6. Let us designate the third plane just described as the projecting plane of an original point. It follows as a corollary from this definition of the plane, that the projecting planes of a series of points t, or, N are parallel to each other, and perpendicular to both coordinate planes, as well therefore as to the common intersection Y z of those co-ordinate planes.
6. Let Y z always designate the common intersection of the two co-ordinate planes ; let the projections t, m, N be termed the plans, and the projections 1,111, a the elevations of the original points a, sr, N. It follows that if an original point lie in either co-ordinate plane, its projecting line will coincide with that plane, and its projection on the other will be a point in v z.
7. Let us next consider a right line cu, supposed to join or pass through two points in space a, m. Then the right line LSI joining or passing through the plans of a and sr, is called the plan of am, and int is the elevation of the same original line.
8. It is obvious, from the preceding definitions, that the plan and devotion of any original right lino c m in space are the intersections with the co-ordinate planes respectively of two planes passing through I the original line perpendicular to the co-ordinate planes.
9. We will distinguish the projecting plane of an original line cm, by which the plan of that line may be conceived as produced, as the plan-projecting plane of am ; and the projecting plane by which lm is produced, as the elevation-projecting plane. But the reader must not confound the projecting plane of an original point, which is necessarily perpendicular to both co-ordinate planes, with the projecting plane of an original line ; which, though necessarily perpendicular to one co-ordinate plane, may be parallel, perpendicular, or oblique to the other, according as the original line is parallel, perpendicular, or oblique to that other co-ordinate plane. Nevertheless the projecting plane of an original line will always intersect that co-ordinate plane, to which it is not necessarily perpendicular, in a line which is perpen dicular to 10. Besides the plan and elevation, there are two other elements regarding an original lino which it is necessary to consider ; these are the points in which that line itself intersects the two co-ordinate planes. The principles of projection furnish us with the following theorems relating to these points, and to the plan and elevation of the lino.
11. If the original line be parallel to both co-ordinate planes, it can intersect neither, and both its plan and elevation are parallel to r z. It is clear, on this supposition, that the original line is itself also parallel to YZ.
12. If the original line be perpendicular to one, and therefore parallel to the other co-ordinate plane, the projection on that other plane will be parallel to the original, and perpendicular to vz, while the projection on the first will be a point, that in which the original lino itself intersects that co-ordinate plane.
13. If the original line he oblique to one, and yet parallel to the other co-ordinate plane, its projection' on that to which the line is not parallel will be parallel to vz; while its projection on the co-ordinate plane, to which the original is parallel, will cut x z in the projection of the point in which the original intersects the former co-ordinate plane.
14. If the original line be oblique to both co-ordinate planes, neither its plan nor elevation can be parallel to T z ; the plan of the line will cut T z In the projection of the point of intersection of the original with the co-ordinate plane in which its elevation lies ; while that elevation will cut Y 2 in the projection of the intersection of the original with the plane in which the plan lies.
15. It a'so follows that the projecting line of the point in which an original lino intersects either coordinate plane coincides with the intersection of the projecting plane of that line.
16. If an original line, oblique to both co-ordinate planes, lie In a plane perpendicular to them both, its plan and elevation will both be perpendicular to T z, since its two projecting planes will coincide with that in which the lino lies. In this case the plan and elevation could not furnish sufficient data for determining the original lines, since they would be common to every line, in the perpendicular plane, which was not parallel to either plane of projection ; if however we have, in addition to the indefinite plan and elevation of the line, those of two points in it, or the two points in which the original line cuts the two co-ordinate planes, then the original line is determined.