In this case we y inches. DVPI PS 50° HL DES 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. 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 500 scale upon which the drawing is made, in the manner we with the ground. have done here, leaving the scale to be constructed if necesAll this will be very evident if we consider that "if any num- sary. The meaning of the fraction as 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 3 Lessons in Geo foot on paper, the metry, V., Vol. I., result is inch page 156.) There to the foot. It fore (Fig. 67), if A Fig. 70. also means that is 30° with the the original ob PP, and B 90° with ject, whether A, then B will be building or piece 600 with the PP, the whole making of machinery, is 48 times larger two right angles. than the drawing With regard to which represents the last supposi it. If the scale tion, we shall see had been written, that the lines of the wall, the roof yards it would DVP3 be the same as or shutter, and inch to represent the ground, form a yard. The way a right-angled tri to arrive at this is angle, the three as follows:interior angles of which are together he equal to two right of y = 4 inch to VPI the foot. angles. Therefore, SE inches. as the angle of the of p = 1 inch to wall with the the yard. ground is 90°, and The above method the shutter or roof of stating the 40° with the wall, scale ought to be the shutter will be understood by at an angle of 50° every one engaged with the horizon (Fig. 68). Conse upon plan-draw ing. quently, this angle of 500 must be To return to the a problem. The constructed for the principal vanishing line, and the subject treated deration relates t Fig. 63. the shutter. Th as an inclined inclination may be plane. (See Problems XXXI., upwards, at an a XXXII., gle of 40 with the and XXXIII.) From wall, or it may be downwards at the all this we deduct a rule for finding same angle. W vanishing points will represent bot for lines or planes cases. First, whe inclined dom which are stated wards. Draw th to be at given урз angles with other HL, which is 4 fer from the ground lines or planes not Fig. 66. Fig. 67. line; from Ps dra parallel with the picture plane : a perpendicular : When the sum of E; this will be th the two angles of the given bbjects is greater than a right angle, the semicircle meeting the al to determine del and Di radius for drawis it is subtracted from the sum of two right angles, and the remain. Find the vanishing point for the wall vpl, and its distance der is the extent of the angle sought. This will explain the re- point dypl; also find the vp? by drawing a line from E to vi sults of the first, second, and fourth suppositions above. at a right angle with the one from E to vpl, because if the 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 fous PROBLEM XLI. (Fig. 69).-A wall at an angle of 40° with whether the line retires to it horizontally or not, because they 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 v 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would hat 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 lis conditions at pleasure. Scale of feet te is found by drawing a line from, in this case, the DVP', accor consi ing to the angle of inclination, to where it cuts a perpendicular wards, establishing its vp above the eye or HL.) Consequently, line drawn through the ypa; thus we find its vanishing point, we must draw the vanishing line for the vp3 downwards from whether its inclination be downwards or upwards; therefore DVP?. The sides of the shutter, t w and mv, must be drawn draw a line from pvp, at an angle of 500 with the HL, cutting in the direction of vps, and cut off from DVP3, first by drawing the perpendicular from vp2 at vp3, the vanishing point. We a line through t to y; make y æ equal to the length of the All the early have made the nearest corner of the window 2 feet to the shutter; draw from * to DVP3, producing w. left of the eye, represented by the distance i to b; a line from part of the problem, relating to the wall and windows, 6 must be ruled to PS, and the remaining lines upon which we wish to cut w v and t m, will be but off 4 feet to find a, the a repetition of the shutter Dearest point within; a line under the first position. Fig. 71. from c, which is 4 feet from We can prove the truth of o, must be drawn to DE', this method of drawing the and where it cuts the line perspective inclination of a ps in a is the point re plane by another method. quired. Draw the perpen Draw the right angle dionlar ahm. Draw from cad (Fig. 68); make a b DVP' through a to p; make equal to the length of the pr equal to the width of shutter, and at an angle of 40° with a cor 500 the window. Draw back again from f, cutting with a d; draw b c paovpl