36 MATHEMATICS. 
points of projection of the upper semicircle have been found, we describe 
the semi-curve a’c° b*, and corresponding to it, the symmetrical half, 
lying beneath. 
Solids are bounded by surfaces, as these are by lines; the problem of 
finding the projection of a solid resolves itself, then, into finding the pro- 
jection of points. In illustration of this, we will explain the method of 
projecting a regular six-sided pyramid, in its various positions. Pl. 4, fig.24, 
exhibits this pyramid in its regular position, in a horizontal and vertical 
plane: acdbef is the polygon forming the base of the pyramid, and which 
we have placed in the same position in the ground plan, with respect to the 
bases xy, that the pyramid is to have inelevation. If we suppose the pyramid — 
to be completed above this base, we shall have a view from above of the 
former. For this purpose, if the pyramid be right, we find the centre of the 
polygon: this will be the projection of the apex g of the pyramid. Lines 
drawn from the point g to the corners of the base, form the projections of 
the edges of the pyramid. ‘To the eye, however, the projection thus obtained 
will suit for any height of the pyramid, as the point g is not determined 
with respect to its distance from the base; we must therefore have a side 
view of the pyramid, since, as already mentioned, two projections, at least, 
are necessary to determine the position of a point. As the lines which stand 
perpendicularly to the ground plane are projected as points, under the same 
conditions surfaces will be projected as lines. The case is the same with 
respect to the plane of elevation: g is the projection of the altitude of the 
pyramid in the plane, which appears as a line in elevation ; acdbef is the 
projection of the base in plan, which appears in the plane of elevation as a 
sgries of lines, whose position and individual extremities are determined by 
drawing the perpendiculars aa’, cc’, &c. The position of the point g’, in the 
perpendicular gg’, is determined by the method already explained. By con- 
necting the point g’ with the points a’,b/——"’, we shall obtain the vertical 
projection of the pyramid. 
Suppose that an oblique section, h’n’, be made through the pyramid, per- 
pendicular to the plane of elevation, and its projection in the ground plane 
required ; the first step will be to indicate the plane f'n’ by a straight line. 
The lines a'g’ and ag, c'g' and cg, &c., are corresponding projections. If, 
then, from the points where the plane /’n’ cuts the different edges of the 
pyramid in the elevation, perpendiculars be let fall upon the corresponding 
edges of the plan, the points of intersection will determine the corners of 
the plane of intersection, h, 1, k, 1, m, n. 
If we suppose the pyramid to rest with one corner, b*, upon the basis zy, 
as in fig. 25, its axis, however, still parallel to the plane of projection, the 
projection on the horizontal plane must be changed, as the altitude of the 
pyramid is no longer perpendicular to this plane. To describe this projec- 
tion, place the elevation obtained in fig. 24 upon the corner 0’, at the re- 
quired angle, and then draw from the point g a perpendicular to the ground 
plan. From the point g, of fig. 24, draw a line parallel to the basis zy, until 
it cuts the perpendicular in g’; then g* is the apex of the pyramid for the 
new projection. It is evident that the line gg’ must be parallel to the basis 
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