39

meet, ought to be lightly drawn,—namely a b or a' b',

a d, and a' /'. Again, the lateral planes represented by

b c, c d, b' e, and e /, are obviously in the shade, as no light

falls upon them directly: and these lines are strengthened,

to express the distinction.

In (Figs. 174 and 175), the lines composing the interior

and exterior contours in the elevation are parallel, and thus

contrast well. It is obvious that the portion of the exte-

rior from b by c to cZ is in the shade, while the rest is light;

and the inverse is the case with the inner edges. A pecu-

liarity, however, occurs at d, for here the edges, inner and

outer, are parallel to the direction of the light. It is plain

that the surfaces which come up to these edges will be in

a medium shade, and that the lines at d should be of

medium thickness.

(Figs. 176 and 177) represent a hollow cylinder in projec-

tion. In the plan, two lines a, c, drawn parallel to the direc-

tion of the light, and touching the exterior of the cylinder,

define the semicircular outline, a b' c, which is thrown in

the shade, and ought to be strengthened. The outlines at

a and c are, like the edges at d (Fig. 174), parallel to the

light, and the contour on each side gradually recedes and

advances to the light. The thickness of the line should,

therefore, be rather gradually reduced at the points a, c,

and it would besides appear awkward to stop the shade

abruptly on a circle. In the elevation, the base-line d f

should be shaded, and b d but half-shaded as it lies in a

curve surface.

If, again, the cylinder be hollow, presenting in plan the

interior contour circle e h, then the semicircle e g h ex-

presses the shady side of the interior, the light striking-

directly upon the opposite semicircle.

These examples illustrate every case of shade-lining

that occurs in outline drawings. The effect is enhanced

by proportioning the thickness of the lines to the depth

of the surfaces to which they belong, below the original

surfaces from which the shadows arise.

The system of shading above described, according to

which the light is supposed in plan to strike towards the

right hand upper corner, falling as it were, in front of the

objects, is mostly peculiar to French practice. In England,

the shadows are, for simplicity, thrown all one way, in ele-

vation and plan, towards the right hand and foot of the

sheet. The illustrative Plates contain examples of shading

in both ways.

The constructions of the following problems in projection

are made independently of the use of the T square. The

subjects of the earlier problems are exhibited in Plate VIII.

SECTION IV.

PROJECTIONS OF SIMPLE BODIES.

Projections of a* Pyramid.—Plate I.

The first subject for projection is a regular hexagonal

pyramid, shown by Figs. 1 and 2, Plate VIII. On in-

specting the figure, it would appear that two distinct

geometrical views are necessary to convey a complete idea

of the form of the object; namely, an elevation to repre-

sent the sides of the body, and to express its height; and a

plan of the upper surface, to express the form horizontally.

It is to be observed that this body has an imaginary

axis or centre-line, about which the same parts are equally

distant. This is an essential characteristic of all symme-

trical figures, or such as may be supposed to consist of two

halves of the same form joined together. A cone, for in-

stance, may be cut in two halves of the same form, down

through the axis or centre line, and so also may the

pyramid.

In the first place, a horizontal straight line L T is to

be drawn through the centre of the sheet; this line will

represent the ground line. Then draw a perpendicular

Z Z' through the middle of the ground line: this may be

done by the geometrical methods pointed out; but, in

case the compasses should not be capable of opening far

enough to admit of the arcs at Z and Z' being described

from the extremities of the ground line, the points m and

n, equidistant from these extremities, may be assumed as

centres, from which, with the largest convenient radius,

portions of circles are to be struck, above and below the

ground line, and their points of intersection joined by the

straight line Z Z', which will be the perpendicular re-

quired. It is to these two straight lines L T and Z Z'

that all the other lines in the drawings are to be referred;

those parallel to the former being considered as horizontal,

and those parallel to the latter as vertical. We should

next set out the border lines of the drawing, that is, the

rectangular space within which all the figures are to be

contained. Having determined its dimensions, take half

the breadth as a radius, and from any points as L and T,

in the ground line describe arcs above and below it, and

draw tangents X U and V Y to these arcs. Then from

the points g and li, where these lines intersect the per-

pendicular Z Z', set off, on each side of the latter, half

the length of the rectangular space required, and draw

XY, U V.

The next step is to draw the axes or centre lines of the

various figures to be represented. The vertical centre

line S S', Figs. 1 and 2, is obtained by describing from

the points g and h, at any convenient distance from the

line Z Z', arcs intersecting the horizontal border lines in

x and y, and joining x y. The other vertical axes may be

set out in the same manner. For the sake of preserving

the symmetry of the drawing, the centres of the lower

range of figures are all in the same straight line M N ;

this is to be drawn parallel to the ground line, by mark-

ing off, from L to M, and from T to N, its proper distance

from that line.

Plate I.—Figs. 1, 2.—In delineating the pyramid, it is

necessary, in the first place, to construct the plan, or hori-

zontal projection. The point S', where the centre line S S'

intersects the line M N, is to be taken as the centre of the

figure, and from this point, with a radius equal to the side

of the hexagon which forms the base of the pyramid,