VPI DVPI PS HL DE2 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 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 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, 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 building or piece 600 with the PP, of machinery, is the whole making X 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 be the same as DNP3 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 h of 4 = 4 inch to equal to two right the foot. angles. Therefore, SEI 50° OVP2 as the angle of the of 31 = { 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 ought to be the shutter will be understood by at an angle of 50° every one engaged with the horizon upon plan-dram. (Fig. 68). Conse ing. quently, this angle To return to the of 500 must be a problem. The constructed for the principal consi vanishing line, and deration relates to the subject treated Fig. 63. the shutter. The as an inclined inclination may be plane. (See Pro upwards, at an anblems XXXI., gle of 40° with the XXXII., and wall, or it may be XXXIII.) From downwards at the all this we deduct same angle. We a rule for finding will represent both Vanishing points cases. First, when for lines or planes inclined downwhich are stated wards. Draw the to be at given VP3 HL, which is 4 feet angles with other from the groundlines or planes not Fig. 66. Fig. 67. line; from is draw parallel with the В. a perpendicular to 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 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 E to vp 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 abont 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 VP 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 VP 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would bare 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 line conditions at pleasure. Scale of feet te is found by drawing a line from, in this case, the DVP", accord 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 DVP2, at an angle of 50° with the hi, cutting in the direction of vps, and cut off from DVP3, first by drawing the perpendicular from vp at vp), the vanishing point. We a line through t to y; make y a equal to the length of the have made the nearest corner of the window 2 feet to the shutter; draw from « to DVP3, producing w. All the early 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 nearest point within; a line under the first position. Fig. 71. from c, 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 b ps in a is the point re plane by another method. quired. Draw the perpen A Draw the right angle dicular a hm. 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 the window. Draw back of 40° with a cor 500 again from r, cutting with a d; draw b c paDvpl 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 vp2 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 f to e; vp3 babove a, viz., ca a 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 mth n. the perspective length Now we must draw of the further side of the shutter; the cor the shutter. ner nearest us is v, A plan of a build. consequently it in ing may be made, clines uproard towards having all its proporthe wall, but down tions, angles, and vards 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 picturemeasure or set off the plane, and from this length of the shutter, plan several perspecThe 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 yp? vpi necessary will be to through s to meet the DEL draw a PP across the ground-line. From t paper in such a posidirected fromvp3 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 t directed from DVP3 wish to take of it. DVP3! Suppose A (Fig. 71) is draw & line to the the plan of a build. line of contact, meet ing, and we wished to ing it in y; make y x have two views of itequal 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 window; draw from back again to pvp, cutting t w in w; view of the front and the opposite side-we should then place draw w v, 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 wall, but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would redifference 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 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 shatter 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, withplone, 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 VP 2 HL DVD/ 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. Fig. 70 f to the foot. It fore (Fig. 67), if A g. U. also means that is 30° with the 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 u 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 write, that the lines of yards #, it would the wall, the roof ground is 90°, and The above method the shutter or roof of stating the 40° with the wall, scale ought to bi 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 problem. Thi constructed for the principal consi vanishing line, and deration relatest the subject treated the shutter. To as an inclined inclination mayb plane. (See Pro- upwards, at anal blems XXXI., gie of 40° withth XXXII., and wall, or it may b XXXIII.) From downwards at th all this we deduct same angle. W 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 ground lines or planes not line; from Ps dra parallel with the a perpendicular: picture plane:— r; this will be th 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 Do 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 ove"; also find the ve” by drawing a line from E to Yo sults of the first, second, and fourth suppositions above. at a right angle with the one from E to ve", because if t 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 ve”; 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 the V 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 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 lit conditions at pleasure, Scale of feet #. is found by drawing a line from, in this case, the Dvr", accor Fig. 69. ing to the angle of inclination, to where it cuts a perpendicular line drawn through the vr"; thus we find its vanishing point, whether its inclination be downwards or upwards; therefore draw aline from DvP*, at an angle of 50° with the HL, cutting the perpendicular from vro 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 tearest 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 him. Draw from DVF'through a top; make pr equal to the width of the window. Draw back again from r, cutting w?" ins; draw the perpendicular st; the base of the window is drawn from f on the line of contact, 5 feet from the fronnd, to the ve"; the height of the window, 4 feet 3 inches, is marked from f to e : a line from e to ve!, tisting the perpendiculars from a and s in * and t, will give the top of the window. The opening of the window is m t h m. Now we must draw wards, establishing its we above the eye or HL.) Consequently, we must draw the vanishing line for the ve" downwards from DVP2. 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 z 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 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 d of contact, meet*it in y; make y z * to the length of the shutter, the same as the length of the *dow; draw from a back again to Dvip”, cutting t w in w; *: wo, directed by vo. and on directed by vo. ** will now draw the shutter at the same angle with the * but inclined upwards from it (Fig. 70). The important in working the problem under these conditions arises o the upward inclination of the shutter from the wall, but *d downwards to meet the wall. This last view of the **on 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 H.L. * 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 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 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 p ...” 48 times larger two right les. than the drawing With re to ^ t- 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 i or shutter, and inch to represent the ground, form 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 h is of f = ′ incho equal to two right --- wpi inch the foot angles. Therefore, se” * = ... to as the angle of the # of * = #. to wall with the e ground is 90°, and The above method the shutter or roof of stating the 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 problem. Th constructed for the principal cons vanishing line, and deration relatest the subject treated the shutter. Th as an inclined inclination mayb plane. (See Pro- upwards, at anal blems XXXI., gie of 40° with to XXXII., and wall, or it may 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 dow which are stated wards. Draw to to be at given HL, which is 4fe angles with other from the ground lines or planes not line; from Ps parallel with the a perpendicular picture plane:— E; this will bet When the sum of radius for drawi: the two angles of the given objects is greater than a right angle, the semicircle meeting the HL to determine DE' and no 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. This will explain the re- point DvP"; also find the vro 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 if t 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 ve”; 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 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 lit conditions at pleasure. Scale of feet #. is found by drawing a line from, in this case, the DVF, accor |