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supposed to pass through the centre of the earth parallel to the plane of the horizon of a spectator, as before mentioned, and by planes intersecting one another in the line drawn through that centre and the station of the observer, that is, perpendicularly to his horizon. The first plane, or its trace in the heavens, is called the rational horizon of the observer, and the circles in which the perpendicular planes cut the celestial sphere are called azimuthal or vertical circles: their circumferences evidently intersect each other in the zenith and nadir. An arc of the rational horizon intercepted between the meridian of a station and a vertical circle passing through a celestial body is called the azimuth, and an arc of a vertical circle between the horizon and the celestial body is called the altitude of that body: also the arc between the body and the zenith point is called the zenith distance; and a plane passing through any point, parallel to the horizon, will cut the celestial sphere in a small circle which is called a parallel of altitude. This last system of co-ordinates is that to which the places of celestial bodies are immediately referred by such observations as are made at sea; and it is also generally employed by scientific travellers who have occasion to make celestial observations on land.

21. An arc of the horizon intercepted between the east or west point and the place of any celestial body at the instant when, by the diurnal rotation, it comes to the circumference of the horizon, (that is, the instant of rising or setting,) is called the amplitude of that celestial body.

CHAP. II.

PROJECTIONS OF THE SPHERE.

NATURE OF THE DIFFERENT PROJECTIONS EMPLOYED IN PRACTICAL ASTRONOMY AND GEOGRAPHY.- PROPOSITIONS RELATING TO THE STEREOGRAPHICAL PROJECTION IN PARTICULAR. EXAMPLES OF THE ORTHOGRAPHICAL, GNOMONICAL, GLOBULAR, AND CONICAL

PROJECTIONS. MERCATOR'S DEVELOPMENT.

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22. THE trigonometrical operations which occur in the investigations of formula for the purposes of practical astronomy, require the aid of diagrams in order to facilitate the discovery of the steps by which the proposed ends may be most readily gained; and representations of the visible heavens or of the surface of the earth, with the circles by which the astronomical positions of celestial bodies or the geographical positions of places are determined, are particularly necessary for the purpose of exhibiting in one view the configurations of stars or the relative situations of terrestrial objects. The immediate objects of research in practical astronomy are usually the measures of the sides or angles of the triangles formed by circles which, on the surface of the celestial sphere, connect the apparent places of stars with each other, and with certain points considered as fixed: and as, for exhibiting such triangles, the formation of diagrams on the surface of a ball would be inconvenient, mathematicians have invented methods by which the surface of a sphere with the circles upon it can be represented on a plane, so that the remarkable points on the former may be in corresponding positions on the latter; and so that with proper scales, when approximative determinations will suffice for the purpose contemplated, the values of the arcs and angles may be easily ascertained.

23. These are called projections of the sphere, and they constitute particular cases of the general theory of projections. The forms assumed by the circles of the sphere on the plane of projection depend upon the position of the spectator's eye, and upon that of the plane; but of the different kinds of projection which may be employed for the purposes of astronomy and geography, it will be sufficient to notice only those which follow.

The first is that in which the eye is supposed to be upon the surface of the sphere, and the plane of projection to pass through the centre perpendicularly to the diameter at the extremity of which the eye is situated. This projection is described by Ptolemy in his tract entitled "The Planisphere," and its principles are supposed to have been known long before his time: it was subsequently called the Stereographical Projection, from a word signifying the representation of a solid body. A modification of this projection was made by La Hire, in supposing the eye of the spectator to be at a distance beyond the surface of the sphere equal to the sine of 45 degrees (the radius being considered as unity); and this method, which has been much used in the formation of geographical maps, is sometimes called the Globular projection. The second is that in which the eye is supposed to be infinitely distant from the sphere, and the plane of projection to be any where between them, perpendicular to the line drawn from the eye to the centre of the sphere: it is employed by Ptolemy in his tract entitled "The Analemma,” and it has been since called the Orthographical Projection. It may be here observed, that when from any given point, or when from every point in a given line or surface, a straight line is imagined to be drawn perpendicularly to any plane, the point, line, or surface, supposed to be marked on the plane by the extremities of the perpendiculars, is said to be an orthographical projection of that point, line, or surface. In the third projection the eye is supposed to be at the centre of the sphere, and the plane of projection to be a tangent to its surface: this projection is called Gnomonical, from a correspondence of the projecting point, or place of the eye, to the summit of the gnomon or index of a sun-dial. It was not used by the ancients. A modification of this projection was proposed by Flamstead, but it has not been adopted; the method which is now distinguished by the name of that astronomer consists in placing the projecting point at the centre of the sphere, and projecting a zone of its surface on the concave surface of a hollow cone in contact with the sphere on the circumference of a parallel of latitude or declination, or on the concave surface of a hollow cylinder in contact with the sphere on the circumference of the equator.

