In presenting these ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY to the notice of the Public, it has been the author's aim to embrace as much of the theory and practice as appeared to be consistent with the time usually devoted to this branch of analysis in a system of liberal education ; and to exhibit the subject agreeably to that systematic reasoning, and those generalizations, which constitute the chief beauty and excellence of mathematical learning The first section contains the usual principles and definitions. In defining the trigonometrical lines, cosine, cotangent, and cosecant*, they have been re * When these terms were first introduced, the Science of Trigonometry was confined to a single object, as may be seen by the words toywsos, and ustgew; and therefore it was seldom or never necessary to consider these terms as applied otherwise than to the solution of triangles, or to arcs and angles less than 180°, in which little difficulty 425503 garded as independent of the complement, principally for the sake of uniformity, and for the purpose of affording greater facility to the learner in acquiring a knowledge of their positions throughout the four quadrants. In the second section, the principal theorems commonly called the Arithmetic of Sines, have been given. They have been arranged in such a manner, that the student, it is presumed, will experience no difficulty in turning immediately to any formula he may want, when the work is used as a reference in the further continuance of his studies. The third section explains the nature of subsidiary angles, and illustrates their use in computing the numerical values of formulæ not immediately adapted to logarithmic calculation. In the solution of plane and spherical triangles contained in the fourth, fifth, and sixth sections, formulæ for every case have been given; those have been pointed out which are generally the most convenient; and care has been exercised that the accompanying examples should enable the student to descend with facility into practical operations. could arise respecting the sine, &c., of the complement. But in the Arithmetic of Sines, which now forms so prominent a feature in the modern acceptation of Trigonometry, it is not considered sufficient that trigonometrical formulæ should be restricted to arcs of limited magnitude. For the purpose, therefore, of generalizing the demonstrations, as well as for the above reasons, it appeared more simple to define the cosine, cotangent, and cosecant independently of the comple The usual definitions will then appear in the light of propositions, to which indeed they strictly belong. ment. The seventh section includes such a view of the higher trigonometrical analysis as accorded with the limited nature of the work. The eighth contains one or two useful articles relating to the circle; and the explanation of the methods of determining the roots of quadratic and cubic equations which are not easily obtained otherwise. The construction of trigonometrical tables occupies the ninth section, and with this concludes the subject of Trigonometry. It is always more interesting to young persons studying the principles of any science, to have early opportunities of exercise in the practical application, so important for many obvious reasons. With this view, a variety of examples on the heights and distances of objects have been selected to exemplify Plane Trigonometry: and immediately succeeding the explanation of the Doctrine of the Sphere, a considerable number of Astronomical Problems have been solved, the accompanying examples to which afford, perhaps, the most obvious exercise ;—whilst the concluding three sections, on the Longitude, Dialling, and Geodetic operations, appeared to be in no respect foreign to a work professing to illustrate the use of Spherical Trigonometry. Such being a short account of the principal subjects in the book, it may as connected with the general execution, to state the plan which was adopted by the author, previously to its being sent to the press. When the manuscript had been nearly finished, each section was neatly copied, and successively placed in the hands of seven or eight young students, but very slightly acquainted with Trigonometry, whose ages varied from thirteen to eighteen years, with a request to each that he would study the work. By this means, such passages and subjects were discovered, in the understanding of which, more than the average number experienced a difficulty. These were consequently examined, and generally either altered or further explained. It was intended in this manner to compile an elementary work for young persons, which would induce them to look more frequently within themselves for assistance in their mathematical studies than has |