The Popular Educator

In this case we

inches.

DVPI

50°
DVP2

DES

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 necesAll this will be very evident if we consider that "if any num- sary. The meaning of the fraction as 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 3 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 ob PP, 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.

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

he equal to two right

of y = 4 inch to VPI

the foot. angles. Therefore, SE

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 ought to be the shutter will be

understood by at an angle of 50°

every one engaged with the horizon (Fig. 68). Conse

upon plan-draw

ing. quently, this angle of 500 must be

To return to the a

problem. The constructed for the

principal vanishing line, and the subject treated

deration relates t Fig. 63.

the shutter. Th as an inclined

inclination may be plane. (See Problems XXXI.,

upwards, at an a 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 vanishing points

will represent bot for lines or planes

cases. First, whe

inclined dom which are stated

wards. Draw th to be at given

урз 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 picture plane :

a perpendicular : When the sum of

E; this will be th the two angles of the given bbjects is greater than a right angle, the semicircle meeting the al to determine del and Di

radius for drawis 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 dypl; also find the vp? by drawing a line from E to vi 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, right angle, the sum will be the angle sought. This answers to would have vanished at vp; 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 fous 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 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 hat 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 lis conditions at pleasure. Scale of feet te

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

consi

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 mv, must be drawn draw a line from pvp, at an angle of 500 with the HL, cutting in the direction of vps, and cut off from DVP3, first by drawing the perpendicular from vp2 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, 6 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 o, must be drawn to DE',

this method of drawing the and where it cuts the line

perspective inclination of a ps in a is the point re

plane by another method. quired. Draw the perpen

Draw the right angle dionlar ahm. 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

of 40° with a cor 500 the window. Draw back again from f, cutting

with a d; draw b c paovpl in s; draw the per

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

equal to the height of b

above a. 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 vpå 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 s to e;

b above a, viz., ca 3 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 on 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 build sonsequently it in

ing may be made, clines upward towards

having all its propor. the wall, but down

tions, angles, and seeds from it; there

other measurements fore, the vp for the

arranged and noted, sbatter must be above

yet nothing may be the IL, which we

said as to its position have explained. To

with the picturemeastire or set off the

plane, and from this length of the shutter,

plan several perspecwe 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 vp

sypi necessary will be to throagh s to meet the

draw a PP across the ground-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 farther

rays, the picture-plane side of the shutter; its

we have chosen may length must be deter

receive the view we sumed thus:From

wish to take of it. I 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 pindow ; draw from < back again to pvp?, cutting t w in w; view of the front and the

opposite side-we should then place baw o o, directed by vp!, 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 all, but inclined upwards from it (Fig. 70). The important the visual rays from the front and the end B; pp would reüzerence 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 kielined dounwards 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 siter a little consideration we shall perceive that it is a retiring when its proportions are given, as is frequently

the case, withplane, but dmonwards ; 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.

DE
1

vp 2

DVD/

Ovp2

In this case we

inches.

VPI

DE2

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 ve 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 600 with the PP,

building or piece

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 = 1 inch to equal to two right

the foot. angles. Therefore, SE

DVPI
50° OVP2

inches. as the angle of the

of 30 = 4 inch to wall with the

the yard ground is 90°, and

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

of stating the

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 th of 500 must be

problem. Th constructed for the

principal consi vanishing line, and

deration relatest the subject treated

Fig. 68.

the shutter. TI as an inclined

inclination may be plane. (See Problems XXXI.,

upwards, at an at

gle of 40° with the XXXII., and

wall, or it may XXXIII.) From

downwards at ti all this we deduct a rule for finding

same angle. W

will represent bot vanishing points

cases. First, whe for lines or planes

inclined dom which are stated

wards. Draw to be at given vp3

HL, which is 4 fe angles with other

from the groun lines or planes not

Fig. 66.

Fig. 67.

line; from Ps dra parallel with the

a perpendicular picture plane :

E; this will be th 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 DI

radius for drawir it is subtracted from the sum of two right angles, and the remain- Find the vanishing point for the wall vpl, and its distan 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 E to vpl, because if ti

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 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 hat 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 lis conditions at pleasure. Scale of feet to

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

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 pvp, at an angle of 50° with the HL, cutting in the direction of vp3, and cut off from DVP3, first by drawing the perpendicular from vp 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 æ 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, 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. nearest point within; a line

Fig. 71. from e, 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 ó ps in a is the point re

plane by another method. quired. Draw the perpen

Draw the right angle

A dionlar ahm. Draw from

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

equal to the length of the pr equal to the width of

shutter, and at an angle

of 40° with a cor 500 the window. Draw back again from r, cutting

with a d; draw b c paDyplin 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 Vpa 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 s to e;

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

(Fig. 68). Draw from catting the perpendi.

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

Fig. 69.

be found to cut the 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 vindow is in th n.

the perspective length Now we must draw

of the further side of the shutter; the cor

the shutter. Der Dearest us is v,

A plan of a builddonsequently it in

ing may be made, d'ines upward towards

having all its propor. the 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 have explained. To

with the picturemeasiire or set off the

plane, and from this length of the shutter,

plan several perspec"we 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

vpi necessary will be to

PLS filarongh s to meet the

draw a PP across the Ezoand-line. From t

paper in such a posidirected from yp3 draw

tion with the plan, line throngh w; this

that by drawing visual will be the further

rays, the picture-plane ride of the shutter; its

we have chosen may length must be deter

receive the view we mined thus From

wish to take of it. i directed from DVPS DVP3!

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

the plan of a build. line of contact, meet

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

have two views of itsenal 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 pindow ; draw from a back again to pvp, cutting t w in w; view of the front

and the

opposite side-we should then place daw na y, 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 real, 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 o. In short, any line bom 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, with. plane

, 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.

DVD/

lovp2

In this case we

VPI

DVPI

DEP

inches.

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 denomiside of it, the sum of the angles which they make with this straight nator. Thus a scale of feet is sigpifies 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,

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 a, 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

inches. which are together

of 4 = 4 inch to equal to two right

the foot angles. Therefore, SEI

50° DVP2 as the angle of the

of * = 4 inch te wall with the

the yard ground is 90°, and

The above method the shutter or roof

of stating the 40° with the wall,

scale ought to be the shutter will be

understood by at an angle of 50°

every one engage with the horizon (Fig. 68). Conse

upon plan-draw

ing. quently, this angle

To return to the of 500 must be

problem. Th constructed for the

principal vanishing line, and

deration relatest the subject treated

Fig. 63.

the shutter. TL as an inclined

inclination may b plane. (See Problems XXXI.,

upwards, at an ar XXXII.,

gle of 40° with the and

wall, or it may ! XXXIII.) From

downwards at th all this we deduct

same angle. W a rule for finding

will represent bot Vanishing points for lines or planes

cases. First, whe

inclined dow which are stated

wards. Drawth to be at given v P3

HL, which is 4 fel angles with other

from the ground lines or planes not

Fig. 66.

Fig. 67.

line; from Ps dra parallel with the picture plane :BA

a perpendicular

E; this will be the When the sum of

radius for drawis the two angles of the given bbjects is greater than a right angle, the semicircle meeting the ul to determine del and DI it is subtracted from the sum of two right angles, and the remain. Find the vanishing point for the wall vpl, and its distans der is the extent of the angle sought. This will explain the re- point pvp?; also find the vp by drawing a line from a to 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, right angle, the sum will be the angle sought. This answers to would have vanished at vpa; that is, if it had been perpet 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 fou 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 ha 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 lir conditions at pleasure. Scale of feet to

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

consi

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The Popular Educator, 第 3～4 巻、第 13 巻