of the nautical almanac and tables of corrections for determining the co-ordinates of the true places of the heavenly bodies, and the solutions in Spherical Trigonometry necessary for converting one set of co-ordinates into another, with all the best methods of determining latitude and longitude, either on land or at sea. App. V. contains the description of the reflecting circle and mural circle, the determination of latitude by circummeridian altitudes, by the method of Littrow, and by an altitude of the pole star out of the meridian. Part VI. contains the necessary instruction for conducting a geodetic survey on a scale of sufficient magnitude to require not merely the spherical figure, but also the spheroidal figure of the earth to be taken into consideration. When the formulas in this part involve the theory of conic sections, they are given and the use taught, but the demonstrations are reserved for the last appendix, in which the calculus is freely introduced when necessary. The subject commences with the modes of measuring bases, with an account of the beautiful improvements in the base apparatus recently made in this country, and the formulas of reduction to the level of the neighboring seas. Then follows a description of the great theodolite, and the methods of conducting the observations of the great or primary triangulation, the modes of verifying and correcting the observed spherical angles, and of computing the elements of the spherical triangles. Then the methods of determining geodetically the differences of latitude, longitude, and azimuth of the stations at the vertices of the triangles, with the construction of maps and the explanation of the necessary tables. Then the best methods of conducting the Astronomical observations for latitude, longitude, and azimuths. The description of the instruments and modes of conducting the magnetic observations, and the use of the formulas for determining the elements of terrestrial magnetism. App. VI. describes the equatorial, the altitude and azimuth instrument, the prime vertical transit, and gives theorems for determining the size and figure of the earth, &c. The methods given in this geodetic treatise are those employed upon the coast survey of the United States.* The tables include a table of logarithms, of numbers, of logarithmic sines, tangents, cosines, cotangents, secants and cosecants ;t a table of natural sines and cosines, a table of difference of latitude and departure for every point and quarter point of the quadrant, a table of Rhumbs, a table of meridional parts, Workman's table for the correction of the middle latitude, a table of refractions, with corrections for the states of the barometer and thermometer, a table for dip or depression of the horizon, a table of the sun's parallax in altitude, of the contraction of the * These are in some respects superior to the latest and best European methods. The author has to acknowledge the politeness of the accomplished superintendent of the coast survey in furnishing every facility for obtaining information. + The last two are not usually found in the best tables. The method of taking out the difference for the seconds in these tables is new and expeditious. sun's or moon's vertical semi-diameter from refraction, of the augmentation of the moon's semi-diameter with its altitude, a table of proportional logarithms, a table of the reductions of the moon's equatorial parallax for the spheroidal figure of the earth, and finally a table of natural versed sines for reducing observations to the meridian. Besides these, other small tables and specimens of tables are scattered throughout the work. Most of the tables are printed from the beautiful and accurate stereotype plates of the tables accompanying Bowditch's Navigator, by permission of the proprietor, Mr. G. W. Blunt. The author has to acknowledge the kindness of Prof. CHAUVENET, of the U. S. Naval Academy, in permitting the use of his valuable paper on Unlimited Spherical Triangles, first introduced by Gauss. It will be found in Appendix II., as contained in the Astronomical Journal, with some slight modifications and explanatory notes. Trigonometric lines have the same signs in pairs, Algebraic notation for trigonometric lines, Expressions for the tangent, cotangent, secant, and cosecant, in terms of the Relations of tangent and cotangent, Exercises in the trigonometric functions, Derivation of formulas for the solution of right angled plane triangles, 22 ib. 9 28 Rules for obtaining log. sec from log. cosine, and by log. cosec for log. sin ib. ib. Description and mode of using the trigonometric tables of Callet, Method of finding the degrees, minutes, and seconds corresponding to any SOLUTION OF RIGHT ANGLED TRIANGLES BY THE AID OF LOGARITHMS, |