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tance of their meridians measured in degrees, &c. upon the equator. 9. The sensible horizon is a circle, the plane of which is supposed to touch the spherical surface of the earth, in the place of the spectator whose horizon it is. The rational horizon is a circle whose plane passes through the centre of the earth, parallel to the plane of the sensible horizon. The radius of the earth being exceedingly minute compared with that of the celestial sphere, the sensible and rational horizon may, in many astronomical inquiries, be supposed, without error, to coincide. 10. Great circles which are drawn as secondaries to the rational horizon, are called vertical circles; they serve to measure the altitude or the depression of any celestial object. 11. The two points in which all the vertical circles that can be drawn to any rational horizon meet, are called, the one above the spectator the zenith, and that which is below him the nadir. 12. Almucantars, or parallels of altitude, are circles parallel to the horizon, or whose poles are the zenith and nadir. All the points of any one almucantar are at equal altitudes above the horizon. 13. The real motion of the earth about the sun once in a year, gives rise to an apparent motion of the sun about the earth in the same interval of time. The circle in which the sun appears to move is called the ecliptic; the angle in which it crosses the equinoctial the obliquity of the ecliptic; * and the two points where it intersects that circle, the equinoxes.

* The obliquity of the ecliptic is a variable quantity, oscillating between certain limits which it never passes. According to the profound investigations of Laplace in physical astronomy, the obliquity may always be determined very nearly by this formula, viz.

23°28'23"-05 – 11917,2184 [1 – cos (t 13".94645)]
–3347”-0496 sin (t 32"-11575)

where t denotes the number of years run over from 1750; it is negative before, positive after, that epoch. This theorem is found to answer very well up to the time of Pytheas, 350 years before

14. The distance of the sun, or of any of the heavenly bodies, from the equator, measured on an are of the meridian, is called the declination; north or south, according as the body is situated north or south of the equator. 15. Secondaries to the celestial equator are called circles of declination; of these, twenty-four, which divide the equator into equal parts of 15° each, are called hour circles; because the sun in his apparent diurnal motion passing over 360° of a circle parallel to the equator, goes through orth of them, or 15°, in an hour. 16. The right ascension of a celestial body is an arc of the equinoctial, intercepted between one of the equinoxes, and a declination circle passing through the body; it is measured according to the order of the sun’s apparent motion through the twelve signs. 17. The longitude of a heavenly body is an arc of the ecliptic, contained between the 1st point of Aries (that is, one of the equinoctial points), and a secondary to the ecliptic, or a circle of latitude passing through the body. †. The latitude of a body is its distance from the ecliptic measured upon a secondary to that circle. And the angle formed at the body by two great circles, one passing through the pole of the equator, the other through the pole of the ecliptic, is called the angle of Sälzo”. 19. The tropics are two circles parallel to the equinoctial, and touching the ecliptic at the two points where it is most remote from the equator; that is to say, the first points of Cancer and of Capricorn; the former

is denominated the tropic of Cancer, the latter the tropic of Capricorn.

the Christian era. The obliquity at the beginning of 1816 is 28° 27, 49'''2.

20. The points where the tropics touch the ecliptic are called solstices, because the sun when in either of them appears to be at a stand with regard to his declination. .. 21. Colures are two secondaries to the equinoctial: one, passing through the equinoctial points, is called the equinoctial colure; the other, passing through the solstices, is called the solstitial colure. - - 22. Small circles drawn at the distance of 23°28' (or correctly 23° 27' 49") from the north and south poles of the equator, are called polar circles; the former the arctic, the latter the antarctic circle. 23. That vertical circle which intersects the meridian of any place at right angles, is called the prime vertical: the points where it cuts the horizon are the east and west points; at the distance of 90° from each of these on the horizon are the north and south; all four being called cardinal points. 24. The distance on the horizon of a vertical circle that passes through any body from the north or south points is the azimuth of that body; the distance of the same circle from the east or west points is the amplitude. . . . .”. 25. In order to represent on a plane the celestial sphere with all its circles great and small, the ancients

invented two kinds of projection. The first, named by

them Analemma, has since received the name of Orthographic Projection. The second, originally denoted by the generic term Planisphere, received from the Jesuit Aguilon the name of Stereographic Projection. The *śe orthographic is given to the former, because it is produced by lines which fall at right angles upon the plane which represents the sphere, That of stereographic was given to the other, because it results from the intersection of two solids, a sphere and a cone. ... - . . . . . . . . . ;

- • * * :- ... it ,

G , - .


Orthographic Projection.

26. In this projection the eye is supposed at an infinite distance: in which case a great circle acDF is apparently reduced to a right line F equal to its diameter BP, the eye * being imagined indefinitely distant in the direction sc. Then,

also, every arc CA which has its B D origin at the apparent centre, has y for its projection a right line sa C A *

equal to the sine of that arc. The quadrant cB or cD will be projected into its sine, or radius. An arc as AE = CE CA, will be projected into ae, = sin CE - sin cA = (by equa. U, chap. iv.) 2 sin (CE - CA) cos ? (CE + cA) = 2 sin \ AE cos (CA -- #AE).... (1. 27. Every arc, as DA, from the edge of the disc towards the centre, is projected into its versed sine Da. The quadrant Dc, therefore, is projected into the radius ps the versed sine of 90°; and the semicircle DcB into the diameter DB, the versed sine of 180°. 28. What is here remarked of the circle BcDF applies equally to all great circles which intersect at s and form the visible hemisphere: each of these semicircles is reduced to its diameter, and the hemisphere is reduced to a disc. *29. In this projection every circle, great or small, , whose plane prolonged does not pass through the eye, will be seen obliquely, and under an elliptical form: for an oblique circle making throughout the same angle with the plane of projection, its several parallel ordinates are all reduced in a constant ratio; therefore, the projected ordinates are all in a constant ratio to the corresponding ordinates of the circle of equal diameter

on the plane of projection, and together constitute an ellipse. (Hutton on Ellipse, prop.3, cor. 1). 50. In general, in the orthographic projection every great circle perpendicular to the plane of projection is represented by its diameter; and every circle perpendicular to that plane is represented by a chord of the primitive circle equal to its diameter. - - - Thus if, by way of showing the use of this projection, we assume the meridian for the plane of projection, the horizon will be represented by its diameter Ho: the rime vertical by its 3. zv, which cuts the former perpendicularly : the six o'clock hour circle will be represented by its diameter which is the Pi g axis PP', making with the horizon the angle PCo = the height of the pole = the latitude = L: the equator will be projected into its diameter eq, making with the horizon an angle Hce = 90° — L: #. parallels to the equator will be represented by chords, such as AB parallel to the diameter of the equator: the almucantars are projected into chords, such as Rs, parallel to the horizontal diameter Ho. 31. AB being the projection of a certain parallel, suppose that the star which in its apparent motion describes that parallel, has its inferior transit of the meridian at B. The point B which is its place on the sphere, is also its place then in the projection. The star being in the horizon at T, or will be the versed sine of its azimuth, and cT the sine of its amplitude. Also, refer

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