Consequently, the tangent pt of an arc of a great circle terminated at the plane of projection, is projected into a right line equal to it. P Let Pt, and Pt, two such tangents, be connected by, the right line tt' which will be in the plane of projection. Let st, st', be the projections of those tangents; the triangles trt', tst', are (from the above) equal in all respects: therefore the angles opposite to the common side tt' will be the same in both: conse quently, the tangents of any two arcs terminated at the plane of projection, are projected into lines which are respectively equal to them, and which form an equal angle. Hence, two circles which intersect in P on the sphere, form on the projection an angle equal to that which they make on the sphere; because, at the point of intersection the elements of the arcs coincide with those of their tangents. Therefore, all great circles intersect mutually on the plane of projection under the same angle as on the sphere; so also do little circles which intersect at the same points, and have, by consequence, common tangents. 43. By way of showing the application of these principles, let us suppose that the eye is at the south pole of the equator. The plane of projection will then be the equator itself; the centre of the equator will represent the north pole; AP (fig. to art. 35) will be = :0; the projections of the parallels to the equator will all have for a common centre that of the projection; and the radii of those circles will be the tangents of the halves of the polar distances. Thus, for the polar circle ....r=tan}( 23°28′)=tan11°44 for the antarctic circle r=tan or, As for the meridians, whose planes all pass through the place of the eye, they all become diameters which divide the equator in its several degrees, and form at the centre of the projection angles equal to the differences of longitude. For these circles d∞, and r = (art. 38). This kind of projection, the easiest of all to describe, serves very conveniently for eclipses of the sun. The meridians and the parallels are herein divided mutually into degrees: those of the parallels are equal; those of the meridians unequal; for the expression for one of their degrees is, ♪= tan } (4 + 1o) sin 30' tan A * 44. On a planisphere of this kind, the stars are placed according to their right ascensions and declinations. Then the ecliptic is traced, as well as its poles, the circles of latitude all intersecting mutually at those two poles, and then the parallels to the ecliptic. Thus, on the radius marked 270°, at a distance cz from the centre = tan 11° 44', we mark the north pole of the ecliptic. From that point, with an opening of the compasses cosec 23° 28′ we mark the point E, or, which amounts to the same, we make CE = cot 23° 28'. Through the point E we draw the indefinite perpendicular VEX, which is evidently the locus of the centres of all the circles of latitude intersecting mutually in П and E will be the centre of that circle of latitude which passes through 0° and 180°, or of the equinoctial colure, which will be the circle vaxп, will be the solstitial colure. = In general, make EAG' longitude: from the centre G' with the radius G' we describe a circle, it will be the circle of latitude which answers to the longitude supposed. Repeating the same operation on the other side of the line E, we shall have the circles of latitude of the other hemisphere. For the circles parallel to the ecliptic we employ the formulæ D being 23° 27′ 49′′, or nearly 23° 28', the distance between the poles of the ecliptic and the equator; and A the polar distance of the parallel. When A > D, the sign of the second term will be changed. 45. The principal defect of this projection is the little resemblance and proportion between the arcs of the sphere and their projections. Thus, the arcs A and All represent arcs of 90°; D represents an arc of 113° 28′, and D, though greater, only represents 66° 32′; the arcs 7, 72, 7, &c. are of 90°; DE represents 23° 28′ DAH inclination of the ecliptic AnQ to the equator ADQ. It is true, however, that the greatest inequalities are out of the circle ADQB which is properly the projection. If we regard the circle vix as a map of half the terrestrial globe, then V, Y, G, TE, will represent arcs of 90°, though v will be the only one of those four which is actually a quadrant. 46. Another inconvenience of this projection, is the difficulty of finding the true distance of two points of which we have the projections. Yet, let м and N in the diagram to art. 42, be two such points: produce cм, CN, to m and n respectively; the arc mn will give us MCN which is the same as on the sphere. CM and CN are the tangents of the half distances from the pole of the projection. The spherical triangle will give (see chap. vi. equa. 2), 1 COS MN = COS CM COS CN + sin CM sin CN cos c tan2 ¿cм) (1 — tan2 CN) + 4 tan Cм tan CN Cos C (1 + tan3 CM) (1 + tan2CN) CM3) (1 — CN2) + 4CM. CN COS C (1 + CM2) (1 + CN2) Take Cм' Cм and draw oм'M", AM" will be equal to the arc represented by CM. Proceed similarly for CN. Then cos MN may be computed from the above. The third member of the equation is obtained from the second by substituting for cos Cм, sin CM, their values in tan Cм, &c. deduced by means of equa. R, chap. iv. 47. The projections here treated serving for the usual purposes of astronomy, we need not enter upon the explication of the other kinds of projection devised by geometers for different purposes. The principal of these is the gnomonic projection, in which the eye is supposed at the centre of the sphere, and the plane of projection a tangent plane to the sphere at any assumed point. All the points within adequate limits have their projections at the extremitics of the tangents of their distances from the point of contact; and those tangents: form respectively the same angles as the arcs that mea sure the several distances from the principal point. In this projection too, a less circle will evidently be projected into an ellipse, a parabola, or a hyperbola, ac cording as the distance of its most remote point is less, equal to, or greater than 90°, from the centre of the plane of projection. But for a farther developement of these properties, and for the geometrical constructions derived from them, such as want to enter more minutely into this subject may consult Emerson's Projection of the Sphere, the treatise in Bishop Horsley's Elementary Treatises on Practical Mathematics, or that in the Traité de Topographie, par Puissant. CHAPTER IX. On the Principles of Dialling. 1. DIALLING, or gnomonics, is the art of drawing on the surface of any given body, whether plane, angular, or curved, a sun-dial, that is, a figure, the different lines of which, when the sun shines, indicate by the shadow of a style or gnomon the time of the day. 2. The general principles which serve as a basis to the theory of dialling, cannot be more aptly illustrated than they have been by Ozanam and Ferguson, in the following contrivance. Suppose a hollow transparent sphere DPBP, of glass, to represent the earth as transparent, and its equator divided into 24 equal parts by so many meridian semicircles a, b, c, d, e, &c. one of which is the geographical |