HARVARD COLLEGE LIBRARY GIFT OF GINN & CO. COPYRIGHT, 1899, 1900, BY ELMER A. LYMAN IAT Norwood Press J. S. Cushing & Co. - Berwick & Smith Norwood Mass. U.S.A. PREFACE. MANY American text-books on trigonometry treat the solution. of triangles quite fully; English text-books elaborate analytical trigonometry; but no book available seems to meet both needs adequately. To do that is the first aim of the present work, in the preparation of which nearly everything has been worked out and tested by the authors in their classes. The work entered upon, other features demanded attention. For some unaccountable reason nearly all books, in proving the formulæ for functions of a ẞ, treat the same line as both positive and negative, thus vitiating the proof; and proofs given for acute angles are (without further discussion) supposed to apply to all angles, or it is suggested that the student can draw other figures and show that the formulæ hold in all cases. As a matter of fact the average student cannot show anything of the kind; and if he could, the proof would still apply only to combinations of conditions the same as those in the figures actually drawn. These difficulties are avoided by so wording the proofs that the language applies to figures involving any angles, and to avoid drawing the indefinite number of figures necessary fully to establish the formulæ geometrically, the general case is proved algebraically (see page 58). Inverse functions are introduced early, and used constantly. Wherever computations are introduced they are made by means of logarithms. The average student, using logarithms for a short time and only at the end of the subject, straightway forgets what manner of things they are. It is hoped, by dint of much practice, extended over as long a time as possible, to give the student a command of logarithms that will stay. The fundamental formulæ of trigonometry must be memorized. There is no substitute for this. For this purpose oral work is introduced, and there are frequent lists of review problems involving all principles and formulæ previously developed. These lists serve the further purpose of throwing the student on his own resources, and compelling him to find in the problem itself, and not in any model solution, the key to its solution, thus developing power, instead of ability to imitate. To the same end, in the solution of triangles, divisions and subdivisions into cases are abandoned, and the student is thrown on his own judgment to determine which of the three possible sets of formulæ will lead to the solutions with the data given. Long experience justifies this as clearer and simpler. The use of checks is insisted upon in all computations. For the usual course in plane trigonometry Chapters I-VII, omitting Arts. 26, 27, contain enough. Articles marked * (as Art. 26) may be omitted unless the teacher finds time for them without neglecting the rest of the work. Classes that can accomplish more will find a most interesting field opened in the other chapters. More problems are provided than any student is expected to solve, in order that different selections may be assigned to different students, or to classes in different years. Do not assign work too fast. Make sure the student has memorized and can use each preceding formula, before taking up new ones. No complete acknowledgment of help received could here be made. The authors are under obligation to many for general hints, and to several who, after going over the proof with care, have given valuable suggestions. The standard works of Levett and Davison, Hobson, Henrici and Treutlein, and others have been freely consulted, and while many of the problems have been prepared by the authors in their class-room work, they have not hesitated to take, from such standard collections as writers generally have drawn upon, any problems that seemed better adapted than others to the work. Quality has not been knowingly sacrificed to originality. Corrections and suggestions will be gladly received at any time. E. A. L., YPSILANTI. October, 1900. |