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PLANE AND SPHERICAL
EDWIN P. SEAVER, A. M.
SUPERINTENDENT OF THE PUBLIC SCHOOLS, BOSTON,
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS IN HARVARD COLLEGE.
TAINTOR BROTHERS & CO
NEW YORK AND CHICAGO
BOSTON: WILLIAM WARE & CO
THIS book is intended to serve two distinct though not inconsistent purposes: first, to furnish a very brief practical course for those schools in which the study can claim but a little time; and, secondly, to present the whole subject of trigonometric functions in the manner in which the student of higher mathematics needs to have learned it.
The first purpose will be served by the first three chapters (pp. 1-65) of the book. In this part the functions of an acute angle only are defined and used. Right triangles are solved by direct reference to the definitions of functions, and oblique triangles by dividing them into right triangles. Not even a knowledge of logarithms is necessary for reading this part of the book, all the methods being illustrated by examples solved with three-place natural functions, a table of which is given on page 9. The application of logarithms is treated as a special matter in separate paragraphs. Among the exercises in the first three chapters occur various applications of trigonometry to the calculation of heights and distances.
The second purpose has governed in the preparation of the next four chapters (pp. 66-160). Here, after the generalized conception of the angle has been introduced, together with some notions of positive and negative lines and of rectangular coördinates, the trigonometric functions
are defined as ratios of coördinates. The advantage of this is seen in laying the foundation for the general proof of the formulas for the sine and the cosine of the sum and of the difference of two angles. The usual method of proving these formulas by a geometrical figure in the first quadrant, the validity of the proof for figures that might be drawn in the other quadrants being assumed, seems, from a teacher's point of view, unsatisfactory; partly because fundamental formulas like these should be solidly established, and partly because this is the first important instance the mathematical student meets where a proposition proved for the first quadrant does hold for the others. Whatever he may afterwards allow himself to assume, he should in this instance at least carefully explore the whole ground of geometric configuration. These considerations have suggested the unusually thorough treatment of this topic in Chapter VI, and the introduction of Chapter V, on projections, to furnish a foundation for this treatment. In Chapter VII are treated the important properties of plane triangles. From these are deduced some methods for logarithmic solution which are simpler than those given earlier. When this chapter is to be studied, it may be found well to postpone the exercises near the end of Chapter III until advantage can be taken of these simpler methods.
In spherical trigonometry, Chapters VIII-XI, it has been found simpler and better to cast aside the usual limitation of the results of solution to such sides and angles as do not exceed 180°, and to accept and interpret all results. One step more removes a similar limitation from the data; and thus we have the general spherical triangle, treated in the last chapter.
Among the methods proposed for the solution of spherical triangles, attention may be called to the elegant ones furnished by the Gaussian equations. Notwithstanding the ambiguity attaching to the square roots of which these equations are composed, the results present but little difficulty of interpretation, as the examples in the last chapter show. In case, however, of any perplexity, the simple and direct method of solution based on the fundamental equations can be employed to remove it. By using three-place natural functions, a rough approximation can be made, which will show the general shape of the required triangle, and, except in extreme cases, give the values of the required parts within a fraction of a degree.
The method of solution which consists in the introduction of an auxiliary angle for the purpose of adapting the fundamental formulas to logarithmic computation is not added to the others, because it is step for step equivalent to the method which divides the triangle into two spherical right triangles by means of a perpendicular; and the latter method has the advantage of a direct geometrical interpretation.
One chapter on the development of trigonometric functions into series, De Moivre's theorem, and other such matters formed a part of the original plan of the book; but the other chapters have so outgrown intended limits that this must, for the present at least, be dropped.
BOSTON, May, 1889.
E. P. S.