482 



I'KKSPECTIVE. 



part marked A m, of Uie line A i, in the ground plan; 

 ami a line joining the points A and e.in fig. 2G, is the 

 perspective of Uie line A i, in the ground plan, when 

 it is indefinitely extended in the direction A i. 



5. Points in contact with the transparent plane 

 must be at the same distance from, and in the same 

 position with respect to, each other in the perspec- 

 tive view, as in the elevation; for the line drawn to 

 the eye, which marks the perspective, can neither 

 converge nor diverge in passing betwixt a point 

 whose perspective is wanted and the picture-sheet, 

 as in ttii* in-tawe there is no distance betwixt the 

 place of the point and the line/g-, in fig. 24. From 

 this it will be evident, that any line or plane surface 

 in contact with the transparent plane, will have the 

 same shape and dimension in the perspective view as 

 in the elevation. And it will also be seen, that the 

 principal reason why the point marked e, in fig. 26, 

 is fixed upon as the place in the perspective view of 

 the point marked p, in fig. 24, is, that as this point e, 

 is found in the place where a perpendicular let fall 

 from the point p, to the llne/g, in figs. 25 and 26, 

 cuts a horizontal line running through the point e, 

 in fig. 25, and the point e, in fig. 26, being the per- 

 spective of a point in contact with/, in fig. 24, set- 

 ting it off in this manner will allow the place of any 

 other point of the objects to be shown in perspective 

 which is in contact with the picture-sheet, to be ob- 

 tained by a similar process, which process is very 

 easily gone through by means of a drawing square. 

 As the part of the line.fg, which is under fig. 25, 

 represents a horizontal surface, which cuts the trans- 

 parent plane, the intersection of the picture-sheet and 

 this horizontal plane being a line in contact with the 

 picture-sheet must be shown at the same distance 

 below e, in the perspective view, that the part of/ 

 g, which is under fig. 25, is below the position of the 

 eye, which is also the position in the elevation of the 

 point marked;), in the ground plan. As we proceed 

 it will become evident that the part of the line fg, 

 which is under the perspective view, is of very great 

 use to set up the height from it which any point in 

 the elevation has above theline/^infigs. 25 and 26, 

 when the elevation cannot conveniently be drawn on 

 the same board with the ground plan and the per- 

 spective view. The line A e, which shows, in fig. 26, 

 the perspective of the line A i, when it runs to an 

 indefinite distance from the picture-sheet, must be a 

 level line, as Uie point A, in fig. 25, at which the line 

 commences, is in a level with the point e, the posi- 

 tion of p, in the same fig., and these points being 

 both in contact with / g, in the ground plan, must 

 have the same position in the perspective view that 

 they have in the elevation. 



6. Suppose the line A i, which runs perpendicular 

 to the picture-sheet, and on a level with the eye, to 

 have its commencement in fig. 24, at the point /, in- 

 stead of the point A. By reasoning in the same 

 way, as in paragraph 3, it will be found that I p is 

 the perspective in the ground plan of the line h i, 

 when it is extended to an indefinite distance beyond 

 the point /, in the ground plan. And it will further 

 be found that the nearer to the point p, that any line, 

 running perpendicular to the picture-sheet, and on a 

 level with the eye, is taken, the indefinite perspective 

 (that is, the perspective of a line when it is indefi- 

 nitely extended.) will always get shorter, so that if 

 a line, such as A i, has its commencement in the point 

 p, its indefinite perspective will be shown in the 

 ground plan by the point p itself. In the same way 

 it may be shown that p is the vanishing point (that 

 is, the point which terminates the perspective of a 

 line when it is indefinitely extended.) of any line 

 running perpendicular to the picture-sheet, and in a 

 level with the point of sight, although the line com- 



mence at a point/, on the different side of the point 

 p, from the line A '; and the reasoning employed in 

 paragraph 3, will also show that Uie perspective of 

 any point, in a line so situated, is obtained in the 

 same way in which the perspective of the point 

 marked m, in fig. 24, or the point i, in the same fig., 

 is found. Or, if a line running perpendicular from 

 the picture-sheet to an indefinite distance beyond it, 

 have its commencement not, as in the above exam- 

 ples, at a part of the picture-sheet on a level with 

 the eye, but in a line passing through the point p, 

 and running at right angles to the plane on which 

 the objects a, b, &c., stand, the vanishing point of a 

 line so situated will be the point p in the ground 

 plan, and the point e, in fig. 26, in this case also: this 

 may be demonstrated in the same way as the vanish, 

 ing point of the line h i, or any of the other lines run- 

 ning parallel to, and on a level with it, was shown 

 to be the points p and c, respectively, in figs. 24 and 

 26. In a similar manner it may be shown, that if any 

 line running perpendicular to the transparent plane, 

 to an indefinite distance beyond it, has its commence- 

 ment at any point v, in fig. 25, which is neither in a 

 horizontal nor a perpendicular line, passing through 

 the position of the eye in the same fig.; the vanish- 

 ing point of a line so placed is, as in the above ex- 

 amples, the point p, in the ground plan; or the point 

 e, in the elevation. 



7. Let the line h f, fig. 24, have its commencement 

 in the elevation at r, one of the corners of the cube 

 a, its perspective view is found as follows: From 

 the point h, in fig. 24, draw a line A s, perpendicular 

 to the linefg, in figs. 25 and 26, and from r, draw a 

 line r q, parallel to/| r , and the point q, where the 

 line r q cuts the line A s, is the commencement of 

 the perspective of the line A iy join q with e, and this 

 line will be the perspective of the line A i, when it 

 is indefinitely extended. A line joining the points 

 s and e is the perspective of the line A i, if it is in- 

 definitely extended, when it has its position in the 

 elevation at the corner which is under r, of the cube 

 a. The point s, where the perspective line s e com- 

 mences, is found in the very same way as the point 

 g was found. As q e is the perspective of the line 

 A i, which runs perpendicular to the picture-sheet, 

 when it is indefinitely extended, from r, in fig. 25, one 

 of the top-corners of the cube a; and as * e is the 

 perspective of h i, when it is indefinitely extended, 

 in a direction perpendicular to the picture-sheet, 

 from the corner under r, of the cube a, in fipr. 26, the 

 triangle q e s is the perspective of a parallel surface, 

 standing perpendicular to the surface on which the 

 objects a, b, &c., stand, and running at right angles 

 to the picture-sheet, to an indefinite distance from it. 

 The side of the cube a, that is towards the centre of 

 the picture, and the same side of the cube under the 

 pyramid, form part of the perspective of this parallel 

 surface. The point t, where the line n t, let fall 

 from the point n, in the ground plan, perpendicular 

 to the \\I\G f g, in figs. 25 and 26, cuts the line s e, is 

 the perspective of the bottom corner at m, of the 

 cube a ; and the place where this same line, n t, 

 cuts the perspective line q e, is the perspective of 

 the top corner m, of the cube marked a, in the 

 ground plan. So now we have got the perspectives 

 of the four corners of one of the sides of the cube, 

 in front of the picture ; and by joining these corners 

 we get the surface q t, and this surface is the per- 

 spective of the side of this cube, which is towards 

 the object d. A perpendicular, let fall upon the 

 line/g 1 , in the elevation and perspective view, from 

 the point /, in fig. 24, will cut the lines q e, and s e, 

 so as to give the perspectives of the top and bottom 

 corners at i, of the cube under the pyramid ; and 

 the other two corners of the side of this cube, which 



