GEOMETRICAL PEBSPEOTIVE. 



73 



tation of an incline. A rod 5 feet long ii inclined to the hori- 

 zon 40". The plan / ih<- rod it 50 with tlie picture plane, 

 the nearest end 1 foul from it. In this case the vanixhing 

 point of (> plan of the rod must bo found, and 

 iinf Unit of the rod itself. Wo intend in a future 

 lesson to show how the vanishing point for an 

 inolino may bo found without a plan, giving only 

 the dimensions and positions, and the method of 

 osing it ; but for the present turn book to Problem 

 IV. , Fig. 1 1 (Vol. II., page 297), where the same sub- 

 ject is shown in orthographic projection ; the rod 

 is there placed at a given angle with the ground, 

 ay, and perpendiculars are drawn from the ex- 

 tremities between 

 which the lino a b, - 

 the plan, is drawn. 

 Now we muat first 

 project the rod or- 

 thographically in or- 

 der to determine the 

 plan preparatory to 

 drawing it perspec- 

 tively. An indefinite 

 lino a b must bo 

 drawn at an angle of 

 50 with the picture 

 plane ; c is the point 

 where the rod touches 

 the ground, draw c e 

 5 feet long at an 

 angle of 40 with 

 a 6 ; draw e d per- 

 pendicularly to a b ; 

 c d will then be tlui 

 plan of the rod ; com- 

 plete the perspective 

 representation of c d, 

 which will be / g (see 

 Fig. 7, Lesson II., 

 Vol. II., page 225). 

 This last observation 

 refers to the perspec- 

 tive only of the plan ; 

 we must now repre- 

 sent the rod in its 

 inclined position. As 

 one end of the rod is 

 on the ground, and 

 the other above it, 

 our attention must be 

 directed to the ele- 

 vated end, because 

 the lower end is al- 

 ready found in g. It 

 must be evident, on 

 turning once more to 

 Fig. 7, that the line 

 f g is the perspective 

 of the line d c ; and 

 since the line d c is 

 the plan of the given 

 line e c, therefore e 

 must be perpendicu- 

 larly over d. The 

 question now comes 



to this : how far above d ? We answer, the 

 length of d e, which must be set off on the 

 lino of contact, namely, h i. From i draw a 

 lino to the VP, and the point m where tho lino 

 from i to VP cuts the visual ray from d will 

 determine the position of the upper end of the 

 rod ; join m g, which will bo the perspective 

 representation of the rod. 



PROBLEM XIX. (Fip. 39). A square board 

 is inclined to the horizon at an angle of 48** ; one edge is hori- 

 zontal, (lie plan nf the inclined edge of the board is 50 ivith the 

 picture plane ; length of side 6 feet. Tho scale may be either 

 4 feet or 2 feet to the inoii. (We give tho pupil the choice, and 



observe that M his knowledge and ""nlHftK* InnioMti he should, 

 when ho repeat* the problems, take other ancle* and other 

 oales of proportion. This kind of repetition will be of great 

 service to him.) We mart fiwt show the ortho- 

 graphic projection of the board (Fig. 38), and 

 then apply it to the perspective projection. When 

 the board is horizontal, or laid upon the ground, 

 the plan will be a square, abed; but if we raise 

 the side a c, allowing the edge b d to remain upon 

 the ground, it would then be inclined to the horwrn 

 as represented by the tine or edge of the board, 

 e f; drop a perpendicular from/, then the 

 plan is projected by a! b tf d. Observe, if the 

 edge of the board 

 were still farther ele- 

 vated, the plan would 

 become narrower, 

 that w, a! c" would 

 approach b d. When 

 the board becomes 

 perpendicular to the 

 ground, then the plan 

 would be a line only 

 (see the observations 

 made upon the circle, 

 Figs.lOandll.Lesson 

 III.,VoLII.,page297). 

 We will now proceed 

 with the perspective 

 projection of the 

 board as given in the 

 question ; it is there 

 stated that the edge 

 of the board is in- 

 clined at 50 tri/ft 

 the ff ; therefore 

 draw on indefinite 

 line d e at that angle, 

 make d b equal to the 

 length of a side, and 

 at a right angle with 

 d e; this is the edge 

 upon which it rests, 

 and is horizontal ; 

 draw b h parallel to 

 d e ; draw b f at an 

 angle of 48 with 

 b h, and make it 

 equal to b d; from 

 / draw perpendicu- 

 larly to b h the line 

 / o' c", we shall then 

 have in the parallelo- 

 gram a' d b d the 

 plan of the board 

 at the given inclina- 

 tion. The angle d 

 of the board touches 

 the picture plane, 

 therefore d is a point 

 of contact ; also the 

 line 6 a' is produced 

 to the picture plane 

 at m ; d u and m a 

 are lines of contact 



upon each of which the height of the incli- 

 nation of the board a'f is set off as o p and 

 u p; from the points p, p draw linos to the VP. 

 Visual rays cutting these lines will give the 

 npper angles of the board, r * ; u t s r will 

 be the perspective view of the board. 



After this, we recommend the pupil to apply 

 the same directions and angles of inclination 

 in representing an equilateral triangle, making 

 the edge equal to d b at 40 with the PP ; bf, the inclination, will 

 be equal to a perpendicular from the centre of tho base to the 

 opposite angle, placed in tho plan half way between c" o'. To find 

 the perpendicular, the triangle most be separately constructed. 