in s; draw the per rallel to a d; ac will be pendicular st; the base equal to the height of b above a. This must now of the window is drawn from f, on the line of be applied to Fig. 70. contact, 5 feet from the Draw a line from vpå groand, to the vpl; the through t to e on the line height of the window, of contact; make ef 4 feet 3 inches, is equal to the height of marked from s to e; b above a, viz., ca 3 line from e to vpl, (Fig. 68). Draw from cutting the perpendi s back to vp; it will culars from a and s in Fig. 69. be found to cut the m and t, will give the corner of the shutter top of the window. in w, proving by both The opening of the methods that t w is window is on th n. the perspective length Now we must draw of the further side of the shutter; the cor the shutter. Der nearest as is v, A plan of a build sonsequently it in ing may be made, clines upward towards having all its propor. the wall, but down tions, angles, and seeds from it; there other measurements fore, the vp for the arranged and noted, sbatter must be above yet nothing may be the IL, which we said as to its position have explained. To with the picturemeastire or set off the plane, and from this length of the shutter, plan several perspecwe have raised a line tive elevations may be of contact for that raised. When such is purpose from o, found the case, all that is by drawing from vp sypi necessary will be to throagh s to meet the draw a PP across the ground-line. From t paper in such a posidirected fromyp3 draw tion with the plan, s line through w; this that by drawing visual will be the farther rays, the picture-plane side of the shutter; its we have chosen may length must be deter receive the view we sumed thus:From wish to take of it. I directed from DVP3 Suppose A (Fig. 71) is draw a line to the the plan of a build. line of contact, meet ing, and we wished to ing it in y; make ya have two views of itsqual to the length of the shutter, the same as the length of the one taken with an end and front in sight, the other with a pindow ; draw from < back again to pvp?, cutting t w in w; view of the front and the opposite side-we should then place baw o o, directed by vp!, and u m directed by vps. the PP at such an angle with the side or front as might be We will now draw the shutter at the same angle with the considered to be the best for our purpose. ppl would receive all, but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would reüzerence in working the problem under these conditions arises ceive those from the front and the end C. In short, any line from the upward inclination of the shutter from the wall , but may be drawn which represents the PP at any angle with the kielined dounwards to meet the wall. This last view of the plan, or opposite any side we may wish to project . This will position of the shatter is the proper one for our purpose, because give a very useful illustration of the way to treat a subject siter a little consideration we shall perceive that it is a retiring when its proportions are given, as is frequently the case, withplane, but dmonwards ; therefore its vp is below the eye or Hl. out any reference to the view to be taken of it; in other words, In the former case the shutter was a retiring plane, but up the angle it forms with the picture-plane. ma 22 DE vp 2 HL DVD/ Ovp2 In this case we inches. VPI PS HL DE2 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. 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 ve with the ground. have done here, leaving the scale to be constructed if necesAll this will be very evident if we consider that "if any num- sary. The meaning of the fraction 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 denomiside 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, V., Vol. I., result is inch page 156.) There to the foot. It fore (Fig. 67), if A Fig. 70. also means that is 30° with the the original obPP, and B 90° with ject, whether a A, then B will be 600 with the PP, building or piece of machinery, is the whole making 48 times larger two right angles. than the drawing With regard 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 DVP3 be the same as or shutter, and inch to represent the ground, form a yard. The way a right-angled tri to arrive at this is angle, the three as follows interior angles of which are together of Y = 1 inch to equal to two right the foot. angles. Therefore, SE DVPI inches. as the angle of the of 30 = 4 inch to wall with the the yard ground is 90°, and The above method the shutter or roof 400 with the wall, of stating the scale onght to be the shutter will be understood by at an angle of 50° every one engaged with the horizon upon plan-draw (Fig. 68). Conse ing. quently, this angle To return to th of 500 must be a