An Introduction to Mathematics by Alfred North Whitehead

Author:Alfred North Whitehead
Language: eng
Format: epub
Publisher: INscribe Digital
Published: 2017-04-11T04:00:00+00:00

Fig. 15.

(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 G2 A2 G2′ and L2 A2′ L2′ 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 B.C., and died about 200 B.C.), 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 going beyond the plane into a solid figure. At the same time, just as in the “solid” 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 16, 17, and 18. The points A and A′ in the figures are called the vertices and the line AA′ the major axis. It will be noted that a parabola (cf. fig. 17) has only one vertex. Apollonius proved1 that the ratio of PM to AM.MA′ remains constant both for the ellipse and the hyperbola (figs. 16 and 18), and that the ratio of PM2 to AM is constant for the parabola of fig. 17; and he bases most of his work on this fact. We are evidently advancing towards the desired uniform definition which does not go out of the plane; but have not yet quite attained to uniformity.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.