PREFACE. THE following compilation originated in the belief that our schools are in want of a Treatise on Surveying, adapted to the methods practised in this country, and freed from the defects of the systems now in use. Notwithstanding the importance of the science, and the large number that make it an object of study, it is believed we are not in possession of a treatise on this subject, suited to the wants of the student. The works of Gibson and Jess are the only ones at present in general use; the former, though much the better of the two, is deficient in many respects. It may be sufficient here, merely to advert to its want of examples, which renders it entirely unsuitable for a school book. From the latter, the student would in vain expect to become acquainted with the principles of the science, or the rationale of any of the rules, necessary in performing the various calculations.* In order to understand the principles of surveying, a previous knowledge of Geometry is absolutely necessary; and this knowledge will be best acquired from a regular treatise on the subject. In the demonstrations, therefore, throughout this work, the student is supposed to be acquainted with the elements of that science. The references are adapted to Playfair's Geometry, but they will in general apply equally well to Simson's translation of Euclid's Elements. As there are many who wish to obtain a practical knowledge of Surveying, whose leisure may be too limited to admit of their * Each of these works has lately gone through a new edition, in which considerable additions are stated to have been made. On examination, however, it does not appear, that those additions are such as to supply the deficiencies. The additions made to Gibson, consist principally of some nautical problems quite foreign to a treatise on Surveying. Those made to Jess, consist of a few extracts from Gibson, in one of which the Pennsylvania method of calculation is introduced, as being quite different from that given by Jess; whereas it is well known to be the method given by that author, and used, as well in the preceding, as in the subsequent part of his work. going through a course of Geometry, the author has adapted his work to this class, by introducing the necessary geometrical definitions and problems, and by giving plain and concise rules, entirely detached from the demonstrations; the latter being placed in the form of notes at the bottom of the page. Each rule is exemplified by one wrought example; and the most of them by several unwrought examples, with the answers annexed. In the laying out and dividing of land, which forms the most difficult part of surveying, a variety of problems is introduced, adapted to the cases most likely to occur in practice. This part of the subject, however, presents such a great variety of cases, that we should in vain attempt to give rules that would apply to all of them. It cannot therefore be too strongly recommended to every one, who has the opportunity, to make himself well acquainted with Geometry, and also with Algebra, previous to entering on the study of Surveying. Furnished with these useful auxiliaries, and acquainted with the principles of the science, the practitioner will be able to perform, with ease, any thing likely to occur in his practice. The compiler thinks proper to acknowledge, that in the arrangement of the work, he availed himself of the advice of his learned preceptor and friend, E. Lewis of New-Garden; and that several of the demonstrations were furnished by him. ADVERTISEMENT TO THE FOURTH EDITION. In preparing this edition for the press, several alterations have been made, which, it is believed, will be found to be real improvements. A number of new Problems has been introduced, and a more methodical arrangement of the whole has been adopted. Instead of three different rules for calculating the content of a Survey, one general rule, including these, is now given. It may be further added, that the rules for solving several of the problems in Division of Land, have been considerably simplified. The Mathematical Tables have been stereotyped, after carefully revising them and comparing them with the most correct European Editions. THE AUTHOR. ADVERTISEMENT TO THE FOURTEENTH EDITION To meet the wants of the Student of Civil Engineering, this edition has been enlarged by the addition of several chapters, in which the Theodolite and Levelling Instrument are described, the methods of adjusting and using them are given, and the principles and practice of Levelling and Topography are explained and illustrated.—The whole has been carefully revised and the few typographical errors existing in former editions have been corrected. 7 J. G. A TREATISE ON SURVEYING. OF LOGARITHMS LOGARITHMS are a series of numbers so contrived, that by them the work of multiplication is performed by addition, and that of division by subtraction. If a series of numbers in arithmetical progression be placed as indices, or exponants, to a series of numbers in geometrical progression, the sum or difference of any two of the former, will answer to the product or quotient of the two corresponding terms of the latter. Thus, 0. 1. 2. 3. 4. 5. 6. 1. 2. 4. 8. 16. 32. 64. Now 2+3=5. 7. &c. arith. series, or indices. 128. &c. geom. series. also 7-3-4. and 128+8=16. Therefore the arithmetical series, or indices, have the same properties as logarithms; and these properties hold true, whatever may be the ratio of the geometrical series. There may, therefore, be as many different systems of logarithms, as there can be taken different geometrical series, having unity for the first term. But the most |