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A

COURSE

OF

MATHEMATICS.

IN TWO VOLUMES.

COMPOSED FOR THE USE OF

THE ROYAL MILITARY ACADEMY.

BY

CHARLES HUTTON, LL.D. F.R.S.

FORMERLY PROFESSOR OF MATHEMATICS IN THAT

INSTITUTION.

CONTINUED AND AMENDED BY

OLINTHUS GREGORY, LL.D. F.R.A.S.

TWELFTH EDITION,

WITH CONSIDERABLE ALTERATIONS AND ADDITIONS,

BY

THOMAS STEPHENS DAVIES,

F.R.S. L. & E. F.S.A.

ROYAL MILITARY ACADEMY.

LONDON:
LONGMAN, BROWN, & CO.; J. M. RICHARDSON; J. G. F. & J. RIVINGTON; HAMILTON

& CO.; WHITTAKER & CO.; DUNCAN & MALCOLM; SIMPKIN, MARSHALL, & CO.;
SHERWOOD & Co.; SOUTER & LAW; COWIE & CO.; SMITH, ELDER, & CO.; ALLEN &
CO.; HARVEY & DARTON; HOULSTON & STONEMAN; H. WASHBOURNE; C. DOLMAN;
DARTON & CLARKE; AND G. ROUTLEDGE. EDINBURGH: STIRLING & CO.

1813.

ADVERTISEMENT.

The present edition of Hutton's Course, Vol. II., is entirely recomposed, so as to be in accordance with the most improved state of the several subjects included within its range: and neither labour nor expense has been spared to render it worthy of the same approbation which has been awarded to the preceding volume.

The section on Spherical Trigonometry has been much enlarged ; and it includes several important original investigations. A new view is taken of Napier's circular parts; and the arrangement of the cases for oblique triangles rendered as simple and easy of remembrance as the cases of Plane Trigonometry. A chapter on Spherical Astronomy is added, in which the whole of the ordinary problems are reduced to the solution of two specific spherical triangles.

The chapters on the Conic Sections are also enlarged, especially in the part relating to the parabola ; and several investigations of the more recondite properties of the curves, not to be found in English treatises, as well as interesting properties of the figures formed by their revolution, are also given. The theorems of Pascal, Desargues, Brianchon, Lambert, and others, are demonstrated in, generally, two or three ways.

A concise treatise on Transversals is also given, comprising the chief and most general results obtained by foreign mathematicians, and demonstrated in a simple, consistent, and perspicuous manner.

The Geometry of Co-ordinates of two dimensions, especially as regards the straight line, the circle, and the conic sections, is laid down in the most general manner; and the use and advantages of the polar equations of the tangent and normal amply illustrated. The most general properties of the conic sections are obtained by this method.

The doctrine of Fluxions has been superseded by a complete treatise on the Differential and Integral Calculus, as far as functions of a single variable are concerned. Numerous applications to Geometry, as well as considerable and varied examples of other kinds, are included in this section of the work : and probably there is no topic of importance which comes under this head which is not discussed with the requisite detail. It has been attempted to render the doctrine of curves, and some classes of curve surfaces, as complete as in an elementary work it was possible to accomplish.

Some subjects of the second volume of the preceding edition have been omitted in the present one. As they were there given, they would be useless in the present state of science ; and their republication in the shape

3

they formerly assumed would neither be considered creditable to an editor, nor prove advantageous to a publisher.

The proof sheets have been read with great care by Messrs. Rutherford and Fenwick, as well as myself; and I hope that there will be few, if any, errata discovered. Should my readers, however, find any in either volume, they will confer a favour by giving me the information.

May, 1843.

VOL. J.

With respect to the first volume, though the remodelling was not so entire as in the second, I may state that considerable and important alterations were made.

The method of using detached coefficients in elementary algebra has been explained and enforced ; and of that most important process, synthetic division, two investigations have been given. Also, to the simpler operations of algebra, where the reason of the step is not apparent at once, investigations are annexed, to secure the student a complete understanding of the logic of his processes.

In the chapter on simple and quadratic equations, the introductory remarks and suggestions, as well as the examples chosen for illustrating the methods by actual working, have been generally exchanged for others better adapted to show the true character of the operations. In the quadratics, the Hindû method of completing the square is enforced, as being generally superior, in respect of facility, to the Italian or common

one.

The chapter on the general resolution of numerical equations has been wholly recomposed ; and I hope it will be found free from those logical defects which are so liable to insinuate themselves into abbreviated treatises on subjects involving so many distinct principles as this does. The theory of equations, is, however, carried no further than is requisite for numerical solution :

: though to this extent, great pains have been taken to render it logically complete. Legitimate proofs, on elementary principles, are given of the criteria of De Gua and Budan, for detecting the imaginary roots of an equation; and as brief a form of investigating Sturm's criterion as I could devise, has also been added.

Upon Horner's method of continuous approximation to the roots of equations, I have dwelt at sufficient length to render it easy of comprehension. As the first attempt ever made to compose an elementary treatise on this subject was made by myself in the previous edition of this work, my attention was naturally directed to it subşeqaont] ya with sufficient precision

to enable me to separate the essential and the useful part of that composition, from the parts which I found superfluous, and make such additions as experience might suggest during my professional use of the volume.

The chapters on indeterminate coefficients, piling of balls, the binomial and exponential theorems, and on logarithms, it will be seen are all written anew, and with especial reference to the order in which the subjects naturally present themselves in a systematic course of study. The same may be said of the chapters on series and finite differences.

The practical geometry has been entirely recomposed, and in especial reference to the circumstances under which the problems themselves occur in practice. A number of constructions of this kind, which are believed to be new, and are adapted to peculiar exigences, have been introduced : and it will, probably, be found to be the most complete system of its kind that has ever been published.

The chapter on practical geometry in the field contains a series of problems of great importance to the military profession, to engineers and surveyors, and which formed the substance of a course of lectures originally delivered at the Royal Artillery Institution, Woolwich.

In the plane trigonometry nothing besides the examples for exercise, of the last edition, remains in this. To give every thing essential to elemenmentary trigonometry investigated in a direct and simple manner, and entirely to exclude all matters of mere scientific curiosity, has been my guiding principle in the composition of these chapters. Trigonometry, therefore, instead of forming two separate treatises in two successive volumes, is now brought entirely into the first; and the examples that are changed in place have been marked by a quotation of the places in which they previously stood, for the convenience of those who wish to make reference to any works founded on the preceding edition.

The figures in this edition are nearly all newly-cut, and every attention has been paid to the arrangement of each page, both for convenience of reading and reference, and of losing no space that could possibly be filled up with useful matter. Much of the phraseology, and the entire notation, of former editions has been modernised, and an attempt has been made to render it, with the exceptions specified in the preface, consistent and systematic throughout.

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