NOTE 47, p. 9. Longitude. The vernal equinox, op, fig. 11, is the zero point in the heavens whence celestial longitudes, or the angular motions of the celestial bodies, are estimated from west to east, the direction in which they all revolve. The vernal equinox is generally called the first point of Aries, though these two points have not coincided since the early ages of astronomy, about 2233 years ago, on account of a motion in the equinoctial points, to be explained hereafter. If S op, fig. 10, be the line of the equinoxes, and the vernal equinox, the true longitude of a planet p is the angle op Sp, and its mean longitude is the angle ∞ Cm, the sun being in S. Celestial longitude is the angular distance of a heavenly body from the vernal equinox; whereas terrestrial longitude is the angular distance of a place on the surface of the earth from a meridian arbitrarily chosen, as that of Greenwich. NOTE 48, pp. 9, 58. Equation of the centre. The difference between Cm and Sp, fig. 10; that is, the difference between the true and mean longitudes of a planet or satellite. The true and mean places only coincide in the points P and A; in every other point of the orbit, the true place is either before or behind the mean place. In moving from A through the arc A Q P, the true place p is behind the mean place m ; and through the arc PDA the true place is before the mean place. At its maximum, the equation of the centre measures C S, the excentricity of the orbit, since it is the difference between the motion of a body in an ellipse and in a circle whose diameter A P is the major axis of the ellipse. NOTE 49, p. 9. Apsides. The points P and A, fig. 10, at the extremities of the major axis of an orbit. P is commonly called the perihelion, a Greek term signifying round the sun; and the point A is called the aphelion, a Greek term signifying at a distance from the sun. NOTE 50, p. 9. Ninety degrees. A circle is divided into 360 equal parts, or degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. It is usual to write these quantities thus, 15° 16' 10", which means fifteen degrees, sixteen minutes, and ten seconds. It is clear that an arc m n, fig. 4, measures the angle m Cn; hence we may say, an arc of so many degrees, or an angle of so many degrees; for, if there be ten degrees in the angle m C n, there will be ten degrees in the arc m n. It is evident that there are 90° in a right angle, m C d, or quadrant, since it is the fourth part of 360°. NOTE 51, p. 9. Quadratures. A celestial body is said to be in quadrature when it is 90 degrees distant from the sun. For example, in fig. 14, if d be the sun, S the earth, and p the moon, then the moon is said to be in quadrature when she is in either of the points Q or D, because the angles QSd and D S d, which measure her apparent distance from the sun, are right angles. NOTE 52, p. 9. Excentricity. Deviation from circular form. In fig. 6, CS is the excentricity of the orbit PQ AD. The less CS, the more nearly does the orbit or ellipse approach the circular form; and, when C S is zero, the ellipse becomes a circle. NOTE 53, p. 9. Inclination of an orbit. Let S, fig. 12, be the centre of the PN An the orbit of a planet moving from west to east in the sun, direction Np. Let E N men be the shadow or projection of the orbit on the plane of the ecliptic, then N Sn is the intersection of these two planes, N m' m NOTE 54, p. 9. Latitude of a planet. The angle p S m, fig. 12, or the height of the planet pabove the ecliptic E N m. In this case the latitude is north. Thus, celestial latitude is the angular distance of a celestial body from the plane of the ecliptic, whereas terrestrial latitude is the angular distance of a place on the surface of the earth from the equator. NOTE 55, p. 9. Nodes. The two points N and n, fig. 12, in which the orbit N An P of a planet or comet intersects the plane of the ecliptic e N En. The part N An of the orbit lies above the plane of the ecliptic, and the part n P N below it. The ascending node N is the point through which the body passes in rising above the plane of the ecliptic, and the descending node n is the point in which the body sinks below it. The nodes of a satellite's orbit are the points in which it intersects the plane of the orbit of the planet. Sp in fig. 12. If op be the of the planet p, m Sp is its When these three quantities NOTE 56, p. 10. Distance from the sun. vernal equinox, then Sp is the longitude latitude, and Sp its distance from the sun. are known, the place of the planet p is determined in space. NOTE 57, pp. 10, 59. Elements of an orbit. Of these there are seven. Let PN An, fig. 12, be the elliptical orbit of a planet, C its centre, S the sun in one of the foci, the point of Aries, and E N e n the plane of the ecliptic. The elements are-the major axis A P; the excentricity CS; the periodic time, that is, the time of a complete revolution of the body in its orbit; and the fourth is the longitude of the body at any given instant— for example, that at which it passes through the perihelion P, the point of its orbit nearest to the sun. That instant is assumed as the origin of time, whence all preceding and succeeding periods are estimated. quantities are sufficient to determine the form of the orbit, and the motion of the body in it. Three other elements are requisite for determining the position of the orbit in space. These are, the angle op SP, the longitude of the perihelion; the angle A Ne, which is the inclination of the orbit to the plane of the ecliptic; and, lastly, the angle op S N, the longitude of N the ascending node. These four NOTE 58, p. 10. Whose planes, &c. The planes of the orbits, as PNAn, fig. 12, in which the planets move, are inclined or make small angles e N A with the plane of the ecliptic E Nen, and cut it in straight lines, NS n passing through S, the centre of the sun. NOTE 59, p. 11. Momentum. Force measured by the weight of a body and its speed, or simple velocity, conjointly. The primitive momentum of the planets is, therefore, the quantity of motion which was impressed upon them when they were first thrown into space. NOTE 60, p. 