Entered, according to the Act of Congress, in the year 1844, by in the clerk's office of the District Court of the United States in and for the Eastern District of Pennsylvania. PREF A C E. It will probably appear, to some, a work of supererogation to add another tract on Plane and Spherical Trigonometry to the great number already before the public; especially as some of those treatises are the productions of men whose talents and attainments were unquestionably of the highest order. Still, it has appeared to me that, however valuable many of those works must be considered, there are none of them exactly suited to the use of schools in which this branch of mathematics is traced to its principles. It is, indeed, no unusual thing to find young men who have studied this science in the way it is commonly taught, who are very imperfectly acquainted with the nature, and almost entirely ignorant of the construction, of the tables which they are continually using. And it must be admitted, that, when the nature and construction of logarithms, and of sines and tangents, are explained by Algebra and common Geometry, the processes are generally either so obscure, or so prolix, as to discourage the majority of students. The Differential Calculus is well known to furnish the most direct, if not the only direct, and simple method of A * (v) |