S the angles Aam, вbn may be computed. But c representing the centre of the earth, if EC be drawn parallel to sa or sb, the angles ACE, BCE will be respectively equal to Aam, Bbn; therefore the difference between these last angles is equal to the angle ACB. Hence, the arc AB being measured, the following proportion will give the length of the earth's circumference: - E m B n ACB (in degrees): 360°: AB circumference (=24850 miles, nearly.) 15. The processes by which the earth's form and magnitude are with precision determined, as well as those which are employed in finding the magnitudes of the sun, moon, and planets, the distances of the moon from the earth, and of the earth and planets from the sun, will be presently explained. It is sufficient to observe here, that since the planetary bodies, when viewed through a telescope, present the appearance of well-defined disks, subtending, at the eye of the spectator, angles of sensible magnitude, while the stars called fixed, though examined with the most powerful instruments, are seen only as lucid points; it will follow that the sun, the earth, and the planets constitute a particular group of bodies, and that a sphere supposed to encompass the whole of the planetary system may be considered as infinitely small when the imaginary sphere of the fixed stars is represented by one of any finite magnitude. Hence, in describing the systems of circles by which the apparent places of celestial bodies are indicated, it is permitted to imagine that either the earth or the sun is a point in the centre of a sphere representing the heavens and, in the latter case, the earth and all the planets must be supposed to revolve in orbits whose peripheries are at infinitely small distances from the sun. P B 16. If the earth's orbit (supposed to be a plane) be produced to the celestial sphere, it will there form the circumference of a circle which is called the trace of the ecliptic (let it be EL): this is represented on the common celestial globes; and on those machines, the representations of the zodiacal stars near which it appears to pass will serve, when the stars are recognised, as indications of its position in E q R L the heavens. A line passing through the sun at c (the centre of the sphere) perpendicularly to the plane of the ecliptic, meets the heavens in the points designated p and q, which are called the poles of the ecliptic. Now, if planes be supposed to pass through p and q, these planes will be perpendicular to that of the ecliptic, and they will cut the celestial sphere in the circumferences of circles which are called circles of celestial longitude: these are also represented on the celestial globes. 17. The trace of the ecliptic in the heavens is imagined to be divided into twelve equal parts called signs, which bear the names of the zodiacal constellations before mentioned. They follow one another in the same order as those constellations, that is, from the west towards the cast; and the movement of any celestial body in that direction is said to be direct, or according to the order of the signs: if the movement take place from the east towards the west, it is said to be retrograde, or contrary to the order of the signs. 18. It has been shown in art. 7. that the path (the ecliptic) of the earth about the sun is inclined to the plane which is perpendicular to the axis of the diurnal rotation; the axis pg must, therefore, be inclined to the latter axis. Now, the centre of the earth being at c infinitely near, or in coincidence with that of the sun, agreeably to the above supposition, let P Q be the axis of the diurnal rotation; then the plane A B passing through c perpendicularly to P Q and produced to the heavens will be the plane last mentioned: it will cut the surface of the earth (which is assumed to be a sphere or spheroid) in the circumference of a circle called the terrestrial equator, and that of the sphere of the fixed stars in the circumference av B of the celestial equator. The plane of this circle will cut that of the ecliptic in a line, as y c^, which is called the line of the equinoxes, of which one extremity in the heavens is called the point of the vernal equinox. If planes pass through PQ they will be perpendicular to the equator, and their circumferences in the celestial sphere form what are called circles of declination: such is the circle P R Q which is made to pass through s, the supposed place of a star. Of these circles that which passes through the line of the equinoxes is called the equinoctial colure, and that which, being at right angles to the former, passes through p, is called the solsticial colure. The last-mentioned planes will cut the surface of the earth in the circumferences of circles, if the earth be a sphere, or in the perimeters of ellipses if it be a spheroid; and these are called the meridians of the stations, or remarkable points which they pass through on the angles Aam, вbn may be computed. But c representing the centre of the earth, if EC be drawn parallel to sa or sb, the angles ACE, BCE will be respectively equal to Aam, Bbn; therefore the difference between these last angles is equal to the angle ACB. Hence, the arc AB being measured, the following proportion will give the length of the earth's circumference: b A m B n E ACB (in degrees): 360°:: AB circumference (=24850 miles, nearly.) 