certain points; secondly, obtaining the projections of lines by joining the points; and thirdly, obtaining the projection of plane figures by joining the lines, thus giving the projections of a solid.

The simple solids are the Cube, Prism, Pyramid, Sphere, Cylinder, and Cone, which may be defined as follows:

Cube.-A cube is a solid having six faces all equal squares (Fig. 68).

Prism.-A right prism is a solid having two equal and similar bases, and a number of equal oblong faces perpendicular to

them. Prisms are distinguished according to the shape of their bases. Fig. 69 represents a hexagonal prism.

Pyramid.-A right pyramid is a solid having one base, and a number of equal triangular faces meeting in a point over the centre of the base. This point is called the apex. Pyramids

ABA

Fig. 70.

Fig. 71.

Fig. 72.

are distinguished according to the shape of their bases. Fig. 70 represents a square pyramid.

Spheres, cylinders, and cones are called solids of revolution, because they are generated by the revolution of certain plane figures about a fixed line, as an axis.

Spheres.-A sphere is a solid generated by the revolution of a semicircle about its diameter, as an axis. All points in the

surface are equidistant from a point within the sphere, called the centre. All plane sections of a sphere are circles.

Cylinder.-A cylinder is a solid generated by the revolution of an oblong about one of its sides, as an axis. It has two equal circular bases, and may be regarded as a right prism having an infinite number of faces (Fig 71).

Cone. A cone is a solid generated by the revolution of a right angled triangle about its perpendicular, as an axis. It has a circular base, and may be regarded as a pyramid having an infinite number of faces. The vertex of the triangle forms the

apex of the cone (Fig. 72).

Axis. The axis of a solid may be regarded as its central line. In a cube, the line joining the centre of any face to the centre of the opposite face, and in a sphere, any diameter may be called the axis. In a prism and cylinder the axis is the line joining the centres of the two bases, and in a pyramid and cone it is the line joining the apex to the centre of the base. The axis is shown by a dotted line in the figures.

A cube and sphere can be drawn if we know the length of side and the diameter respectively. In drawing the other four solids, we require to know the length of the axis, and the shape and size of the base.

The drawing of these solids may be rendered much easier and of much greater value to the student if he uses small models, which can be placed in different positions relative to the planes of projection. In this way a very complete conception of the solids may be acquired, which will prove of immense benefit in more advanced problems and especially in machine design, where the shape of so many common parts are identical with the simple solids just defined.

Very satisfactory models of these solids, except the sphere, can be made of stout paper, by first developing the surfaces of the solid then cutting the pattern to the pattern of the figure thus drawn, and afterwards folding upon the lines representing the edges of the solid, and gumming together.

Development of Surfaces.-By the development of a surface is understood the drawing upon a flat plane, such as a sheet of paper, the true shape of the complete surface, such that when the paper is cut to the figure thus obtained it would completely cover the solid of which it is the development. In practical engineering work, such as the construction of boilers, funnels, and in other iron and tin plate work, it is necessary to develop the surfaces of the structures before the plates of which they are made can be marked off, hence development is a very important part of engineering drawing.

Cube. The development of a cube is a figure made up of six equal squares, and conveniently drawn as in Fig. 73, where the

squares 1 and 3 form the top and bottom, when square 3 is the base, and squares 2, 4, 5, 6 the sides.

Parts are shown by dotted lines as extensions of squares 3, 4, and 6. These are merely strips to enable the sides to be joined together—that is, the slip on 3 is gummed to the top edge of 5, and the slips on 4 and 6 to the edges of square 3. Further strips

Fig. 75.

may be added on the other sides if desired. The student should arrange to allow for similar strips in the development of the other solids.

Prism. The development of a prism is a figure made up of a number of equal oblongs and two equal figures representing the bases. Fig. 74 shows a convenient arrangement of the development of a hexagonal prism.

Pyramid. The development of a pyramid is a figure made up of a number

of equal triangles, and of one regular figure to represent the base. Fig. 75 shows the development of a hexagonal pyramid, obtained as follows:-Draw the base of the solid and a diagonal of the base, and then draw a right-angled triangle

having the axis of the solid a' b' as a perpendicular, and half the diagonal of the base of the solid a'c' as a base, then the hypotenuse b'c' is the length of the sloping edges of the solid. With the length of the sloping edges as radius, draw an arc of a circle from any centre as at C, mark off along this arc (as chords of the arc) the length of the sides of the base of the solid as a b, b c, c d, de, and join a to b, b to c, and c as shown. This is the development of the faces of the pyramid. Draw the hexagon of the base on any of the lines as d e for a side, and the development will be complete.

Cylinder. The development of a cylinder is made up of an oblong, the length of which is equal to the circumference of the base, and the height of which is equal to the axis, and of two circles equal in diameter to the bases. It is shown in Fig. 76.

Cone. The development of a cone is made up of a segment of a circle, the radius of which is the hypotenuse of a right

Fig. 76.

Fig. 77.

angled triangle, having the axis of the cone as a perpendicular, and the radius of the base, as a base, and the arc of which is equal to the circumference of the cone base; and of a circle, of diameter equal to the base. It is shown in Fig. 77. To obtain the radius of the arc draw a right-angled triangle, having the radius of the cone base a' c' as a base, and the axis of the cone a'b' as a perpendicular; then the hypotenuse b'c' is the radius required. To set off the correct length around the arc (see p. 73).

EXAMPLES.

EX. 26.-Draw the development of the surfaces of the following solids: (a) a cube, 21′′ edge; (b) hexagonal prism, 11" edge of base, axis 3"; (c) square pyramid, 13" edge of base, axis 3";

(d) cylinder bise, 1" diameter, axis 31"; (e) cone base, 1" diameter, axis 34". Make models of the solids.

Plan and Elevations of Simple Solids.-The projections of the solids just developed, in simple positions, relative to the HP and V P, are shown in Fig. 78. The student should follow the drawings by placing the solids represented in the given positions, using a model of the planes as before.

Fig. 78 (a).-A cube having one edge in HP, one face inclined at 30° to H P, and one face parallel to the V P 0.75" in front.

The elevation must be drawn first, as that shows the inclination and a true shape of one face.

Fig. 78 (6).-A hexagonal prism, having one base in HP, one face inclined at 20° to V P, nearest edge " in front.

First draw the lines st, and the hexagon having one side b c in the line. Then draw a second X Y 1" from the edge c.

Fig. 78 (c).-A square pyramid with base in H P, one edge at 20° to VP and touching V P.

Fig. 78 (d).-A cylinder with one base in H P.

Draw the plan first, and the diameter ab parallel to X Y, to give the correct points for the projection to the elevation.

Fig. 78 (e).-A cone with one base in HP, axis 2" in front of VP. Draw the plan first having its centre on the line ab parallel to the XY. It is shown dotted in Fig. 78 (d). The projections of a sphere need no description, they are circles equal in diameter to the diameter of the sphere in all positions.