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JOHN NARRIEN, F.R.S. AND R.A.S.

PROFESSOR OF MATHEMATICS, ETC. IN THE INSTITUTION.

LONDON:

PRINTED FOR

LONGMAN, BROWN, GREEN, AND LONGMANS,

PATERNOSTER-ROW.

1846.

LONDON : Printed by A. SPOTTISWOODE,

New-Street-Square,

ADVERTISEMENT.

A TREATISE of Analytical Geometry, with the Properties of the Conic Sections, being required for immediate use in this Institution, it has been found convenient so far to depart from the original arrangement as to publish a work under that title alone. The following Treatise must, therefore, be considered as the fourth of the series which is to constitute a General Course of Mathematics for the use of the gentlemen cadets and the officers in the senior department; and the Course, when completed, will now comprehend the subjects whose titles are subjoined: - I. Arithmetic and Algebra. * II. Geometry.* III. Plane Trigonometry with Mensuration.* IV. Analytical Geometry with the Properties of Conic Sections. * V. The Differential and Integral Calculus. VI. Practical Astronomy and Geodesy, including Spherical Trigonometry.* VII. The Principles of Mechanics; and VIII. Physical Astronomy.

Royal Military College,

1846.

* Published.

PREFACE.

THE properties of straight lines, as well as of plane figures bounded by straight lines or circles, and of solids bounded by planes, or by surfaces produced by the revolutions of straight lines or circles, have, in a part of the course already published for this Institution (Elements of Geometry), been demonstrated by the synthetical method of the ancient geometers; but in the present work, which contains investigations relating to the properties of curve lines and surfaces of more complex kinds, processes of an analytical nature have been employed. For such subjects the analytical method has great advantages over the other, both from the simplicity with which the steps occurring in the researches may be represented, and from the comprehensiveness of the results obtained; this last being such that the solutions of the different cases of a proposition are usually included in one general formula.

Descartes was the first who applied the processes of algebra to geometrical propositions : he observed that lines, straight or curved, and the curved surfaces of solids, afford relations between the co-ordinates of points in those lines and surfaces ; and he was led to express lines and surfaces by algebraio equations; those which involve two variable quantities being capable of representing the positions of an infinite number of points in a straight line, or a plane curve, while equations involving three variables are capable of representing the positions of an infinite number of points in a line or surface of any kind, and situated in any manner in space : he, also, first arranged curves in orders according to the degrees of the variables in the equation. These discoveries gave rise to the branch of science called analytical geometry, which, in its actual state, is of the highest value as a means of investigating the properties of curve lines and surfaces.

In the present work it has been attempted to render the investigations as easy as possible for students who, in mathematics, are familiar with the processes of elementary geometry and algebra only ; and it will, perhaps, be thought that, in some cases, this has been attended with a sacrifice of elegance. It is hoped, however, that the reason here given

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