one chain; which will be found as ready, by a little practice, and perhaps more exact, than those already published. Truth calls upon me to acknowledge, that the methods by calculation, herein set forth, got their rise from those of the late Thomas Burgh, esq. who first discovered an universal method for determining the areas of right lined figures, and for which he obtained a parliamentary reward. I hope therefore it cannot be construded as an intention in me to take from his great merit, when I say, that the methods herein contained are much more concise and ready than his. Section the sixth contains the nature of off-sets, and the method of casting them up by the pen the nature and application of enlarging, diminishing, and connecting of maps: variation of the compass by amplitudes and azimuths, with some of its uses; to which is added, a table of the sun's declination how to find by what scale a map is laid down, having the map and area given: how to find the content of ground that is surveyed by a chain that is too long or too short the method of dividing lands: And the whole concludes with some necessary directions and remarks on surveys in general. PRINCIPLES OF SURVEYING. SECT. I. Containing Decimal Fractions, the Square Root, Geon metrical Definitions, Theorems and Problems; with the Nature and Use of the Tables of Logarithm Numbers, Sines, Tangents, and Secants. DEFINITION. URVEYING is that art which enables us SUR to give a plan, or just representation, of any piece or parcel of land, and to determine the content thereof, in such measure as is agreeable and customary to the country or place where the land is. This science depends on some parts of the mathematics, which must be known before we can treat of it, wherefore we shall begin with 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. В The numerator in the vulgar form, is always wrote over the denominator, and these are separa ted by a small line thus or 1 Denominator 7 Numerator the first of these called 5 twelfths, and the latter 7 twelfth's of an inch, yard, perch, &c. or of whatever the whole thing originally was. Fractions are expressed in two forms, that is, either vulgarly or decimally. All Fractions whose denominators do not consist of a cypher or cyphers set after unity, are called vulgar ones, and their denominators are always wrote under their numerators. The treating of these would be foreign to our present purpose. But fractions whose denominators consist of an Unit prefixed to one or more cyphers, are called decimal fractions; the numerators of which are written without their denominators, and are distinguished from integers by a point prefixed: thus and in the decimal form, are expressed by .2 .42 .172 30 1OO 20 1000 4 2 172 1000 The denominators of such fractions always consisting of an unit, prefixed to as many cyphers as there are places of figures in the numerators, it follows, that any number of cyphers put after those numerators, will neither increase nor lessen their value: for and 30% are all of the same value and will stand in the decimal form thus .3 .30 .300; but a cypher or cyphers prefixed to those numerators, lessen their value in a tenfold proportion: for 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 than the third. 03 1000 greater Hence it appears, that as the value and denomination of any figure or numbers 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 cyphers 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 cyphers to it and that any number of cyphers, before an integer, or after a decimal fraction, has no effect in changing their values, Having placed those figures which are equi-distant from the point (as well integers as fractions) under each other, add them as if they were integers. EXAMPLES. Add 4.7832 3.2543 7.8251 6.03 2.857 and 3.251 together. Place them thus, Add 6.2 121.306 .75 2.7 and .0007 to gether. 121.306 .75 2.7 .0007 Answer 130.9567 What is the sum of 6.57 1.026 .75 146.5 8.7 526. 3.97 and .0271? Answer 693.5431. What is the sum of 4.51 146.071 .507 .0006 132. 62.71 .507 7.9 and .10712? Answer 354.31272. Subtraction of DECIMALS. Having, placed the figures which are equi-distant from the point, under each other, deduct as if they were integers. EXAMPLES. From 38.765 take 25.3741 25.3741 Answer 13.3909 From 2.4 take .8472 .8472 1.5528 From 71.45 take 8.4837248 Answer 62.9662752. |