88 



THE POPULAE EDUCATOE. 



at an angle of 120, because we always prefer to make use of 

 the angle formed by the nearest approach of the projection to 

 the line of our position, or the picture plane. 



4th. Again, suppose an inclined shutter, or a roof which is 

 united horizontally with a wall, 'is said to be at an angle of 40 

 with the wall, the shutter or roof would be at an angle of 50 

 with the ground. 



All this will be very evident if we consider that " if any num- 

 ber of straight lines meet in a point in another straight line on one 

 side of it, the sum of the angles which they make with this straight 

 line, and with each other, is equal to two right angles." (See 

 Lessons in Geo- 

 metry, V., Vol. I., 

 page 156.) There- 

 fore (Fig. 67), if A 

 is 30 with the 

 pp, and B 90 with 

 A, then B will be 

 60 with the PP, 

 the whole making 

 two right angles. 

 With regard to 

 the last supposi- 

 tion, we shall see 

 that the lines of 

 the wall, the roof 

 or shutter, and 

 the ground, form 

 a right-angled tri- 

 angle, the three 

 interior angles of 

 which are together 

 equal to two right 

 angles. Therefore, 

 as the angle of the 

 wall with the 

 ground is 90, and 

 the shutter or roof 

 40 e> with the wall, 

 the shutter will be 

 at an angle of 50 

 with the horizon 

 (Fig. 68). Conse- 

 quently, this angle 

 of 50 must be 

 constructed for the 

 vanishing line, and 

 the subject treated 

 as an inclined 

 plane. (See Pro- 

 blems XXXI., 

 XXXII., and 

 XXXIII.) From 

 all this we deduct 

 a rule for finding 

 vanishing points 

 for lines or planes 

 which are stated 

 to be at given 

 angles with other 

 lines or planes not 

 parallel with the 

 picture plane : 

 When the sum of 



the two angles of the given objects is greater than a right angle, 

 it is subtracted from the sum of two right angles, and the remain- 

 der is the extent of the angle sought. This will explain the re- 

 sults of the first, second, and fourth suppositions above. 



When two angles of the given objects are together less than a '. 

 right angle, the sum will be the angle sought. This answers to 

 the third supposition. We now propose a problem to illustrate 

 our remarks about the wall and the shutter. 



PROBLEM XLI. (Fig. 69). A wall at an angle of 40 with 

 our position is pierced by a window of 4 feet 3 inches high and '. 

 4 feet broad ; a shutter projects from the top of the window at an 

 cmgle of 40 with the wall ; the window is 5 feet from the ! 

 ground, and its nearest corner is 4 feet vMhin the picture ; other j 

 conditions at pleasure. Scale of feet & 



Before proceeding to work this problem, we wish to give the 

 student some directions about the scale. In this case we 

 have given the representative fraction of the scale, and not 

 the number of feet to the inch. It is a common practice 

 with architects and engineers to name the proportion of the 

 scale upon which the drawing is made, in the manner we 

 have done here, leaving the scale to be constructed if neces- 

 sary. The meaning of the fraction ^ is that unity is divided 

 into the number of equal parts expressed by the denomi- 

 nator. Thus a scale of feet signifies that one standard 

 foot is divided into 48 equal parts, each part representing a 



foot on paper, the 

 result is inch 

 to the foot. It 

 also means that 

 the original ob- 

 ject, whether a 

 building or piece 

 of machinery, is 

 48 times larger 

 than the drawing 

 which represents 

 it. If the scale 

 had been written, 

 yards ^, it would 

 be the same as 3 

 inch to represent 

 a yard. The way 

 to arrive at this ia 

 as follows : 



inches. 



iof I = linchto 

 the foot, 

 inches. 



Ji g of 3 f = 2 inch to 

 the yard. 



The above method 

 of stating the 

 scale ought to be 

 understood by 

 every one engaged 

 upon plan -draw- 

 ing. 



To return to the 

 problem. The 

 principal consi- 

 deration relates to 

 the shutter. The 

 inclination may be 

 upwards, at an an- 

 gle of 40 with the 

 wall, or it may be 

 downwards at the 

 same angle. We 

 will represent both 

 cases. First, when 

 inclined down- 

 wards. Draw the 

 HL, which is 4 feet 

 from the ground- 

 line ; from PS draw 

 a perpendicular to 

 E ; this will be the 

 radius for drawing 



the semicircle meeting the HL to determine DE 1 and DE 2 . 

 Find the vanishing point for the wall VP 1 ., and its distance 

 point DVP 1 ; also find the VP 2 by drawing a line from E to VP S 

 at a right angle with the one from E to VP 1 , because if the 

 shutter had projected from the wall in a horizontal position, it 

 would have vanished at VP 2 ; that is, if it had been perpen- 

 dicular or at right angles with the wall. In short, the vanishing 

 point for the horizontal position of a line must always be found 

 whether the line retires to it horizontally or not, because the VP 

 for an inclined retiring line is always over or under the VP 

 (according to the angle of inclination) to which it would have 

 retired if in a horizontal position. (See Prob. XXXI., Fig. 53.) 

 Consequently, the vanishing point for an inclined retiring line 

 is found by drawing a line from, in this case, the DVP 2 , accord- 



