4S INTRODUCTION. 



185. Theorem. When a picture is projected on a 

 plane, by right lines supposed to be drawn from each point 

 to the eye, the whole image of every right line, produced 

 without limit, is a right line drawn from its intersection 

 with the plane of projection, to its vanishing point, or the 

 point where a line drawn from the eye, parallel to the 

 given line, meets the plane of projection ; and this image 

 is divided, by the image of any given point, in the ratio of 

 the portion of the line, intercepted by that point and the 

 picture, to the line drawn from the eye to the vanishing 

 point; so that if any two parallel lines be drawn from the 

 ends of the whole image, and the distances of the eye and 

 of the given point be laid off on them respectively, the line, 

 joining the points thus found, will determine the place of 

 the required image of the point. 



For A being the eye, and B the va« 

 nishing point of the line CD ; AB and 

 CD, being parallel, are in the same 

 plane, and AD is also in that plane 

 C- (62) J and BC is the intersection of this 



plane with that of-the picture; therefore E, the image of the point D, 

 is always in the line BC ; and AB : CD: :BE : EC ; and taking the 

 parallel lines BF, CG, in the same ratio, FG will also cut BC in E. 

 When AB is perpendicular to the plane, B is called the point of 

 sight, and is the vanishing point of all lines perpendicular to the plane 

 of the pictiu-e : and the vanishing point of any other line may be 

 found by setting oflF from B a line equal to the tangent of its inclina- 

 tion to the perpendicular line, tlie radius being AB. 



Scholium. When a line becomes parallel to the plane of the pic- 

 ture, the distance of its vanishing point becomes infinite, and tlie 

 image is, tlierefore, parallel to the original. Jn this case, the magni- 

 tude of the image may be determined by means of lines drawn in any 

 other direction through the extremities of the original line. In the 

 orthographical projection, the images of all parallel lines whatever 



