Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1911 edition. Excerpt: ...abstract conceptions which correspond to the fundamental facts of the physical world. CHAPTER X 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 B.c. and died 325 B.c.); 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...