ing to the angle of inclination, to where it cuts a perpendicular wards, establishing its yf above the eye or HL). Consequently, he drawn through the vo"; thus we find its vanishing point, we must draw the vanishing line for the ve” downwards from whether its inclination be downwards or upwards; therefore | DvP*. The sides of the shutter, t w and m v, must be drawn inwaline from Dvee, at an angle of 50° with the HL, cutting in the direction of ve", and cut off from DVP", first by drawing the perpendicular from ve" at ve”, the vanishing point. We a line through t to y; make y a equal to the length of the hare made the nearest corner of the window 2 feet to the shutter; draw from a to Dve", producing w. All the early lost of the eye, represented by the distance i to b; a line from part of the problem, relating to the wall and windows, h must be ruled to Ps, upon which we wish to cut of 4 feet to find a, the nearest point within; a line from c, which is 4 feet from l, must be drawn to DE', and where it cuts the line brs in a is the point required. Draw the perpendicular a hom. Draw from pro'through a top; make pr equal to the width of the window. Draw back again from r, cutting BWP' ins; draw the perpendicular st; the base of the window is drawn from f, on the line of contact, 5 feet from the ground, to the ve"; the height of the window, # feet 3 inches, is marked from f to e : 1 line from e to ve!, miting the perpendialars from a and s in mand t, will give the top of the window. The opening of the window is on th m. Now we must draw and the remaining lines w v and t m, will be but a repetition of the shutter under the first position. We can prove the truth of this method of drawing the perspective inclination of a plane by another method. Draw the right angle c a d (Fig.68); make a b equal to the length of the shutter, and at an angle of 40° with a c or 50° with a d, draw b c parallel to a d, a c will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from VPthrough t to e on the line of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from fback to ve”; it will be found to cut the corner of the shutter in w, proving by both methods that t w is the perspective length of the further side of draw a PP across the paper in such a position with the plan, that by drawing visual rays,the picture-plane we have chosen may receive the view we wish to take of it. Suppose A (Fig. 71) is ** a line to the of contact, meet#in y; make y z * to the length of the shutter, the same as the length of the *::: draw from a back, again to Dve”, cutting t w in w; ***, directed by ve", and v m directed by ves. We will now draw the shutter at the same angle with the *ill, but inclined upwards from it (Fig. 70). The important forence in working the problem under these conditions arises * the upward inclination of the shutter from the wall, but *d downwards to meet the waii. This last view of the otion of the shutter is the proper one for our purpose, because **little consideration we shall perceive that it is a retiring * but downwards; therefore its vris below the eye or no. * the former case the shutter was a retiring plane, but up the plan of a building, and we wished to have two views of it— one taken with an end and front in sight, the other with a view of the front and the opposite side—we should then place the PP at such an angle with the side or front as might be considered to be the best for our purpose. FP" would receive the visual rays from the front and the end B; PP” would receive those from the front and the end c. In short, any line may be drawn which represents the PP at any angle with the plan, or opposite any side we may wish to project. This will give a very useful illustration of the way to treat a subject when its proportions are given, as is frequently the case, without any reference to the view to be taken of it; in other words, the angle it forms with the picture-plane. 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 number 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.” Lessons in Geometry, W., Wol. I., page 156.) Therefore (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 les. With re to the last supposition, we shall see that the lines of the wall, the roof or shutter, and the ground, form a right-angled triangle, 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° with the wall, the shutter will be at an angle of 50° with the horizon (Fig. 68). Consequently, this angle of 50° must be constructed for the vanishing line, and the subject treated as an inclined plane. (See Problems 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 Fig. 70. 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 : inch to represent a yard. The way to arrive at thisis as follows:– Ing. To return toth problem. Th principal cons deration relatest the shutter. To inclination mayo upwards, at ana: gie of 40° withth wall, or it mayb downwards atti same angle. W will representbo cases. First, who inclined down wards. Draw to HL, which is 4fe from the ground line; from Pso: a perpendicn E; this will be: radius for drawin the two angles of the given bbjects is greater than a right angle, the semicircle meeting the HL to determine D.E. and Dr. it is subtracted from the sum of two right angles, and the remain. Find the vanishing point for the wall ve", and its distan" der is the ertent of the angle sought. right angle, the sum will be the angle sought. the third supposition. We now propose a problem to illustrate | dicular or at right angles with the wall. 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 angle of 40° with the walls the window is 5 feet from the ground, and its nearest corner is 4 feet within the picture; other conditions at pleasure, This will explain the re- point Dve"; also find the ve” by drawing a line from E to Yo at a right angle with the one from E to wr', because # * When two angles of the given objects are together less than a shutter had projected from the wall in a horizontal position. retired if in a horizontal position. This answers to would have vanished at vp”; that is, if it had been perpo In short, the vanishin point for the horizontal position of a line must always be foun PRoBLEM XLI. (Fig. 69).