Conic Sections
When the Greek geometers had exhausted,
as they thought, the more obvious and interesting properties of figures made up of straight lines and circles, they turned to the study of other curves; and, with their almost infallible instinct for hitting upon things worth thinking about, they chiefly devoted themselves to conic sections, that is, to the curves in which planes would cut the surfaces of circular cones. The man who must have the credit of inventing the study is Menaechmus (born 375 and
died 325 ); he was a pupil of Plato and one of the tutors of Alexander the Great. Alexander, by the by, is a conspicuous
example of the advantages of good tuition, for another of his tutors was the philosopher Aristotle. We may suspect that
Alexander found Menaechmus rather a dull teacher, for it is related that he asked for the
proofs to be made shorter. It was to this request that Menaechmus replied: "In the
country there are private and even royal roads, but in geometry there is only one road for all." This reply no doubt was true enough in the sense in which it would have been immediately understood by Alexander. But if Menaechmus thought that his proofs could not be shortened, he was grievously mistaken; and most modern mathematicians would be horribly bored, if they were compelled to study the Greek proofs of the properties of conic sections. Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea. There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of "important" problems. The history of the science is entirely against him. There are royal roads in science; but those who first tread them are men of genius and
not kings.
The way in which conic sections first presented themselves to mathematicians was as follows: think of a cone (cf. [fig.]15), whose vertex (or point) is , standing on a circular base . For example, a conical shade to
an electric light is often an example of such a surface. Now let the "generating" lines which pass through and lie on the surface be all produced backwards; the result is a double cone, and is another circular cross section on the opposite side of to the cross section . The axis of the cone passes through all the centres of these circles and is perpendicular to their planes, which are parallel to each other. In the diagram the parts of the curves which are supposed to lie behind the plane of the paper are dotted lines, and the parts on the plane or in front of it are continuous lines. Now suppose this double cone is cut by a plane not perpendicular to the axis , or at least not necessarily perpendicular to it. Then three cases can arise:–-