33? 



THE POPULAE EDUCATOE. 



GEOMETRICAL PERSPECTIVE. X. 



PROBLEM XXIX. (Fig. 51). A cube 4 feet side has one 

 of its faces at an angle of 50 with the PP, its nearest edge 

 touches the picture plane 1 foot to the left of the eye ; height of 

 eye 5 feet ; distance from, the PP 8 feet; scale 1 inch to the foot, 



It will be seen that as the nearest angle touches the PP, it 

 will commence at b, 

 1 foot to the left of 

 a ; and because b is 

 a point of contact, 

 its height, b c, may 

 be measured from b ; 

 6 d is equal to the 

 edge of the cube, 4 

 feet; its perspective 

 length, b TO, is cut 

 off the vanishing line 

 b vp 2 by its distance 

 point DVP 2 . The 

 other face of the 

 cube must be treated 

 in the same way ; 

 it vanishes at VP 1 , 

 therefore the line 

 from e to cut off the 

 perspective length 

 b n must be drawn to 

 DVP 1 ; the lines of 

 the horizontal and 

 upper face of the 

 cube will be ruled to 

 their respective va- 

 nishing points, as in 

 Fig. 33, Lesson V., 

 Vol. III., page 9. 



PROBLEM XXX. 

 (Fig. 52). Draw by 

 this method the flight 

 of steps given in 

 Lesson VIII., page 

 203. There are three, 

 each 4 feet long, 

 1 foot wide, and 9 

 inches high ; tJieir 

 front making an 

 angle of 40 ivith the 

 picture plane. The 

 distance of the eye of 

 the observer from the 

 picture plane is 6 

 feet; from the plane 

 to the nearest point 

 of the object 1 foot; 

 the height of the eye 

 4'5 feet ; scale 1 inch 

 to the foot. 



We will merely go 

 through the order 

 of procedure, until 

 we come to some- 

 thing especially sug- 

 gested by this pro- 

 blem. Draw the 

 TP ; the HL ; place 

 the station point, 

 marked E ; draw the 



line from E to find the VP 1 for the angle of inclination of the 

 face with the PP. As the base of the object forms a right 

 angle, the line E vp 2 must be drawn at a right angle with E vp 1 

 for the VP of the ends of the steps. Produce E PS to the PP at 

 a ; the nearest point within is 1 foot ; make a b equal 1 foot, 

 and a line from 6 drawn to r>E will cut PS a in c, the nearest 

 point within ; draw lines from c to each VP, and find their dis- 

 tance points. A line from DVP 2 must be drawn through c to 

 the PP at e ; the widths of the steps will be marked off at /, g, h. 

 froouce vp 2 c to the PP at m, draw the perpendicular m n for 

 a measuring line, and upon it mark off the heights of the three 



PP 



3 



e a v ml 



steps, o, p, n; rule from these points to vp 3 . Prom the widths of 

 the steps e, f, g, h, draw lines towards DVP' 2 , stopping at the 

 vanishing line from c, from which perpendicular lines, made to 

 cut the retiring lines from o p n, will give the respective ends 

 and heights of the steps ; from the angles of the steps draw 

 lines towards vp 1 . To cut off the lengths of the steps upon the 

 vanishing line c VP 1 , draw the line c v, directed by DVP 1 ; make 



v w equal to 4 feet, 

 the length of the 

 steps ; from w draw 

 back again towards 

 DVP 1 , cutting the 

 vanishing line from 

 c in k ; draw from fc 

 to r, directed by vp 2 , 

 from r raise another 

 measuring line for 

 the opposite ends 

 of the steps. Make 

 s t u equal to o p n, 

 draw lines from them 

 to vp 2 ; these last 

 lines,intersecting the 

 retiring lines from 

 the tops of the steps, 

 will give the further 

 ends. These slight 

 directions will be 

 quite sufficient for 

 the guidance of 

 those who have tho- 

 roughly studied Pro- 

 blem XXVII. 



One of the greatest 

 difficulties in geo- 

 metrical perspective 

 is the treatment of 

 inclined lines and 

 planes. The plan 

 method we have al- 

 ready given is, no 

 doubt, as useful as 

 any, but in some 

 cases the method we 

 are about to explain 

 in this lesson will 

 be found easier and 

 more satisfactory. 

 If the pupil will turn 

 back to Lesson VI., 

 Problem XVIII., 

 Fig. 37, page 72, he 

 will there be re- 

 minded how the per- 

 spective of an in- 

 clined line or plane 

 is obtained by tho 

 help of orthographic 

 projection ; that is, 

 from a given posi- 

 tion of the inclined 

 plane, to produce 

 its plan and ele- 

 vation, and after- 

 wards from both 

 produce the perspec- 

 tive projection. We 

 now propose to draw the perspective of inclinations without 

 previously constructing a plan. We must start once more 

 from one of the leading principles of perspective belonging 

 to every system, and which is well known to our pupils 

 that all horizontal retiring lines and planes have their 

 vanishing points upon the line of sight ; to this must now be 

 added : directly a line or a plane ceases to be horizontal, having 

 one of its ends raised or lowered, its vanishing point is raised 

 or lowered also, for, notwithstanding its inclination, it retires, 

 and has a vanishing point; therefore the vanishing point of 

 an inclined line or plane is perpendicularly above the point to 



