Now as the section plane is parallel to the axis, it is clear that the plan of the section will be a rectangle. In order to determine the

position of this rectangle on the plan, drop perpendiculars from A' and B', cutting the ends of the plan in a and c; and in b and d; then abdc is the plan of the required section.

NOTE. If a cylinder be cut by a plane parallel to its base, the section will be a circle.

Problem 53.

To project the section of a triangular prism when cut by an oblique plane.

Let ABC be the plan, and D'EF'G' the elevation of a prism which is cut by the section a'b'c' oblique to its axis. From points a' and b', where the section passes through the angles of the prism, also from point c', where the section passes through the end, draw the projectors a'a, b'b, c'c, to any line ac parallel to a'c'. Next from c' draw the projector c'n at right angles to xy, and crossing the sides of the prism on the plan at the points m and n.

From a as centre, draw the arcs bb and cc to xy; and from the points abc, where these arcs cut xy, draw projectors at right angles to xy, which intersect the projectors drawn from the points A, C, n, and m, on the plan of the oblique section parallel to xy. Then the point A, where the projector from A and a intersect, will be one angle of projection, and B, where the projectors from b and c inter

sect will be another; whilst the other points of projection will be C and C, where the projector from c intersects those from m and n.

Join the angles of intersection A, B, C, and C by straight lines, and the trapezium ABCC is the required projection of the oblique section.

Problem 54.

To construct the sectional elevation of a tetrahedron upon a vertical plane parallel to the section plane, the trace of which is at right angles to one of the lateral edges of the solid.

Take AB, and upon it describe an equilateral triangle ABC. Find D, the centre of the triangle, and join DA, DB, DC. We have thus the plan of the tetrahedron when its base is horizontal.

Next, draw mn, the trace of the section plane, at right angles to DC, and assume this plane to be vertical. We now proceed to find the elevation of the whole solid. In order to do this, we must know the height of D above A, B, C. As each of the triangles DAB, DBC, and DAC is equilateral, the real length of DA, DB, DC, is expressed by any of the edges of the base as AB, AC; consequently from D raise a perpendicular to DC indefinitely, and with centre C, and radius CB, describe an arc, cutting this perpendicular in e, and join Ce.

The points A, C, B, are projected on xy in A'C'B'; then since D is

elevated above these points a distance De; make C'D' equal to this distance, and join D'A', D'C', and D'B'.

Now, the section plane cuts AC, CB, in a and c, and the projections of these are a' and c'. It now remains to find the elevation

of b, the point in which the plane cuts DC. It will be seen that b is elevated above C, a distance bd; therefore, make C'b' equal to bd, and join b'a', b'c', which will complete the sectional elevation.

Problem 55.

To construct the sectional plan and elevation of a pentagonal pyramid standing on its base on the horizontal plane.

Let AB be one edge of the base inclined to xy at any angle 8, on AB describe the regular pentagon ABCDE, and find O the centre of the circumscribed circle. Join OA, OB, OC, OD, and OE; this will be the plan of the required pyramid. Draw 00' at right angles to xy, and make O'a' equal to the perpendicular height of the pyramid; draw Aa', Bb', Cc, Dd', and Ee' perpendicular to xy, and join O'a', O'b', O'c, O'd', and O'e', which will complete the elevation of the pyramid.

We now proceed to construct the plan of a section made by a plane perpendicular to the vertical plane, its trace m'n' making with

xy the angle 6. Let the trace m'n' meet the elevations e'O', d'O, a'O', c'O', and b'O' in the points f', g, h, k, l, respectively; then the plans f, g, h, k, l, will be found by drawing f'f, g'g, h'h, k'k, and l'l perpen

dicular to xy; and meeting EO, DO, AO, CO, and BO in fghkl respectively. Then the figure fgklh will be the sectional plan required.

Problem 56.

To construct the sectional elevation and plan of a pentagonal pyramid standing on its base on the horizontal plane.

Let AB be one edge of the base inclined to xy at any angle ; on AB describe the regular pentagon ABCDE, and find O the centre of the circumscribed circle; join OA, OB, OC, OD, and OE; this will be the plan of the required pyramid. Draw 00' at right angles to xy, and make O'F" equal to the perpendicular height of the pyramid;

draw Aa", BB', Cc", Dd", and EE perpendicular to xy, and join a"O', B'O, c"O, d"O', and EO, which will complete the elevation of the pyramid.

We next proceed to construct the elevation of a section made by a vertical plane whose trace mn makes with xy an angle ; let this trace cut ED, DO, OC, and CB in the points a, b, c, d, respectively; then a and d being points in the horizontal plane, their elevations will be in the base line; draw aa' and dd' at right angles to xy; then

a' and d' will be the elevations of a and d; b and c also are the plans of points in the straight lines whose elevations are c'O' and d'O'; if therefore from b and c straight lines be drawn at right angles to xy, and meeting d'O' and c′′O in b′ and c', then b' and c' will be the elevations corresponding to b and c. Join a'b', b'c', and c'd'; then the figure a'b'c'd' will be the sectional elevation required.

Problem 57.

To draw the vertical projection of the section of a square pyramid parallel to the section plane, the trace of the plane, which