the question as will enable us to discover the mathematical conditions upon which it depends. With respect to shadows, for example, if the luminous body is reduced to a point, it is evident that they will be determined by the space comprehended by a conical surface tangent to the opaque body, and having for its summit the luminous point. Consequently, to determine the shadow thrown upon any surface whatever, is to find the portion which this cone embraces of the surface in question, that is, the portion circumscribed by the curve which is the intersection of this cone with the proposed surface. We cannot here discuss those considerations which require knowledge foreign from Geometry; they are mentioned only to show of what use, in the arts, may be a familiarity with the geometry of space. Spherical Projections. 107. In spherical representations for geographical, astronomical, and nautical purposes, a perspective of the outline with other remarkable lines and points in the body is made upon a plane supposed to pass through the centre of the sphere and perpendicular to a straight line drawn through this centre to the eye. This section of the sphere by the plane of delineation is called the primitive circle of the perspective projection. The plane itself is called the primative plane. For these purposes two kinds of projections are used, the orthographic and the stereographic. 108. Orthographic Projection. In the orthographic projection the eye is supposed to be situated at an infinite distance from the body to be projected; so that the rays of light which proceed to the eye from the various points in the body, may be considered as making an infinitely small angle with each other; that is, they are parallel. And the sphere being transparent as well as the plane of delineation, these rays will project orthographically upon this plane, the various lines and points belonging to the farther hemisphere. 108. As all great circles perpendicular to the plane of the perspective, are in planes passing through the eye, they will be projected in straight lines drawn through the centre of the primitive. And every small circle perpendicular to the primitive, will have for its projection, that chord of the primitive which results from the intersection of the plane of this circle with the plane of the primitive, (76). 109. All circles of the sphere, which are parallel to the primitive, have for their projections, circles equal to themselves; as the visual rays proceeding from the original circle to the eye, form a cylinder of which the circle and its perspective are parallel sections, (El. 275). 110. [The orthographic projections of a circle inclined to the primitive, will be an ellipse (77) whose transverse axis is equal to the diameter of the circle, and whose conjugate axis is equal to this diameter multiplied by the cosine of the inclination. If the circle be a great circle, the centre of the eliptical projection will be in the centre of the primitive. Any ordinate of the ellipse will be equal to the corresponding semi-chord of the circle multiplied by the cosine of inclination.] 111. PROBLEM. To find the conjugate axis of the orthographic projection of an inclined great circle. Let the arc M m" measure the inclination of the proFig. 46. posed circle to the plane of the primitive (fig. 46); draw m'm parallel to the horizontal diameter of the primitive (which is also the transverse axis of the ellipse) till it meets the vertical diameter in m; take O m' equal to O m, and m m' will be the conjugate axis required." To find the ordinate on, corresponding to the semichord o N; make the angle Non" equal to the inclination of the proposed circle to the plane of the primitive; take on" equal to o N, and draw n'n parallel to Ao; on will be the ordinate required. By finding a sufficient number of ordinates the curve may be approximately traced. 112. Stereographic Projection. The stereographic projection of a sphere is a perspective in which the eye is situated at the pole of the primitive circle. The sphere and the plane of delineation being here also supposed to be transparent, the various lines and points upon the farther hemisphere, will be projected upon the primitive circle by the visual rays passing from the several points Fig. 46. to the eye; and the lines and points of the nearer hemisphere will be projected upon the same plane beyond the circumference of the primitive. 113. All great circles perpendicular to the primitive, as they pass through the axis of the primitive, will have their planes pass through the eye, and will therefore be projected in straight lines passing through the centre of the primitive. 114. Every circle parallel to the primitive will have a circle for its projection; for the visual rays, in this case, constitute a right cone whose base is the original circle and of which the projection is a section parallel to the base (El. 268). As the axis of this cone of rays passes through the centre of the primitive, The centre of the projection of circles parallel to the primitive, will be in the centre of the primitive circle. 