inches. DVPI PS HL DEO 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. 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 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 2 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 , 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 = 4 inch to equal to two right VP the foot. angles. Therefore, SE 50° OVP2 as the angle of the of 38 = 1 inch to wall with the the yard. ground is 90°, and The above method the shutter or roof of stating the 400 with the wall, scale onght to be the shutter will be understood by at an angle of 50° every one engaged with the horizon upon plan- drar (Fig. 68). Conse ing. quently, this angle To return to the of 500 must be a problem. The constructed for the principal vanishing line, and deration relates to the subject treated Fig. 68. the shutter. The as an inclined inclination may be plane. (See Problems XXXI., upwards, at an an XXXII., gle of 40° with the and wall, or it may be XXXIII.) From downwards at the all this we deduct a rule for finding same angle. We will represent both Vanishing points for lines or planes cases. First, when inclined downwhich are stated wards. Draw the to be at given HL, which is 4 feet angles with other from the ground lines or planes not Fig. 66. Fig. 67. line; from Ps dras parallel with the picture plane : a perpendicular to E; this will be the When the sum of radius for drawing the two angles of the given bbjects is greater than a right angle, the semicircle meeting the ul to determine del and Da 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 Dvpl; also find the vp by drawing a line from a to vrl 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, it right angle, the sum will be the angle sought. This answers to would have vanished at vp®; that is, if it had been perpenthe third supposition. We now propose a problem to illustrate dicular or at right angles with the wall. In short, the vanishing our remarks about the wall and the shutter. point for the horizontal position of a line must always be found PROBLEM XLI. (Fig. 69).--A wall at an angle of 40° with whether the line retires to it horizontally or not, because the vi 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 ye 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would have angle of 400 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 ling conditions at pleasure. Scale of feet te is found by drawing a line from, in this case, the Dyr*, accord a. VP3 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 m v, must be drawn draw a line from DVP, at an angle of 50° with the HL, cutting in the direction of vps, and cut off from DVP3, first by drawing the perpendicular from vpå 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 a 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, b 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. Dearest point within; a line 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 6 Ps in a is the point re plane by another method. Draw the right angle quired. Draw the perpendionlar a hm. Draw from ca d (Fig. 68); make a b DVP' through a to p; make equal to the length of the shutter, and at an angle pr equal to the width of of 40° with a cor 500 the window. Draw back again from r, cutting with a d; draw b c padypl in 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 VP ground, 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 f to e; b above a, viz., ca 2 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 E 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 in 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 buildconsequently it in ing may be made, slines upward towards having all its proporthe wall, but down tions, angles, and terds from it; there other measurements lore, the vp for the arranged and noted, shutter must be above yet nothing may be the HL, which we said as to its position kare explained. To with the picturemeasure or set off the plane, and from this length of the shutter, plan several perspecwe have raised a line tive elevations may be V contact for that raised. When such is purpose from o, found the case, all that is by drawing from vp svot necessary will be to pls farough s to meet the draw a PP across the poand-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 further rays, the picture-plane side of the shutter; its we have chosen may length must be deter receive the view we mined thus From wish to take of it. A 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 indow ; draw from a back, again to dvps, cutting t w in w; view of the front and the opposite side we should then place deze w s, 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 I, 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 C. In short, any line on 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, withMore, 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, La the former case the shutter was a retiring plane, but up- the angle it forms with the picture-plane. m 22 DI Vp 2 HL DVD lovp2 DVP3 In this case we n DVPI PS HL DEP 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 400 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 necesAl 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 to 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 2 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 yards, it would the wall, the roof DNVP3 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 inches. h! of = 1 inch to equal to two right VPI the foot angles. Therefore, SE Vpz inches. as the angle of the of 39 = inch to wall with the the yard. ground is 90°, and The above method the shutter or roof of stating the 400 with the wall, scale onght to be the shutter will be understood by at an angle of 50° every one engaged with the horizon la upon plan-draw (Fig. 68). Conse ing. quently, this angle To return to the of 500 must be problem. The constructed for the principal vanishing line, and deration relates te the subject treated Fig. 68. the shutter. The as an inclined inclination may by plane. (See Problems upwards, at an an XXXI., 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 will represent bot vanishing points for lines or planes cases. First, whe inclined down which are stated wards. Draw th to be at given vp3 HL, which is 4 fee angles with other from the ground lines or planes not Fig. 66. Fig. 67. line; from Ps dras parallel with the picture plane : a perpendicular When the sum of E; this will be the radius for drawin the two angles of the given objects is greater than a right angle, the semicircle meeting the HL to determine del and DE it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vpl, and