Project Gutenberg’s Conic Sections Treated Geometrically, by W.H. Besant docx

Project Gutenberg’s Conic Sections Treated Geometrically, by W.H. Besant This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Conic Sections Treated Geometrically and, George Bell and Sons Educational Catalogue Author: W.H. Besant Release Date: September 6, 2009 [EBook #29913] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK CONIC SECTIONS *** Produced by K.F. Greiner, Joshua Hutchinson, Nigel Blower and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by Cornell University Digital Collections) This file is optimized for screen viewing, with colored internal hyperlinks and cropped pages. It can be printed in this form, or may easily be recompiled for two- sided printing. Please consult the preamble of the L A T E X source file for instructions. Detailed Transcriber’s Notes may be found at the end of this document. George Bell & Sons’ Mathematical Works. CAMBRIDGE MATHEMATICAL SERIES. Crown 8vo. ARITHMETIC. With 8000 Examples. By Charles Pendlebury, M.A., F.R.A.S., Senior Mathematical Master of St. Paul’s, late Scholar of St. John’s College, Cambridge. Complete. With or without Answers. 7th edition. 4s. 6d. In two Parts, with or without Answers, 2s. 6d. each. Part 2 contains Commercial Arithmetic. (Key to Part 2, 7s. 6d. net.) 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ANALYTICAL GEOMETRY FOR BEGINNERS. Part 1. The Straight Line and Circle. By the Rev. T. G. Vyvyan, M.A. 2s. 6d. CONIC SECTIONS, An Elementary Treatise on Geometrical. By H. G. Willis, M.A., Clare College, Cambridge, Assistant Master of Manchester Grammar School. 5s. CONICS, The Elementary Geometry of. By C. Taylor, D.D., Master of St. John’s College, Cambridge. 7th edition. Containing a New Treatment of the Hyperbola. 4s. 6d. SOLID GEOMETRY, An Elementary Treatise on. By W. Steadman Aldis, M.A., Trinity College, Cambridge; Professor of Mathematics, University College, Auckland, New Zealand. 4th edition, revised. 6s. ROULETTES AND GLISSETTES, Notes on. By W. H. Besant, Sc.D., F.R.S., late Fellow of St. John’s College, Cambridge. 2nd edition. 5s. GEOMETRICAL OPTICS. An Elementary Treatise. By W. Steadman Aldis, M.A., Trinity College, Cambridge. 4th edition, revised. 4s. RIGID DYNAMICS, An Introductory Treatise on. By W. Steadman Aldis, M.A. 4s. 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George Bell & Sons’ Mathematical Works. MECHANICS, A Collection of Problems in Elementary. By W. Walton, M.A., Fellow and Assistant Tutor of Trinity Hall, Lecturer at Magdalene College. 2nd edition. 6s. PHYSICS, Examples in Elementary. Comprising Statics, Dynamics, Hydro- statics, Heat, Light, Chemistry, Electricity, with Examination Papers. By W. Gallatly, M.A., Pembroke College, Cambridge, Assistant Examiner at London University. 4s. MATHEMATICAL EXAMPLES. A Collection of Examples in Arithmetic Algebra, Trigonometry, Mensuration, Theory of Equations, Analytical Ge- ometry, Statics, Dynamics, with Answers, &c. By J. M. Dyer, M.A. (As- sistant Master, Eton College), and R. Prowde Smith, M.A. 6s. CONIC SECTIONS treated Geometrically. By W. H. Besant, Sc.D., F.R.S., late Fellow of St. John’s College. 8th edition, fcap. 8vo. 4s. 6d. ANALYTICAL GEOMETRY for Schools. By Rev. T. G. Vyvyan, Fellow of Gonville and Caius College, and Senior Mathematical Master of Charter- house. 6th edition, fcap. 8vo. 4s. 6d. CAMBRIDGE MATHEMATICAL SERIES CONIC SECTIONS GEORGE BELL & SONS LONDON: YORK STREET, COVENT GARDEN AND NEW YORK, 66, FIFTH AVENUE CAMBRIDGE: DEIGHTON, BELL & CO. CONIC SECTIONS TREATED GEOMETRICALLY BY W. H. BESANT Sc.D. F.R.S. FELLOW OF ST JOHN’S COLLEGE CAMBRIDGE NINTH EDITION REVISED AND ENLARGED LONDON GEORGE BELL AND SONS 1895 Cambridge: PRINTED BY J. & C. F. CLAY, AT THE UNIVERSITY PRESS. PREFACE TO THE FIRST EDITION. In the present Treatise the Conic Sections are defined with reference to a focus and directrix, and I have endeavoured to place before the student the most important properties of those curves, deduced, as closely as possible, from the definition. The construction which is given in the first Chapter for the determination of points in a conic section possesses several advantages; in particular, it leads at once to the constancy of the ratio of the square on the ordinate to the rectangle under its distances from the vertices; and, again, in the case of the hyperbola, the directions of the asymptotes follow immediately from the construction. In several cases the methods employed are the same as those of Wallace, in the Treatise on Conic Sections, published in the Encyclopaedia Metropolitana. The deduction of the properties of these curves from their definition as the sections of a cone, seems `a priori to be the natural method of dealing with the subject, but experience appears to have shewn that the discussion of conics as defined by their plane properties is the most suitable method of commencing an elementary treatise, and accordingly I follow the fashion of the time in taking that order for the treatment of the subject. In Hamilton’s book on Conic Sections, published in the middle of the last century, the properties of the cone are first considered, and the advantage of this method of commencing the subject, if