" Now. to find their projections in the position which they ought naturally to have, we must suppose the perpendicular plane to he restored to its original position, by turning it back upon the horizontal line, or hinge, A n. and it will carry with it the point J, the two tangents J c, J n, produced till they cut A It in the points a a% and the chord c D, which will likewise cut A n, in the point N. In this movement. it is evident, that the points K. N, which are upon the hinge. will he fixed, and that the two points of contact c, D, ill describe arcs of circles, which will be in planesperpendieular to the hinge. and whose horizontal projections will be obtained by dropping from the points C D, the indefinite perpendiculars C P, n a, upon A ir. The horizontal projections of the two points of contact will therefore he found upon the two right lines C D Q. But in the retrograde movement, of the pe• pendicular plane, the two tangents J C J K n do not cease to pass through the respective points of contact ; and when this plane is returned to its primitive position, the point J is projected anew in o, and the two tangents are projected according to the right lines o G K. The two latter, there fore, must each contain one of the points of contact ; and, in fine, the intersections of these two right lines with the respective lines c r, a Q, will determine the horizontal pro icetiolls, It, s, of the two points of contact, which are found in the same line with the point N.
To obtain the vertical projections of the same points. first, draw the indefinite perpendiculars e r, s .5, upon L M ; then by projecting the points K le, to A:. A'', and drawing the lines!, A', 1' k', from the point y, we shall have the vertical projections of two similar tangents. These lines, therefore, will contain the projections of the respective point: of contact ; and the points, r, .5, of their intersection with the vertical a r, S s. will be the projections required.
The horizontal and vertical projections of the two point; of contact being to construct, upon the horizontal plane the traces of the two tangent planes, lines parallel to the given right line must be conceived to pass through each of the points of contact. These lines will be in the respective
tangent planes. and their horizontal and vertical projections will be obtained by drawing a u, s v, parallel to E F, and r u, s r, parallel to e 1: On the horizontal plane, construct the trace, r, of the given right line, and the traces, u. v, of the two last lines ; and the lines T u, T Will be the traces of the two tangent planes.
"Instead of supposing fresh lines to pass through the points of contact, we may find the traces of the two tangents a rt, o s, which will answer the same purpose. As to the traces of two similar planes with the vertical plane, they may be obtained by the method already so often alluded to.
This solution may be rendered much more elegant by making the two planes of projection pass through the centre of the sphere itself. By this mode the two projections of the sphere would be mingled ill the same circle, and the prodne tilms of the right line: would not be so long. We have only separated the two projections for the sake of perspicuity in the exposition : for it is easy to give to the construction all the conciseness of which it is susceptible.
" Second Method.—Figure 17. Let .5, a, be the two pro jections of the centre of the sphere; A B, or I), its radius; B c D. the projection of its great horizontal circle ; and E e f, the projections of the given right line. Conceive the plane of the great horizontal circle produced till it cut the given right line in a certain point, and we shall have the vertical projection of the plane, by drawing the indefinite horizontal line it ct g throng)) the point a ; the point g. where this horizontal line cuts ei will be the vertical projection of the point of coincidence of the plane with the given right line; and we shall have the horizontal projection, o, of this point by projecting y upon E F.