The point n being the trace of the generator, if it be projected on the vertical plane, by means of the perpendicular D d, and if from the point d, the line d f be drawn parallel to a b, we shall obtain the vertical projection of the generator. We are thus in possession of the two projections of one of the right lines, through which the required tangent plane must pass. The vertical projection of the point of contact ought to be on the line c c'. drawn from the given point perpendicularly to L ; it should also be on d f; consequently it will he in the intersecting point, c, of those two lines.
" If the line E cut the trace, E P 13, of the cylindric sur thee in several points, as D and E, we must proceed for each in the same manner as that just directed for the point a, considered by itself, and we shall obtain the vertical projec tions, d_); of as many generating lines, and the vertical projections, c c', of as many points of contact, as there are points of intersection between the line E F and the trace E p b.
" In the instance of Figure 1•2, the trace of the cylindric surface in the circumference of a circle, which has the pro perty of being cut by a right line in two points; so that the vertical line elevated from the given point c, must meet the surface twice : first, in a point whose vertical projection is c, through which the generator passes when it touches upon the point D ; and, secondly, in another point, whose vertical projection is c', through which the generator passes when it rests on tit; point E of the trace. These two points, although they have the simie horizontal projection, are nevertheless very distinct. and should each have a particular eorres pon lent tangent plane. Tucked. for each of the two of costa •1. we must find the second right line, by which position of the tangent plane is to be determined. Were we strictly to follow the general method, by consider ing the trace as a second generating line, it would be necessary to conceive it as passing successively through each of the points of contact, and to construct a tangent for each of such points; but in the present case of cylindric surfaces. a much alone simple process may be pursued. For example: the plane tangent to the point c, c, touches the snrfaee, through out its extent, of the generating right line which passes from this point ; it touches it, then, in 0, which is a point of such generating line, and ought therefore to pass through the tangent to the trace at the point D. By parity of reasoning,
we shall find that the plane tangent, c. c', ought to pass along the tangent to the trace in E. Therelbre. if from the two points n, E., we draw the trace of the two tangent planes D K. E G, pradlleed till they cut the line in two points K. G. WC shall have, in the horizontal plane, the traces ofthe two tangent planes.
" It only remains to discover the traces of the same planes on 1110 vertical ; and having alrealy for one of these traces the 1.01111 K. and for the other, the point a, we have only to determine a single point for each of them.
•• With this \ iew, operating for the first of the two tangent planes, imagine the point to be constructed, to be one in which a horizontal line drawn upon the plane throtedt the point of contact, would meet the vertical plane; we shall have the horizontal projection of such line, by drawing from the point c, a line parallel to the trace n K, which may be produced till it meet the line L NI in the point a ; and we may obtain its verth•al projection by drawing from the point e an indefinite horizontal line. The point of coincidence of the vertical plane with the horizont:LI, will be tbund at once both upon the vertical line i i and the horizontal line c i. and w ill be at the point, i, of their iitersection ; therefore, if' the points i and K a line he drawn, it will give the trace of the first plane tangent to the vertical plane. By a similar process for the second tangent plane. we shall find its trace on the vertical plane, by drawing from the point c a line, c parallel to the horizontal trace E G, which may be produced till it cut i. m in the point H, upon which the vertical line n h may be raised; from the point c' draw a horizontal line to cut the vertical line a h, in the point h, from which, and from the point a, by drawing the line c h, we shall obtain the trace required.
" Second chte8tion.—From an imaginary point of a COBB; surface, of N1 hick the horizontal projection is given, to draw a tangent plane to such surface.