Prefixed to the tables will be found a brief explanation of the nature and use of logarithms. It has been a defect in former editions, that this subject has not been more fully treated. A very distinguishing feature, likewise, of the present edi. tion, is the introduction of a new and elegant set of tables, more extensive and on plainer type than those which have before been attached to the work. The decimals are carried to six figures, instead of five, as formerly, and a column of differences is added, for the purpose of finding intermediate numbers. For many purposes, sufficient accuracy may be perhaps attained by the use of logarithms, extending only to five, or even four decimal places, but others will likewise occur, in the experience of every surveyor, in which such logarithms will not answer the purpose. But if logarithms are carried to six places, they may be taken to as few, or as many, within that limit, as is desired. The above constitute the principal peculiarities of the present edition. Hartford, August 2nd, 1830. PREFACE TO THE FIRST EDITION. cases. The following work is chiefly a compilation from other Books ; and but very little new is added, except a more full explanation, than has yet been published, of RECTANGULAR SURVEYING, or the method of calculating the area of fields arithmetically, without drawing a plot of them and measuring with a scale and dividers, as has been the common practice; and also a more particular explanation of the use of natural sines than is contained in most mathematical books. The compiler has endeavoured to render this work so easy and intelligible that a learner will require but little assistance from an Instructer, except with regard to the construction and use of mathematical and surveying instruments. Before, however, he enters on the study of this book, he must be well acquainted with common Arithmetic, with decimal fractions, and the square root; and he must also know the various characters or marks used in Arithmetic. A surveyor will doubtless find many questions arise in the course of his practice, for the solution of which, no particular directions are here given; nor is it possible to give directions for every case that may occur. In all practical sciences much must be left to the judgment of the practitioner, who, if he is well acquainted with the general principles of his art, will readily learn to apply those principles to particular The primary design of this treatise is to teach common Field Surveying; at the same time it contains the elements of Surveying upon a larger scale; and the system of Geometry and Trigonometry with which it is introduced, with the problems for the mensuration of superficies, as also the mathematical tables at the end, will be found useful for many other purposes. It would be well, therefore, for those who do not intend to become practical surveyors to acquaint themselves with what is here taught; and with this view the following work is very proper to be introduced into academies and those higher schools which are designed to fit young men for active business in life. Indeed every person who frequently buys and sells land should learn to calculate the contents of a field arithmetically; a knowledge which may be acquired in a very little time, from the particular explanation here given of that method. Notwithstanding the many books already published on the subjects here treated upon, it was thought a work of this kind was really wanted, and that if judiciously executed it would be useful. It is more particularly necessary at the present time in Connecticut, as the legislature of the State have lately enacted a law on the subject of surveying, in consequence of which more attention must be paid to the theory of that art than has been common. These considerations induced the compiler to select from various publications what appeared to him important; and to arrange the whole in a method best adapted, in his view, for teaching that useful art. How far he has succeeded in his endeavours to simplify the subject, and render it easy to the learner, must be submitted to the test of The system of Geometry is divided into two parts. The first contains geometrical definitions respecting lines, angles, superficies, &c. The second part contains a number of geometrical problems necessary for Trigonometry and Surveying. The system of Trigonometry is also divided into two parts : and teaches the solution of questions in right and oblique angled trigonometry, by logarithms and also by natural sines. The treatise on Surveying is divided into three parts. Part first treats of measuring land, and is divided into three sections. The first contains several problems respecting mensuration, and for finding the area of various right-lined figures and circles. The second section teaches different methods of taking the Survey of fields; also to protract them, and find their area in the manner commonly practised, and likewise by arithmetical and trigonometrical calculations, without measuring diagonals and perpendiculars with a scale and dividers ; interspersed with sundry useful rules and directions. The third section is a particular explanation and demonstration of RECTANGULAR SURVEYING, or the method of computing the area of fields from the field notes, by mathematical tables, without the necessity of plotting the field. To this section is added a useful problem for ascertaining the true area of a field which has been measured by a chain too long or too short. Part second treats of laying out land in various shapes. Part third contains sundry problems and rules for dividing land and determining the true course and distance of dividing lines, or from one part of a field to another. To this is added an appendix concerning the variation of the compass and attraction of the needle ; also, a rule to find the difference between the present variation, and that at a time when a tract was formerly surveyed, in order to trace or run out the original lines. The mathematical tables, are a traverse table, or table of difference of latitude and departure, calculated for every degree and quarter of a degree, and for any distance up to 50; a table of natural sines calculated for every minute; a table of logarithms comprised in four pages, yet sufficiently extensive for common use; and a table of logarithmic or artificial sines, tangents, and secants, calculated for every 5 minutes of a degree. To these tables are prefixed particular explanations of the manner of using them. * * This view of the contents was drawn up by Mr. Flint, and refers of course to the original edition. ERRATA. Page 18, line 14 from top, for obtruse, read obluse. 19, fig. 30, transpose letters A and C. GEOMETRY. GEOMETRY is a Science which treats of the properties of magnitude. PART I. Geometrical Definitions. 1. A point is a small dot; or, mathematically considered, is that which has no parts, being of itself indivisible. 2. A line has length but no breadth. 3. A superficies or surface, called also area, has length and breadth, but no thickness. 4. A solid has length, breadth, and thickness. 5. A right line is the shortest that can be drawn between two points. Fig. 1. 6. The inclination of i wo lines meeting A one another, or the opening between them, is called an angle. Thus at B. Fig. 1. is an angle, formed by the meeting of the lines AB and BC. B2 Fig. 2. 7. If a right line CD. Fig. 2. fall upon D another right line AB, so as to incline to neither side, but make the angles on each side equal, then those angles are called right angles; and the line CD is said to be perpendicular to the other line. B Fig. 3. 8. An obtuse angle is greater than a right angle ; as ADE. Fig. 3. 9. An acute angle is less than a right angle; as EDB. Fig. 3. A -B D Note. When three letters are used to express an angle, the middle letter denotes the angular point. 10. A circle is a round figure, bounded Fig. 4. by a single line, in every part equally distant from some point, which is called the centre. Fig. 4. 11. The circumference or periphery of a circle is the bounding line; as ADEB. B Fig. 4. 12. The radius of a circle is a line drawn from the centre to the circumference; as CB. Fig. 4. Therefore all radii of the same circle are equal. Fig. 5. 13. The diameter of a circle is a right line drawn from one side of the circumfer. ence to the other, passing through the centre; and it divides the circle into two equal parts, called semicircles; as AB or DE. Fig. 5. B 14. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds; and these into thirds, &c. NOTE. Since all circles are divided into the same number of degrees, a degree is not to be accounted a quantity of any determinate length, as so many inches or feet, &c. but is always to be reckoned as being the 360th part of the circumference of any circle, without regarding the size of the circle. 15. An arc of a circle is any part of the circumference ; as BF or FD. Fig. 5 ; and is said to be an arc of as many de. |