interesting to stimulate discovery, they would also the study of what is already known. The analytic method, though not always practicable before the mind is somewhat furnished, is doubtless, by far the best method of training. Besides this general reason for introducing astronomical problems here, it was deemed useful thus to prepare the way for the study of astronomy, whilst the formulæ and rules of trigonometry were fresh in the memory, and to prevent that neglect of the trigonometrical solutions of astronomy, which is apt to result from the trouble of recalling what has been long laid aside. It was thought, too, that this foretaste of astronomy might excite a relish for that study. Part III. is a recapitulation of the formulæ and rules demonstrated in the previous parts, collected for convenience of reference or committing to memory. Part IV.exhibits tlre application of trigonometry to the principles of navigation and nautical astronomy, and is chiefly from the admirable treatise of Mr. Young. This affords a pleasing and useful exercise in the formulæ and rules of the preceding parts. It will occupy the student but a few days, and has always been considered an essential part of a polite education. Should it, however, be omitted, it will be necessary, previously to the study of Analytical Geometry, to go over the Addenda contained in Part V. Part VI. is a sort of supplement which the general student may well omit, but which will be found to contain matter useful to practical men, and interesting to those more exclusively devoted to the mathematics. It was the author's intention to have added some formulæ applicable to geodesy and other branches of practical science, but considering that these are always demonstrated in the treatises upon the subjects to which they refer, it was thought advisable not to increase the size of this work, already sufficiently large for its main object, viz., the use of colleges and the purposes of a general education. A valuable collection of logarithmic, nautical and astronomical tables, will be found at the end. The minute exhibition of the processes in the numerical examples has been regarded by the author as essential to the imparting of an available knowledge of trigonometry. These examples are few in the first and second Parts, but the deficiency is abundantly supplied in Part IV., where, since they could be made most useful and interesting, it was thought that they might best be multiplied. In the preparation of the present work nothing has been sacrificed to the desire of originality. Free use has been made of the comprehensive treatise of Mr. Young. Recourse has been had for some minor portions, to the recent work of M. Francoeur upon Geodesy. The idea of the simple instruinent for taking angles, described at Art. 10, was derived from a small work on geometry for beginners, by Dr. Ritchie, late of the London University. The greater part of Article 130 is from the Edinburgh Encyclopedia. B 31. Algebraic notation of the trigonometrical lines, 32. Expression for the tangent in terms of the sine and cosine, FORMULE FOR THE SOLUTION OF RIGHT-ANGLED TRIANGLES. 35 36 36 44. Logarithms of the base and unity, 45. Definition of a table of logarithms, 46. Method of calculating tables of logarithms, 47. Theory of the characteristic, 48. Rule to find the logarithm of any number between 1 and 10,000, 49. Multiplication and division by logarithms, 50. Logarithms of decimal numbers, 51. Rule to find the logarithms of numbers greater than 10,000, 52. Rule to find the number corresponding to any given logariihm, 53. Examples in multiplication and division by logarithms, 54. Formation of powers by logarithms, 55. Extraction of roots by logarithms, 56. Table of logarithmic sines, tangents, &c., 57. Rule to find from the table the logarithmic sine, tangent, &c., of any given number of degrees, minutes, and seconds, 58, 59. To find the degrees, minutes, and seconds corresponding to any given logarithmic sine, tangent, &c., 60. Rules to find the logarithmic secant and cosecant of any given arc, - Solution of right-angled triangles with the aid of logarithms. 62. Use of the arithmetical complement, 63. Example in the measurement of distances, Solution of oblique-angled triangles by logarithms. 64. A side and the opposite angle being two of the given parts, 65. Two angles and the interjacent side being given, 66. Example in the measurement of heights, the bases of which are in- 67. Two sides and the angle opposite one of them being given; an am- 68. Derivation of a formula for the cosine of an angle in terms of the 69 and 70. Derivation of formulæ for the sine and cosine of the sum 71. Derivation of formula for the sine and cosine of an arc in terms of |