CASE V.

Given angles B and C and a side b opposite to one of them: to determine sides a and c, and the third angle A.

Let the side b be drawn on the horizontal plane, and revolved about its edge Om until it makes the angle C with that plane.

The indefinite line fø will then be the vertical trace of side b on the plane xy of the profile angle of C; which is, of course, perpendicular to Om: and fp on this trace, taken equal to fP, gives the elevation of a point P in the second edge of side b. p in xy, obtained from p', determines the point pp' in this second edge: and Op is the plan of that edge.

Determine a plane to contain the line OP, in space, just found, and inclined at the known angle of B, by Problem 9, Chapter I.: Ol, the horizontal trace of this plane, is the third edge of the required spherical triangle.

The side c can now be 'constructed' on the plane of a about Ol. The angle A may be determined as before, by a plane at right angles to OP through point P.

CASE VI.

Given the three angles A, B and C: to determine the three sides a, b and c.

If the dihedral angle C has one face in the horizontal plane, the traces of the other face can be readily drawn; and of these the horizontal trace is an indefinite edge of the required spherical triangle.

A third plane making the angle of A with the horizontal, and of 180-B with the inclined plane, determined by Problem 26, Chap. I., will complete the required spherical triangle. The intersections Ol, OP of this third plane with the planes of the angle C form the second and third edges of the spherical triangle. The face b will thus be determined in the horizontal plane; into which the remaining faces a and can be 'constructed' about their respective edges or horizontal traces in the face b.

XIII.

ISOMETRIC PROJECTION.

IN Isometric Projection solid bodies the chief planes of which are mutually at right angles are so represented, that from one drawing, a plan, the principal measurements as to length, breadth, and height can be obtained.

The solid, the cube for instance, is to be so placed that the three planes which meet together at one corner are equally inclined to the horizontal plane, the plane of projection. The three lines which meet at that corner will then be projected so as to form three equal angles of 120° each, and will be the plans of the three edges of the cube. The plans of the opposite edges will be parallels to these, and hence it follows that in Isometric Projection all angles which are in reality right angles are projected into angles of either 12c° or 60°.

Commence therefore an Isometric Problem by drawing three lines, making angles of 120° with one another, for the plans of the three lowest edges of the solid.

Then construct an Isometric scale, so as to be able to find the Isometric lengths of the required edges; measure