AN ELEMENTARY TREATISE IN WHICH THE CONIC SECTIONS ARE DEFINED AS THE PLANE SECTIONS OF A CONE, AND TREATED BY THE METHOD OF PROJECTIONS. BY J. STUART JACKSON, M.A., LATE FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE. PREFACE. THE following pages have been written with a view to give the student the benefit of the Method of Projections as applied to the Ellipse and Hyperbola. This Method is calculated to produce a material simplification in the treatment of those curves and to make the proof of their properties more easily understood in the first instance and more easily remembered. It is also a powerful instrument in the solution of a large class of problems relating to these curves. When the Method of Projections is admitted into the treatment of the Conic Sections there are many reasons why they should be defined, not as has been the case of late years with reference to the focus and directrix, but according to the original definition from which they have their name, as plane Sections of a Cone. First and principally, because this definition gives an immediate proof of the relation by projection of the Ellipse and Circle and of the General and Rectangular Hyperbolas : and in the second place, it naturally divides the properties that may be proved by projection from those connected with the focus and directrix, and thus introduces a valuable simplification into the treatment of the subject. It is also a consideration of some importance that we can see at once from the form of the cone, the general form of the curves that may be cut from it by a plane in different positions; and, by turning the plane about a certain line, we see how the curves pass from one form into another. It is hoped these may be thought sufficient reasons for departing from the definition which has been in use of late years. I have retained the proof of the constancy of the ratio of the rectangles under the segments of intersecting chords in constant directions (Chap. v. § 15), given by Dean Hamilton in his treatise published in 1773, as an improving and interesting study and one that is in harmony with the scheme of this work. I have to acknowledge the great kindness of Rev. Chas. Taylor, Fellow of St John's College, Cambridge, in permitting me the use of his excellent proof of the intersection of tangents at the extremities of a chord of any Conic Section on the diameter of the chord; with the proofs of CV.CT CP (Chap. v. § 8), and PV=PT (Chap. VII. § 20), depending on it: and also of his proof of SQP=HQP' (Chap. IV. § 13). Professor Adams' property of the tangent is indispensable in any geometrical treatment of the Conic Sections, and I have his kind permission to make use of it. I shall be greatly obliged by any corrections and hints towards improvement in case this work should be so fortunate as to reach a second edition. 9, BROOKSIDE, CAMBRIDGE, Christmas, 1871. |