets the ecliptic..

294. The mean distance of a planet is the line drawn from that focus of its elliptical orbit in which the sun is placed, to either end of the conjugate axis of its orbit: it is manifest from the nature of the ellipsis that this line is equal to half the transverse axis. Some authors call a mean proportional between the two axes, the mean distance.

295. The distance from the centre of the orbit I to either focus is called its excentricity.

296. The apsides, or apses, are the two points in the orbit of a planet, where it is at its greatest and least distances from the sun. The point at the greatest distance is called the higher apsis, that at the least distance, the lower apsis. The higher apsis is also called the aphelion; the lower, the perihelion. The diameter which joins these two points is called the line of the apsides, and is supposed to pass though the centre of the sun. But it is not true that the apsides are always in the same straight line passing through the sun; for they are sometimes out of a

right line, making an angle greater or less than 180°; and the difference from 180° is called the motion of the line of the apsides. When the angle is less than 180°, the motion is said to be in antecedentia, or contrary to the order of the signs; when it is greater than 180°, the motion is in consequentia, or according to the order of the signs.

297. The sun and moon are in perigee, when at their nearest distance from the earth; and in apogee, when their distance from the earth is greatest.

298. The argument of latitude is the angle at the sun between the planet and its ascending node, estimated in the planet's orbit.

299. The true anomaly, or equated anomaly, as it is sometimes called, is the angle at the sun which is formed by the radius vector, or line drawn from the sun to the planet, and the line drawn from the sun to the aphelion of the planet: the mean anomaly is the angular distance of a planet from its aphelion (taken at the same time with the true anomaly), supposing it to move uniformly with its mean angular velocity. The difference between the true and mean anomaly is called the equation of the centre, or the prosthapheresis.

300. If a circle be supposed drawn on the line of the apsides as a diameter, and through the place of the planet a perpendicular to the line of the apsides be drawn till it meet the circumference of the circle; then the angle formed by two lines, one drawn from the centre of the planet's orbit to the aphelion, and the other to the point where the perpendicular through the planet's place intersects the circumference of the circle, is called the excentric anomaly, or the anomaly of the centre,

301. PROB. I. To find the time in which the earth performs a revolution about the sun.

A solution to this problem may be found in Chap. III. where we determined the length of the sidereal