11. Unstable equilibrium. A body is said to be in equilibrium when it is so balanced as to remain at rest. But there are two kinds of equilibrium, stable and unstable. If a body balanced in stable equilibrium be slightly disturbed, it will endeavour to return to rest by a number of movements to and fro, which will continually decrease till they cease altogether, and then the body will be restored to its original state of repose. But, if the equilibrium be unstable, these movements to and fro, or oscillations, will become greater and greater till the equilibrium is destroyed. NOTE 61, p. 14. Retrograde. Going backwards, as from east to west, contrary to the motion of the planets. NOTE 62, p. 14. Parallel directions. Such as never meet, though prolonged ever so far. NOTE 63, pp. 14, 16. The whole force, &c. Let S, fig. 13, be the sun, Nm n the plane of the ecliptic, p the disturbed planet moving in its orbit np N, and d the disturbing planet. Now, d attracts the sun and the planet p with different intensities in the directions d S, dp: the difference only of these forces disturbs the motion of p; it is therefore called the disturbing force. But this whole disturbing force may be regarded as equivalent to three forces, acting in the directions p S, p T, and p m. The force acting in the radius vector p S, joining the centres of the sun and planet, is called the radial force. It sometimes draws the disturbed planet p from and sometimes brings it nearer to him. The force which acts in the sun, direction Np. Let E N men be the shadow or projection of the orbit on the plane of the ecliptic, then N Sʼn is the intersection of these two planes, N SC m' m for the orbit rises above the plane of the ecliptic towards Np, and sinks below it at N P. The angle p Nm, which these A two planes make with one another, is the inclination of the orbit PNPA to the plane of the ecliptic. NOTE 54, p. 9. Latitude of a planet. The angle p S m, fig. 12, or the height of the planet pabove the ecliptic E N m. In this case the latitude is north. Thus, celestial latitude is the angular distance of a celestial body from the plane of the ecliptic, whereas terrestrial latitude is the angular distance of a place on the surface of the earth from the equator. NOTE 55, p. 9. Nodes. The two points N and n, fig. 12, in which the orbit N An P of a planet or comet intersects the plane of the ecliptic e N En. The part N An of the orbit lies above the plane of the ecliptic, and the part n P N below it. The ascending node N is the point through which the body passes in rising above the plane of the ecliptic, and the descending node n is the point in which the body sinks below it. The nodes of a satellite's orbit are the points in which it intersects the plane of the orbit of the planet. Sp in fig. 12. If op be the of the planet p, m Sp is its When these three quantities NOTE 56, p. 10. Distance from the sun. vernal equinox, then Sp is the longitude latitude, and Sp its distance from the sun. are known, the place of the planet p is determined in space. NOTE 57, pp. 10, 59. Elements of an orbit. Of these there are seven. Let PN An, fig. 12, be the elliptical orbit of a planet, C its centre, S the sun in one of the foci, the point of Aries, and E N e n the plane of the ecliptic. The elements are-the major axis A P; the excentricity CS; the periodic time, that is, the time of a complete revolution of the body in its orbit; and the fourth is the longitude of the body at any given instant— for example, that at which it passes through the perihelion P, the point of its orbit nearest to the sun. That instant is assumed as the origin of time, whence all preceding and succeeding periods are estimated. These four quantities are sufficient to determine the form of the orbit, and the motion of the body in it. Three other elements are requisite for determining the position of the orbit in space. These are, the angle op SP, the longitude of the perihelion; the angle A Ne, which is the inclination of the orbit to the plane of the ecliptic; and, lastly, the angle op S N, the longitude of N the ascending node. NOTE 58, p. 10. Whose planes, &c. The planes of the orbits, as PNAn, fig. 12, in which the planets move, are inclined or make small angles e N A with the plane of the ecliptic E Nen, and cut it in straight lines, N S n passing through S, the centre of the sun. NOTE 59, p. 11. Momentum. Force measured by the weight of a body and its speed, or simple velocity, conjointly. The primitive momentum of the planets is, therefore, the quantity of motion which was impressed upon them when they were first thrown into space. NOTE 60, p. 11. Unstable equilibrium. A body is said to be in equilibrium when it is so balanced as to remain at rest. But there are two kinds of equilibrium, stable and unstable. If a body balanced in stable equilibrium be slightly disturbed, it will endeavour to return to rest by a number of movements to and fro, which will continually decrease till they cease altogether, and then the body will be restored to its original state of repose. But, if the equilibrium be unstable, these movements to and fro, or oscillations, will become greater and greater till the equilibrium is destroyed. NOTE 61, p. 14. Retrograde. Going backwards, as from east to west, contrary to the motion of the planets. NOTE 62, p. 14. Parallel directions. Such as never meet, though prolonged ever so far. NOTE 63, pp. 14, 16. The whole force, &c. Let S, fig. 13, be the sun, Nm n the plane of the ecliptic, p the disturbed planet moving in its orbit np N, and d the disturbing planet. Now, d attracts the sun and the planet p with different intensities in the directions d S, dp: the difference only of these forces disturbs the motion of p; it is therefore called the disturbing force. But this whole disturbing force may be regarded as equivalent to three forces, acting in the directions p S, p T, and p m. The force acting in the radius vector p S, joining the centres of the sun and planet, is called the radial force. It sometimes draws the disturbed planet p from the sun, and sometimes brings it nearer to him. The force which acts in |