find it, take the distances from the corners of the square to the horizontal trace when the plane is folded down, and set them along the vertical trace from its junction with xy. This will give these points in elevation. Their plans will lie in the perpendiculars to xy from their elevations (Theorem I.) and also in the perpendiculars to the horizontal trace from the points as "constructed" (Theorem X.). Thus let the point of which a' is the elevation when folded down or constructed' be marked A. A perpendicular from a to xy contains the plan a, as also does the perpendicular from A to the horizontal trace. The plan a therefore lies in the intersection of these perpendiculars; in other words, these perpendiculars are loci of the point a, and their intersection gives the point itself.

(4) The axis will be drawn from the centre of the base in plan and elevation at right angles to the traces, and being parallel to the vertical plane will be set in true length in elevation. It only remains to join the proper points and the projections of the solid are completed.

2.

Plan and elevation of a cube of 2.5 inches edge, when one face is inclined 60°, and no side horizontal.

3. Plan and elevation of a pyramid, 4 inches high, with regular hexagonal base of 2.5 inches side inclined 70°, no side horizontal, and its lowest corner 1 inch above the ground.

4. Plan and elevation of the pyramid in the last problem when the base is inclined 70° and one side of the base 30o.

In this problem there is given not only the inclination of the plane of the base, but also of one edge in that base.

LIMITS. The problem is therefore to draw a line of given inclination in a plane of given inclination, and this is always possible if the inclination of the line is not greater than that of the plane.

(1) Determine by its traces the required plane inclined 70° and at right angles to the vertical plane. Assume any point A in the plane; its elevation a' is by the position of the plane in the vertical trace; its plan a on a perpendicular to xy from a', and for convenience it may be assumed at the point where this perpendicular meets xy; and from ď draw a'm to meet the ground line at the given angle 30°. Rotate the triangle thus formed on the vertical line a'a, the projector of the point A, until the point m meets the horizontal trace. As A and m are now both in the given plane, the line joining a and m will be the plan of Am and will fulfil the conditions by being the plan of a line wholly in plane 70° and itself inclined 30o.

(2) 'Construct' plane 70° on its horizontal trace into the horizontal plane. On the line Am, thus brought into the horizontal plane, describe the required hexagon, and complete plan and elevation of the whole as in former problems.

5. Plan and elevation of a cube of 2.5 inches edge when the plane of one face is inclined 65° and one diagonal of that face 25°.

Determine

6. Plan and elevation of a tetrahedron of 3 inches edge with one face 70° and one edge in that face 40°. also the inclinations of the other edges of the solid.

7. Plan and elevation of hexagonal right prism when the plane of one of its faces ABCD is inclined 50° and the edge AB inclined 35°

(1) Determine the plan of rectangular face ABCD as in preceding problems.

(2) It will next be necessary to determine the plane of the base, which, by the definition of the solid, is perpendicular to the plane of the face. The line AB already determined is in the base, being common to both planes, and its horizontal trace will therefore be a point in the horizontal trace of the required plane of base. And as the projections of a perpendicular to a plane are perpendicular to its traces, the horizontal trace of the plane of the base may be at once drawn through this point at right angles to the long edges of the prism, that is, in this case to be or ad.

(3) Take a ground line at right angles to the horizontal trace of plane of base, and set up the height of A or B (both known points) in the new elevation. By the position of the new plane of elevation, the vertical trace will contain the elevations of both A and B, and therefore can at once be drawn.

Fold the plane of the base about its horizontal trace to bring AB down into the horizontal plane. On the AB thus constructed describe hexagon and complete as in former problems.

V.

SOLIDS WITH THE

INCLINATIONS OF TWO

ADJACENT EDGES GIVEN.

LIMITS. It must be remembered that the sum of these inclinations with the angle contained by the two edges must not exceed 180°.

For these problems refer to Chapter I., Problem 20.

I. Plan and elevation of a cube when the inclinations of two adjacent edges as BA, BC, 20o and 48° are given.

(1) Draw the square ABCD to represent the face 'constructed' into the horizontal plane. Then to find a horizontal in the plane of this face. At any point E in BA, or BA produced, make the angle BEF 20°. Then if a perpendicular be drawn from B to EF, it will be the length of the line expressing the height to which В will be raised above horizontal plane containing point E when in required position. From B as a centre with this perpendicular for radius, describe a circle. The tangent to this circle to make an angle of 48° (i.e. the inclination of the other edge) with BC will give a point H where it meets BC in the horizontal required. E being another point in it, the straight line through H and E will be the horizontal sought in plane of face ABCD.

(2) Assume a plane of elevation at right angles to this inclined plane. The height to which point B is to be raised being known, a line of level that distance above xy will contain the elevation of B. The rest of the work is similar to that in preceding problems.

NOTE. To explain the principles of this construction, the teacher had better cut a card-board model, by which to illustrate the folding up and down of the plane of the angles and of the lines.

2. Plan and elevation of a tetrahedron when two of the adjacent edges in one face are inclined 28° and 62°.

3. Plan and elevation of hexagonal right prism when the two edges AB, BC in one face are inclined respectively 25° and 40°.

Determine the plan of the rectangular face ABCD as in Problem 1, and refer to explanations of Problem 8, Chapter IV., to find the plane of base and complete the projections of the solid.