Conic Sections: Treated Geometrically - and, George Bell and Sons Educational Catalogue

W. H. (William Henry) Besant

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• Publié le: 08 décembre 2010
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• Langue: English
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Sections Treated Geometrically, by W.H. Besant

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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

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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 aNew 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. ELEMENTARY DYNAMICS, A Treatise on, for the use of Colleges and Schools. By William Garnett, M.A., D.C.L. (late Whitworth Scholar), Fellow of St. John’s College, Cambridge; Principal of the Science College, NewcastleonTyne. 5th edition, revised. 6s. DYNAMICS, A Treatise on. By W. H. Besant, Sc.D., F.R.S. 2nd edition. 10s. 6d. HYDROMECHANICS, A Treatise on. By W. H. Besant, Sc.D., F.R.S., late Fellow of St. John’s College, Cambridge. 5th edition, revisedI.. Part Hydrostatics. 5s. ELEMENTARY HYDROSTATICS. By W. H. Besant, Sc.D., F.R.S. 16th edition. 4s. 6d.Key, 5s. HEAT, An Elementary Treatise on. By W. Garnett, M.A., D.C.L., Fellow of St. John’s College, Cambridge; Principal of the Science College, Newcastle onTyne. 6th edition4, revised. s. 6d.

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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 theEncyclopaedia Metropolitana. The deduction of the properties of these curves from their definition as the sections of a cone, seemsà priorito 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 onConic 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. (156have made). I 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 ChapterVI., 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 ChapterXI., and also to Mr E. Hill, Fellow of St John’s College, for his kindness in looking over the latter half of the proofsheets. I venture to hope that the methods adopted in this treatise will give a clear view of the properties of Conic Sections, and that the numerous Examples appended to the various Chapters will be useful as an exercise to the student for the further extension of his conceptions of these curves.

Cambridge, March, 1869.

W. H. BESANT.