ing to the angle of inclination, to where it cuts a perpendicular line drawn through the VP; thus we find its vanishing point, whether its inclination be downwards or upwards; therefore draw a line from DVP2, at an angle of 50° with the HL, cutting the perpendicular from vps at VP3, the vanishing point. We have 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 b must be ruled to PS, upon which we wish to cut off 4 feet to find a, the

we

ave explained. To leasure or set off the ength of the shutter, re have raised a line contact for that urpose from o, found y drawing from VP2 hrough s to meet the round-line. From t irected fromvp3 draw line through w; this ill be the further de of the shutter; its ngth must be deterined thus:- -From directed from DVP3 aw a line to the

e of contact, meet

git in y; make y z

DVP 3

wards, establishing its VP above the eye or HL.) Consequently, we must draw the vanishing line for the VP3 downwards from DVP2. The sides of the shutter, tw and m v, must be drawn in the direction of VP3, and cut off from DVP3, first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to DVP3, producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines wv 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 cad (Fig. 68); make ab 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 VP2 through t to e on the line

al to the length of the shutter, the same as the length of the indow; draw from a back again to DVP3, cutting tw in w; awwv, directed by VP', and v m directed by VP3. We will now draw the shutter at the same angle with the all, but inclined upwards from it (Fig. 70). The important ference in working the problem under these conditions arises om the upward inclination of the shutter from the wall, but dlined downwards to meet the wall. This last view of the sition of the shutter is the proper one for our purpose, because ter a little consideration we shall perceive that it is a retiring ane, but downwards; therefore its VP is below the eye or HL. the former case the shutter was a retiring plane, but up

Pp1

Vp1

of contact; make ef equal to the height of b above a, viz., c a (Fig.68). Draw from f back to VP2; 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 may be 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 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. PP1 would receive the visual rays from the front and the end B; PP2 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.

Fig. 70.

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." (See Lessons in Geometry, V., Vol. 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 angles. With regard to the last supposition, we shall see that the lines of

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 ХХХІІІ.) 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

VP3

Fig. 66.

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 remainder is the extent of the angle sought. This will explain the results 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 angle of 40° with the wall: the window is 5 feet from the ground, and its nearest corner is 4 feet within the picture; other conditions at pleasure. Scale of feet t

d

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 necessary. 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 object, 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 this is as follows:

x

inches.

of pinch 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 deration relates t the shutter. Th inclination may b upwards, at an an gle of 40° with th wall, or it may b downwards at th same angle. W will represent bot cases. First, whe inclined dow wards. Draw the HL, which is 4 fee from the ground line; from Ps dra

a perpendicular t
E; this will be th
radius for drawin

the semicircle meeting the HL to determine DE and DE Find the vanishing point for the wall vp, and its distan point DVP; also find the vp2 by drawing a line from E to V at a right angle with the one from E to VP', because if th shutter had projected from the wall in a horizontal position, would have vanished at vp2; that is, if it had been perper dicular or at right angles with the wall. In short, the vanishin point for the horizontal position of a line must always be foun whether the line retires to it horizontally or not, because the v for an inclined retiring line is always over or under the v (according to the angle of inclination) to which it would hav retired if in a horizontal position. (See Prob. XXXI., Fig. 53 Consequently, the vanishing point for an inclined retiring li is found by drawing a line from, in this case, the DVP2, accord

Fig. 71.

Pp2

C

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 a line from DVP2, at an angle of 50° with the HL, cutting the perpendicular from VP at VP3, the vanishing point. We have 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 b must be ruled to PS, upon which we wish to cut off 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 bps in a is the point required. Draw the perpendicular a hm. Draw from DVP through a to p; make pr equal to the width of the window. Draw back again from r, cutting DVP in s; 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 vp'; the height of the window, 4 feet 3 inches, is marked from ƒ to e; a line from e to vp1, cutting the perpendicalars from a and s in 3 and t, will give the top of the window. The opening of the window is m thn. Now we must draw the shutter; the corner nearest us is v, consequently it indlines upward towards the wall, but downwards from it; therefore, the VP for the hutter must be above the HL, which have explained.

we

To

measure or set off the length of the shutter, we have raised a line f contact for that purpose from o, found by drawing from vp2 hrough a to meet the round-line. From t irected fromyp3 draw line through w; this ill be the further ide of the shutter; its ngth must be deterained thus:-From directed from DVP3 aw a line to the ne of contact, meet

g it in y; make y z

VP3

m

E

wards, establishing its VP above the eye or HL.) Consequently, we must draw the vanishing line for the vp3 downwards from DVP2. The sides of the shutter, t w and m v, must be drawn in the direction of VP3, and cut off from DVP3, first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to DVP3, producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines wv 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 cad (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 VP2 through t to e on the line