115. PROBLEM. To find the distance from the centre of the primitive, of the projection of any point whose distance from the farther pole is given. Let AP'BO' (fig. 47) be the horizontal projection of the Fig. 47. sphere; AB the horizontal trace of the plane of delineation, or the horizontal projection of the primitive circle; and M' the horizontal projection of the proposed point (supposed in the circumference of the horizontal great circle), O' being the horizontal projection of the eye. Draw O'M' meeting AB in the point m; and C'm will be the distance of the perspective of the proposed point from the centre of the primitive circle. 116. [The angle P'O'M' is measured by half the arc P'M' (El. 116); and by drawing the arc C'E it will be perceived that C'm is the tangent of this angle, that is, the tangent of half the arc P'M', which measures the distance of the point M from the farther pole and is called the polar distance of the point. We therefore say,The projection of every point in the surface of a sphere, is at a distance from the centre of the primitive circle, equal to the tangent of half the polar distance of that point. 117. This gives us the radius of the projection of every circle parallel to the primitive; for every part of its circumference being at the same distance from the Fig. 47. pole, The tangent of half the polar distance of a circle parallel to the primitive, is the radius of the projection of that circle.] 118. PROBLEM. To find the projection of a circle inclined to the plane of the primitive. Let the circle in question be a small circle; the visual rays proceeding to the eye from the several points in this circle will constitute a scalene cone. The intersection of this cone with the plane of the primitive will be the projection required. Suppose the sphere to be so situated that the plane determined by the axis of the cone and the axis of the primitive circle shall be horizontal; Fig. 48. and let figure 48 represent the horizontal projection of the sphere, AB being the horizontal trace of the plane of the picture, and M'N' the horizontal projection of the circle proposed. The angle nm O', made by the two chords AB and O'M', is measured by half the sum of the arcs AO', BM', (El. 115) or (as AO' is equal to BO') by half of the arc O'M'; but this is the measure of the angle O'N'M' (El. 116), therefore the angles n m O' and M'N'O' are equal. The angle N'O'M' is common to the two triangles nm O' and N'M'O'; and having two angles of the one equal to two angles of the other, the other angles must be equal, that is, the angle O'M'N' is equal to the angle O'n m; and this intersection of the cone of rays by the perspective plane, is a sub-contrary section, and is therefore a circle, the base of the scalene cone being a circle. To show that this section is a circle, suppose a section parallel to this, made by a plane whose horizontal trace is n' m', and whose intersection with the base of the cone is a vertical line having its horizontal projection in the point c'. This line is a common chord of the two curvilinear sections of this cone; and as the section whose horizontal projection is M'N' is a circle, the square of half this common chord is equal to the product of the two segments of the diameter; that is, equal to 'N' X c' M', (El. 126). The angle c' m' M' has its sides parallel to those of the angle cm O' which was shown to be equal to c′ N'n' ; therefore the angle c' m' M' is equal to c ́N'n'; and as the two angles at c' are vertical angles, the two triangles are equiangular and therefore similar, and give the pro c' N' c' m' portion = ; and, by multiplying by the de- Fig. 48. c'n' c'M' nominators, we obtain c'N' × c′ M′ = c'′ m' X c'n'. This last product, therefore, equals the square of half the common chord of the curves; this last curve is consequently a circle; and as this is a section parallel to the perspective in question, the perspective is therefore a circle. And we say,-The stereographic projection of a small circle inclined to the plane of the primitive, is also a circle. 119. If the circle in question were a great circle, the line M'N' would pass through the point C; and a process similar to the above would show, that the intersection by the plane of the primitive circle, of a cone of which this is the base and O the summit, would be a circle. We thence conclude that,―The stereographic projections of all circles of the sphere are circles. 120. As the process in article 118, gives us the two extremities m and n of the diameter of the projection of the inclined circle proposed; c the middle of m n will be the centre of the projection, and cm the radius. We can therefore find the projection of any circle inclined to the plane of the primitive. 121. If the small circle be perpendicular to the primitive, so that its horizontal projection may be EF (fig. 47); the polar distance O'E or PF will give two points of Fig. 47. the projection, E and F; and by drawing the straight line O'F we obtain the distance gC' from the centre of the picture to the nearest point of the curve; which take from the centre of the primitive, on the horizontal line. Having now three points in a circular curve, the curve is readily drawn, (El. 110). 122. We have, in figure 49, an orthographic projec- Fig. 49. tion of the sphere upon the plane of the equator. The meridians, being perpendicular to the plane of the projection, are straight lines; the parallels of latitude arc circles concentric with the primitive. In figure 50 we have a stereographic projection of the Fig. 50. same. The meridians, as they pass through the eye, are projected in straight lines; all the other circular lines upon the sphere have circular curves for their projections. |