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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 inch page 156.) There

to the foot. It fore (Fig. 67), if A * Fig. 70.

also means that is 300 with the

the original ob PP, and B 90° with

ject, whether a 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

th equal to two right

of Y = 4 inch to Vpi

the foot angles. Therefore, SE

50° ovp2

inches. as the angle of the

of = 1 inch to wall with the ground is 90°, and

The above method the shutter or roof 400 with the wall,

of stating thi

scale onght to be the shutter will be

understood by at an angle of 50° with the horizon

every one engage (Fig. 68). Conse

upon plan-draw

ing. quently, this angle

To return to the of 500 must be constructed for the

problem. Th vanishing line, and

principal the subject treated

deration relates t Fig. 68.

the shutter. T. as an inclined

inclination may be plane. (See Problems XXXI.,

upwards, at an at XXXII.,

gle of 40° with th and

7 XXXIII.) From

wall, or it may all this we deduct

downwards at the a rule for finding

same angle. W

will represent bot vanishing points for lines or planes

cases. First, whe

inclined dow which are stated

wards. Draw th to be at given

vp3 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

B picture plane :

a perpendicular

E; this will be the 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 DE

radius for drawin it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vpl, and its distane 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 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, right angle, the sum will be the angle sought. This answers to would have vanished at vp?; that is, if it had been perper 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 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 4 feet broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would hav angle of 40° with the wall: the window is 5 feet from the retired if in a horizontal position. (See Prob. XXXI., Fig. 53 ground, and its nearest corner is 4 feet within the picture ; other Consequently, the vanishing point for an inclined retiring lin conditions at pleasure. Scale of feet te

is found by drawing a line from, in this case, the DVP, accort

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ing to the angle of inclination, to where it cuts a perpendicular wards, establishing its ve above the eye or HL.) Consequently, line drawn through the vps; 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 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 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,

and the remaining lines must be ruled to PS,

w v and t m, will be but upon which we wish to cut off 4 feet to find a, the

a repetition of the shutter

under the first position. Dearest point within; a line Fig. 71.

We can prove the truth of from c, which is 4 feet from

this method of drawing the b, must be drawn to DE', and where it cuts the line

perspective inclination of a bps in a is the point re

plane by another method. quired. Draw the perpen

Draw the right angle

A dicular ahm. Draw from

ca d (Fig. 68); make a b DEPthrough a to p; make

equal to the length of the

shutter, and at an angle Pf equal to the width of the window. Draw back

of 40° with a cor 500

with a d; draw b c paagain from 1, cutting wpl in s; draw the per

rallel to a d; a c will be pendicular st; the base

equal to the height of b

above 0. 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 yp2 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 f to e;

b above a, viz., ca i line from e to vpl,

(Fig. 68). Draw from cutting the perpendi

f back to ypa; it will clars from a and s in

Fig. 69.

be found to cut the 21 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. her nearest as is v,

A plan of a build. consequently it in

ing may be made, cines upacard towards

having all its proporBe 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 bare 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 contact for that

raised. When such is porpose from o, found

the case, all that is by drawing from vp

vpi necessary will be to klarongh s to meet the DEL

draw a PP across the ground-line. From t

paper in such a posi. directed 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 DVP3

Suppose A (Fig. 71) is draw a line to the

the plan of a build. boe of contact, meet

ing, and we wished to ing it in y; make y z

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 window ; draw from a back, again to bvpa, cutting t w in w; view of the front and the

opposite side-we should then place draw z v, 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 ll, 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 o. In short, any line roma 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 raition 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, withplaze, bat downwards ; therefore its vp is below the eye or ul. 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.

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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 a 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 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 = 1 inch to equal to two right

VP2

the foot angles. Therefore, SE

DVPI
50° DVP2

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 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 the of 500 must be

a

problem. The constructed for the

principal consi vanishing line, and

deration relates to the subject treated

Fig. 68.

the shutter. The as an inclined

inclination may be plane. (See Pro

upwards, at an av blems XXXI., XXXII.,

gle of 40° with the and XXXIII.) From

wall, or it may be

downwards at the all this we deduct

same angle. W a rule for finding

will represent bota vanishing points for lines or planes

cases. First, wher inclined

down which are stated

wards. Draw the to be at given vp3

HL, which is 4 feel angles with other A

from the ground lines or planes not

Fig. 66.