its distaze der is the extent of the angle sought. This will explain the re- point dypl; also find the vp by drawing a line from a to vi 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, i right angle, the sum will be the angle sought. This answers to would have vanished at vp; that is, if it had been perpen 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 the 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 har angle of 400 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 the is found by drawing a line from, in this case, the Dyrs, accord consi с B 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 m v, must be drawn draw a line from pvp, at an angle of 50° with the HL, cutting in the direction of vp), and cut off from DVP3, first by drawing the perpendicular from vp3 at vp.), 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 a 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, b 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 0, must be drawn to DE', this method of drawing the and where it cuts the line perspective inclination of a • pg in a is the point re plane by another method. quired. Draw the perpen Draw the right angle dionlar a hm. Draw from ca d (Fig. 68); make a b DUP' throngh a to p; make equal to the length of the snutter, and at an angle pr equal to the width of of 40° with a cor 500 the window. Draw back again from T, cutting with a d; draw bc paByplin s; draw the per rallel to a d; ac 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 Vpo groand, to the vpl; the through t to e on the line height of the window, of contact; make ef feet 3 inches, is equal to the height of marked from f to e; b above a, viz., ca 2 line from e to vp', (Fig. 68). Draw from catting the perpendi s back to vp'; it will clars from a and s in Fig. 69. be found to cut the In 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 in th n. the perspective length Now we must draw of the further side of the shutter; the cor the shutter. Der nearest tis is v, A plan of a build. Bonsequently it in ing may be made, cines upward towards having all its propor. the scall, but down tions, angles, and words from it; there other measurements lore, the VP for the arranged and noted, shutter must be above yet nothing may be the ri, which we said as to its position Lave explained. To with the picturemeasure or set off the plane, and from this length of the shutter, plan several perspecwe have raised a line tive elevations may be ut contact for that raised. When such is purpose from o, found the case, all that is by drawing from vp? DV.PL/ vpi necessary will be to through s to meet the pls draw a PP across the grand-line. From t paper in such a posidirected fromyP3 draw tion with the plan, a line through w; this that by drawing visual will be the further rays, the picture-plane side of the shutter; its we have chosen may length must be deter receive the view we mined thus From wish to take of it. directed from DVP3 ovp.3 Suppose A (Fig. 71) is inaw a line to the the plan of a build. lize of contact, meet ing, and we wished to log it in y; make y z have two views of itgal 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 indow ; draw from æ back, again to pvp, cutting t w in w; view of the front and the opposite side-we should then place traw wv, directed by vp!, 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 all , but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would reeference in working the problem under these conditions arises ceive those from the front and the end c. In short, any line born the upward inclination of the shutter from the wall , but may be drawn which represents the PP at any angle with the clined 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 fter a little consideration we shall perceive that it is a retiring when its proportions are given, as is frequently the case, withMore, 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. 20 DE1 vp 2 HL lovp2 In this case we inches. n VPI DVPI PS HL the yard. 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 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 1 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 2 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, 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 h of Y= inch to equal to two right the foot angles. Therefore, SEI Vpz 50 OVP2 inches. as the angle of the of 39 = { inch to wall with the ground is 90°, and The above method the shutter or roof 40° with the wall, of stating the scale ought to be the shutter will be understood by at an angle of 50° with the horizon every one engaged d upon plan- draw (Fig. 68). Conse ing. quently, this angle To return to the of 500 must be d. problem. The constructed for the principal vanishing line, and deration relates to the subject treated Fig. 68. the shutter. The as an inclined inclination may be plane. (See Problems XXXI., upwards, at an an XXXII., gle of 40° with the and XXXIII.) From wall, or it may be all this we deduct downwards at the a rule for finding same angle. We will represent both vanishing points for lines or planes cases. First, wher inclined down which are stated wards. Draw th to be at given HL, which is 4 fed angles with other from the ground lines or planes not Fig. 66. Fig. 67. line; from ps dras parallel with the a perpendicular picture plane : E; this will be the When the sum of radius for drawing the two angles of the given bbjects is greater than a right angle, the semicircle meeting the ul to determine DEl and Dr it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vp!, and its distant der is the extent of the angle sought. This will explain the re-point pvpl; also find the vp by drawing a line from s to v sults of the first, second, and fourth suppositions above. at a right angle with the one from a 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, i right angle, the sum will be the angle sought. This answers to would have vanished at vp; that is, if it had been perpen the third supposition. We now propose a problem to illustrate dicular or at right angles with the wall. In short, the vanishing our remarks about the wall and the shutter. point for the horizontal position of a line must always be found PROBLEM XLI. (Fig. 69).- A wall at an angle of 40° with | whether the line retires to it horizontally or not, because the 11 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 vi 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would har angle of 400 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 bin conditions at pleasure. Scale of feet the is found by drawing a line from, in this case, the DVP", accord a consi B |