The demonstrations of the principal properties relating to the different projections, and the rules for constructing the representations of circles of the sphere on a plane surface, constitute the subjects of this chapter.

DEFINITION.

24. The eye of the spectator being upon or beyond the surface of the sphere, the circle on whose plane, produced if necessary, that surface is represented, is called the Primitive Circle; its plane is called the plane of projection.

Cor. 1. The projecting point being in the direction of a diameter of the sphere perpendicular to the plane of projection, either extremity of the diameter is one of the poles of the primitive circle, and it is evident that the centre of the latter is the point in which the pole opposite to the projecting point is projected.

Cor. 2. Since every two great circles of the sphere intersect one another in a diameter of the sphere, it follows that the projection of any great circle intersects the circumference of the primitive circle in two points which are in the direction of a diameter of the latter; and that when any two great circles are projected, the line which joins their points of intersection will pass through the centre of the primitive circle.

Note. The chords, sines, tangents, and secants of the angles or arcs, which, in the projections of the sphere, are to be formed or measured, may be most conveniently taken from the scales on a sector; the arms of the instrument being opened, so that the distance between the chord of 60°, the sine of 90°, the tangent of 45°, or the secant of 0°, may be equal to the radius of the primitive circle.

PROPOSITION I.

25. In any projection, if the plane of a circle of the sphere pass through the projecting point or the eye of the spectator, the representation of its circumference on the plane of jection will be a straight line.

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For imagine lines to be drawn from all points in that circumference to the projecting point, and to be produced in directions from thence, if necessary, they will be in the plane of the circle (Geom. 1. Planes); and the plane of projection is cut by the same lines, that is, by the plane in which they are. But the intersection of two planes is a straight line, therefore the representation of the circumference is a straight line. Q. E. D.

PROPOSITION II.

26. If from a point on the exterior of a sphere lines be drawn to the circumference of any circle of the sphere, whose plane does not pass through the point, and those lines, pro

duced if necessary, cut a plane passing through the centre of the sphere perpendicular to the diameter in which produced is the given point; the figure projected on the plane will, except when the circle is parallel to the plane of projection, be an ellipse.

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Let c be the centre, and let A G B D, supposed to be perpendicular to the plane of the paper, and having its centre in the line AFB, represent the circle of the sphere; also let the plane of projection passing through the line Q CT, perpendicularly to the plane of the paper, cut the plane of the circle in some line, as DFG: this line will be perpendicular to AB and to QT. (Geom., Planes, Prop. 19.) Let E, in the

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produced diameter PCE at right angles to QT, be the given point; and through that diameter imagine a great circle APT, in the plane of the paper, to be described: this circle will be perpendicular to the planes of projection and of the circle

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Now, if lines be drawn from E to every point in the circumference A G B D, they will constitute the curve surface of a cone whose base A G B D is a circle making any angle with its axis, that is, with a line drawn from E to the centre of A G B D: let this surface be produced if necessary, and let it be cut by the plane of projection; the section N G M D of the oblique cone is the projection of the circle A G B D, and it is required to prove that it is an ellipse.

In the triangles AFN, MFB (Plane Trigonometry, Art. 57.), FN FA sin. FAN: sin. FNA,

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FM FB sin. F B M sin. FMB.; whence FN.FM: FA. FB :: sin. FAN sin. FBM : sin FNA sin. FM B. Let the two last terms of the proportion be represented by p and q respectively; then

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but A G B D being a circle, F A. F B = F G2 (Euc. 35. 3.); therefore FN.FMF G2. But again, with the same circle of the

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