0.5

1 cm

40

ENGINEER AND MACHINIST’S DRAWING-BOOK.

describe a circle, cutting M N in A' and D\ From these

points with the same radius, draw four arcs of circles,

cutting the primary circle in four points. These six

points being joined by straight lines, will form the figure

A' B' O' D' E' F', which is the base of the pyramid ; and

the lines A' D', B' E', and C' F', will represent the projec-

tions of its edges fore-shortened as they would appear in

the plan. If this operation has been correctly performed,

the opposite sides of the hexagon should be parallel to

each other, and to one of the diagonals; this should be

tested by the application of the square or other instru-

ment proper for the purpose.

By the help of the plan obtained as above described,

the vertical projection of the pyramid may be easily con-

structed. Since its base rests upon the horizontal plane,

it must be projected vertically upon the ground line;

therefore, from each of the angles at A', B', C', and D',

raise perpendiculars to that line. The points of intersec-

tion A, B, C, and D, are the true positions of all the angles

of the base; and it only remains to determine the height

of the pyramid, which is to be set off from the point G to

S, and to draw S A, S B, SC, and S D, which are the

only edges of the pyramid visible in the elevation. Of

these it is to be remarked that S A and S D alone, being

parallel to the vertical plane, are seen in their true length;

and moreover, that from the assumed position of the solid

under examination, the points F and E' being situated in

the lines B B' and C G, the lines S B and S C are each

the projections of two edges of the pyramid.

Figs. 3 and 4.—To construct the projections of the same

pyramid, having its base set in an inclined position,

but with its edges S A and S D still parallel to the ver-

tical plane.

It is evident, that, with the exception of the inclina-

tion, the vertical projection of this solid is precisely the

same as in the preceding example, and it is only necessary

to copy Fig. 1. For this purpose, after having fixed the

position of the point D, upon the ground line, and at the

most suitable distance from the vertical Z Z', draw through

this point an indefinite straight line D A, making with

L T an angle equal to the desired inclination of the base

of the pyramid. Then set off the distance DA, Fig. 1,

from D to A, Fig. 3 ; and, since the axis must still, as in

the former case, pass perpendicularly through the centre

of the base, draw two circular arcs from the points D and

A as centres, and with the distance D A as radius ; join

their points of intersection, and set off G S equal to the

height of the pyramid. Transfer also from Fig. 1 the dis-

tance B G or C G to the corresponding points in Fig. 3,

and complete the figure by drawing the straight lines

A S, B S, C S and D S, representing, as before, the outer

edges of the pyramid.

In proceeding to construct the horizontal projection of

the pyramid in this position, it is first to be remarked

that since the edges S A and S D are still parallel to the

vertical plane, and the point D remains unaltered, the

projection of the point A will still be in the line M N.

Its position at A', Fig. 4, is determined by the intersec-

tion of the perpendicular A A' with that line. The re-

maining points B', O', &c., in the projection of the base,

are found in a similar manner, by the intersections of

perpendiculars let fall from the corresponding points in

the elevation, with lines drawn parallel to M N, at a dis-

tance, (set off at o, p), equal to the width of the base. By

joining all the contiguous points, we obtain the figure

A' B' G D' E' F', representing the horizontal projection

of the base, two of its sides, however, being dotted, as

they must be supposed to be concealed by the body of

the pyramid. The vertex S having been similarly pro-

jected to S', and joined by straight lines to the several

angles of the base, the projection *of the solid is completed.

Figs. 5 and 6.—To find the horizontal projection of a

transverse section of the same pyramid, made by a

plane perpendicular to the vertical, but inclined at

an angle to the horizontal plane of projection ; and let

all the sides of the base be at an angle with the ground

line.

Having drawn, as before, the vertical S S', the centre

line of the figures, its point of intersection with the line

MN is the centre of the plan. From this point, then, as

a centre, draw, as in Fig. 2, a circle, and inscribe in it a

regular hexagon; but since none of the sides of the base

are to be parallel with the ground line, draw a diameter

A' D' making the required angle with that line, and from

the points A' and D', proceed to set out the angular points

of the hexagon as already explained. Then, in order to

obtain the projections of the edges of the pyramid, join

the angular points which are diametrically opposite ; and,

following the method pointed out in reference to Fig. 1,

project the figure thus obtained upon the vertical plane,

as shown at Fig. 5.

Now, if the cutting plane be represented by the line

a d, in the elevation, it is obvious that it will expose, as

the section of the pyramid, a polygon whose angular

points, being the intersections of the various edges with

the cutting plane, will be projected in perpendiculars

drawn from the points where it meets these edges respec-

tively. If, therefore, from the points a, f, b, &c., we let

fall the perpendiculars a a ,f f, b b , &c., and join their con-

tiguous points of intersection with the lines A' D', F C',

B' G, &c., we shall form a six-sided figure, which shall

represent the section required. This hexagon, it is almost

unnecessary to remark, is irregular, being formed by a

plane which is inclined to the base of the pyramid. The

edges F S and E S being concealed in the elevation, but

necessary for the construction of the plan, have been

expressed in dotted lines, as also the portion of the pyra-

mid situated above the cutting plane, which, though sup-

posed to be removed, is necessary in order to draw the

lines representing the edges. We have here, also, intro-

duced the ordinary method of expressing sections in purely

line-drawings, and in engraving on metal and wood; and

to which we shall adhere throughout this work, namely,

by filling up the spaces comprised within their outlines,

with a quantity of parallel straight lines drawn at equal

distances.

