FIG.

3.

drawn to the earth's center. Thus, the small circle Ax By, in the heavens, is referred to LvMw on the earth. Hence, if AxBy be the tropic or the polar circle in the heavens, LuMw will be the tropic or polar circle on the earth. These circles therefore retain the same relative situations, that is, the former is as far from the pole in the heavens, as the latter is from the pole of the earth. The planes of these corresponding small circles do not coincide; but when they become great circles, then the planes become coincident.

46. A body is in Conjunction with the sun, when it has the same longitude; in Opposition, when the difference of their longitudes is 180°; and in Quadratures, when the difference of their longitudes is 90°. The conjunction is marked thus, the opposition thus 8, and quadratures thus.

47. Syzygy is either conjunction or opposition.

48. The Elongation of a body is its angular distance from the sun when seen from the earth.

49. The diurnal parallax is the difference between the apparent places of the bodies in our system when referred to the fixed stars, if seen from the center and surface of the earth. The annual parallax is the difference between the apparent places of a body in the heavens, when seen from the opposite points of the earth's orbit.

50. The Argument is a term used to denote any quantity by which another required quantity may be found. For example, the argument of that part of the equation of time which arises from the unequal angular motion of the earth in its orbit about the sun, is the sun's anomaly, because that part of the equation depends entirely upon the anomaly; and the latter being given, the former is found from it. The argument of a star's latitude is its distance from its node, because upon this the latitude depends.

51. The Nodes are the points where the orbits of the primary planets cut the ecliptic, and where the orbits of the secondaries cut the orbits of their primaries. That node is called ascending where the planet passes from the south to the north side of the ecliptic, and the other is called the descending node. The ascending node is marked thus 8, and the descending node thus 8. The line which joins the nodes is called the line of the nodes.

52. If a perpendicular be drawn 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.

53. The angle of Position is the angle at an heavenly body formed by two great circles, one passing through the pole of the equator and the other through the pole of the ecliptic.

54. Apparent noon is the time when the sun comes to the meridian.

55. True or mean noon is 12 o'clock, by a clock adjusted to go 24 hours in a mean solar day.

56. The Equation of Time is the interval between true and apparent time. 57. A star is said to rise or set cosmically, when it rises or sets at sun rising; and when it rises or sets at sun setting, it is said to rise or set achronically.

58. A star rises heliacally, when, after having been so near to the sun as not to be visible, it emerges out of the sun's rays and just appears in the morning; and it sets heliacally, when the sun approaches so near to it, that it is about to immerge into the sun's rays and become invisible in the evening.

59. Curtate distance of a planet from the sun or earth, is the distance of the sun or earth from that point of the ecliptic where a perpendicular to it passes through the planet.

60. Aphelion is that point in the orbit of a planet which is furthest from the

sun.

61. Perihelion is that point in the orbit of a planet which is nearest the sun. 62. Apogee is that point of the earth's orbit which is furthest from the sun, or that point of the moon's orbit which is furthest from the earth.

63. Perigee is that point of the earth's orbit which is nearest the sun, or that point of the moon's orbit which is nearest the earth.

The terms aphelion and perihelion are also applied to the earth's orbit. 64. Apsis of an orbit, is either its aphelion or perihelion, apogee or perigee; and the line which joins the apsides is called the line of the apsides.

65. Anomaly (true) of a planet is its angular distance at any time from its aphelion, or apogee-(mean) is its angular distance from the same point at the same time if it had moved uniformly with its mean angular velocity.

66. Equation of the center is the difference between the true and mean anomaly; this is sometimes called the prosthapheresis.

67. Nonagesimal degree of the ecliptic is that point which is highest above the horizon..

68. The mean place of a body is the place where a body, not moving with an uniformly angular velocity about the central body, would have been, if the angular velocity had been uniform. The true place of a body is the place where the body actually is at any time.

69. Equations are corrections which are applied to the mean place of a body in order to get its true place.

70. A Digit is a twelfth part of the diameter of the sun or moon.

71. Those bodies which revolve about the sun in orbits very nearly circular, are called Planets, or primary planets for the sake of distinction; and those bodies which revolve about the primary planets are called secondary planets, or satellites.

72. Those bodies which revolve about the sun in very elliptic orbits are called Comets. The sun, planets and comets, comprehend all the bodies in what is called the Solar system.

73. All the other heavenly bodies are called fixed stars, or simply Stars. 74. Constellation is a parcel of stars contained within some assumed figure, as a ram, a dragon, an Hercules, &c. the whole heaven is thus divided into constellations. A division of this kind is necessary, in order to direct a person to any part of the heavens which we want to point out.

Characters used for the Sun, Moon and Planets.

CHAP. II.

ON THE DOCTRINE OF THE SPHERE.

