Excerpt from The Cone and Its Sections: Treated Geometrically
Of the works of the ancient Geometricians that hare descended to us, none perhaps has been more justly celebrated than the Treatise of Apllonius, of Perga, on the Conic Sections - a work that has, apparently, maintained its superiority over every other Treatise that has since been written on the subject.
According to his Commentator, Eutocius, Apollonius was the first to derive the Sections from the Scalene Cone by different inclinations of the Cutting Plane, and in accordance with this derivation of the Curves he deduces the Primary Properties of the Sections from the Cone itself.
Modern writers have however to a considerable extent departed from this method - commencing their works by definitions of the Curves described in plano, such definitions being adapted to or derived from Properties of the Curves themselves; a strong tendency has however been shown, and especially by one Geometer of the last century, Hamilton, to revert to the Apollonian method. Accordingly in the present work the Primary Properties of the Sections are derived from the Cone, and it is believed that this is the case to a greater extent than in any previous Treatise. The reasons that have principally led to this course of treatment, are first, that the elementary Properties alluded to, and contained in Prop. V., are by the help of the most elementary Properties of the Cone demonstrated as it were together, and their connection and mutual dependence shown in the clearest manner; and secondly, it is presumed that by becoming acquainted with these principles as thus deduced, a deeper impression is obtained of their great and essential importance than could possibly be the case were that acquaintance made by deductions from any definition of the Curves in plano, seeing that according to the definition adopted the process of deduction must necessarily vary.
These Properties will then be found demonstrated both from the Scalene and Right Cone, and although their derivation from the former involves considerable prolixity, it is believed that the reader will be well repaid for the time and patience expended in the investigation.
The description of the Curves in plano and the derivation of their Properties thence, has however not been omitted; the method adopted in this part of the work being that of the Generating Circle, by which means not only the elementary, but many other Properties of the Sections can be proved both easily and elegantly. In the Articles on the Anharmonic Properties of the Sections, demonstrations of two important Theorems have been specially added in further illustration of the utility of this principle.
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