Fig. 67.

line; from Ps drar parallel with the

a perpendicular picture plane :When the sum of

B; this will be the

radius for drawing the two angles of the given bbjects is greater than a right angle, the semicircle meeting the al to determine del and Dr it is subtracted from the sum of two right angles, and the remains Find the vanishing point for the wall vp!, and its distane der is the extent of the angle sought. This will explain the re- point pvpl; also find the vp by drawing a line from E to vr 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, il right angle, the sum will be the angle sought. This answers to would have vanished at vp; that is, if it had been perpent 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 ve 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 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 line conditions at pleasure. Scale of feet te

is found by drawing a line from, in this case, the Dyp, accord

VP3

ing to the angle of inclination, to where it cuts a perpendicular wards, establishing its ve above the eye or HL.) Consequently, line drawn through the vps; 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 vps, and cut off from DVP3, first by drawing the perpendicular from vpå 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 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, must be ruled to PS,

and the remaining lines

w v and t m, will be but upon which we wish to cut off 4 feet to find a, the

a repetition of the shutter

under the first position. Dearest point within; a line Fig. 71.

We can prove the truth of from c, which is 4 feet from

this method of drawing the O, must be drawn to DE', and where it cuts the line

perspective inclination of a bps in a is the point re

plane by another method. quired. Draw the perpen

Draw the right angle dicular a hm. Draw from

cad (Fig. 68); make a b Dip through a to p; make

equal to the length of the pt equal to the width of

shutter, and at an angle the window. Draw back

of 40° with a cor 500

with a d; draw b c paagain from ", cutting ovpl 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 0. This must now from f, on the line of

be applied to Fig. 70. contact, 5 feet from the

Draw a line from vp2 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 f to e;

b above a, viz., ca a line from e to vp,

(Fig. 68). Draw from cutting the perpendi

s back to vp?; it will calars from a and s in

Fig. 69.

be found to cut the -1 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 n t h n.

the perspective length Now we must draw

of the further side of the shutter; the cor

the shutter. ter nearest as is v,

A plan of a build. consequently it in

ing may be made, dines upward towards

having all its proporthe wall, bret 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 ul, which we

said as to its position hare explained. To

with the picture measure or set off the

plane, and from this length of the shutter,

plan several perspecwe 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

vot necessary will be to through s to meet the

draw a PP across the ground-line. From t

paper in such a posi. directed from ypj draw

tion with the plan, aline 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

wish to take of it. directed from DVP3

Suppose A (Fig. 71) is draw a line to the

the plan of a build. ise of contact, meet

ing, and we wished to ing it in y; make ya

have two views of itqual 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 ovpă, cutting t w in w; view of the front and the

opposite side-we should then

place drav vo v, directed by vpl, 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 ell, but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would reäference in working the problem under these conditions arises ceive those from the front and the end C. In short, any line krom the upward inclination of the shutter from the wall

, but may be drawn which represents the PP at any angle with the felined doinwards 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, withplaze, 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.

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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 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 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 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 & A, then B will be

building or piece 600 with the PP, the whole making

of machinery, is

48 times larger two right angles. With regard to

than the drawing

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 #s, it would DNP3

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 interior angles of

as follows : which are together

inches. n.

of y = 1 inch to equal to two right

the foot angles. Therefore, se Vpz

inches. as the angle of the

of 30 = 1 inch to wall with the ground is 90°, and

The above method the shutter or roof 40° with the wall,

of stating the

scale onght to be the shutter will be

understood by at an angle of 50° with the horizon

every one engaged a (Fig. 68). Conse

upon plan.dra

ing. quently, this angle

To return to the of 500 must be

problem. The constructed for the vanishing line, and

principal

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 vanishing points

will represent both for lines or planes

cases. First, when

inclined down which are stated

wards. Draw the to be at given VP3 angles with other

HL, which is 4 feet

from the ground lines or planes not

Fig. 66.

Fig. 67.

line; from Ps dras parallel with the

B picture plane :

a perpendicular to When the sum of

E; this will be the the two angles of the given objects is greater than a right angle, the semicircle meeting the ul to determine de and Dr

radius for drawing it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vpl, and its distane der is the extent of the angle sought. This will explain the re- point dupl; also find the vp by drawing a line from 1 to FF 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, i right angle, the sum will be the angle sought. This answers to would have vanished at vpo; 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 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 w 4 fest broad; a shutter projects from the top of the window at an (according to the angle of inclination) to which it would hay angle of 40° with the wall: the window is 5 feet from the retired if in a horizontal position. (See Prob. XXXI., Fig. 53. ground, and its nearest corner is 4 feet within the picture; other Consequently, the vanishing point for an inclined retiring lin conditions at pleasure. Scale of feet the

is found by drawing a line from, in this case, the DYP", accord

a

consi

c

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