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PREFACE.

It is the design of this treatise to establish the theory of Plane and Spherical Trigonometry analytically, and to present that theory, together with some of its most interesting and valuable applications, in a form fitted for elementary instruction.

Of late years several analytical works on Trigonometry have been published in this country; but, as they are confined almost entirely to the theory of the subject, it may be questioned whether, to many young students, they prove much else than so many collections of mere algebraical exercises. Yet a book upon so practical a subject as Trigonometry ought undoubtedly to be something more than this, and ought not to be considered as complete when the various calculations which the science involves, and which its name implies, are wholly omitted.

The symbolical expression of a practical rule, in algebraic language, will often, to the young student, but indistinctly point out the numerical operation. Those much occupied

in mathematical instruction know full well that a learner may readily yield his assent to every step of an algebraic process, be fully satisfied as to the truth of the result to which leads, may even clearly see a valuable truth involved in it, and may yet be very far from perceiving how to turn it to account in any case of actual calculation. Indeed, algebraical formulas, transform them as we will, cannot always be made to indicate the best mode of arithmetical arrangement; and yet much, as regards facility of operation, depends upon this arrangement in many parts of practical mathematics, but especially in Trigonometry.

In the present volume, therefore, both the theory and the practice of the science have been introduced, every practical formula being illustrated by examples of the numerical calculation, arranged in the proper form. This plan of combining practice with theory, in works like the present, was always adopted by the earlier English writers, and it is to be regretted that recent authors have, in their admiration of foreign methods, departed so widely, in this respect, from the example of their predecessors, dwelling so much as they do upon the symbols, and so little upon the things signified.

In addition to the practical illustration of formulas, a distinct part of the work is devoted to the principles of Navigation and Nautical Astronomy, in which will be found a very short and convenient method of clearing the Lunar Distance, for the purpose of ascertaining the Longitude at

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Sea. This method is probably new, although, as the analytical expression for it occurs during the investigation of the well known formula of Borda, it is equally probable that it has been noticed before.

The supplement appended to the treatise is from the pen of my valued and accomplished friend, T. S. Davies, Esq. Fellow of the Royal Society of Edinburgh, and of the Royal Astronomical Society of London. It will be found to contain several new and interesting researches, which cannot fail to prove acceptable both to the inquiring student and to the more advanced analyst.

J. R. YOUNG.

Recl.o 12-20-40. N.

January 1, 1833.

1. ELEMENTS of the INTEGRAL CALCULUS; with its Applications to Geometry, and to the Summation of Infinite Series, &c. 9s. in cloth.

This volume forms one of an Analytical Course. " More elegant Textbooks do not exist in the English Language, and we trust they will speedily be adopted in our Mathematical Seminaries. The existence of such auxiliaries, will, of itself, we hope, prove an inducement to the cultivation of Analytical Science; for, to the want of such Elementary Works, the indifference hitherto manifested in this country on the subject, is, we apprehend, chiefly to be ascribed. Mr. Young has brought the science within the reach of every intelligent student, and, in so doing, has contributed to the advancement of Mathematical Learning in Great Britain.— The Presbyterian Review, Jan. 1832.

2. The ELEMENTS of the DIFFERENTIAL CALCULUS: comprehending the General Theory of Curve Surfaces and of Curves of Double Curvature. 8s. in cloth.

3. An ELEMENTARY TREATISE on ALGEBRA, Theoretical and Practical; with Attempts to simplify some of the more difficult Parts of the Science, particularly the Demonstration of the Binomial Theorem, in its most general form; the Solution of Equations of the higher orders; the Summation of Infinite Series, &c. 8vo. boards, 10s. 6d.

4. An ELEMENTARY TREATISE on the COMPUTATION of LOGARITHMS. Intended as a Supplement to the various books on Algebra 12mo. 2s. 6d.

5. ELEMENTS of GEOMETRY; containing a New and Universal Treatise on the Doctrine of Proportion, together with Notes, in which are pointed out and corrected several important Errors that have hitherto remained unnoticed in the Writings of Geometers. 8vo. 8s.

6. The ELEMENTS of ANALYTICAL GEOMETRY; comprehending the Doctrine of the Conic Sections, and the general Theory of Curves and Surfaces of the second order, with a variety of local Problems on Lines and Surfaces. Intended for the use of Mathematical Students in schools and universities. 9s. cloth.

7. ELEMENTS of MECHANICS; comprehending the Theory of Equilibrium and of Motion, and the first Principles of Physical ASTRONOMY, together with a variety of Statical and Dynamical Problems, Illustrated by numerous plates. 10s. 6d. cloth.

“ If works like the present be introduced generally into our Schools and Colleges, the continent will not long boast of its immense superiority over the country of Newton, in every branch of modern analytical science.”The Atlas, July 25, 1830.

“ Mr. Young is already favorably known to the Public by his writings; and the Treatise on Mechanics, which we now propose briefly to notice, will add considerably to his reputation as the author of Elementary Works of Science. Το read the works of Laplace, Ivory, Somerville, &c., a knowledge of the methods employed by these writers is previously required, and render such preparatory works as those of Mr. Young's absolutely necessary.”—Presbyterian Review, July 1832.

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