Math 801844 1882, Nov. 27, Gift of W. H. Tillinghost, of Cambridge. 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. J. Fagan, Stereotyper. (2) PREFACE. 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 investigating the formula by which those tables are most expeditiously computed. But that calculus itself, as commonly exhibited, presents so many refined speculations, that very few, except those who have a taste for mathematical studies, can avail themselves of its advantages. As it is evidently unscientific to erect a system, either in theory or practice, upon unknown principles, it has been my object, in the following work, to trace every process which is required to be adopted, to principles which are supposed to be previously understood. The student is supposed to be already acquainted with Algebra and Geometry. If the student is master of the first six, and the eleventh books of Euclid; or, which is nearly the same thing, of the first six, and the second supplementary book of Playfair's Geometry; and of as much algebra as is contained in the first ninety pages of my treatise; he may proceed with confidence to the study of the following tract. This work was intended to include as much only of the Differential Calculus, as the elucidation of the science of Trigonometry required. I have therefore confined myself to differentials of the first order; and, by the use of proper expedients, have deduced the requisite formulæ from those differentials. Some of the methods used in this work are supposed to be new; and, if so, they may be considered as improvements upon the labours of my predecessors. Of this character are the investigations of Gregory's theorems for computing an arc in terms of its tangent, and for computing logarithms. The treatises on Spherical Trigonometry with which our schools are supplied, are nearly all of them destitute of anything on the subject of Spherical Projections. This appears to me an important defect. A small tract on that subject, which I added, more than thirty years ago, to a Philadelphia edition of Thomas Simpson's Plane and Spherical Trigonometry, is the only one, so far as I know, which is to be found in our schools; unless we consider Davies's Descriptive Geometry as one. Simpson's work being now out of print, and the work of Davies, notwithstanding its merits, not appearing calculated to supply the place of the appendix, I have revised, or rather written anew, that part of my early labours, and subjoined it to this work. The present treatise being designed as an introduc-tion or preliminary to Astronomy, a concise tract on the Conic Sections, including all the properties of the ellipse and parabola which are usually cited by writers on that science, has been introduced. This appeared |