OF FRIGONOMETRY, PLANE AND SPHERICAL. ADAPTED TO THE PRESENT STATE OF ANALYSIS. TO WHICH IS ADDED, THEIR APPLICATION TO THE PRINCIPLES OF NAVIGATION AND NAUTICAL ASTRONOMY. WITH LOGARITHMIC, TRIGONOMETRICAL, AND NAUTICAL TABLES. FOR THE USE OF COLLEGES AND ACADEMIES. BY THE REV. C. W. HACKLEY, PUBLISHED BY WILEY & PUTNAM; COLLÍNS, KEESE & Co., NEW-YORK-THOMAS, COWPER- 1838. Entered according to Act of Congress, A. D. 1838, by Rev. C. W. Hackley, in the Clerk's office of the District Court of the United States for the NEW-YORK: PRINTED BY WILLIAM OSBORN, 88 WILLIAM-STREET. Gres in Zoon rent count . PREFACE. Analytical Trigonometry has always been to the 1. A too sudden transition from Geometry to Tri- 2. A tedious succession of general formulæ at the commencement, the use and application of which is so long delayed as to produce weariness and discouragement before there is any apparent fruit to reward labor. 3. Too much abridgment in the demonstration, and particularly in the derivation of the algebraic results. The author is aware of the importance attached to the exercise of intellect required to discover the connexion between propositions whose mutual dependence is shown by intermediate links which the mind must supply unaided, but it will be admitted on the other hand, that the ordinary term of study is too limited, and the field of knowledge in this department too extensive, to afford the loss of time which such a mode occasions. Besides, there will be abundant scope for this kind of exercise, in a more matured familiarity with mathematical reasoning, for which the shortening of labor here, will leave additional room. The following work begins with some constructions of triangles according to the rules given in geometry, followed by others in which scales of equal parts and protractors are employed, showing at once and distinctly, what is to be understood by the solution of a triangle, and the value of trigonometry in the measurement of inaccessible heights and distances, The evident inaccuracy in the use of instruments, leads the learner to perceive the necessity of a more exact and certain method, and prepares him to enter with satisfaction upon the study of Analytical Trigonometry. The explanation of the Trigonometrical Lines \ias has been labored with great care, and it is believed, a that considerable improvement in the method of exhibiting their changes will be observed. Their application to the solution of triangles, is immediately shown in a few cases, with the help of a table of natural sines and cosines at the end. Then follows a full exposition of the theory and use of logarithms, with every variety of example. Part I. concludes with the application of logarithms and logarithmic sines, tangents, &c., to a number of practical examples involving every case in the solution of plane triangles. Part II. contains Spherical Trigonometry. Particular care has been taken to render the demonstrations here, plain and easy, and to avoid all unnecessary repetition and complication. It was found that the introduction of a few celestial circles, such for the most part as the study of geography may be supposed to have already rendered familiar, would afford an opportunity for making all the examples of Spherical Trigonometry, Astronomical. This, together with the practical character of those in plane trigonometry, most of which are problems in the measurement of heights and distances, is a peculiar feature in the plan of the present work. The consideration which led to it was, that since trigonometry had grown out of the practical wants of men in these very particulars, if they were sufficiently |