PERSPECTIVE. 



483 



la next the object c. is obtained in a similar manner ; 

 and by joining these corners \ve get the surface o w, 

 which shows, in perspective, the side of the base 

 of the object b, marked o i, in the ground plan. 



8. It may not be plain to every one how that the 

 line, m w, in the ground plan, as well as every other 

 line in the objects to be shown in perspective, which 

 runs in a horizontal direction, and parallel to the 

 picture sheet, should be shown by a level line in fig. 

 26. That this is the case, can very readily be proved, 

 in a line, as above described, but in the position x y, 

 in the ground plan, with the points x and y, which 

 terminate the line, each at the same distance from 

 the line, e p, produced. For whether the line x y 

 be above, or below, or on the same level with, the 

 eye, the rays of light, proceeding from the whole 

 line x y, to the point of sight, form a plane of the 

 shape of an isoceles triangle, having x y for its base ; 

 and a line joining the points x and e, will be the 

 one side, while a line joining the points y and c, will 

 form the other side. But this triangular plane is the 

 surface which gives, by its intersection with the 

 picture sheet, the perspective of the \\nexy; and 

 as x y is a level line, and parallel to fg, in fig. 24, 

 the line which forms the intersection of the trian- 

 gular plane with the picture-sheet, must be a hori- 

 zontal line, which every person, at all acquainted 

 with the properties of parallel lines and the inter- 

 section of planes, will see at once. Now, it will be 

 evident that the perspective of any line in a position 

 as m w, must be a level line, for m w forms a part 

 of a line similarly situated to the line x y. Some 

 people think that a line as x y, should appear in 

 the perspective view, bent upwards or downward at 

 the ends, from the point in it exactly under or above 

 the point e, according as x y happens to be below 

 or above the level of the point of sight in the eleva- 

 tion ; but this is a mistake ; for the perspective of 

 every straight line must be a straight line, as it is 

 formed by the intersection of two planes. By rea- 

 soning in the same way as in the former part of 

 this paragraph, it will be seen that every line which 

 stands in a perpendicular direction in the objects to 

 be represented in perspective, will be shown by a per- 

 pendicular line in fig. 3. The upright corners of the 

 cubes, and some other lines in the tigs, illustrate this. 



9. If what is written in the preceding paragraphs 

 be well understood, it will be seen that the different 

 figs, are placed in such a way, that when the per- 

 spective of a line, which stands perpendicular to the 

 horizontal surface, passing through the lowest point 

 of the objects to be shown in perspective, and of no 

 particular length, is wanted, we have just to draw 

 a line to the place of the eye, in the ground plan, 

 from the point which marks the position of the per- 

 pendicular line in the same fig., and at the point in 

 the picture-sheet where the line, passing betwixt 

 the place of the perpendicular line and the eye, cuts 

 it, let fall a perpendicular line upon / g, in figs. 25 

 and 26, and this line will be the perspective of the 

 line whose perspective is wanted. And when we 

 want to find the perspective of a line running per- 

 pendicular to the picture-sheet, from any point in it, 

 we have first to let fall nponfg, in figs. 25 and 26, 

 a perpendicular line from the point in the picture- 

 sheet, in fig. 24, where the line, whose perspective is 

 wanted, commences, then we have to draw a hori- 

 zontal line, to cut this perpendicular line, from the 

 point which marks the place of the line to be shown 

 in perspective, in fig. 26, and the place where this 

 horizontal line cuts the perpendicular line, is the 

 point in fig. 26, where the perspective of the line 

 commences ; and joining this point with the point e, 

 in the same fig., will give the perspective of the 

 line whose perspective is wanted, when it is inde- 



finitely extended from the point where it commences 

 in the picture-sheet. The following rule to find 

 the perspective of any point rests upon the principle, 

 that a point to be shown in perspective, which does 

 not happen to be in the intersection of two lines, 

 the one running perpendicular to the picture-sheet, 

 and the other at right angles to the horizontal surface 

 on which the objects to be drawn in perspective 

 stand, may be supposed to be so situated, and then 

 if the perspectives of these lines, cutting each other 

 in the point whose perspective is wanted, be found, 

 the point where they cross, in fig. 26, will give the 

 perspective of the point. 



RULE. From the place of the point in the ground 

 plan, draw a line to the point of sight, and from 

 the point where this line cuts the picture-sheet, let 

 fall a perpendicular upon the line/ g, in figs. 25 and 

 26. After this, from the place of the point in the 

 ground plan, whose perspective is wanted, let fall 

 another perpendicular upon the line / g, in figs. 25 

 and 26, on this perpendicular set up the height that 

 the point stands at in the elevation above the line 

 f g ; measuring this height from part of fg, which 

 is under the perspective view ; then, from the height 

 so set up, draw a line to the point e, in the perspec- 

 tive view, and the place where this line cuts the 

 perpendicular let fall from the point in the picture- 

 sheet where the line drawn to the eye in the ground 

 plan cuts it, is the perspective of the point 

 wanted. Thus: suppose that we want to find the 

 perspective of the top point k, of the pyramid 6. 

 From k , in the ground plan, draw a line k e, to the 

 eye, and from the point n, where this line cuts the 

 picture-sheet, let fall a line M k, perpendicular tofg, 

 in figs. 25 and 26. Then from the point A.infig. 24 

 let fall a line k z, perpendicular tofg, in figs. 25 and 

 26, on this line set up the point z, above the linefg, 

 at a distance equal to the height that the top k, in 

 the elevation of the pyramid, is above the part of f g, 

 which is under fig. 25, and from the point z, draw a 

 line to e, in the perspective view, and the point k, 

 where the lines z e and n k intersect, is the perspec- 

 tive of the top point of the pyramid. As all the 

 lines that run up the sides of the pyramid meet at 

 the top, the perspective view of the pyramid is com- 

 pleted by finding the perspectives of the bottom ends 

 of these lines, and joining as many of the perspective 

 points as are not hid by surfaces in front of them, 

 with the points/ and then join these perspective 

 points, the one with the other. The method of 

 drawing the cube in front of the picture, and also 

 the cube on which the pyramid stands, is fully 

 sketched out in the engraving. The six-sided prism 

 c, is drawn in perspective, in the very same way as the 

 pyramid, by finding the perspectives of the points 

 at the ends of all the lines in it, and joining these 

 perspective points. 



To find the perspective of a circle or any other 

 curve. Mark off, at random, a number of points in 

 the ground plan of the curve, after this, mark off the 

 positions of the same point in the elevation, then find 

 by the rule the perspective of each point, and when 

 that is done, connect the perspective points by a line, 

 and this line will be the perspective of the curve. The 

 line which shows the perspective of a curve will be 

 a straight line, when the curve to be shown in per- 

 spective is placed in a plane, which if it was produc- 

 ed, would pass through the point of sight. If a cir 

 cle is placed in a plane, parallel to the picture-sheet, 

 its perspective is a circle. In any other position 

 than the two now mentioned the perspective ot a 

 circle is an ellipse, and not two segments of a circle 

 meeting at the ends, which is the way that persons 

 who do not understand the subject draw a circle in 

 perspective. 



