The fifth section contains a new, and much more concise method of determining the areas of surveys from the field-notes, or by calculation than any hitherto published; and Eventure to assert that it is impossible (from the nature of right-lined figures) that any method or methods more concise than this, can be investigated.

To these methods is annexed a short table of differ ence of latitude and half departure, to every degree and quarter of a degree of the quadrant, the stationary distance being 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 construed 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.

SECTION I.

Containing Decimal Fractions, the Square Root, Geometrical Definitions, Theorems and Problems: with the Nature and Use of the Tables of Logarithm Numbers, Sines, Tangents, and Secants.

DEFINITION.

SURVEYING, is that art which enables us to give a plan, or just representations, 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 mathematicks, 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.

The numerator in the vulgar form is always wrote over the denominator, and these are separated by a small line thusor Denominator 7 Numerator 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 and 17% in the decimal form, are expressed by .2 .42.172.

2 42

100

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 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 100. S 03 and

30

1000

00 which in the decimal form we denote by .3 .03/

1000

and .003, are fractions, of which the first is ten times greater than the second; and the second ten times

レ

1

Hence it appears that as the value and denomination of any figure or numbers of figures in common arithmetick 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 arithmetick, 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.