S

qual to the length of the shutter, the same as the length of the indow; draw from a back again to DVP3, cutting tw in w; aw wv, directed by vp1, and v m directed by VP3. We will now draw the shutter at the same angle with the all, but inclined upwards from it (Fig. 70). The important ference in working the problem under these conditions arises om the upward inclination of the shutter from the wall, but lined downwards to meet the wall. This last view of the osition of the shutter is the proper one for our purpose, because fter a little consideration we shall perceive that it is a retiring ase, but downwards; therefore its VP is below the eye or HL. In the former case the shutter was a retiring plane, but up

ovp2

Pp1

Vp1

of contact; make eƒ equal to the height of b above a, viz., c a (Fig. 68). Draw from f back to vp2; 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 may be 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 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. rp1 would receive the visual rays from the front and the end B; PP2 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.

Fig. 70.

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." (See Lessons in Geometry, V., Vol. 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 angles. With regard to the last supposition, we shall see that the lines of

or shutter, and the ground, form a right-angled triangle, the three interior angles of which are together equal to two right

angles. Therefore, SE 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

inch

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 necessary. The meaning of the fraction is that unity is divided into the number of equal parts expressed by the denominator. 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 to the foot. It also means that the original object, 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 this is as follows:

x

vp3

Fig. 66.

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 results 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 angle of 40° with the wall: the window is 5 feet from the ground, and its nearest corner is 4 feet within the picture; other conditions at pleasure. Scale of feet t

of inch to the yard.

The above method of stating the scale ought to be understood by every one engaged upon plan-drawing.

consi

To return to the problem. The principal deration relates to the shutter. The inclination may be upwards, at an angle of 40° with the wall, or it may be downwards at the same angle. We will represent both cases. First, when inclined downwards. Draw the HL, which is 4 feet from the groundline; from PS draw

a perpendicular to E; this will be the radius for drawing

the semicircle meeting the HL to determine DE and DE Find the vanishing point for the wall VP1, and its distance point DVP; also find the vp by drawing a line from E to V at a right angle with the one from E to VP, because if the shutter had projected from the wall in a horizontal position, it would have vanished at vp2; that is, if it had been perpendicular 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 VF for an inclined retiring line is always over or under the vi (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, accord

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 a line from DVP2, at an angle of 50° with the HL, cutting the perpendicular from VPS at VP3, the vanishing point. We have 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 b must be ruled to PS, upon which we wish to cut off 4 feet to find a, the nearest point within; a line from c, which is 4 feet from b, must be drawn to DE', and where it cuts the line

bps in a is the point required. Draw the perpendicular a hm. Draw from DVP' through a to p; make pr equal to the width of the window. Draw back again from r, cutting DVpl in s; 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 VP'; the

height of the window,

4 feet 3 inches, is

To

g it in y; make y z

DVP 3

VP3

Vp 2

Fig. 71.

wards, establishing its VP above the eye or HL.) Consequently, we must draw the vanishing line for the vp3 downwards from DVP2. The sides of the shutter, t w and m v, must be drawn in the direction of VP3, and cut off from DVP3, first by drawing a line through t to y; make y a equal to the length of the shutter; draw from a to DVP3, producing w. All the early part of the problem, relating to the wall and windows, and the remaining lines wv 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 cad (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; ac will be equal to the height of b above a. This must now be applied to Fig. 70. Draw a line from VP2 through t to e on the line

al to the length of the shutter, the same as the length of the indow; draw from a back, again to DVP3, cutting tw in w; awwv, directed by ve1, and v m directed by VP3. We will now draw the shutter at the same angle with the all, but inclined upwards from it (Fig. 70). The important fference in working the problem under these conditions arises om the upward inclination of the shutter from the wall, but clined downwards to meet the wall. This last view of the peition of the shutter is the proper one for our purpose, because ter a little consideration we shall perceive that it is a retiring e, bat downwards; therefore its VP is below the eye or HL. the former case the shutter was a retiring plane, but up

Pp1

of contact; make ef equal to the height of b above a, viz., c a (Fig. 68). Draw from f back to VP2; it will be found to cut the corner of the shutter in w, proving by both methods that tw 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 may be 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 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. Fr1 would receive the visual rays from the front and the end B; PP2 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.