one chain; which will be found as ready; by a little practice, and perhapsinore 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. THE PRINCIPLES OF SURVEYING SECT. I. Containing Decimal Fractions, the Square Root, Grün metrical Definitions, Theorems and Problemis ; with the Nature and Use of the Tables of Logarithm Numbers, Sines, Tangents, and Secants. DEFINITION. URVEYING is that art which enables us 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. B Numerator The numerator in the vulgar form, is always wrote over the denominator, and these are separated by a small line thus - or 17 Denominator the first of these called 5 twelfths, and the latter 7 twelfths 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 i 1.6 and 1 in the decimal form, are expressed by .2 .42 .172 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 10 and 103. 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 ou and do which in the decimal form we denotě 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 numbers of figures in common arithmetic is enlarged, and becomes ten Or 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. E x A M P L E S. Add 4.7832 3.2543 7.8251 6.03 2.857 and 3.251 together. Place them thus, 4.7832 Answer 28.0006 Add .75 2.7 and .0007 to Add 6.2 121.306 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. EX A M P L E ş. From 38.765 take 25.3741 25.3741 Answer 13.3909 From 2.4 take .8479 .8472 1.5528 From 71.45 take 8.4837248 Answer 62.9662752. From |