The Project Gutenberg Ebook #29913: Conic Sections
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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 LATEX 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.) In use at Winchester; Wellington; Marlborough; Rugby; Charterhouse; St. Paul's; Merchant Taylors'; Christ's Hospital; Sherborne; Shrewsbury; Bradford; Bradfield; Leamington College; Felsted; Cheltenham Ladies' Col- lege; Edinburgh, Daniel Stewart's College; Belfast Academical Institution; King's School, Parramatta; Royal College, Mauritius; &c. &c. EXAMPLES IN ARITHMETIC, extracted from the above, 5th. edition, with or without Answers, 3s.; or in Two Parts, 1s. 6d. and 2s. CHOICE AND CHANCE. An Elementary Treatise on Permutations, Combi- nations, and Probability, with 640 Exercises. By W. A. Whitworth, M.A., late Fellow of St. John's College, Cambridge. 4th edition, revised. 6s. EUCLID. Books I.{VI. and part of Book XI. Newly translated from the orig- inal Text, with numerous Riders and Miscellaneous Examples in Modern Geometry. By Horace Deighton, M.A., formerly Scholar of Queen's Col- lege, Cambridge; Head Master of Harrison College, Barbados. 3rd edition. 4s. 6d. Or Books I.{IV., 3s. Books V. to end, 2s. 6d. Or in Parts: Book I., 1s. Books I. and II., 1s. 6d. Books I.{III., 2s. 6d. Books III. and IV., 1s. 6d. A Key, 5s. net. In use at Wellington; Charterhouse; Bradfield; Glasgow High School; Portsmouth Grammar School; Preston Grammar School; Eltham R.N. School; Saltley College; Harris Academy, Dundee, &c. &c. EXERCISES ON EUCLID and in Modern Geometry, containing Applications of the Principles and Processes of Modern Pure Geometry. By J. McDow- ell, M.A., F.R.A.S., Pembroke College, Cambridge, and Trinity College, Dublin. 3rd edition, revised. 6s. ELEMENTARY TRIGONOMETRY. By J. M. Dyer, M.A., and the Rev. R. H. Whitcombe, M.A., Assistant Mathematical Masters, Eton College. 2nd edition, revised. 4s. 6d. INTRODUCTION TO PLANE TRIGONOMETRY. By the Rev. T. G. Vyvyan, M.A., formerly Fellow of Gonville and Caius College, Senior Math- ematical Master of Charterhouse. 3rd edition, revised and corrected. 3s. 6d. George Bell & Sons' Mathematical Works. 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. 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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 `apriori 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 ChapterI., 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, PREFACE TO THE FIRST EDITION. vi including the theory of reciprocal polars for the particular case in which the circle is employed as the auxiliary curve. For the more general methods of projections, of reciprocation, and of an- harmonic properties, the student will consult the treatises of Chasles, Pon- celet, Salmon, Townsend, Ferrers, Whitworth, and others, who have recently developed, with so much fulness, the methods of modern Geometry. I have to express my thanks to Mr R. B. Worthington, of St John's College, and of the Indian Civil Service, for valuable assistance in the con- structions of Chapter XI., and also to Mr E.