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the most eminent mathematicians, while this seems to have been neglected; insomuch that I have not been able to meet with one author, who has sufficiently explained the whole art in its theory and practice ; for the most part, it has been treated of in a practical manner only ; and the few who have undertaken the theory, have in a great measure omitted the practice.

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These considerations induced me to attempt a methodical, easy, and clear course of Surveying ; how far I have succeeded in it, must be determined by the .impartial reader: the steps I have taken to render the whole evident and familiar are as follow :

In section the first, you have decimal fractions, the square root, geometrical definitions, some necessary theorems and problems; with the nature and use of the tables of logarithm numbers, sines, tangents, and secants.

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The second section contains plane trigonometry right angled and oblique, with its application in determining the measures of inaccessible heights and distances.

The third section gives an account of the chains and measures used in Great Britain an Ireland; methods of surveying and of taking inaccessible distances by the chain only, with some necessary problems; also a particular description of the several instruments used in surveying, with their respective uses.

The fourth section contains two methods of finding the areas of maps from their geometrical construction, more concise than any heretofore made publick.

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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 I venture to assert that it is impossible (from the nature of right-lined figures) that any method or methods more concise than this, can be investi. gated.

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 sur

veyed by a chain that is too long or too short: the me. thod 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, Geo

metrical 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 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 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 thus i' or 's Demoeritarar , 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 to tur 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 is it and

300 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 ið 10% and 100which in the decimal form we denote by .s .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 arith. metick is enlarged, and becomes ten or an hundred, or

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