Treatise on Conic Sections: Edited in Modern Notation, With Introductions Including an Essay on the Earlier History of the Subject (Classic Reprint)

Excerpt from Appolonius of Perga, Treatise on Conic Sections: Edited in Modern Notation With Introductions Including an Essay on the Earlier History of the SubjectAgain, the form of the propositions is an additional difficulty, because the reader finds in them none of the ordinary aids towards the comprehension of somewhat complicated geometrical work, such as the conventional appropriation, in modem text-books, of definite letters to denote particular points on the various conic sections. On the contrary, the enunciations of propositions which, by the aid of a notation once agreed upon, can now be stated in a few lines, were by Apollonius invariably given in words like the enunciations of Euclid. These latter are often sufficiently unwieldy; but the inconvenience is greatly intensified in Apollonius, where the greater complexity of the conceptions entering into the investigation of conics, as compared with the more elementary notions relating to the line and circle, necessitates in many instances an enunciation extending over a space equal to (say) half a page of this book. Hence it is often a matter of considerable labour even to grasp the enunciation of a proposition. Further, the propositions are, with the exception that separate paragraphs mark the formal divisions, printed continuously; there are no breaks for the purpose of enabling the eye to take in readily the successive steps in the demonstration and so facilitating the comprehension of the argument as a whole. There is no uniformity of notation, but in almost every fresh proposition a different letter is employed to denote the same point: what wonder then if there are the most serious obstacles in the way of even remembering the results of certain propositions? Nevertheless these propositions, though unfamiliar to mathematicians of the present day, are of the very essence of Apollonius' system, are being constantly used, and must therefore necessarily be borne in mind.