15. The processes by which the earth's form and magnitude are with precision determined, as well as those which are employed in finding the magnitudes of the sun, moon, and planets, the distances of the moon from the earth, and of the earth and planets from the sun, will be presently explained. It is sufficient to observe here, that since the planetary bodies, when viewed through a telescope, present the appearance of well-defined disks, subtending, at the eye of the spectator, angles of sensible magnitude, while the stars called fixed, though examined with the most powerful instruments, are seen only as lucid points; it will follow that the sun, the earth, and the planets constitute a particular group of bodies, and that a sphere supposed to encompass the whole of the planetary system may be considered as infinitely small when the imaginary sphere of the fixed stars is represented by one of any finite magnitude. Hence, in describing the systems of circles by which the apparent places of celestial bodies are indicated, it is permitted to imagine that either the earth or the sun is a point in the centre of a sphere representing the heavens and, in the latter case, the earth and all the planets must be supposed to revolve in orbits whose peripheries are at infinitely small distances from the sun. P p B 16. If the earth's orbit (supposed to be a plane) be produced to the celestial sphere, it will there form the circumference of a circle which is called the trace of the ecliptic (let it be EL): this is represented on the common celestial globes; and on those machines, the representations of the zodiacal stars near which it E q Q R L the heavens. A line passing through the sun at c (the centre of the sphere) perpendicularly to the plane of the ecliptic, meets the heavens in the points designated p and q, which are called the poles of the ecliptic. Now, if planes be supposed to pass through p and q, these planes will be perpendicular to that of the ecliptic, and they will cut the celestial sphere in the circumferences of circles which are called circles of celestial longitude: these are also represented on the celestial globes. 17. The trace of the ecliptic in the heavens is imagined to be divided into twelve equal parts called signs, which bear the names of the zodiacal constellations before mentioned. They follow one another in the same order as those constellations, that is, from the west towards the east; and the movement of any celestial body in that direction is said to be direct, or according to the order of the signs: if the movement take place from the east towards the west, it is said to be retrograde, or contrary to the order of the signs. 18. It has been shown in art. 7. that the path (the ecliptic) of the earth about the sun is inclined to the plane which is perpendicular to the axis of the diurnal rotation; the axis pg must, therefore, be inclined to the latter axis. Now, the centre of the earth being at c infinitely near, or in coincidence with that of the sun, agreeably to the above supposition, let PQ be the axis of the diurnal rotation; then the plane A B passing through c perpendicularly to P Q and produced to the heavens will be the plane last mentioned: it will cut the surface of the earth (which is assumed to be a sphere or spheroid) in the circumference of a circle called the terrestrial equator, and that of the sphere of the fixed stars in the circumference A y B of the celestial equator. The plane of this circle will cut that of the ecliptic in a line, as C, which is called the line of the equinoxes, of which one extremity in the heavens is called the point of the vernal equinox. If planes pass through PQ they will be perpendicular to the equator, and their circumferences in the celestial sphere form what are called circles of declination: such is the circle P R Q which is made to pass through s, the supposed place of a star. Of these circles that which passes through the line of the equinoxes is called the equinoctial colure, and that which, being at right angles to the former, passes through p, is called the solsticial colure. The last-mentioned planes will cut the surface of the earth in the circumferences of circles, if the earth be a sphere, or in the perimeters of ellipses if it be a spheroid; and these are called the meridians of the stations, or remarkable points which they pass through on the earth. This is, however, only the popular definition of a terrestrial meridian: if from every point in the circumference of a circle of declination in the celestial sphere lines be let fall in the directions of normals, or perpendiculars, to the earth's surface, a curve line supposed to join the points in which the normals meet that surface will be the correct terrestrial meridian; and if the earth be not a solid of revolution this meridian is a curve of double curvature. If any point, ass, be the place where a perpendicular raised from any station on the surface of the earth meets the celestial sphere, s will be the zenith of that station; R will express its geographical longitude, and R S its geographical latitude. The arc T or the angle CT on the plane of the ecliptic is designated the longitude of any star s, through which and the axis pq the plane of a circle is supposed to pass; and the arc T s, or the angle T C s, is called the latitude of such star: ps or q s is called its ecliptic polar distance. A plane passing through any point, parallel to the ecliptic E L, will cut the celestial sphere in the circumference of a circle which is called a parallel of celestial latitude. The arc R, or the angle CR on the plane of the equator, is designated the right-ascension of any star s, through which and the axis PQ the plane of a circle of declination is supposed to pass; and the arc RS or the angle R C S is called the declination of such star: PS or Qs is called the polar distance. A plane passing through any point, parallel to that of the equator AB, will cut the celestial sphere in the circumference of a small circle which is called a parallel of declination. 19. The ecliptic and the circular arcs perpendicular to it form one system of co-ordinates: the equator and its perpendicular arcs form another system; and the knowledge of the number of degrees in the arcs y T and T S, VR and RS, whether obtained by direct observation, or from astronomical tables, is sufficient to determine the place s of a star in the celestial sphere. It may be necessary to observe that the system of the equator and its perpendiculars is continually changing its position by the annual movement of the earth about the sun; but on account of the smallness of the orbit when compared with the magnitude of the celestial sphere, and the axis PQ being always parallel to itself (art. 8.), omitting certain deviations which will be hereafter mentioned, that change of position creates no sensible differences, except such as depend on the deviations alluded to, in the situations of the stars with respect to these co-ordinates. 20. A third system of co-ordinates is formed by a plane |