OF GEOMETRY AND TRIGONOMETRY, WITH AN EASY AND CONCISE SYSTEM OF LAND SURVEYING. BY JAMES HALE. BELLOWS FALLS, VT. PUBLISHED BY JAMES 1. CUTLER AND CO. James I. Cutler & Co. Printers, 1829. DISTRICT OF NEW-HAMPSHIRE, to wit: District Clerk's Office. L.S. BE IT REMEMBERED, That on the fifth day of October, A. D. 1828, in the fifty-third year of the Independence of the United States of America, JAMES HALE, of the said District, has deposited in this Office the Title of a Book, the right whereof he claims as author, in the words following, to wit : "Elements of Geometry and Trigonometry, with an easy and concise System of Land Surveying. By James Hale. In conformity to an Act of the Congress of the United States, entitled " An Act for the Encouragement of Learning, by securing the Copies of Maps, Charts and Books, to the Authors and Proprietors of such Copies, during the times therein mentioned:" and also to an Act entitled an Act supplementary to an Act, entitled, An Act for the encouragement of Learning, by securing the Copies of Maps, Charts and Books to the Authors and Proprietors of such Copies during the times therein mentioned; and extending the benefits thereof to the Arts of Designing, Engraving, and Etching, Historical and other Prints. CHARLES W. CUTTER, Clerk 23-44.BAP Tuttle 3-22-43 47577 PREFACE. THE principal design of the following work is to present to the student the common principles of Field Surveying, in a concise, attractive, and intelligible manner. It does not pretend to much originality of matter, but every thing extraneous or foreign to the subject is omitted. It was supposed a work of this description would be profitable to the youth of this country, who wish to improve their minds, by extending their knowledge of the Mathematics, beyond the rules of common Arithmetic. It is hoped the execution of the work is such, that, it will be a convenient, and useful assistant, to the practical Surveyor. The work is divided into two parts. PART I.-Contains GEOMETRY and TRIGONOMETRY, with various explanations, and the use of these sciences, as connected with SURVEYING. PART II.-Treats wholly on SURVEYING, and contains partieular directions for taking the Survey of Fields, differently situated. Rules for calculating their Area, GEOMETRICALLY or ARITHME TICALLY, and directions for laying out and dividing land. While treating on SURVEYING, instead of prolix and abstruse demonstrations, frequent references are made to the system of GEOMETRY and TRIGONOMETRY in the first part of the work. It was thought, that shewing existing analogies would, more effectually, elicit the intellectual powers of the student. Should this little volume meet the approbation of a candid and enlightened public, whose opinion is ever heard with respect, the author will consider himself amply rewarded for his exertions to promote useful education; but should it be otherwise, the satisfaction, taken in its compilation, cannot be considered of small account. PART I. ELEMENTS OF GEOMETRY. SECTION 1. DEFINITIONS. 1. GEOMETRY is a Science, by which the measure and properties of Magnitude are determined. 2. A Point is considered as a mark only, without any regard to dimensions. 3. A Line has length, but not breadth. 4. A Right Line is the shortest that can be drawn between two Points. 5. A Superficies or Surface has length and breadth, but no thick ness. 6. The measure or content of a Surface is called an Area. Fig. 1. 7. Parallel Lines are such as are equally dis- Atant from each other as A. B.-C. D. Fig, 1. 8. An Angle is the opening between two Lines, which begin at a Point, and recede from each other. At B is an An- B gle formed by the opening of the Lines BA and BC. Fig. 2. 9. If a Right Line CD, fall upon another Right Line AB, so as to incline to neither side, but make the Angles on each side qual, then those Angles are Right Angles, and the Line CD Perpendicular to AB, Fig. 3. Fig. 3. D |