the use of solid figures be not objected to, is especially shewn in the very general theorem of Art. (156). I have made much use of this treatise, and, in fact, it contains most of the theorems and problems which are now regarded as classical propositions in the theory of Conic Sections. I have considered first, in Chapter I., a few simple properties of conics, and have then proceeded to the particular properties of each curve, commencing with the parabola as, in some respects, the simplest form of a conic section. It is then shewn, in Chapter VI., that the sections of a cone by a plane produce the several curves in question, and lead at once to their definition as loci, and to several of their most important properties. A chapter is devoted to the method of orthogonal projection, and another to the harmonic properties of curves, and to the relations of poles and polars, [...]... now given a general method of constructing a conic section, and we have explained the nomenclature which is usually employed We proceed to demonstrate a few of the properties which are common to all the conic sections For the future the word conic will be employed as an abbreviation for conic section Prop II If the straight line joining two points P , P of a conic meet the directrix in F , the straight... Reciprocal Polars 217 CHAPTER XIII The Construction of a Conic from Given Conditions 231 CHAPTER XIV The Oblique Cylinder, the Oblique Cone, and the Conoids 245 CHAPTER XV Conical Projection 257 II 269 Miscellaneous Problems CONIC SECTIONS Introduction DEFINITION If a straight line and a point be given in position... traced out by the moving point is called a Conic Section The fixed point is called the Focus, and the fixed line the Directrix of the conic section When the ratio is one of equality, the curve is called a Parabola When the ratio is one of less inequality, the curve is called an Ellipse When the ratio is one of greater inequality, the curve is called an Hyperbola These curves are called Conic Sections, ... continuity with which the curves pass into each other will appear from the definition of a conic section as a Locus, or curve traced out by a moving point, as well as from the fact that they are deducible from the intersections of a cone by a succession of planes CHAPTER I PROPOSITION I The Construction of a Conic Section 1 Take S as the focus, and from S draw SX at right angles to the directrix, and... axis, and intersecting the curve in R, and R , is called the Latus Rectum It is hence evident that the form of a conic section is determined by its eccentricity, and that its magnitude is determined by the magnitude of the latus rectum, which is given by the relation SR : SX :: SA : AX CONICS 6 3 Def The straight line P N (Fig Art 1), drawn from any point P of the curve at right angles to the axis,... Hyperbola 125 CHAPTER VI The Cylinder and the Cone 135 CONTENTS ix CHAPTER VII The Similarity of Conics, the Areas of Conics, and the Curvatures of Conics 152 CHAPTER VIII Orthogonal Projections 165 CHAPTER IX Of Conics in General 174 CHAPTER X Ellipses as Roulettes and Glissettes 181 I ... first, when the conic is a circle, and secondly, when it consists of two straight lines 2 Having given two points of a conic, the directrix, and the eccentricity, determine the conic 3 Having given a focus, the corresponding directrix, and a tangent, construct the conic 4 If a circle passes through a fixed point and cuts a given straight line at a constant angle the locus of its centre is a conic 5 If P... be a focal chord of a conic, and P any point of the conic, and if QP , Q P meet the directrix in E and F , the angle ESF is a right angle For, by Prop II., SE bisects the angle P SQ , and SF bisects the angle P SQ; hence it follows that ESF is a right angle This theorem will be subsequently utilised in the case in which the focal chord Q SQ is coincident with the axis of the conic FOCAL CHORDS 8 7... methods of geometry as applied to the conic sections A new edition, the fourth, of the book of solutions of the examples and problems has been prepared, and is being issued with this new edition of the treatise, with which it is in exact accordance W H BESANT December 14, 1894 CONTENTS page Introduction 1 CHAPTER I The Construction of a Conic Section, and General Properties... and if the common tangent to the conic and circle touch the conic in P and the circle in Q, the angle P SQ is bisected by the latus rectum (Refer to Cor 2 Art 14.) 29 Given two points, the focus, and the eccentricity, determine the position of the axis 30 If a chord P Q subtend a constant angle at the focus, the locus of the intersection of the tangents at P and Q is a conic with the same focus and directrix . properties which the curves possess in common, and also the special characteristics wherein they differ from each other; and the continuity with which the curves pass into each other will appear from the. Project Gutenberg’s Conic Sections Treated Geometrically, by W. H. Besant This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You. I have also inserted a new chapter, on Conical Projections, dealing however only with real projections. The first nine chapters, with the first set of miscellaneous problems, now constitute the