THE POPULAR EDUCATOR. 



the line r o through o to fc. Produce i fc and its parallels from 

 s, z, x indefinitely; through the point m, where i k produced inter- 

 sects the ray r n, draw u v in the direction of PS, also through p 

 draw the line t iv to PS. Draw the diagonals u w and t v, through 

 their intersection fc draw the line e f towards PS ; we shall then 

 have the rectangle in perspective, within which is to be drawn 

 by hand the elliptical form of the shadow as in Fig. 31. For ob- 

 serve, in proportion as the sun's rays are inclined to the plane on 

 which the shadow falls, so will the diameter h m become longer 

 than the diameter of the circle. 



PROBLEM LVII. (Fig. 95). 

 An archway parallel to the 

 picture. The sun's inclination 

 40 and elevation 30. 



Because the arch is parallel 

 to the picture, draw a line A 

 if rom VPSE to PS ; this repre- 

 sents a plane perpendicular 

 to the picture and passing 

 through the sun. Draw the 

 line a b tangential to the arch 

 and parallel to the line A, 

 also any number of normals 

 anywhere at pleasure, c c, d d, 

 e e, etc. Commencing at the 

 tangent, the point where the 

 shadow begins, draw lines from 

 it and c, d, e, f, etc., to PS, and 

 from the opposite correspond- 

 ing points in the arch draw 

 lines to VPSE ; the intersec- 

 tions of these last with the 

 former will give the points 

 through which to draw the 

 form of the shadow. The sha- 

 dow appears to be convex, it 

 really is not so ; it is only the 

 effect produced from having a 

 front view of it as it lies upon 

 the interior of the arch. If 

 we had a side view of this 

 shadow, we should then sea it 

 was concave. 



PROBLEM LVIII. (Fig. 9G). 

 Two cylin- 

 ders, one ho- 

 rizontal, the 

 other perpen- 

 dicular. The 

 base of the 

 horizontal 

 cylinder is 

 at an angle 



O f 40 with Fig. 94. 



the PP. The 

 sun's incli- 

 nation 50, 

 elevation ^^ 



28. The 



proportions, angles of sight, etc., at pleasure. 



It will be noticed that we have drawn the 

 semicircle of distance from the PS through E 

 (the position of the eye) below the HL ; 

 hitherto we have drawn it above. It is of 

 no importance on which side of the HL it 

 may be drawn ; the process of working is the same in both 

 oases. Our present reason for placing it where we have is 

 f Dr the sake of economising space, and it gives us the oppor- 

 tunity of introducing this convenient arrangement to our 

 pupils. As the bases of both cylinders retire, it will be 

 necessary to construct them according to the rule given in 

 Lesson V., Vol. III., page 9, already referred to in Problem 

 LVI. The principle upon which the shadow of the upright 

 cylinder crosses the horizontal one is, that in perspective pro- 

 jection its form takes that of the object receiving it, and is in 

 this case almost the repetition of a section of the cylinder 

 parallel to the base. We say almost, because the rays of the 

 sun's inclination are not quite parallel with the base of the 



I cylinder. If the rays and the base of the horizontal cylinder 

 i had been parallel, then both would have retired to the same 

 i vanishing point, and then the shadow of the base would have 

 : been a straight line, but their not being parallel causes the 

 ' shadow of the base on the ground to be slightly curved. To 

 I draw this curve, the shadow of the base a b c d e, lines must be 

 i ruled from a, the part of the cylinder that is upon the ground, 

 ! from b, the projection of the point/, and c, the projection of the 

 j point g, each to VPSI. Bays drawn from / and g towards VPSE 



to intersect those lines respec- 

 tively in d and e will determine 

 the points through which the 

 curve is to be drawn by hand ; 

 the remaining portion of the 

 edge of the shadow from e is 

 straight and directed towards 

 the vanishing point of the cy- 

 linder vp 2 . In the same way 

 the curve of the shadow across 

 cylinder B is not parallel to 

 the curve of the base : there- 

 fore, to obtain it, produce the 

 tangent in h at the base of 

 cylinder A to the base of the 

 picture in o, draw the per- 

 pendicular, and make the dis- 

 tances oo 1 , oo 2 , etc., equal vv l , 

 v v 2 , etc. ; rule from each point 

 o 1 , to VPSI, to intersect lines 

 drawn from i, Ic, m, n towards 

 the VP 2 ; through the intersec- 

 tionsat xx l , etc., drawthecurve 

 of the shadow by hand. The sha- 

 dow which falls on the ground 

 beyond the cylinder B will not 

 need an explanation. The mode 

 of construction has been al- 

 ready given in Lesson XVII. 



PROBLEM LIX. (Fig. 97). A 

 column supporting a horizon- 

 tal square slab at right angles. 

 with the picture plane, A 

 pole leans against the wall be- 

 hind, and casts its shadow 

 on the co- 

 lumn. Sun's 

 inclination. 

 40, eleva- 

 tion 35. 



Shadows 

 on curved 

 surfaces are 

 for the most 

 part pro- 

 duced by pro- 

 jecting lines. 

 Let A be the 

 pole ; mark 

 any number 



of points in the pole, at any distance apart, 

 and dra.w perpendicular lines from these 

 points to intersect the horizontal projection 

 of the pole, a VP I . From the points of in- 

 tersection draw lines towards VPSI as far aa 

 the base of the column ; afterwards they 

 must be taken perpendicularly to meet the rays drawn from 

 the points in the pole to VPSE. Where these rays intersect the 

 perpendiculars will be found the points of the shadow of the 

 pole projected on the column, and through which the curve of 

 the shadow must be drawn by hand. The same principle obtains 

 with regard to the shadow of the slab on the column of which 

 b is the shadow of the corner c. Draw a line from c to VPSI to 

 the edge of the column beneath the slab, from which draw a 

 perpendicular line to cut the sun's rays from c to VPSE. The 

 intersection of these two lines will give the shadow of c at b. 

 The same process from d will give the shadow of d at c ; any- 

 other point in the course of the shadow may be thus projected; 

 through these projected points draw by hand the curve of the 



