THE HE word Surveying, in the Mathematics, signifies the art of measuring land, and of delineating its boundaries on a map. The Surveyor, in the practice of this art, directs his attention, at first, to the tracing and measuring of lines; secondly, to the position of these lines in respect to each other, or the angles formed by them; thirdly, to the plan, or representa tion of the field, or tract, which he surveys; and fourthly, to the calculation of its area, or superficial content When this art is employed in observing and delineating Coasts and Harbours, in determining their variation of the Compass, their Latitude, Longitude and soundings, together with the bearings of their most remarkable places from each other, it is usually denominated Maritime-Surveying. This branch of Surveying, however, demands no other qualifications than those, which should be thoroughly acquired by every Land-Surveyor, who aspires to the character of an accomplished and skilful practitioner. Surveying, therefore, requires an intimate acquaintance with the several parts of the Mathematics, which are here inserted as an introduction to this trea tise. B PART I. Containing Decimal Fractions, Imolution and Evo lution, the Nature and Use of Logarithms, Geometry and Plane Trigonometry. SECTION I. DECIMAL FRACTIONS. If we suppose unity or any one thing to be divided into any assigned number of equal parts, this number is called the denominator ; and if we chuse to take any number of such parts less than the whole, this is called the numerator of a fraction. The numerator, in the vulgar form, is always written over the denominator, and these are separated by a small line thus }, or 8; the first of these is called three-fourths, and the latter five-eighths of an inch, yard, &c. or of whatever the whole thing originally consisted: the 4 and the 8 are the denominators, showing into how many equal parts the unit is divided; and the three and the five are the numerators, showing how many of those parts are under consideration. Fractions are expressed in two forms, that is, either vulgarly or decimally. All fractions whose denominators do not consist of a cipher, or ciphers, set after unity, are called vulgar; and their denominators are always written under their numerators. The treatment of these, however, would be foreign to our present purpose. But fractions whose denominators consist of an unit prefixed to one or more ciphers, are called decimal fractions; the numerators of which are written without their denominators, and are distinguished from integers by a point prefixed: thus to, i no and do, in the decimal form, are expressed by .2.42 .172. or; and if we can Darts less than the rator of a fractions ar form, is alrus The denominators of such fractions consisting always of an unit, prefixed to as many ciphers as there are places of figures in the numerators, it follows, that any number of aphers put after those numerators, will neither increase nor lessen their value : forum do and so are all of the same value, and will stand in the decimal form thus.3.30.300; but a cipher, or ciphers prefixed to those numerators lessen their value in a tenfold proportion: for io, io and which in the decimal form we denote by .3.03. and .003, are fractions, of which the first is ten times greater than the second; and the second, ten times greater than the third. Hence it appears, that as the value and denomination of any figure, or number of figures, in common arithmetic is enlarged, and becomes ten, or an hundred, or a thousand times greater, by placing one, or two, or three ciphers after it; so in decimal arithmetic, the value of any figure, or number of figures, decreases, and becomes ten, or a hundred, or a thousand times less, while the denomination of it increases, and becomes so many times greater, by prefixing one, or two, or three Giphers to it: and that any number of ciphers, before an integer, or after a decimal fraction, has no effect in changing their values. nd these are som the first d tors latter five_isbris aterer the shake and the 8 are the many equalpat and the fres 3 of those parti forms, that is lo not consis nity, are cel - are always e treatment to our pre nominators pre ciphers nerators of ominators Sy a point decimal Write the numbers under each other according to the value or denomination of their places ; which position will bring all the Decimal points into a column, or vertical line, by themselves. Then, beginning at the right hand column of figures, add in the same manner as in whole numbers, and put the decimal point, in the sum directly beneath the other points. EXAMPLES Add 4.7832 3.2543 7.8251 6.03 2.857 and 3.251 together. Place them thus, 4.7832 3.2543 7.8251 6.03 2.857 3.251 Sum=28.0006. Add 62 121.306.75 27 and 2007 to gether. 121.06 .75 2.7 .0007 Sum=130.9567 What is the sum of 6.57 1.026.75 145 8.7 526. 3.97 and 0271? Answer 693.5131. What is the sum of 451 146.071 507 .0006 132. 62.71 507 7.9 and .10712? Answer 351.31972. SUBTRACTION OF DECIMALS. Write the figures of the subtrahend beneath those of the minuend according to the denom nation of their places, as directed in the rule of addition; then, beginning at the right hand, subtract as in whole numbers, and place the decimal point in the difference exactly under the other two points. EXAMPLES 25.3741 Difference=13.3909. From 2.4 take .8472 Diff.= 1.5528 |