AN INTRODUCTION TO THE THEORY AND PRACTICE OF PLANE AND SPHERICAL TRIGONOMETRY, AND THE STEREOGRAPHIC PROJECTÍON OF THE SPHERE; INCLUDING THE THEORY OF NAVIGATION: COMPREHENDING A Variety of Rules, Formulæ, &c. with their Practical Applications to the BY THOMAS KEITH, 757-1824 PRIVATE TEACHER OF MATHEMATICS. THE SECOND EDITION, CORRECTED AND IMPROVED. La Trigonométrie est sans contredit une des plus utiles applications de la Géométrie Discours sur les Progrès des Sciences, rendu par l'Institut de France, LONDON: PRINTED FOR THE AUTHOR; AND LONGMAN, HURST, REES, ORME, AND BROWN, 1810. PREFA CE. TRIGONOMETRY is an important branch of the mathematical sciences: the speculative parts, like the Elements of Euclid, habituate the mind to close and demonstrative reasoning; and the practical parts are of extensive use in the common concerns of life. By Trigonometry we determine the magnitudes of the earth and planets; and the positions of the fixed stars with respect to each other, by which we are enabled to depict the appearance of the heavens in a small compass. The distances of the planets from the sun, their motions, eclipses, and other phænomena, are calculated by Trigonometry, as are likewise the distances and positions of places on the earth, with their latitudes and longitudes; it may therefore justly be considered as the basis of Astronomy and Geography. Navigation, with all its modern improvements, depends entirely on Trigonometry, which is likewise the foundation of maritime surveying, and of almost every branch of practical mathematics; accordingly we find this subject has been studied in the earliest ages of mathematical learning. Among the ancients were Hipparchus, Theodosius, Menelaus, Ptolomey, &c. who contributed to the advancement of this science. The improvements in Astronomy, Navigation, and Trigonometry, nearly kept pace with each other. The invention of Logarithms by Baron Napier was an invaluable acquisition to these sciences; and the improvements made by this illustrious person, in Spherical Trigonometry, will be a lasting monument of his penetration and judgment. From this period the history of Logarithms and that of Trigonometry are closely connected, and there is scarcely a writer on one of these subjects, who has not likewise introduced the other. The first authors of note, after the invention of Logarithms, were Briggs and Gellibrand.-Dr. Charles Hutton, in the Introduction to his Logarithms, has given a very complete account of the different writers on that subject, which likewise includes the principal authors on Trigonometry; to this valuable work the Reader is therefore referred for further information in the history of the science. The authors on Trigonometry may be divided into two classes, theoretical and practical; for none of them have combined the theory with the practice, in such a manner as to render the subject plain and intelligible to a learner: the most valuable and scientifical are, in general, too abstruse, and the prac tical scarcely furnish the student with the rationale of a single rule or operation. The object of the ensuing treatise is to simplify the theory, yet to retain a methodical and accurate mode of investigation, and to exemplify this theory by a variety of important and useful examples. The demonstrations are frequently founded on principles strictly Geometrical, especially where those principles can be rendered very plain and perspicuous by the assistance of simple diagrams; and, sometimes the process of demonstration is conducted by algebraical signs, particularly where the Geometrical method would require a complicated figure, or a long and tedious process. In the construction of various formulæ the algebraical mode, of deduction tends greatly to simplify the subject; yet the definitions and the elementary parts of the science must be acquired from Geometrical principles illustrated by diagrams; otherwise a student will never obtain a clear and satisfactory knowledge of the subject. Should any person attempt to teach the |