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THE POPULAR EDUCATOE. 



the paper, to mark them on the PP for the purpose of cutting 

 them off their respective vanishing-lines, guided by their true 

 distance-point. When such is the case, we have recourse to the 

 use of half-distance points. Our pupils are aware how a distance- 

 point is found for any given vanishing-point. If the space on 

 the HL between the VP and its DVP be bisected, tlie middle point 

 thus found will be the half-distance point. To explain and illus- 

 trate the construction and application of this very useful prin- 

 ciple in perspective, we have employed only a single line. 



PROBLEM XXXVIII. (Fig. 62). On reference to the figure, it 

 will be seen that c d is the perspective view of a line at an angle 

 of 35 with the PP, the real length of which is a b, from which 

 lines are drawn in the usual way to the DVP, to determine c d on 

 the vanishing-line. Find the half-distance point by the bisection 

 explained above, mark it J DVP, and draw from it a line through 

 c to n ; take half the length of the given line to be represented, 

 and set it off from n to /, rule from / to i DVP. It will be seen 

 that the two lines from n and / pass through the same points 

 c and d to the DVP, which were originally found by the two 

 lines from a and b to the DVP. Suppose it were necessary to 

 represent a line double, or of a greater length than a b ; in this 

 instance we will take double the length to show the advantage of 



parallel with the HL) to meet the corresponding retiring lines of 

 the opposite retiring wall ; thus will be determined the further 

 end upon which are fixed the folding doors A and B. How to 

 find their vanishing-points and cut off their widths, we trust it 

 will not be necessary to repeat, bnt merely remark that VP 1 is 

 the VP for the door A, VP S for the door B, and VP 4 for c. To 

 ascertain the vanishing-point for the retiring thickness of a 

 door, it will be found by drawing a line from E to the HL at a 

 right angle with the line of its VP ; for example, VP 2 is the VP 

 for the retiring thickness of the door A. With regard to draw- 

 ing the true position of the door at the side, there may be a 

 difficulty not yet explained. Here is a case, which frequently 

 occurs, of a line or plane at an angle or inclination with some- 

 thing else than the picture-plane. In the case before us, a door 

 is stated to be at a given angle with its ivall, whilst at the same 

 time the wall is at a right angle with the PP. The difficulty is 

 how to find the VP for the door. The proposition states that it 

 is at an angle of 40 with its own wall. The difficulty will not 

 be great if we know the angle to the PP of the intermediate 

 plane to which the given object is inclined ; because, if the 

 wall D (see Fig. 64) upon which the door swings is at a right 

 angle with the wall F, and c, the door, is at an angle of 40 with 



this principle of construction. Make / m equal to / n, and rule 

 from m to the i DVP, it will cut the vanishing line in e ; c e will 

 then be the perspective length of a line equal to twice a b. Our 

 pupils will see that it is impossible, from want of space, to 

 double the length of a b on the PP, and so carry a line from the 

 extreme to the DVP ; had there been sufficient room to mark 

 the full length, x would have been the line to the DVP to deter- 

 mine the length of c e. As we shall have occasion to avail 

 ourselves of the half-distance point in some of onr future ques- 

 tions, we advise our pupils to exercise themselves in' this 

 problem, employing various lengths of lines at various angles. 



PROBLEM XXXIX. (Fig. 63). The interior of a room inparallel 

 perspective; the retiring portion in vieiv is 16 feet long, 19 feet 

 tvide, and 12 feet high. Distance of the eye from the picture- 

 plane 12 feet, and its height from the ground 4 feet . At the further 

 end are folding doors 10 feet high, and 4 feet wide ; also a single 

 door at the side, the height and width of which are the same. The 

 door A is at an angle of 32 with the connecting wall, the door B 

 at an angle of 67, and c at an angle of 40 with its wall, and 

 5 feet from the further corner of the room. In this case the PS 

 will be the VP for the retiring walls on both sides ; the width of 

 the room is marked off from a to b on the PP and ruled to the 

 PS ; the height is a d and b f; the depth to bo represented, 

 viz., 16 feet, is set off from a to c, and a line from c to DE will cut 

 off the length of the room in the point n on the line from a to 

 PS ; from this point n a perpendicular line is to be drawn to re- 

 present the corner of the room, to meet the lines from d and / 

 to the PS ; from this perpendicular draw lines across (that is, 



D, therefore c will be at an angle of 50 with F; but F is 

 parallel with the PP, therefore the door c will be at an angle of 

 50 with the PP. Consequently, we shall find the VP of the 

 door (Fig. 63) by drawing a line from E at 50 with the PP, 

 producing VP*. To find its distance from the corner of the 

 room at n, mark the point e 5 feet from c, rule from e to DE, 

 and where this line cuts the line from a to PS will be found the 

 position of that side of the doorway upon which the door 

 swings : the heights of the doors are set off from o. 



PROBLEM XL. (Fig. 65). A box 6 feet long, 3 feet wide, and 

 1 foot 6 inches high, inclined to the picture at an angle of 37. 

 The lid is open and thrown back at an angle of 45 with the per- 

 pendicular. Thickness of wood, 2 inches. Depth of lid, 6 inches 

 Distance of the eye from the picture-plane, 6 feet, and its- heiglA. 

 from the ground 2 feet 6 inches. The nearest angle to touch the 

 picture-plane. Scale, inch to the foot. 



If the lid is at an angle of 45 with the perpendicular, it will 

 be at the same angle with the horizon ; therefore, as VP 2 is the 

 VP for the end of the box, the angle of inclination must be made 

 from DVP 2 . To cut off the retiring length of the lid, the line of 

 contact must be drawn from c n to b, and then from DVP 3 draw 

 a line through the corner of the box joining the lid to a ; make 

 a b equal to the width of the box, and rule from b back again to 

 the DVP 3 . For the depth of the lid draw from DVP 4 to n on the 

 line of contact ; make n c equal to the depth 6 inches, and draw 

 back again as before. As the other parts of the construction 

 are the same which have been repeatedly explained in previous 

 problems, we leave the remainder as an exercise for practice. 



