draw its plan. From the angular points A', B', C', &c., draw lines at right angles to the line of intersection (xy), and upon these set off the length of the prism; i.e., make Aa, &c., equal to the length of the solid. Through the points A and a draw lines parallel to xy, to show the ends of the prism. Each of the points A', B', C', &c., represents a line perpendicular to the vertical plane. In the plan these lines will be shown at right angles to xy (Pr. 2).

NOTE 1.-The line represented by D' will be dotted in the plan, because under the given circumstances it is not seen.

NOTE 2.-The ends of prisms may be triangles, squares, or polygons; a prism is said to be triangular when its ends are triangles, square when its ends are square, &c., &c.

NOTE 3.-The axis of a prism is a line joining the centres of the bases. NOTE 4.-When the base is a regular figure, it is called a regular prism but when the base is an irregular figure, the solid on it is termed irregular.

Problem 12.

To draw the plan of a pentagonal prism, its elevation being given.

Let A'B'C'DE' be the elevation of a pentagonal prism. It is required to draw its plan. From the angular points, A', B', C', &c., draw lines at right angles to the ground line (xy), and upon these set off the length of the prism; i.e., make Aa, &c., equal to the

length of the solid. Through the points A and a, draw lines parallel to xy to show the ends of the prism. Each of the points A', B, C, &c., represents a line perpendicular to the vertical plane. In the plan these lines will be shown at right angles to xy (Pr. 2).

NOTE. The line represented by E' will be dotted in the plan, because under the given circumstances it is not seen.

Problem 13.

To find the elevation of a hexagonal prism, its plan being given. Let ABCDEF be the plan of a hexagonal prism. It is required to find its elevation.

From A, raise a perpendicular to xy, and from a', the point in which it cuts xy, set off a'A' equal to the length of the prism. From A', draw A'D' parallel to xy, and from F, E, D, raise perpendiculars to meet this line. We have then drawn the elevation of the hexagonal prism.

Problem 14.

To find the plan of a hexagonal pyramid, its elevation being given.

Let A'B'C'D'E' be the elevation of a hexagonal pyramid. It is required to find its plan.

First, from C' draw a projector CC perpendicular to xy, also from D' draw a projector D'D at right angles to xy and equal in length to C'C. Join C and D, and it will be the projection of the front edge of the base of the pyramid.

Next, from B' drop a perpendicular to xy, and from C, with CD as radius, describe an arc cutting the perpendicular last drawn in B. Join B and C, and it will be the plan of the side edge B'C'.

Again, from E', drop a perpendicular to xy, and from D, with DC as radius, describe an arc cutting the perpendicular last drawn in E. Join D and E, and it will be the plan of the side edge D'E'.

Now, as the points immediately behind C and D' are covered in the elevation by C and D', it follows that their projections will coincide; therefore, from B and E as centres, with radius BC or DE, cut those projectors in G and F respectively. Join BG, GF, and FE; then these will be the plans of the remaining edges of the base. Now, join BE, a projector drawn from A' to meet it will give the point A, the plan of the apex of the pyramid, so that BA and AE are the plans of the lines B'A', A'E'. Draw the lines CF and

DG passing through point A, so that CA, AD, will be respectively the plans of C'A' and A'D'.

NOTE 1.-AG and AF are the plans of the lines not seen in elevation. NOTE 2.-Pyramids take their names from their bases, like prisms. NOTE 3.-The axis of a pyramid is a line joining the centre of its base to the apex.

Problem 15.

To find the elevation of a square pyramid, its plan being given. Let ABCD be the plan of a square pyramid. It is required to find its elevation.

From the point B draw a line at right angles to xy, meeting it in b, also from C, D, A, draw lines perpendicular to xy, meeting it in the points c', d', u'.

Now, since C, D, A are points on the horizontal plane, and as xy represents that plane seen in elevation, the points b', a', c', d' on it are the elevations of those points. E is the plan of the apex of the pyramid, to find the elevation of which we draw a line at right angles to xy from E, and elevate it above the line the required height of the point E above the horizontal plane. Join E'', E'c', E'd', for the angles of the figure.

NOTE-The line E'a' being covered by the surface b'c'E' is represented by a dotted line.

Problem 16.

To find the plan of a cylinder, its elevation being given. Let A'B'a'b' be the elevation of a cylinder. It is required to find its plan.

In this case, the surface represented by the line A'B' is a circle whose plane is parallel to xy; the points A and B will represent the diameter and C' the centre. Draw projectors from A', B', and C', to the points A, B, and C. Draw AC, CB, parallel to xy. From C, with radius CA or CB, describe the required circle ABD.

NOTE.-A cylinder may be defined as a prism having an infinite number of faces.