-A wall at an angle of 40° with whether the line retires to it horizontally or not, because the * for an inclined retiring line is always over or under the w (according to the angle of inclination) to which it would ho Consequently, the vanishing point for an inclined retiring in is found by drawing a line from, in this case, the DVP", accor ing to the angle of inclination, to where it cuts a perpendicular | he drawn through the vr"; thus we find its vanishing point, whether its inclination be downwards or upwards; therefore inwaline from Dwr', at an angle of 50° with the HL, cutting the perpendicular from vp” at ve", the vanishing point. We hare made the nearest corner of the window 2 feet to the left of the eye, represented by the distance i to b; a line from } must be ruled to Ps, upon which we wish to cut of 4 feet to find a, the nearest point within; a line from c, which is 4 feet from !, must be drawn to DE', and where it cuts the line *Ps in a is the point required. Draw the perpendicular a hom. Draw from pop"through a top; make pr equal to the width of the window. Draw back again from r, cutting by?" ins; draw the perpendicular st; the base of the window is drawn from f on the line of contact, 5 feet from the fromd, to the ve"; the hight of the window, feet 3 inches, is marked from f to e, 1 inefrom e to ve", miting the perpendialars from a and s in sandt, will give the top of the window. The opening of the window is m t h m. Now we must draw wards, establishing its vp above the eye or HL.) Consequently, we must draw the vanishing line for the ve" downwards from DvP*. The sides of the shutter, t w and m v, must be drawn in the direction of ve", and cut off from DvP", first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to Dve”, producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines w v and t m, will be but a repetition of the shutter under the first position. We can prove the truth of this method of drawing the perspective inclination of a plane by another method. Draw the right angle c a d (Fig. 68); make a b equal to the length of the shutter, and at an angle of 40° with a c or 50° with a d: draw b c parallel to a d, a c will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from VP” through t to e on the line of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from f back to ve”; it will be found to cut the corner of the shutter in w, proving by both methods that t w is the perspective length of the further side of paper in such a position with the plan, that by drawing visual rays,the picture-plane we have chosen may receive the view we wish to take of it. Suppose A (Fig. 71) is ** a line to the * of contact, meet. **i; in y; make y z * to the length of the shutter, the same as the length of the ; draw from a back, again to Dve”, cutting t w in w; ***, directed by ve", and v m directed by ves. **ill now draw the shutter at the same angle with the o, but inclined upwards from it (Fig. 70). The important *nce in working the problem under these conditions arises o the upward inclination of the shutter from the wall, but * downwards to meet the wall. This last view of the *ion of the shutter is the proper one for our purpose, because ** little consideration we shall perceive that it is a retiring * but downwards; therefore its vris below the eye or HL. * the former case the shutter was a retiring plane, but up d the plan of a building, and we wished to have two views of it— one taken with an end and front in sight, the other with a view of the front and the opposite side—we should then place the PP at such an angle with the side or front as might be considered to be the best for our purpose. rp' would receive the visual rays from the front and the end B; PP” would receive those from the front and the end c. In short, any line may be drawn which represents the PP at any angle with the plan, or opposite any side we may wish to project. This will give a very useful illustration of the way to treat a subject when its proportions are given, as is frequently the case, without any reference to the view to be taken of it; in other words, the angle it forms with the picture-plane. at an angle of 120°, because we always prefer to make use of Before proceeding to work this problem, we wish to give the the angle formed by the nearest approach of the projection to student some directions about the scale. In this case we the line of our position, or the picture plane. have given the representative fraction of the scale, and not 4th. Again, suppose an inclined shutter, or a roof which is the number of feet to the inch. It is a common practice united horizontally with a wall, is said to be at an angle of 40° with architects and engineers to name the proportion of the with the wall, the shutter or roof would be at an angle of 50° scale upon which the drawing is made, in the manner we with the ground. have done here, leaving the scale to be constructed if neces. All this will be very evident if we consider that “if any num- sary. The meaning of the fraction is is that unity is divided ber of straight lines meet in a point in another straight line on one into the number of equal parts expressed by the denomi. side of it, the sum of the angles which they make with this straight | nator. Thus a scale of feet is signifies that one standard line, and with each other, is equal to two right angles.” (See foot is divided into 48 equal parts, each part representing a Lessons in Geo- foot on paper, the metry, W., Wol. I., result is ; inch page 156.) There- - - - to the foot. It fore (Fig. 67), if A Fig. 70. f also means that is 30° with the 2^ the original ob. PP, and B 90° with E. ject, whether a A, then B will be building or piece 60° with the PP, of machinery, is the whole making ar 48 times larger two right les. than the drawing With re to ^ to- which represents the last supposi- it. If the scale tion, we shall see had been written, that the lines of yards #, it would the wall, the roof D m be the same as : or shutter, and inch to represent the ground, form s a yard. The way a right-angled tri- y to arrive at this is angle, the three as follows:interior angles of - inches. which are together h fo # of f = | incho equal to two right | / - wpi. inch the foot angles. Therefore, se” * , ; as the angle of the * of * = #: wall with the eyard ground is 90°, and The above method the shutter or roof of stating thi 40° with the wall, scale ought to b the shutter will be understood by at an angle of 50° every one en with the horizon upon plan-draw (Fig. 68). Conse- ing. quently, this angle To return toth of 50° must be o problem. Th constructed for the principal cons vanishing line, and deration relatest the subject treated the shutter. To as an inclined inclination mayb plane. (See Pro- upwards, at ana: blems XXXI., gle of 40° with th XXXII., and c wall, or it may b XXXIII.) From downwards atti all this we deduct same angle. a rule for finding will representbol vanishing points cases. First, who for lines or planes inclined down which are stated wards. Draw to to be at given HL, which is 4fe angles with other from the groun lines or planes not Fig. 67. line; from Ps dra parallel with the a perpendicular picture plane:— E; this will be to When the sum of radius for drawin the two angles of the given objects is greater than a right angle, the semicircle meeting the HL to determine DE' and po it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall ve", and its distan" der is the extent of the angle sought. This will explain the re. point Dve"; also find the ve” by drawing a line from E to . sults of the first, second, and fourth suppositions above. at a right angle with the one from E to ve", because # th When two angles of the given objects are together less than a shutter had projected from the wall in a horizontal position, right angle, the sum will be the angle sought. This answers to would have vanished at vp”; that is, if it had been perpe the third supposition. We now propose a problem to illustrate | dicular or at right angles with the wall. In short, the vanishin our remarks about the wall and the shutter. point for the horizontal position of a line must always be foun PRobLEM XLI. (Fig. 69).-A wall at an angle of 40° with whether the line retires to it horizontally or not, because *: our position is pierced by a window of 4 feet 3 inches high and for an inclined retiring line is always over or under the 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would has angle of 40° with the wall; the window is 5 feet from the retired if in a horizontal position. (See Prob. XXXI., Fig. 53. ground, and its nearest corner is 4 feet within the picture; other Consequently, the vanishing point for an inclined retiring lin conditions at pleasure. Scale of feet #. is found by drawing a line from, in this case, the DVP, accord * to the length of the shutter, the same as the length of the ; draw from a back, again to Dve”, cutting t w in w; ***. directed by ve", and v m directed by ves. *will now draw the shutter at the same angle with the *all, but inclined upwards from it (Fig. 70). The important *nce in working the problem under these conditions arises * the upward inclination of the shutter from the waii, but wards, establishing its ve above the eye or HL.) Consequently, we must draw the vanishing line for the ve" downwards from The sides of the shutter, t w and m v, must be drawn in the direction of ve”, and cut off from DVP", first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to Dve”, producing w. part of the problem, relating to the wall and windows, All the early and the remaining lines w v and t m, will be but a repetition of the shutter under the first position. We can prove the truth of this method of drawing the perspective inclination of a plane by another method. Draw the right angle c a d (Fig. 68); make a b equal to the length of the shutter, and at an angle of 40° with a c or 50° with a d, draw b c parallel to a d, a c will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from VPthrough t to e on the line of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from f back to ve”; it will be found to cut the corner of the shutter in w, proving by both methods that t w is the perspective length of the further side of the shutter. A plan of a building may be made, having all its proportions, angles, and other measurements arranged and -noted, yet nothing may be said as to its position with the pictureplane, and from this plan several perspective elevations maybe raised. When such is the case, all that is necessary will be to draw a PP across the paper in such a position with the plan, that by drawing visual rays,the picture-plane we have chosen may receive the view we wish to take of it. Suppose A (Fig. 71) is *ed downwards to meet the wan. --- This last view of the Position of the shutter is the proper one for our purpose, because ** little consideration we shall perceive that it is a retiring * but downwards; therefore its vp is below the eye or HL. * the former case the shutter was a retiring plane, but up the plan of a building, and we wished to have two views of it— one taken with an end and front in sight, the other with a view of the front and the opposite side—we should then place the PP at such an angle with the side or front as might be considered to be the best for our purpose. rp' would receive the visual rays from the front and the end B; PP” would receive those from the front and the end c. In short, any line may be drawn which represents the PP at any angle with the plan, or opposite any side we may wish to project. This will give a very useful illustration of the way to treat a subject when its proportions are given, as is frequently the case, without any reference to the view to be taken of it; in other words, the angle it forms with the picture-plane. |