PAGE Elements of Plane Trigonometry Properties and Relations of Trigonometrical Lines Application of Trigonometry, &c. Rules of Trigonometrical Calculation Observations preparatory to the Practical Solution of Solution of the Cases of Right-angled Triangles Solation of the Cases of Oblique-angled Triangles Mensuration of Heights and Distances Problems for the Exercise of Learners Signs of Trigonometrical Lines Formulæ for the Solution of Spherical Triangles Napier's Rules of the Circular Parts Quadrantal Spherical Triangles Affections and Properties of Quadrantal Triangles Solution of Right-angled Spherical Triangles Solution of Oblique-angled Spherical Triangles Numerical Solution of Right-angled Triangles 39 X6774 CORRECTIONS. 8 b. The reader is desired to make the following corrections before he begins to read the book. PAGE, LINE. 12 9 bot. For C read A 37 For 70 read 87. 45 After 1. 17 add of the bottom, and CAE 51° 52 18 For on read or 67 18 Add the theorems in the last four articles are exhibited in one view as follows. 72 16 For s. 1'=read s. 1'+ 76 4 b. For 29 read 19 91 3 b. After IV add and VI. 96 9 For 82 read 83. 97 5 b. For 84 read 85. 98 13 After the second of add the sine of 105 8 b. For C read AC 106 8 b. For Pread Z. 118 19 Add Plate, fig. 4. 118 21 For and read x 118 27 For and read x 121 For Cot. read Cos. 121 26 For Cot. read Cos. 25 Articles which may be omitted. The following articles may be omitted, or deferred till the second reading of the book. Art. 46, 47, 63 to 66, 75, latter part of 86 beginning at line 10, page 37, 88 to the end of Plane Trigonometry. Spherical Trigonometry, art. 50 to 57, 59 to 97, 103 to 120, 127, Ex. 4, page 119. of the best Mathematical and Philosophical Books FOR THE USE OF STUDENTS. FREND'S Algebra, 8vo. These four are the best books of algebra for beginners. Wood's Algebra, 8vo. This is the best elementary treatise of algebra for those who intend to study the higher parts of mathematics, but is too concisely written. Simpson's Mathematical Exercises, 8vo. Webber's Mathematics, 2 vols. 8vo. This is extracted, with improvements, from the best English books, and contains Arithmetic, Algebra, Mensuration, Navigation, Conic Sec. tions, &c. Playfair's Elements of Geometry, 8vo. This is a new edition of Euclid's Elements, corrected, enlarged, and greatly im proved. Legendre's Elements of Geometry and Trigonometry, 8vo, in French. Trigonometry, 8vo, published by Nichols. Kelly's Spherics, 8vo. Horsley's Mathematics, 8vo, vol. iii, containing the different Projections of the Sphere, the Elements of Trigonometry,&c. Robertson's Conic Sections, 8vo. The Conic Sections in Hutton's or Webber's Mathematics. Bonnycastle's Mensuration, 12mo. Hutton's Mensuration, 8vo. Vince's Fluxions, 8vo. Wood's Mechanics, 8vo. Vince's Hydrostatics, 8vo. Mackay's Navigation, 8vo. Bowditch's do. 8vo. Hutton's Mathematical Tables, 8vo. Mackay's do. 8vo. Brown's do. 8vo. Abridgment of Vince's Astronomy, 8vo. Fit only for adepts in mathematics. Young's Analysis of Natural Philosophy, 8vo. The following books may be read by persons who have little mathematical knowledge. Compend of Astronomy, 12mo, published by Nichols. Bonnycastle's Astronomy, 8vo. Bryan's do. 8vo. Ferguson's do. 8vo. Biot's Elements of Physical Astronomy, 2 vols. 8vo, in French. La Place's System of the World, by Pond, 2 vols. 8vo. Young's Lectures on Natural Philosophy, vol. i. 4to. Gregory's Economy of Nature, 3 vols. 8vo. Elements of Art and Science, by Webster, 2 vols. 8vo. Hauy's Physics, by Gregory, 2 vols. 8vo. Ferguson's Lectures on Select Subjects, by Brewster, 2 vols. 8vo. Adams's Lectures on Natural and Experimental Philosophy, by Patterson, 4 vols. 8vo. Gregory's Dictionary of Arts and Sciences, 2 vols. 4to. Nicholson's Encyclopedia, 6 vols. 8vo. Note. Many of the books toward the end of the list treat of the same subjects, and the last two comprise short treatises on the various branches of mathematics, natural and experimental philosophy, the useful and ornamental arts, &c. PREFACE. TRIGONOMETRY is a branch of the general science of geometry. It treats of the properties and relations of certain straight lines described in and about a circle, and also teaches to compute the dimensions of the sides and angles of any tri- . angle. It supplies fundamental rules for ascertaining the distances and altitudes of objects both terrestrial and celestial. Without the aid of this science, the form and magnitude of the earth, and the magnitudes, distances, motions, and eclipses of the heavenly bodies would be utterly unknown. In geography, navigation, astronomy, and the military art, its assistance is absolutely necessary. By its unerring rules we are able to determine the latitudes and longitudes of places upon the surface of the earth, and the varying positions and appearances of the heavenly bodies; to survey countries, and to travel by sea or land to any part of the world, that is accessible to the adventurous spirit of curiosity or avarice. One main object of the publisher is to furnish a treatise of trigonometry, plane and spherical, of a middle size between the concise elementary tracts containing the definitions and a few theorems barely necessary to the solution of the common cases of triangles, and the larger and more comprehensive works, designed for the use of mathematicians, or of students who have time, ability, and inclination to enter deeply into such curious and difficult speculations. This treatise is the production of different authors of approved merit, whose shares may be nearly assigned as follows. Some articles have been supplied by Legendre, Horsley, and Lacroix, which, as they are not numerous, need not be specified. Plane trigonometry consists mostly of the lectures of an eminent mathematician, W. Ludlam, A, M., formerly a felTow and tutør of St. John's college, Cambridge. These, with |