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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 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

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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.

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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. 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

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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.

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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 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

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