HEIGHTS AND DISTANCES, PROJECTIONS OF THE
SPHERE, DIALLING, ASTRONOMY, THE SOLUTION
OF EQUATIONS, AND GEODESIC OPERATIONS.

BY

OLINTHUS GREGORY, LL.D.

Of the Royal Military Academy,

VICE-PRESIDENT OF THE LONDON PHILOSOPHICAL SOCIETY, AND
HONORARY MEMBER OF THE LITERARY AND PHILOSOPHICAL, AND
OF THE ANTIQUARIAN, SOCIETIES AT NEWCASTLE UPON TYNE,

LONDON:

PRINTED FOR BALDWIN, CRADOCK, AND JOY,
47, PATERNOSTER-ROW;

W. BLACKWOOD, EDINBURGH; AND J. CUMMING,

DUBLIN.

PREFACE.

HE who wishes to acquire any other reputation than that of being useful, will not devote his time to the preparation of an elementary work. Such, however, is the humble honour after which I here aspire: and if the small volume I now lay before the public shall contribute, in any degree, to increase the knowledge and improve the taste of the mathematical student in the earlier stages of his progress, I shall have attained my object.

Of late years the authors of elementary treatises on different branches of mathematics, and especially Dr. Hutton and Mr. Bonnycastle, in their compendious manuals for popular use, have shown that it is possible, by judicious arrangement and selection, to compress much interesting, valuable, and scientific matter, within the compass of a small duodecimo volume. There can, I think, be no doubt that to the extensive circulation of this class of books, together with the stimulus furnished by the annual problems in the Ladies' and Gentleman's Diaries, and those proposed in the Senate House at Cambridge, must principally be ascribed the circumstance that mathematical knowledge, to a certain extent, is more widely diffused over the middle classes of society in this country, than in any other part of Europe.

What, however, has been so successfully effected in other departments of mathematics has not yet been attempted with regard to Trigonometry. We have some excellent works on this subject, whose value it would ill become me to depreciate. But such of them as go extensively into the business of Trigonometry and its applications are too large and expensive for general circulation; while others, being confined almost entirely to the elements, exclusive of the applications, must of necessity be restricted, both in point of circulation and utility. There is one treatise, that of Emerson, which is a most copious store-house of curious and elegant theorems: but they are so obscured by a defective notation, that the perusal of greater part of the book must, to a mathematical student, be as perplexing as the solution of a perpetual string of enigmas.

It has been my aim to steer into a middle course, between that in which is presented a mere commonplace book of principles and theorems, and that which, by leading far into the detail of multifarious processes and methods, precludes the study of the science, except by the sacrifice of much time and expense. In order to this I have endeavoured to be select in my materials, and have, for the most part, observed unity of method. A book of three times the size and price might have been drawn up with far less intellectual labour (for the fatigue of selecting from a fund of valuable materials, and casting the result into one mould, is not slight); but such a work would not have tended to accomplish the purpose I have in view. By adopting a small type and a full page, and confining myself chiefly to the analytical mode of investigation, I have been able to introduce, and I shall rejoice if it be thought I have treated perspicuously, a greater variety of the applications of plane and spherical Trigonometry than are to be found in any other work on the subject with which I am acquainted. In the first three chapters I have exhibited the theory of

Plane Trigonometry geometrically, and have shown the application of that theory to the logarithmic solutions of the usual cases into which this portion of the subject is conveniently distributed. I have endeavoured to conduct these introductory inquiries with the utmost perspicuity; that the student, by obtaining a thorough comprehension of the principal topics of research, and by seeing a little of their utility, may enter with the greater relish upon the subsequent investigations; and by tracing the correspondence of these results with such as will afterwards appear in the analytical theory of Plane Trigonometry, may be prepared to lean with full confidence upon the analytical formulæ that are in other places to be laid before him. The deductions from theory in the succeeding chapters are usually obtained by analytical processes; and their utility is shown in the logarithmic and trigonometric solution of a great number of problems, classified under the particular heads of the several applications, as specified in the table of contents. The whole, except what relates to the minute variations of the sides and angles of triangles, and the differential analogies which apply to them, may, I am persuaded, be readily comprehended by any person who is tolerably conversant with the elements of Geometry and Algebra.

In the composition of the work I have freely availed myself of all such matter as was likely to answer my purpose, especially in the productions of foreign mathematicians. The plan and method are of course my own: the materials have been collected, almost of necessity, from all quarters. In addition to the acknowledgments which will occur in different parts of this little volume, it would be unjust not to say here, that the theory of Projections, the general problem in reference to Dialling, and the comprehensive table of Differential Equa tions for the variations of triangles, are taken, simply with such alterations as fitted them better for general