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Then follows the solution of triangles, right and oblique, the general relations of the circular functions, the functions of the sum or difference of two angles, and a variety of interesting practical applications.

It is hoped that Spherical Trigonometry has been made intelligible to the diligent student. More than ordinary care has been given to the development of Napier's principles, and to the discussion of the species of the parts of both right and oblique spherical triangles, Arts. 126, 129, 145, 148, 151.

Mensuration, a subject at once interesting and practically important, has been discussed at length, and formulas have been developed instead of rules for the solution of the problems.

In the Surveying, the instruments are first represented and described, and the methods of making the adjustments given in detail.

The Author takes this opportunity to express his obligations to Messrs. W. & L. E. Gurley, Manufacturers of Surveying and Engineering Instruments, Troy, N. Y., who have kindly granted him the use of their Manual for the delineation and description of the instruments. In consequence of this courtesy, much better drawings and descriptions have been made than would otherwise have been possible.

The instruments themselves should, however, be accessible to the student, who should study them in connection with the descriptions in the book, and learn to use them in practical work, guided by a competent instructor.

The Rectangular method of surveying the Public lands, now brought to great perfection under the direction of the Government, has been minutely explained, and illustrated by field notes of actual surveys. In this portion of the work, the United States Manual of Surveying Instructions has been taken as authority, and thus the authorized methods, which must form the basis for subsequent surveys, have been made accessible to the student.

The methods of finding the true meridian and the variation of the needle have been given at length; also specific direc

tions for finding corners, taking bearings, measuring lines, recording field notes, and plotting.

In addition to the ordinary method of finding the area, a new method, developed by E. M. Pogue, of Kentucky, is given in Art. 304. This method has the merit of giving always a uniform result from the same field notes, and thus avoids disputes about the different results of the ordinary method, unavoidably attending the various distribution of errors by different calculators.

The methods of supplying omissions are explained and illustrated by examples.

Laying out and dividing land, operations admitting of an unlimited variety of applications, have been treated in view of the wants of the practical surveyor. The subject is also full of interest to the student, who can not fail to receive from it new views of the resources of mathematical science.

Leveling, the construction of railroad curves, embankments and excavations, the method of making Topographical surveys, with the authorized conventional symbols, Barometric heights, etc., have been explained and illustrated by diagrams and examples.

It has been thought best to give a clear, elementary treatment of Navigation, not only on account of those who may desire to pursue the subject further, but for the sake of gratifying the wishes of intelligent persons who may desire to know something of Navigation. The limits of the work, however, forbid the discussion of Nautical Astronomy. The examples in Navigation have been selected from the English work of J. R. Young.

The tables of Logarithms, Natural and Logarithmic sines, etc., have been carried only to five decimal places, and for the purposes intended will be found practically better than tables to six or seven places.

The Traverse table has been thrown into a new form, at once condensed and convenient.

These tables have been compiled by Mr. Henry H. Vail, and

by him compared with Babbage's and Wittstein's tables, then by the Author with Vega's tables to seven decimal places. It is hoped that by this double comparison perfect accuracy has been attained.

The table of Meridional Parts, taken from "Projection Tables for the use of the United States Navy," prepared by the Bureau of Navigation, and issued from the Government Printing office, was calculated in the Hydrographic office for the terrestrial spheroid, compression 7.1525. This table, now for the first time published in a text-book, is believed to be more correct than those in general use.

The Author takes pleasure in acknowledging his obligations to Prof. E. H. Warner for critical suggestions and acceptable aid in reading proof and testing the accuracy of the answers.

With the hope that the book will be attractive and useful to the student, teacher, and practical surveyor, it is sent forth to accomplish its work.

BALDWIN UNIVERSITY, BEREA, O., June 12, 1873.

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A. SCHUYLER.

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