PROBLEM 42.

Two sides of a triangle respectively 75′′ and 1" long meet on the horizontal plane. The first has its extremity at 25", and the second at 5" above the horizontal plane, and are both contained by a plane which is inclined at an angle of 50° to the horizon. Complete the triangle, and determine the real angle contained by the two lines as well as their horizontal projection.

Draw the vertical trace V' g at an angle of 50° to X Y, and the horizontal trace perpendicular to X Y.

At any point in X Y make the given heights 25" and 5" from X Y, and through the points draw lines parallel to the ground line until they meet the vertical trace in c'and b'. Revolve the vertical trace on to X Y.

From the points in which c'b' moet X Y when revolved, drop perpendiculars.

From any point a make a B, a C equal to 75" and 1′′* respectively, cutting the perpendiculars let down from X Y in B and C.

Proceed as before to obtain points b c, which, on joining with each other and point a, complete the plan required.

The real angle contained by the lines the angle B a C. The lines fo (·75") and r p (1") show the inclination of the respective sides of the triangle to the horizontal.

PROBLEM 43.

A plane making an angle of 45° with the horizontal contains an isosceles triangle of 1.5" side, whose base of one inch is inclined to the horizon at an angle of 30°; show its projection.

The student is recommended to work this out from his knowledge of the previous problems.

PROBLEM 44.

The side of a pentagonal prism 1" long and 5" high is inclined at an angle of 25° with the horizon, and stands upon a plane forming an angle of 50° with the horizontal; draw its plan.

In all problems in which the determination of the plan and elevation depends upon the position of a line in the figure, when this line is determined, it matters not what may be the shape of the figure the line will always serve as a base of construction.

Having then obtained the pentagon, which is the base of the solid, as in previous problems, and drawn the figure in elevation, it is hoped the student will see how to finish the construction by dropping projectors from a' b'c' d'e' until they meet the horizontal lines drawn from the corresponding corners of the pentagon in plan, and then completing the plan.

*a B, a C may be inclined both in the same direction towards X Y, the direction not being given by problem.

CHAPTER V.

ON THE PROJECTION OF OBLIQUE SURFACES, WHEN THE INCLINATION AND ANGLE BETWEEN TWO STRAIGHT LINES CONNECTED WITH THE SURFACE ARE GIVEN.

In all previous problems X Y, or ground of level, has been given at the commencement of the problem. Now, however, X Y will be determined after some part of the construction has been made. The reason that X Y is in the following problems constructed from the figure is, that because we can have an indefinite number of elevations, but only one plan; therefore we choose the elevation determined by its line of level, that is at right angles to the line or axis of revolution on which the object could have revolved in order to attain the position given by conditions of problem, and this axis of revolution requires to be determined.

PROBLEM 45.

Two lines A B, B C are inclined at 35° and 40° respectively to the horizontal, and contain an angle of 90°; determine by its traces the plane containing the two lines, and the apparent plan of the lines.

These lines A B, B C may be supposed to be the hypotenuse of two right-angled triangles of 35° and 40°, having their apex at the same height above the horizontal plane, and being mutually at right angles to each other. The plane may be made in reality by taking a piece of cardboard and drawing two lines at right angles, as A B, BC (Fig. 46), and on A B constructing a rightangled triangle of 35°, and on B C, with an altitude B d equal to B d, construct a right-angled triangle of 40°. Then join A C, and cut out with a knife the figure. After this is done fold the model upon the lines B A and B C until the points d and d coincide. If now we place the model on the table, resting on the two lines A d, C d, we shall have an inclined surface representing the plane, containing the required line A B, B C; it will also be