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THE old edition has been entirely remodelled, and vast additions of valuable matter which have been some years in collecting, or are the results of recent improvements in science, have been made. The present work begins with some constructions of triangles according to the rules given in geometry, followed by others in which scales of equal parts and protractors are employed, showing at once and distinctly, what is to be understood by the solution of a triangle, and the value of trigonometry in the measurement of inaccessible heights and distances.

The evident inaccuracy in the use of instruments leads the learner to perceive the necessity of a more exact and certain method, and prepares him to enter with satisfaction upon the study of Analytical Trigonometry.

The explanation of the Trigonometrical Lines has been prepared with great care, and it is believed that considerable improvement in the method of exhibiting their changes will be observed. Their application to the solution of triangles is immediately shown in a few cases, with the help of a table of natural sines and cosines at the end.

Then follows a full exposition of the theory and use of logarithms, with every variety of example, including an explanation of the Tables at the end. The use is also taught of the tables of Callet, the tables in highest repute, an American edition of which is known as Hassler's tables.*

Part I. concludes with the application of logarithms and logarithmic sines, tangents, &c., to a number of practical examples in heights and distances, involving every case in the solution of plane triangles.

After this a few pages of miscellaneous exercises occur, which, with those in fine print scattered through the 1st Part, will serve to give greater skill to the better class of students. Appendix I., which follows next in order, contains a vast variety of general formulas, succinct methods of solution, and methods advantageous in particular cases, methods of treating small arcs, resolutions of Algebraic equations by the aid of Trigonometry, various expressions for the area of a triangle in terms of its angles and sides, effects of errors of observation on results; in short, everything necessary for a comprehensive knowledge of Trigonometry.†

The German tables most in use are those of Vega and those of Köhler. Navigation and Surveying, if to be studied, should be taken up immediately after Plane Trigonometry.

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Part II. contains Spherical Trigonometry. Particular care has been taken to render the demonstrations here, plain and easy, and to avoid all unnecessary repetition and complication.

It was found that the introduction of a few celestial circles, such for the most part as the study of geography may be supposed to have already rendered familiar, would afford an opportunity for making all the examples of Spherical Trigonometry Astronomical.

The use of the hour angle and of different kinds of time has led also to the introduction of a full description of the transit instrument and its various adjustments, the theory of which depends on Spherical Trigonometry, and is given in all its details.

The practical character of the problems is a peculiar feature in the plan of the present work. The consideration which led to it was that since Trigonometry had grown out of the actual wants of men in these very particulars, if they were sufficiently interesting to stimulate discovery, they would also incite to 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 formulas 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.

The examination questions will be found convenient for students preparing for examination on Trigonometry, or for those studying without the aid of a teacher.

Appendix II., which follows Spherical Trigonometry, is of a character analogous to Appendix I.

Part III. exhibits a pleasing and useful application of Plane Trigonometry to the principles of navigation. This will be found a very complete treatise on the subject in small compass. The appendix to this part, App. III., includes great circle sailing, a method not usually treated in works on navigation, nor much used at present at sea; but as it serves to shorten voyages, and has no practical inconveniences in the case of steamers, which class of vessels becoming numerous on the ocean, it cannot longer with propriety be omitted. Sumner's method is also here introduced. Part IV. is a very complete treatise of surveying, which, by reducing the subject rigorously to its essential elements, is brought within a small space. Besides what is contained in ordinary treatises, including a full description of all the surveying instruments, will be found the methods of surveying railroads and canals, the principles of Topography, and a new method of Hydrographic Surveying.

Part V., which treats of Nautical and Practical Astronomy, contains a complete description of all the Astronomical instruments used at sea and in observatories, a thorough investigation of the theory of their adjustments, and of the corrections to be applied to the observations for errors of adjustment, the use of nearly every part

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;† 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.

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