411 



PERSPECTIVE. 



PERSPECTIVE. 



411 



which it it not neeeMuily perpendicular, in a line which la 

 diculartoTl. 



10. Besides the plan and elevation, there are two other elements 

 regarding an original line 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 them points, and to the plan and elevation of the line. 



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 T 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 YZ, while the projection 

 on the first will be a point, that in which the original line itself 

 intersects that co-ordinate plane. 



13. If the original line be obUque 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 YZ; 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 

 IU plan nor elevation can be parallel to Y z ; the plan of the line will 

 cut T x 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 Z 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 line intersects either co-ordinate 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 Y z, since iU two projecting planes will coincide with 

 that in which the line 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. 



17. Let us 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 poult, or 

 which i> 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 traett. 



19. The traces of a plane on either co-ordinate plant will obviously 

 p*M through the points in which two or more lines lying in the origina 

 plane intersect that co-ordinate plane. 



80. If an original plane be parallel to one co-ordinate plane, its 

 trace on the other will be parallel to Y z. 



21. If an original plane be perpendicular to either co-ordinate plane 

 It* trace on the other will be perpendicular to Y 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 Y z will be the measure of the dihedra 

 ode formed by the original with the other co-ordinate planes. If an 

 original plane be perpendicular to a co-ordinate plane, it* trace on tha 

 plane will be the common projection of all linen in the original plane 

 and will paw through the projection of all points in that origina 

 plane. 



82. If the original plane be parallel to T z, its trace* on the oo- 

 rdinate planes will both be parallel to T z, and therefore to each other ; 

 but in every other case, If the original be oblique to both, or meet 

 both co-ordinate planes, its trace* on them must intersect in a point 

 of T I. And if the plane be perpendicular to both co-ordinate planes, 

 both its trace* will be perpendicular to T I. 



28. If two original line* are parallel, the plans of those lines will be 

 inrnllel, a* will also be their elevations ; but the plans or the elevations 

 inly of two line* may be parallel, although the lines themselves are 

 not so, the parallelism of either the plans or elevations simply arising 

 rum the accidental parallelism of the plan or elevation projecting 

 ilanes of the original lines. 



24. An analogous theorem applies to two original planes : If these be 

 Htrollel, their traces on both co-ordinate planes will be parallel ; but if 

 .heir traces are parallel on one plane only, it simply indicates that the 



original planes intersect each other in a line parallel to that coordinate 

 done. 



25. The plane* of two lines may cut one another, as may also 

 ,he two elevations, and yet the originals may not lie in one plane, and 

 herefore cannot meet each other. If two loriginal lines really inter- 

 sect, the points in which the plans and elevations cut each other 

 must lie in the projecting plane of the point in which the original* 

 meet 



26. The projections of equal parallel lines will be equal parallels, 

 n the ratio to the originals of the cosine of the angle in which those 

 riginal* are inclined to the plane of projection, to radius. If two 

 ines forming an angle be parallel to two others, whether lying in 

 he same or different planes, the projections of each two lines will 

 onu equal angles. 



27. The 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 wlu'ch 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 be perpendicular to the trace of the plane on each 

 :o-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. 



31. These theorems on projections would be useless to the practical 

 geometrician so long as the co-ordinate planes are supposed to retain 

 their relative situation in space ; to enable him to make the requisite 

 constructions on the projections, and to determine the unknown mag- 

 nitudes entering into the original solids by means of the projections of 

 the known ones, he supposes the one co-ordinate plane turned round 

 on the common intersection Y z till the two planes coincide in one and 

 the same plane : by this supposition the relations to Y z of the lines, 

 points, and traces, on the plane which is supposed to be turned round, 

 remain unaltered ; while the principles on which the projections are 

 made allow" of the correct interpretation of the new relations which 

 the projections of original points and lines on one plane assume 

 with regard to the projections of the same points and lines on the 

 other plane, when these two co-ordinate planes are supposed to coincide 

 in one. 



32. The same method of bringing two planes into one may be 

 applied, or rather conceived to be applied, to the prnjtii"j /Jane of 

 any original point or line, this projecting plane being supposed turn, -I 

 round on the projection of the line till the plane coincide with the co- 

 ordinate plane, that is to say, a construction can bo made with the 



n'ectlou of a line founded on this supposition, by which a lim 

 ound representing the original line as brought into the co-ordinate 

 plane ; and by an analogous construction, an original piano may be 

 constructed as if turned round on its trace till it coincide with the co- 

 ordinate plane. 



33. This principle may be carried utill farther : thus a construction 

 can be made, founded on the supposition that an original plane has 

 been turned round on its intersection with another such plane till they 

 coincide, and this compounded plane, if we may use such an expression, 

 lias been again turned round on its trace till it has been brought into 

 the common plane of projection. 



34. It must hence be understood that the practical application of 

 the theory of projections is entirely synthetical, that is, the draughts- 

 man, first drawing a line to represent Y z, proceeds from this simple 

 assumption to draw the projections of certain points and lines of a 

 olid, on which he propose* to operate, from their known, assumed, or 



