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# Conic Sections if

## cube, plane, cone, ratio, lines, god and figure

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CONIC SECTIONS IF a cone indefinitely extended be cut by a plane in any manner, the common section of its surface and the plane will be a geometrical line, which will be a curve in every case in which the plane does not pass through the ver tex. The curves which may be formed in this way, although agreeing in some of their properties, will yet differ in others. There can only, however, he three varieties ; an Ellipse, which is formed when the cutting plane passes in any direction across the cone ; a Para bola, when it is parallel to one side of the cone ; and an Hyperbola, when it has any other position. The cone may also be so cut that the section may be a Circle, but this curve may be considered as a kind of ellipse ; so that, upon the whole, there are only three distinct curves. Their properties constitute a very extensive mathemati• cal theory, a brief view of which is to form the subject of the present article. But before we proceed to the theory itself, it will be proper to give a short account of its origin.

It is well known that almost all the discoveries and improvements in the mathematics have had their origin in the efforts which have been made to resolve problems. It cannot be doubted but that the attempts which have been made to square the circle, although abortive, have led to the discovery of many interesting properties of that figure : Another problem of far less difficulty, is commonly supposed to have called the attention of ma thematicians to the conic sections, namely, the duplica tion of the cube, or, its equivalent, the finding of two mean proportionals between two given magnitudes.

'When the ancient mathematicians had succeeded in making a figure similar to any given plane figure, and having to it a given ratio, they would be led by analogy to extend the problem to similar solids : and as these are to one another as the cubes of their corresponding lineal dimensions, the whole difficulty would be reduced to the making a cube that should have any given ratio to a given cube. The cast when the ratio was that of 2 to I might be expected to be most easily resolved, and hence the duplication of the cube would occupy the attention of the first cultivators of geometry.

An ancient writer has, however, assigned a less natural origin to this problem. A pestilence is said to have ravaged Attica, and in the time of this calamity a deputa tion was sent to Delos, to consult the oracle by what means the celestial anger might be assuaged. The god was very moderate in his demands ; he only required that his altar, which was in the form of a cube, should be doubled. This was thought easy, and another of double the lineal dimensions was constructed. The true meaning of the god, however, was mistaken ; for the new altar was evidently eight times greater than the old one : no wonder then that the plague raged as as ever. Upon a second application to the god, his order was exactly comprehended, and the affair was referred to Plato, in whose school geometry was at that time held in the highest estimation.

There can be no doubt but that the abstract geome trical problem has been interwoven with the fable, to give it a greater degree of interest ; but it is certain, that this very problem was greatly agitated in the Pla tonic school ; and as, from its nature, it cannot he resolv ed merely by straight lines and circles, the only lines at first admitted into geometry, it became necessary to inquire what other lines next in order of simplicity to these Nr ould afford a solution of this and similar pro blems, and this investigation would naturally lead to the conic sections.

It is impossible now to say exactly who had the merit of the first discovery. Some attribute it to Alemeehmus, a disciple of Eudoxus, and a friend and cotemporary of Plato. This opinion rests on his being the first on record that resolved the problem of finding two mean propor tionals by the conic sections ; and on some verses sub joined by Eratosthenes to his epistle to King Ptolemy, where they arc called the curvet of Alrnechinus. How ever this may be, it is certain, that of eleven geometers, whose solutions of the problem have been recorded by Eutocius, two only have employed the conic sections, Lamely, Memechmus and Apollonius Pergxus, the latter of ‘vhom lived at a later period than the former.

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