

CON 



132 



CON 



contributed to its importance. The peasants of Cara- 

 glio mid other places, fled from the tyranny of their 

 lords, and built a small fortress in the spot where Coni 

 now stands. In 1125, it was attacked in vain by 

 Francis Stampa, the general of the Swiss army ; in 1 1 42, 

 by the famous Claude Annebaud, who was afterwards 

 admiral of France; and in 1557, by the Mareschal de 



Brissac. In 1795 it was taken by the French, and was 

 added to the French empire in 1803. Population 

 16,500. See Deninas Tableau Historique, Staiistique 

 et Morale de la Haute Italie, p. 48, Paris, 1805; and 

 Breton's Voyage en Piemont, p. 229, Paris, 1803. This 

 last work contains a view of the town, (sr) 



Coni, 



CONIC SECTIONS. 



Conic 



Sections. 



Definition. 



History-. 



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 vertex. 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, 

 be three varieties; an Ellipse, which is fonned when the 

 cutting plane passes in any direction across the cone ; 

 a Parabola, 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 ma- 

 thematical 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 ac- 

 count 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 pro- 

 blems. 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 at- 

 tention of mathematicians to the conic sections, name- 

 ly, the duplication 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 woidd be led by 

 analogy to extend the problem to similar solids : and 

 as these are to one another as the cubes of their corre- 

 sponding lineal dimensions, the whole difficulty would 

 be reduced to the making a cube that should have any 

 given ratio to a given cube. The case when the ratio 

 was that of 2 to 1 might be expected to be most easily 

 resolved, and hence the duplication of the cube would 

 occupy the attention of the first cultivators of geo- 

 metry. 



An ancient writer has, however, assigned a less na- 

 tural origin to this problem. A pestilence is said to 

 have ravaged Attica, and in the time of this calamity a 

 deputation 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 re- 

 quired that his altar, which was in the form of a cube, 

 should be doubled. This was thought easy, and an- 

 other 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 

 fiercely as ever. Upon a second application to the 



god, his order was exactly comprehended, and the affair Conic 

 was referred to Plato, in whose school geometry was Section 

 at that time held in the highest estimation. ■""■"V"' 



There can be no doubt but that the abstract geome- Plat0 - 

 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 be re- 

 solved merely by straight lines and circles, the only 

 lines at first admitted into geometry, it became neces- 

 sary to inquire what other lines next in order of simpli- 

 city to these woidd afford a solution of this and similar 

 problems, and this investigation would naturally lead 

 to the conic sections. 



It is impossible now to say exactly who had the me- MenacL 

 rit of the first discovery. Some attribute it to Me- nius. 

 naechmus, a disciple of Eudoxus, and a friend and co- 

 temporary of Plato. This opinion rests on his being 

 the first on record that resolved the problem of finding 

 two mean proportionals by the conic sections; and on 

 some verses subjoined by Eratosthenes to his epistle to 

 King Ptolemy, where they are called the curves of Me- 

 nwchmus. However 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, namely, Menaechmus and Apollonius 

 Pergaeus, the latter of whom lived at a later period 

 than the former. 



The interest which mankind in general take in the 

 mathematical sciences, is but little in comparison to 

 that which is excited by works of poetry, oratory, his- 

 tory, and the like, and hence it has happened that the 

 writings of the ancients on these subjects have had a 

 far better chance than their mathematical theories, of 

 descending to the present times. However much we 

 may regret the circumstance, it will not therefore ap- 

 pear wonderful that we now know nothing more than 

 the names of the early cultivators of the conic sections : 

 and of these Aristaeus deserves to be particularly men- 

 tioned. Pappus of Alexandria informs us, in his Ma- 

 thematical Collections, that this geometer composed 

 five books De Locis Solidis, and as many on conic sec- 

 tions, all which are now entirely lost. The celebrated 

 geometer Euclid is supposed to have been a disciple of 

 Aristaeus, at any rate he must have been his very par- 

 ticular friend. We learn from Pappus, that Euclid 

 composed a treatise in four books on conic sections, but 

 that also has been lost. 



Of several other geometers who appear to have cul- Concn; 

 tivated this theory, we shall only mention Conon the 

 friend of Archimedes ; but he is better known as an 

 astronomer than as a geometer. 



The writings of Archimedes shew, that before his Archi- 

 time, considerable progress had been made in the dis- mcdcf » 

 covery of the properties of the conic sections, as he re- 

 fers to many of them incidentally, and speaks of others 





