From either of the points, as B, draw the straight line BR at right angles to the given plane (Problem 13), meeting it in S, determined by Problem 14.

=

Make SR SB, point R being on the opposite side of the plane to B; join R, C, cutting the given plane in point Q: QB, QC are the required lines.

EXERCISES ON CHAPTER I.

1. Determine the projections of four points, each 2 inches from the planes of projection, viz. one in each dihedral angle.

2.

Determine four points each in a plane of projection and 3 inches from xy; no two of them to be on the same side of that line.

Supposing the loci of these points (Theorem I.) to be 2 inches apart, determine the distance between any two points.

3. A point is 2.5 inches from the plans of projection, draw two lines through it inclined 50°; their plans making angles of 35° with xy. Problem 1, converse (2).

Determine the traces of the lines: also the angle which the lines make with each other. Problem 10.

4. Through a point 2 inches from each plane of projection draw lines inclined 30°, making angles of 40° with the vertical plane. Problem 3 and Theorem IV.

5. A straight line in the horizontal plane makes an angle of 45° with xy and is a trace of planes which are inclined 50°; find their vertical traces. Problem 3, "tan

gent planes to cones," Chap. IX.

6. A regular hexagon, side 2 inches, has its diameter and adjacent side inclined 30 and 45° respectively: draw its plan and determine the inclination of its plane. Problem 20.

7. Draw two parallel planes 2 inches apart and inclined 60°. Problems 1 and 13, and Problem 25, note.

8. A plane is inclined 40°, and is at right angles to another which is inclined 70°: draw the traces of the planes. Problem 25.

9. The horizontal traces of two planes, inclined 45° and 30° respectively, make an angle of 60°: determine a third plane at right angles to the given ones. Problems 12

and 13.

IO. Two planes are inclined 60° and 70° respectively and make an angle of 50° with each other: determine their vertical traces on a plane not perpendicular to them.

II. A regular pentagon, side 2.5 inches, has three of its angular points respectively I, 2, and 3 inches high. Draw its plan. Problem 21.

12. A square, side 3 inches, lies in a plane inclined 45°, one side of the figure makes an angle of 40° with the horizontal trace of its plane. Draw its plan. Problem 18.

13. A plane is inclined 80° and makes an angle of 60° with a plane which is inclined 50°. Draw the traces of the planes. Problem 26.

14. The two traces of a plane make an angle of 50° with each other and make equal angles with xy: determine these latter angles and the inclination of the plane.

15. Three straight lines are perpendicular to one another and two of them are inclined 25° and 40°: determine the inclination of the third line. Problems 20 and 13.

16. The horizontal and vertical traces of a plane make angles of 30° and 55° respectively with xy; a straight line parallel to xy is 2 inches from the vertical and 3 inches from the horizontal plane : find the intersection of the given line and plane. Problem 14.

17. Two planes contain a right angle; one of them is inclined 60° and their intersection is inclined 50o. Draw the traces of the planes and find the inclination of the second. Problems 8 and 13.

18. The hypothenuse of a right-angled triangle is 4 inches long, and horizontal; the plans of the sides make an angle of 115° with each other: determine the lengths and inclinations of the sides. Can more than one set of answers be given? Eucl. 111. 33, and Problem 20.

19. A straight line inclined 40° lies in a plane inclined 60°; determine a plane containing the given line and perpendicular to the given plane. Problems 8 and 13.

20. Through a point 3 inches high draw three planes, inclinations 35°, 45°, and 60° respectively; forming a trihedral angle at the given point. Show the real magnitude of the angle of each face of the solid angle. Problems 12 and 10.

MEMORANDA

FOR WORKING THE PROBLEMS IN ENSUING CHAPTERS.

All the solids given are assumed to be right,' unless otherwise expressed.

All dimensions and arrangements, not expressly mentioned and limited, may be assumed at pleasure.

The inclinations of all lines and planes must be understood, unless otherwise mentioned, to be to the horizontal plane. So likewise with all 'constructions;' see page 7.

Invisible edges. In plan the invisible edges are those which are hidden by the rest of the solid to the eye looking through the solid at right angles to the horizontal plane. In elevation they are those which are similarly concealed from the eye looking through the solid at right angles to the vertical plane. Invisible edges must be drawn in dotted lines.

Sections. The teacher should give directions for sections with each problem, as far as possible.

II.

I.

SOLIDS IN SIMPLE POSITIONS.

Draw plan and elevation of a pyramid, 3'5 inches high, with square base of 2.5 inches side, when resting with its base on the horizontal plane, and with one side of the base making an angle of 30° with the vertical plane.

Commence with the plan.

This will be the square of the base with the opposite corners joined for the plans of the slant edges of the solid. Take the ground line xy inclined 30° with one side of the square and draw perpendiculars to xy (Theorem I.) from the four corners and the centre of the square. The elevations of the four corners of the square will be in the ground line, because the base of the solid rests on the horizontal plane. The height of the pyramid, 3'5 inches, must be set up from the point where the perpendicular from the centre of the square meets the ground line, and at right angles to the ground line (Theorem II.). The whole elevation will be completed by joining the elevation of the vertex to the four points determined on the ground line for the elevations of the four corners of the base.