the V.P. Show the real shape of the section of the solid made by a plane parallel to the V.P. and passing through the vertex.

Hint.-Commence a plan of the solid with one edge of the base inclined at 50° to ay, and proceed as in fig. 142K. The real shape of the section is a scalene triangle.

6. Draw the projections of a regular octahedron when one of its diagonals (which is 21′′ long) is horizontal, and perpendicular to the V.P. (position at pleasure).

7. A right cone (base 2" diameter, axis 2.8" long) is placed with base in the V.P. and touching the H.P. The elevation of the cone is likewise the elevation of a sphere touching the cone in the vertex. Draw their projections, together with a vertical right cylinder (base 11" diameter, axis 21" long), which stands on the H.P. and touches the cone and sphere. Indicate their points of contact.

Hint. The plan of the solids consists of a triangle and two circles all in contact.

The diagrams for Q's 9, 10, 11, are to be copied four times the given size.

8. An equilateral triangle a'b'c' of 11" side is the end elevation of a prism 2" long. Draw its plan when the long edge through A is in the horizontal plane, and the face containing the edge AB is inclined at an angle of 40° to the horizontal plane.

Hint.-See fig. 138.

(1890)

9. The figure represents the plans of two ordinary bricks placed one on top of the other, the lower brick resting on the horizontal plane. The height of each brick is 23". Make an elevation of the bricks on the line ab. Scale, full size. (1884) 10. Draw the plan of one of the bricks (Question 9), when one of its shortest edges is in the horizontal plane, and the plane of its end is inclined at 52° to that plane.

Hint.-See the principle of fig. 138.

(1884)

11. The elevation of a letter X is given. Supposing it is cut out of wood of thickness equal to the breadth of the bars, draw its plan when standing 1" in front of the vertical plane.

(1885)

12. Draw the plan of a regular tetrahedron, the apex of which is 2" above the base, which rests on the horizontal plane.

(April 1898) Hint.-Draw any regular tetrahedron, say, of 2" edge, with the base in the H.P., as in fig. 142; raise the apex to 2" above the base, and complete by drawing similar figures by the principle of Chap. VII.

13. A right equilateral prism, axis 2", edge of base 1.75", lies with one rectangular face in the horizontal plane, and one triangular base in the vertical plane of projection. Draw its plan, and also the plan of a sphere of 1" radius which touches the horizontal plane, the vertical plane, and the prism.

(1895)

Hint. The elevation (which should be first drawn) of the solids consists of a circle touching the xy and a side of an equilateral triangle, the latter having its base in xy. 14. The base (2′′ radius) of a right cone, 2" high, rests on the horizontal plane. A sphere of 1" radius touches the cone at a point 2" from the apex. Draw the plan of the solids showing their point of contact. (June 1899)

15. Draw the projections of two spheres (of 1" and " radius respectively), touching one another and the horizontal plane.

Hint. See the spheres A and c in Prob. 91.

(1896)

16. The figure fghl (4 size) represents the plan of a cylinder resting on the ground. Draw the elevation of the cylinder, and show the projections of a sphere of 1" diameter in contact with it on one side, and also the projections of a cube with an edge (" long) touching the cylinder on the other side; both the sphere and cube are to be on the ground.

8

16

Show carefully in plan the point and line of contact with the cylinder of the sphere and cube respectively. In plan the portions of the solids which are not seen are to be denoted by a dotted boundary line. (April 1900)

Hint.-The elevation of the cylinder is a circle. Proceed on the lines of fig. 147.

PR. GEO.

CHAPTER XIII.

ALTERATION OF GROUND LINE, PROJECTIONS, AND SECTIONS OF SOLIDS.

PLAN and elevation of an object have hitherto been obtained by means of only one horizontal and one vertical plane. In practice, however, more than two projections are frequently required in drawings, in order to clearly depict configurations which would otherwise be difficult to follow from a plan and elevation only. Thus we meet with Side Views, End Views, and various Sections in Civil and Mechanical Engineering; and North Elevation, West Elevation, etc., in Architectural and other drawings. To obtain these additional projections auxiliary planes are introduced, without disturbing the relative positions of the original Co-ordinate planes and the object to be reprojected. A new vertical plane may be placed inclined at any angle we please to the original V.P., and intersecting the H.P. in a second ground line, x, y, thereby enabling the draughtsman to evolve a new elevation of the object, the latter at the same time remaining stationary. Similarly, a new plane may be placed (at right angles to the original V.P.) inclined at any angle to the original H.P., and intersecting the V.P. in a new ground line, making it possible for a new plan to be projected, again without moving the object. In the former case there would be one

horizontal plane and two vertical planes; in the latter, one vertical plane and two other projecting planes at right angles to it.

The number of additional ground lines that may be requisitioned to solve a problem is of course not limited.

Though the use of the alteration of the ground line in Solid Geometry problems is to facilitate the solutions, it will be comprehended that in order to define completely the position of an object in space, relatively to neighbouring objects, it is necessary to know the distance of the object from each of three reference planes. Thus, take a small object in a room, where the planes of reference are two adjacent walls and the floor meeting in a corner c, as shown by the pictorial view in fig. 150, and represent the object in space by a point P.

The distance of P from the 1st wall is represented by rp, or po'; its distance from the second wall by Pp" or po", and its height above the floor by Pp or op' or o′′p." When the vertical planes of the walls are conventionally brought into the horizontal plane of the floor, the orthographic projections to correspond are as shown in fig. 151.

In this case the three planes of reference are mutually at right angles, and are then called the three rectangular co-ordinate planes; the corner c, where they meet, is the origin or pole, and the xy's together with the intersection of the walls are the three co-ordinate axes.

NEW
V.P

2"

Fig. 152.

PROBLEM 92.

Given the original xy, the plan a and elevation a' of a point A (1" from the V.P., and 13" from the H.P.), to 3" determine a new elevation a" on a new ground line x2 y2, which makes 90° with xy (fig. 154.)

Before attempting the problem, cut out two pieces of cardboard of the shapes

and dimensions shown in fig. 152. Bend one of them at the dotted line A a through 90°, and write the given letters on the pieces cut out.

Fig. 153.-Showing projections of a point on three co-ordinate planes when in their natural position.