I We have now to examine the model of a cone, of which a drawing is given above.

Its surface consists of two parts; first a plane circular base, then a curved surface which tapers from the circumference of the base to a point above it called the vertex. Thus the form of a cone suggests a pyramid standing on a circular instead of a rectilineal base.

Let us take a triangle ABC right-angled at B (Fig. 2), and suppose it to rotate about one side AB as a fixed axis. What will BC trace out as the triangle revolves? Notice that AC will always pass through the fixed point A, and move round the curve traced out by C. As AC moves, it will generate a surface. What sort of surface?

We now see that the kind of cone represented in the diagram is a solid generated by the revolution of a rightangled triangle about one side containing the right angle.

Ex. 9. Why must the ▲ ABC, rotating about AB, be rightangled at B, in order to generate a cone?

What would be generated by the revolution of an obtuse-angled triangle about one side forming the obtuse angle?

Ex. 10. What would be generated by an oblique parallelogram revolving about one side?

The curved surface of a cone may be represented by a plane figure thus:

C

A

Taking the slant height AC of the cone as radius, draw a circle. Cut it out from your paper; call its centre A; and cut it along any radius AC. If you now place the centre of the circular paper at the vertex of the cone, you will find that you can wrap the paper round the cone without fold or crease. Mark off from the circumference of your paper the length CD that will go exactly once round the base of the cone; then cut through the radius AD. We have now a plane figure ACD (cailed a sector of a circle) which represents the curved surface of the cone, and has the same area.

(Spheres.)

The last solid we have to consider is the sphere, whose shape is that of a globe or billiard ball.

We shall realise its form more definitely, if we imagine a semi-circle ACB (Fig. 2) to rotate about its diameter as a fixed axis. Then, as the semi-circumference revolves, it generates the surface of a sphere.

10

APPENDIX ON SOLID FIGURES.

Now since all points on the semi-circumference are in all positions at a constant distance from its centre O, we see that all points on the surface of a sphere are at a constant distance from a fixed point within it, namely the centre. This constant distance is the radius of the sphere. Thus all straight lines through the centre terminated both ways by the surface are equal: such lines are diameters.

Ex. 11. We have seen that on the curved surfaces of a cylinder and cone it is possible (in certain ways only) to rule straight lines. Is there any direction in which we can rule a straight line on the surface of a sphere?

Ex. 12. Again we have cut out a plane figure that could be wrapped round the curved surface of a cylinder without folding, creasing, or stretching. The same has been done for the curved surface of a cone. Can a flat piece of paper be wrapped about a sphere so as to fit all over the surface without creasing?

Ex. 13. Suppose you were to cut a sphere straight through the centre into two parts, in such a way that the new surfaces (made by cutting) are plane, these parts would be in every way alike. The parts into which a sphere is divided by a plane central section are called hemispheres. Of what shape is the line in which the plane surface meets the curved surface? If the section were plane but not central, can you tell what the meeting line of the two surfaces would be?.

Ex. 14. If a cylinder were cut by a plane parallel to the base, of what shape would the new rim be?

Ex. 15. If a cone were cut by a plane parallel to the base,

what would be the form of the section?

13

3