17. Let ua next consider in what manner an original plane may be conceived to be referred to two co-ordinate planes. It is clear that as only one plane can be drawn through a straight line and a point, or, which is the same thing, through the two legs of a plane angle, the plans and elevations of any two lines through which the plane passes will determine it. But the intersections of the original plane with the two co-ordinate planes furnish a datum regarding it, of more direct application.
18. The intersections of an original plane with the co-ordinate planes are termed its traces.
19. The traces of a plane on either co-ordinate plane will obviously pass through the points In which two or more lines lying in the original plane intersect that co-ordinate plane.
20. If an original plane be parallel to one co-ordinate plane, its trace on the other will be parallel to ir z.
21. If an original plane be perpendicular to either co-ordinate plane, Its trace on the other will be perpendicular to T z, at the point In which the trace on the first plane meets that line; and the plane oblique angle formed by the trace and T z will be the measure of the dihedral angle formed by the original with the other co-ordinate planes. If an original plane be perpendicular to a co-ordinate plane, Its trace on that plane will be the common projection of all lines in the original plane, and will pass through tho projection of all points in that original plane.
22. If the original plane be parallel to T 2, Its traces on the co ordinate planes will both be parallel to i t, and therefore to each other; but in every other ease, if the original be oblique to both, or meet both co-ordinate planes, Its traces on them must intersect in a point of T z. And if the plane be perpendicular to both co-ordinate planes, both its traces will be perpendicular to 29. If two original lines are parallel, the plans of those lines will be parallel, as will also be their elevations ; but the plans or the elevations only of two lines may be parallel, although the lines themselves are not ao, the parallelism of either the plans or elevations simply arising from the incidental parallelism of the plan or elevation projecting planes of the original lines.
24. An analogous theorem applies to two original planes : If these be parallel, their traces on both co-ordinate planes will be parallel ; but if their traces are parallel on one plane only, it simply indicates that the original planes Intersect each other in a line parallel to that co-ordinate plane.
25. The planes of two lines may cut one another, as may also the two elevations, and yet the originals may not lie In one plane, and therefore cannot meet each other. If two original lines really Inter sect, the points in which the plies and elevations cut each other must lie in the projecting plane of the point in which the originals meet.
26. The projections of equal parallel lines will be equal parallels, in the ratio to the originals of the cosine of the angle in which those originals are inclined to the plane of projection, to radius. If two lines forming an angle be parallel to two others, whether lying in the same or different planes, the projections of each two lines will form equal angles.
27. '1 he e plane angles, which are the projections of equal angles, will be equal, provided the original angles are similarly placed with respect to the traces of the planes in which those originals lie; or else when the original angles lie in a plane parallel to either co ordinate plane, and then the projected angles must be equal to the originals.
28. Hence, since the projection of every parallelogram is a parallelo gram (23 ), the angles of the projection corresponding to the adjacent angles of the original figure will also be complementary to each other.
29. If an original plane and line be mutually perpendicular, the pro jection of the line will he perpendicular to the trace of the plane on each co-ordinate plane. For since the projecting plane of the line must, on this supposition, be perpendicular both to the original and to the co-ordinate plane, and consequently so to their common intersection, which common intersection is the trace of the plane, this projecting plane will cut the co-ordinate plane in a line, namely the projection of the original, perpendicular to the trace of the plane.
30. If a line in an original plane be parallel to a co-ordinate plane, the projection of that line will be parallel to the trace of the plane ; and conversely, if the projection of a line situated in a plane be parallel to the trace of that plane, the original line is parallel to the co-ordinate plane in which that trace lies.