X

(1) The plane may cut the cone in a closed

oval curve, such as $A B A^{'} B^{'}$ which lies entirely on one of the two half-cones. In this case the plane will not meet the other half-cone at all. Such a curve is called an ellipse; it is an oval curve. A particular case of such a section of the cone is when the plane is perpendicular to the axis $C V C^{'}$ , then the section, such as $S T U$ or $P Q R$ , is a circle. Hence a

circle is a particular case of the ellipse.

(2) The plane may be parallelled to a tangent plane touching the cone along one of its "generating" lines as for example the plane of the

curve $D_{1} A_{1} D_{1}^{'}$ in the diagram is parallel to the tangent plane touching the cone along the generating line $V S$ ; the curve is still confined to one of the half-cones, but it is now not a closed oval curve, it goes on endlessly as long as the generating lines of the half-cone are produced away from the vertex. Such a conic section is called parabola.

(3) The plane may cut both the half-cones,

so that the complete curve consists of two detached portions, or "branches" as they are called, this case is illustrated by the two branches $G_{2} A_{2} G_{2}^{'}$ and $L_{2} A_{2}^{'} L_{2}^{'}$ which together make up the curve. Neither branch is closed, each of them spreading out endlessly as the two half-cones are prolonged away from the vertex. Such a conic section is called a hyperbola.

There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas. It is easy to see that, in a sense, parabolas are limiting cases lying between ellipses and hyperbolas. They form a more special sort and have to satisfy a more particular condition. These three names are apparently due to Apollonius of Perga (born

about 260 , and died about 200 ), who wrote a systematic treatise on conic sections which remained the standard work till the sixteenth century.

It must at once be apparent how awkward

and difficult the investigation of the properties of these curves must have been to the Greek geometers. The curves are plane curves, and yet their investigation involves the drawing in perspective of a solid figure. Thus in the diagram given above we have practically drawn no subsidiary lines and yet the figure is sufficiently complicated. The

curves are plane curves, and it seems obvious that we should be able to define them without 16 going beyond the plane into a solid figure. At the same time, just as in the "solid" 17 definition there is one uniform method of definition–-namely, the section of a cone by

a plane–-which yields three cases, so in any "plane" definition there also should be one uniform method of procedure which falls into three cases. Their shapes when drawn on their planes are those of the curved lines in the three figures [fig:16]16, [fig:17]17, and [fig:18]18. The points $A$ and $A^{'}$ in the figures are called

18 the vertices and the line $A A^{'}$ the major axis. It will be noted that a parabola (cf. [fig.]17)