" If instead of a horizontal right line in the plane sought for, a line parallel to the vertical plane were conceived, the construction following, by parity of reasoning, would be obtained : " Draw from the point a. parallel to T. the indefinite right line n n ; front the point g, draw g it parallel to c a, producing it till it cut L at in the point it, whence draw IT n perpendicular to L M : this latter will cut o D in D. whence if a parallel be drawn to a n, one of the traces of the plane sought for will be obtained; and if haying produced this trace to meet L M in E. E F be drawn parallel to B c, we shall have the truce on the vertical plane.
"Third 6. A plane whose two traces are A B and B c, and a point whose two projections are n. d, being given ; to construct, 1, the projections of the right line filling perpendicularly from the point upon the plane; 72. the projection of the point of coincidence of the right line with the plane ^ Solution. The perpendieulars D o, dg, filling from the points n and d upon the respective traces of the plane, will be the indefinite projections of the right line required : if along the perpendicular, a vertical plane be conceived, such plane will cut both the horizontal and the given planes, in two right lines, both of them perpendicular to the common intersection, A B, of the two planes ; the first of these lines, being the projection of the vertical plane, is also that of the perpendicular which is included ; therefore the pro jection of this perpendicular must pass through the point D, and the perpendicular to A B.
"The same demonstration will serve for the vertical projection.
" As to the point of coincidence of the perpendicular with the plane, it is evident that it must be found at the intersec tion of this plane, with the vertical plane drawn along the perpendicular ; such intersection being projected indefinitely upon E F. By obtaining the vertical projection, f e, of this intersection, we shall lind it to contain that of the point required ; and as this point must be projected upon the right line d y, it will be found at the intersection. g, of the lines fe and d g. It remains, therefore, only to discover the right line fe: now, the intersection of the given plane with the vertical plane, which are perpendicular to each other, will meet the horizontal plane in the point E, whose vertical projection, e, will be found by dropping E e perpendicularly upon L at ; and it will meet the vertical plane of projection in a point, whose horizontal projection is the intersection of the line L at with n o, produced, if necessary, and whose vertical projection must be at once upon the vertical line Ff and the trace c B of course, it will be at the point, j; of their intersection.
"The vertical projection. g, of the foot of the perpendicular being found, the construction of its horizontal projection will he easy ; for by dropping the indefinite perpendicular g upon I. at, a right line will be obtained, which will contain the point required: and as the line D F must also contain it, it will be found at the point, a, of intersection of these two right lines.
" Fourth 7. A right line whose two projections arc A IT, a 6, and a point whose two projections are D. d, being given ; to construct the traces of a plane drawn from the point, perpendicularly to the right line, "Sobaion. We have seen from the preceding question, that the two traces must be perpendicular to the respective projections of the two right lines : it remains to be discovered what points each of them ought to pass through. For this purpose, let an horizontal line from the given point be con ceived in the plane required, produced so as to meet the vertical plane of projection, and We shall find its vertical projection, by drawing the indefinite horizontal dr. through the point d, and its horizontal projection by drawing a per pendicular to A B, through the point n, pro.lneed till it cut Let in which will be the horizontal projection of the point of coincidence of the horizontal with the vertical plane of projection. This point of coincidence, which must he found in the vertical line u c, and the horizontal line d a, and eon sequently at the point. a, of the of these two lines, will be one of the points of the trace on the vertical plane; we shall then find this trace by drawing the line F c, from the point o, perpendicular to a 6; and if from the point c, where the first trace meets L NI, c E he drawn per pendicular to A B, we shall have the second trace required.
" The same process would discover the point of coincidence of the plane with the right line.