"A Section of a solid by a plane is a plane figure bounded by the lines (straight or curved) in which that plane cuts the surfaces (plane or curved) by which the solid is contained." It is indicated by section-lines inclined at 45°, or by light shading at the contour lines.

A Sectional Elevation is a section together with that part of the elevation of the solid which is visible beyond the section plane.

A Sectional Plan is a section together with that part of the plan of the solid which is visible beyond the section plane.

The student will now be instructed to draw the projections of solids when placed in simple positions. With his models before him in the positions indicated, he should carefully reproduce all the drawings, taking the dimensions from the models and draw full size. It will likewise be found interesting to cut out each drawing when finished (see the outer border line above xy, in fig. 147), fold up the piece of paper through 90° along xy, and place the models directly over the plans. Any difficulty in realising the projections will then usually disappear, and, as the solutions are verified, interest in the subject will in that way be enhanced.

Fig. 137 is a plan and elevation of a cube, 14′′ edge, standing on the H.P., but with no face parallel to the V.P. The square, abcd, is the plan of the cube, and is drawn first, assuming it to be in any position relatively to the V.P. The corners, a, b, c, d, are the plans of the vertical edges of the cube, so that the true lengths of these edges are set off in elevation. Project them up with T and set-squares, and make a'e', b'f', c'g', and d'h' each equal to the edge of the cube. Draw the line, f'h', to complete the elevation. c'g' is drawn dotted, because to an observer situated in front of the solid and the V.P. it is invisible.

Fig. 138 is a plan and elevation of a right prism, 24" long, with equilateral triangles of 14" side at the ends. It is placed with one end parallel to the V.P. and " from it, a long edge in the H.P., and one of its rectangular faces inclined at 15° to the H.P. Commence by drawing a 13′′ line ab' in elevation inclined to xy at 15°, and then the equilateral triangle a'b'c',

Project down and draw the plan acb parallel to xy and 3" from it. The plan is then completed as shown. ce being the topmost line is visible, and is therefore drawn continuous. Fig. 139

22

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15°

Fig. 138.

shows projections of a right prism, 24" long, having a regular hexagon of 1" side for base. It is placed with the base elevated "above the

a circle with centre 1" from xy, radius 1", the plan is drawn first. The hexagon is inscribed in the circle, such that ab and ed, the plans of two faces, are perpendicular to xy. The elevation, a'f'e', of the base, is drawn " above xy, after which the elevation of the solid is readily completed. If a part of the solid be cut off by a plane as represented by the knife blade, k'n', the plan of the section of the solid is that represented by the portion which. is section-lined. This is not the true shape of the section, as it is inclined. to the H.P. The method for determining true shapes of sections will be described in the next chapter.

Fig. 139.

Fig. 140 is a plan and elevation of a right square pyramid standing on the H.P., with one edge AB (14") of the base inclined at 60° to the V.P., and touching it in A, the axis (24" long) being, of course, vertical. Commence by drawing, with the set-square, the plan ab = 11⁄2" inclined at 60° to xy; then draw the square and diagonals ac, bd, intersecting in v, the plan of the vertex. av, bv, cv, and dv, are the sloping edges of the solid. Project up the plans of the five points, putting the elevation a'b'c'd' in xy, and v' 2" above it; join as shown. v'a' is invisible.

d

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Fig. 140.

The plan shows a vertical cutting plane represented by a knife-blade, kn, parallel to the V.P., and cutting off a portion of the solid. Evidently the knife would cut cd in e, and vc in f. Find e' in xy, and f' in c'v' by projection, and join e'f', f'b'. The triangle, e'f'b' is the elevation of the section made by the vertical cutting plane kn, and in this case it is the true shape of the section, because the triangle EFB is parallel to the V.P.

Fig. 141 is a photograph of an actual square pyramid in the same relative position as in the preceding figure.

Fig. 142 shows projections of a regular tetrahedron standing with a face ABC on the H.P.; one face ABV is at right angles to the V.P., and one edge cv is parallel to the latter. Commence by drawing the equilateral triangle abc, of 2" side, with ab perpendicular to xy, and ascertain the centre, v, the plan of the vertex. This is quickly accomplished with the aid of the 60° set-square. The sloping edge vc, being parallel to xy, is seen in true length in elevation, a fact which enables us to obtain the altitude of the solid. With c' as centre, radius ab, cut the projector through v in v', then complete the figure.

section plane.

It will be found on trial that the elevation o'v' is equal to the true length of the altitude oc of the equilateral triangle abc. This is so because ov is parallel to the V.P. and is actually the altitude of the sloping equilateral triangle, ABV. The small supplementary diagram (K) shows the method adopted when the tetrahedron is otherwise placed, but with the base still on the H.P. The right-angled triangle vov is constructed, making ov oc, and the altitude of the solid is then transferred from vv. Note that the altitude or axis of the tetrahedron is always less than the altitude of the triangular face,

=

Fig. 143 is a plan and elevation of a regular octahedron placed in the simplest position. The square abcd of 13′′ side, and its diagonals ac, bd, are first drawn to represent the plan of the solid. One of the diagonals of the octahedron is then vertical-its plan being represented by v. Project v up, making v' v1 = ac or bd, and bisect it with the horizontal a'b'c'd'. The elevation is then easily completed.

ABCD is the square base common to the two square pyramids previously referred to, and v', v are their vertices in elevation, whilst their common plan is v. The supplementary diagrams, (L) and (M), are other views of the octahedron when a diagonal is vertical. The length of the elevation of the diagonal, and the plans of the solid are found exactly as before, but it is seen that the

(N)

(0)

Fig. 144.

elevations are not the same shape, and the plans are differently situated relatively to the V.P. It should be noted that both plan and elevation of view (L) are equal squares. Use the 45° set-square for plans.

Fig. 144 is a plan and elevation of a right cylinder of diameter 13", and length 24′′, resting on the H.P., with its