problem. Th constructed for the principal consi vanishing line, and deration relatest the subject treated Fig. 68. the shutter. TI as an inclined inclination may be plane. (See Problems XXXI., upwards, at an at gle of 40° with the XXXII., and wall, or it may XXXIII.) From downwards at ti all this we deduct a rule for finding same angle. W will represent bot vanishing points cases. First, whe for lines or planes inclined dom which are stated wards. Draw to be at given vp3 HL, which is 4 fe angles with other from the groun lines or planes not Fig. 66. Fig. 67. line; from Ps dra parallel with the a perpendicular picture plane : E; this will be th When the sum of the two angles of the given bbjects is greater than a right angle, the semicircle meeting the al to determine del and DI radius for drawir it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vpl, and its distan der is the extent of the angle sought. This will explain the re- point dypd; also find the vp by drawing a line from a to v sults of the first, second, and fourth suppositions above. at a right angle with the one from E to vpl, because if ti 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 four PROBLEM XLI. (Fig. 69).-A wall at an angle of 40° with whether the line retires to it horizontally or not, because they 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 hat 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 lis conditions at pleasure. Scale of feet to is found by drawing a line from, in this case, the Dyp, accon A ing to the angle of inclination, to where it cuts a perpendicular wards, establishing its vp above the eye or HL.) Consequently, line drawn through the vp; thus we find its vanishing point, we must draw the vanishing line for the vp3 downwards from whether its inclination be downwards or upwards; therefore DVP?. The sides of the shutter, t w and mv, must be drawn draw a line from pvp, at an angle of 50° with the HL, cutting in the direction of vp3, and cut off from DVP3, first by drawing the perpendicular from vp at vp3, the vanishing point. We a line through t to y; make y a equal to the length of the All the early have made the nearest corner of the window 2 feet to the shutter; draw from æ to DVP), producing w. left of the eye, represented by the distance i to b; a line from part of the problem, relating to the wall and windows, must be ruled to PS, and the remaining lines w v and t m, will be but upon which we wish to cut a repetition of the shutter off 4 feet to find a, the under the first position. nearest point within; a line Fig. 71. from e, which is 4 feet from We can prove the truth of b, must be drawn to DE', this method of drawing the and where it cuts the line perspective inclination of a ó ps in a is the point re plane by another method. quired. Draw the perpen Draw the right angle A dionlar ahm. Draw from cad (Fig. 68); make a b Dup' through a to p; make equal to the length of the pr equal to the width of shutter, and at an angle of 40° with a cor 500 the window. Draw back again from r, cutting with a d; draw b c paDyplin s; draw the per rallel to a d; a c will be pendicular st; the base equal to the height of b of the window is drawn above a. This must now from f, on the line of be applied to Fig. 70. contact, 5 feet from the Draw a line from Vpa ground, to the vpl; the through t to e on the line height of the window, of contact; make e f 4 feet 3 inches, is equal to the height of marked from s to e; b above a, viz., ca a line from e to vpl, (Fig. 68). Draw from catting the perpendi. s back to vp?; it will enlars from a and s in Fig. 69. be found to cut the and t, will give the corner of the shutter top of the window. in w, proving by both The opening of the methods that t w is vindow is in th n. the perspective length Now we must draw of the further side of the shutter; the cor the shutter. Der Dearest us is v, A plan of a builddonsequently it in ing may be made, d'ines upward towards having all its propor. the wall, but down tions, angles, and wards from it; there other measurements fore, the vp for the arranged and noted, shutter must be above yet nothing may be the HL, which we said as to its position have explained. To with the picturemeasiire or set off the plane, and from this length of the shutter, plan several perspec"we have raised a line tive elevations may be contact for that raised. When such is purpose from o, found the case, all that is by drawing from VP vpi necessary will be to PLS filarongh s to meet the draw a PP across the Ezoand-line. From t paper in such a posidirected from yp3 draw tion with the plan, line throngh w; this that by drawing visual will be the further rays, the picture-plane ride of the shutter; its we have chosen may length must be deter receive the view