Art. 75. A SPECTATOR upon the earth's surface conceives himself to be placed in the center of a concave sphere in which all the heavenly bodies are situated; and by constantly observing them, he perceives that by far the greater number never change their relative situations, each rising and setting at the same interval of time and at the same points of the horizon, and are therefore called fixed stars; but that a few others, called planets, together with the sun anl moon, are constantly changing their situations, each continually rising and setting at different points of the horizon and at different intervals of time. Now the determination of the times of the rising and setting of all the heavenly bodies; the finding of their position at any given time in respect to the horizon or meridian, or the time from their position; the causes of the different lengths of days and nights, and the changes of seasons; the principles of dialling, and the like, constitute the doctrine of the sphere. And as the apparent diurnal motion of all the bodies have no reference to any particular system or disposition of the planets, but may be solved, either by supposing them actually to perform those motions every day, or by supposing the earth to revolve about an axis, we will suppose this latter to be the case, the truth of which will afterwards appear.

76. Let pep'q represent the earth, O its center, b the place of a spectator, HZRN the sphere of the fixed stars; and although the fixed stars do not lie in the concave surface of a sphere of which the center of the earth is the center, yet, on account of the immense distance even of the nearest of them, their relative situations from the motion of the earth, and consequently the place of a body in our system referred to them, will not be affected by this supposition. The plane abc touching the earth in the place of the spectator is called (21) the sensible horizon, as it divides the visible from the invisible part of the heavens; and a plane HOR parallel to abc, passing through the center of the earth, is called the rational horizon; but in respect to the sphere of the fixed stars, these may be considered as coinciding, the angle which the arc Ha subtends at the earth becoming then insensible from the immense distance of the fixed stars. Now if we suppose the earth to revolve daily about an axis, all the heavenly bodies must successively rise and set in that time, and appear to describe circles whose planes are perpendicular to the earth's axis, and consequently parallel to each other; thus all the stars would appear to revolve daily about the earth's axis, as if they were placed in the concave surface of a sphere having the earth in

VOL. I.

FIG.

4.

FIG.

5.

the center. Let therefore pp' be that diameter of the earth about which it must revolve in order to give the apparent diurnal motion to the heavenly bodies, then P, p', are called its poles; and if pp' be produced both ways to P, P' in the heavens, these points are called (18) the poles in the heaven, and the star nearest to each of these is called the pole star. Now, although the earth, from its motion in its orbit, continually changes its place, yet as the axis always continues parallel to itself, the points P, P' will not, from the immense distance of the fixed stars, be sensibly altered; we may therefore suppose these to be fixed points*. Produce Ob both ways to Z and N, and Z is the zenith and N the nadir (23). Draw the great circle PZHNR, and it will be the celestial meridian (25), the plane of which coincides with the terrestrial meridian pbp' passing through the place of the spectator. Let eq represent a great circle of the earth perpendicular to its axis pp', and it will be the equator (15), and if the plane of this circle be extended to the heavens it marks out a great circle EQ called the celestial equator (18). Hence, for the same reason that we may consider the points P, P' as fixed, we may consider the circle EQ as fixed. Now as the latitude of a place on the earth's surface is measured by the degrees of the arc be (16), it may be measured by the arc ZE; hence as the equator, zenith, and poles in the heaven, correspond to the equator, place of the spectator, and poles of the earth, we may leave out the consideration of the earth in our further enquiries upon this subject, and only consider the equator, zenith and poles in the heavens, and HR the horizon to the spectator.

77. Let therefore figure the fifth represent the position of the heavens to Ż the zenith of a spectator in north latitude, EQ the equator, P, P' its poles, HOR the rational horizon, PZHP'R the meridian, and draw the great circle ZON perpendicular to ZPRH, and it is the prime vertical (28); R will be the north point of the horizon and H the south (26), and O will be the east or west points (28) according as this figure represents the eastern or western hemisphere. Draw also a great circle POP' perpendicular to the meridian. Now as each circle HR, EQ, ZN, PP' is perpendicular to the meridian, its pole must be in each (8, 9), therefore their common intersection O is the pole of the meridian. Draw also the small circles wH, mt, ae, Rv, yr parallel to the equator; and as the great circle POP' bisects EQ in O, it must also bisect the small circles mt, ae, in r and c; for as EO=90°, therefore (13) tr and ec each 90°; and as QO90°, mr and ac each=90°; hence, ac=ce, = and mr=rt.

78. As all the heavenly bodies, in their diurnal motion, describe either the equator, or small circles parallel to the equator, according as the body is in or out of the equator, if we conceive this figure to represent the eastern hemisphere, QE, ae, mt, may represent their apparent paths from the meridian under *This is not accurately true, the earth's axis varying a little from its parallelism, as will be explained in the proper place.