ENGINEER AND MACHINIST’S DRAWING-BOOK.

describe a circle, cutting M N in A' and D\ From these

points with the same radius, draw four arcs of circles,

cutting the primary circle in four points. These six

points being joined by straight lines, will form the figure

A' B' O' D' E' F', which is the base of the pyramid ; and

the lines A' D', B' E', and C' F', will represent the projec-

tions of its edges fore-shortened as they would appear in

the plan. If this operation has been correctly performed,

the opposite sides of the hexagon should be parallel to

each other, and to one of the diagonals; this should be

tested by the application of the square or other instru-

ment proper for the purpose.

By the help of the plan obtained as above described,

the vertical projection of the pyramid may be easily con-

structed. Since its base rests upon the horizontal plane,

it must be projected vertically upon the ground line;

therefore, from each of the angles at A', B', C', and D',

raise perpendiculars to that line. The points of intersec-

tion A, B, C, and D, are the true positions of all the angles

of the base; and it only remains to determine the height

of the pyramid, which is to be set off from the point G to

S, and to draw S A, S B, SC, and S D, which are the

only edges of the pyramid visible in the elevation. Of

these it is to be remarked that S A and S D alone, being

parallel to the vertical plane, are seen in their true length;

and moreover, that from the assumed position of the solid

under examination, the points F and E' being situated in

the lines B B' and C G, the lines S B and S C are each

the projections of two edges of the pyramid.

Figs. 3 and 4.—To construct the projections of the same

pyramid, having its base set in an inclined position,

but with its edges S A and S D still parallel to the ver-

tical plane.

It is evident, that, with the exception of the inclina-

tion, the vertical projection of this solid is precisely the

same as in the preceding example, and it is only necessary

to copy Fig. 1. For this purpose, after having fixed the

position of the point D, upon the ground line, and at the

most suitable distance from the vertical Z Z', draw through

this point an indefinite straight line D A, making with

L T an angle equal to the desired inclination of the base

of the pyramid. Then set off the distance DA, Fig. 1,

from D to A, Fig. 3 ; and, since the axis must still, as in

the former case, pass perpendicularly through the centre

of the base, draw two circular arcs from the points D and

A as centres, and with the distance D A as radius ; join

their points of intersection, and set off G S equal to the

height of the pyramid. Transfer also from Fig. 1 the dis-

tance B G or C G to the corresponding points in Fig. 3,

and complete the figure by drawing the straight lines

A S, B S, C S and D S, representing, as before, the outer

edges of the pyramid.

In proceeding to construct the horizontal projection of

the pyramid in this position, it is first to be remarked

that since the edges S A and S D are still parallel to the

vertical plane, and the point D remains unaltered, the

projection of the point A will still be in the line M N.

Its position at A', Fig. 4, is determined by the intersec-

tion of the perpendicular A A' with that line. The re-

maining points B', O', &c., in the projection of the base,

are found in a similar manner, by the intersections of

perpendiculars let fall from the corresponding points in

the elevation, with lines drawn parallel to M N, at a dis-

tance, (set off at o, p), equal to the width of the base. By

joining all the contiguous points, we obtain the figure

A' B' G D' E' F', representing the horizontal projection

of the base, two of its sides, however, being dotted, as

they must be supposed to be concealed by the body of

the pyramid. The vertex S having been similarly pro-

jected to S', and joined by straight lines to the several

angles of the base, the projection *of the solid is completed.

Figs. 5 and 6.—To find the horizontal projection of a

transverse section of the same pyramid, made by a

plane perpendicular to the vertical, but inclined at

an angle to the horizontal plane of projection ; and let

all the sides of the base be at an angle with the ground

line.

Having drawn, as before, the vertical S S', the centre

line of the figures, its point of intersection with the line

MN is the centre of the plan. From this point, then, as

a centre, draw, as in Fig. 2, a circle, and inscribe in it a

regular hexagon; but since none of the sides of the base

are to be parallel with the ground line, draw a diameter

A' D' making the required angle with that line, and from

the points A' and D', proceed to set out the angular points

of the hexagon as already explained. Then, in order to

obtain the projections of the edges of the pyramid, join

the angular points which are diametrically opposite ; and,

following the method pointed out in reference to Fig. 1,

project the figure thus obtained upon the vertical plane,

as shown at Fig. 5.

Now, if the cutting plane be represented by the line

a d, in the elevation, it is obvious that it will expose, as

the section of the pyramid, a polygon whose angular

points, being the intersections of the various edges with

the cutting plane, will be projected in perpendiculars

drawn from the points where it meets these edges respec-

tively. If, therefore, from the points a, f, b, &c., we let

fall the perpendiculars a a ,f f, b b , &c., and join their con-

tiguous points of intersection with the lines A' D', F C',

B' G, &c., we shall form a six-sided figure, which shall

represent the section required. This hexagon, it is almost

unnecessary to remark, is irregular, being formed by a

plane which is inclined to the base of the pyramid. The

edges F S and E S being concealed in the elevation, but

necessary for the construction of the plan, have been

expressed in dotted lines, as also the portion of the pyra-

mid situated above the cutting plane, which, though sup-

posed to be removed, is necessary in order to draw the

lines representing the edges. We have here, also, intro-

duced the ordinary method of expressing sections in purely

line-drawings, and in engraving on metal and wood; and

to which we shall adhere throughout this work, namely,

by filling up the spaces comprised within their outlines,

with a quantity of parallel straight lines drawn at equal

distances.