we mined thus From wish to take of it. i directed from DVPS DVP3! Suppose A (Fig. 71) is trae a line to the the plan of a build. line of contact, meet ing, and we wished to ing it in y; make y z have two views of itsenal to the length of the shutter, the same as the length of the one taken with an end and front in sight, the other with a pindow ; draw from a back again to pvp, cutting t w in w; view of the front and the opposite side-we should then place daw na y, directed by vpl, and v m directed by VP3. the PP at such an angle with the side or front as might be We will now draw the shutter at the same angle with the considered to be the best for our purpose. ppl would receive real, but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would rediference in working the problem under these conditions arises ceive those from the front and the end o. In short, any line bom the upward inclination of the shutter from the wall, but may be drawn which represents the PP at any angle with the inclined downwards to meet the wall. This last view of the plan, or opposite any side we may wish to project. This will position of the shutter is the proper one for our purpose, because give a very useful illustration of the way to treat a subject after a little consideration we shall perceive that it is a retiring when its proportions are given, as is frequently the case, with. plane , but downwards ; therefore its vp is below the eye or Hl. out any reference to the view to be taken of it; in other words, In the former case the shutter was a retiring plane, but up the angle it forms with the picture-plane. 22 HL DVD/ lovp2 In this case we m y n VPI DVPI PS HL DEP inches. 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. 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 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 denomiside of it, the sum of the angles which they make with this straight nator. Thus a scale of feet is sigpifies 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, V., Vol. I., result is inch page 156.) There. to the foot. It fore (Fig. 67), if A Fig. 70. also means that is 30° with the the original obPP, and B 90° with ject, whether & A, then B will be building or piece 600 with the PP, of machinery, is the whole making 48 times larger two right angles. than the drawing With regard to which represents the last supposi it. If the scale tion, we shall see had been written, that the lines of yards a, it would the wall, the roof DVP3 be the same as or shutter, and inch to represent the ground, form a yard. The way a right-angled tri to arrive at this is angle, the three as follows: interior angles of inches. which are together of 4 = 4 inch to equal to two right the foot angles. Therefore, SEI 50° DVP2 as the angle of the of * = 4 inch te wall with the the yard ground is 90°, and The above method the shutter or roof of stating the 40° with the wall, scale ought to be the shutter will be understood by at an angle of 50° every one engage with the horizon (Fig. 68). Conse upon plan-draw ing. quently, this angle To return to the of 500 must be d problem. Th constructed for the principal vanishing line, and deration relatest the subject treated Fig. 63. the shutter. TL as an inclined inclination may b plane. (See Problems XXXI., upwards, at an ar XXXII., gle of 40° with the and wall, or it may ! XXXIII.) From downwards at th all this we deduct same angle. W a rule for finding will represent bot Vanishing points for lines or planes cases. First, whe inclined dow which are stated wards. Drawth to be at given v P3 HL, which is 4 fel angles with other from the ground lines or planes not Fig. 66. Fig. 67. line; from Ps dra parallel with the picture plane :BA a perpendicular E; this will be the When the sum of radius for drawis the two angles of the given bbjects is greater than a right angle, the semicircle meeting the ul to determine del and DI it is subtracted from the sum of two right angles, and the remain. Find the vanishing point for the wall vpl, and its distans der is the extent of the angle sought. This will explain the re- point pvp?; also find the vp by drawing a line from a to sults of the first, second, and fourth suppositions above. at a right angle with the one from a to vpl, because if 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 vpa; that is, if it had been perpet 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 fou PROBLEM XLI. (Fig. 69).--A wall at an angle of 40° with whether the line retires to it horizontally or not, because they 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 ha 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 lir conditions at pleasure. Scale of feet to is found by drawing a line from, in this case, the DVP*, DUCON a consi 6 |