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The following treatise on Plane and Spherical Trigonometry and Mensuration first appeared in 1873 as an Introduction to the author's work on Surveying and Navigation.

The general favor with which the Trigonometry has been received renders it probable that its publication in a separate volume would greatly extend its usefulness.

The many studying Trigonometry, who do not care to learn Surveying, ought not to be compelled to purchase a more expensive book than is necessary.

In the application of Logarithms to the processes of Multiplication and Division, Involution and Evolution, the order of treatment is, first, the proposition and its demonstration ; next, the rule, then the solution of examples, thus giving the application of the principle in immediate connection with its statement.

The trigonometrical functions are defined, not as ratios, but as linear functions of the angle, thus giving the student clear geometrical conceptions instead of abstract relations, and enabling him the more readily to grasp the laws of the algebraic signs of the functions. The advantages in analytic investigations resulting from defining these functions as ratios have been secured in the principles relating to the Right Triangle, Art. 64.

Each of the circular functions has, in the first place, been considered by itself, and its value traced for all arcs, from 0° to 360°.

Trigonometry is naturally divided into Plane and Spherical. In Plane Trigonometry triangles are discussed in the order, Right Triangles and Oblique Triangles. Then, under the general head, Relations of the Circular Func

tions, follow, fundamental formulas, each function in terms of each of the others, functions of negative arcs, functions of (n 90° + a), values of functions of particular arcs, inverse functions, functions of the sum and difference of two angles, functions of double and half angles, consequences of the formulas (a), (b), (c), (d), a variety of interesting practical applications, and the computation of the natural and logarithmic functions.

In Spherical Trigonometry, as in Plane, Right Triangles are first discussed, then Oblique.

More than ordinary care has been given to the development of Napier's principles and to the discussion of the species of the parts of both right and oblique triangles, Arts. 126, 129, 145, 148, 151.

Special attention is invited to Arts. 64, 89, 91, 126, 129, 145, 148.

Mensuration, a subject at once interesting and practically important, has been discussed at length, and formulas, instead of rules, have been developed for the solution of problems.

Hoping that the work, as a whole, will prove a contribution to the wants of the student, and render him efficient aid in acquiring a correct mathematical taste, and that its publication in a separate volume will greatly extend its usefulness, it is submitted to the favorable consideration of those who have the responsibility of selecting the text-book on this important branch of mathematical science.


Berea, O., June 12, 1875.

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