AN

INTRODUCTION

ΤΟ

THE THEORY AND PRACTICE

OF

PLANE AND SPHERICAL

TRIGONOMETRY,

AND

THE STEREOGRAPHIC PROJECTION OF

THE SPHERE;

INCLUDING

THE THEORY OF NAVIGATION:

COMPREHENDING 1

A Variety of Rules, Formulæ, &c. with their Practical Applications to the
Mensuration of Heights and Distances; to determining the Latitude by
two Altitudes of the Sun, the Longitude by the Lunar Observations,
and to other important Problems on the Sphere, and on Nautical
Astronomy.

BY THOMAS KEITH.

THE FOURTH EDITION, CORRECTED AND IMPROVED.

La Trigonométrie est sans contredit une des plus utiles applications de la Géométrie
élémentaire: elle est la base de la Geodésie, de la Geographie, de l'Astronomie et de
la Navigation.
Discours sur les Progrès des Sciences, rendu par l'Institut
de France, 1809, page 6.

LONDON:

PRINTED FOR LONGMAN, HURST, REES, ORME, AND BROWN,

PATERNOSTER-ROW;

AND FOR THE AUTHOR.

PREFACE.

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 appear. ance 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, Ptolemy, &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.