PRE FACE T HE Word Geometry imports no more than to measure the Earth, or to measure Land; yet in a larger and more proper Sense, it is applied to all sorts of Dimensions. It is generally supposed to have had its Rise among the Egyptians, rom the River Nile's destroying and confounding all their Land-marks by its annual Inundations, which laid them under the Neceflity of inventing certain Methods and Measures, to enable them to distinguish and adjust the Limits of their respective Grounds, when the Waters were withdrawn. And this Opinion is not entirely to be rejected, when we consider that Mofes is said to have acquired this Art, when he resided at the Egyptian Court. And Achilles Tatius in the Beginning of his Introduction to Aratus's Phænomena, informs us, that the Egyptians were, the first who measured the Heavens and the Earth (and of course the Earth first and that their Science in this Matter, was engraven on Columns, and by that means delivered to Pofterity, It is a Matter of some Wonder, that though Surveying appears to have been the first, or at least one of the first of the Mathematical Sciences, that the rest have met with much greater Improvements from the Pens of the most eminent Mathe maticians, maticians, while this seems to have been ne glected ; infomuch 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. There Confiderations 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, Geornetrical Definitions, some necessary. Theorems and Problems, with the Nat, ture and Use of the Tables of Logarithm Num, bers, Sines, Tangents, and Secants. The second Section contains Plane Trigonometry right angled and oblique, with its Application in determining the Measures of inacceslīble Heights and Distances. The third Section gives an Account of the Chains and Measures used in Great Britain and Ireland. Methods of Surveying and of taking inaccessible Distances by the Chain only, with some necessary Problems; also a particular Description of the se , veral Instruments used in Surveying, with their respective Uses. The fourth Section contains five various Methods of finding the Areas of Maps, from their Geome trical Construction: two of which more concise than the reft, were never before made public, The fifth Section contains four new, and much more concise Methods 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 impoflible (from the Nature of rightlined Figures) that any Method or Methods more concise than these can be investigated. To these Methods is annexed a short Table of Difference 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. To this is annexed a Table for reducing Degrees of the Circumferentor to those of the Quarter Compass, and the contrary; also the Method of changing Angles of the Field, taken by Theodolite, Semicircle, or Plane Table, to those from the Meridian; for the greater Readiness and Accuracy in Protraction, as well as to prepare them for Calculation. 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; wlio 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 Section the sixth contains the Nature of Off-sets, and the Method of casting them up by the Pen : The Nature and Application of Intersections : The Methods of enlarging, diminishing, and connecting of Maps: The Method of tracing defaced Mearings from the Down (or any other) Surveys : The Variation of the Compass by Amplitudes and Azi.muths, with some of its Uses; to which is added, a Table of the Sun's Declination: The Method of reducing one Measure to another ; 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 oo short : The Method of dividing Lands: And the whole concludes with some necessary Directions and Remarks on Surveys in general. 1 THE THE PRINCIPLES OF SUR VE YIN G. SECT. I. Containing Decimal Fractions, the Square Root, Geo metrical Definitions, Theorems and Problems ; with the Nature and Use of the Tables of Logarithin Numbers, Sines, Tangents, and Secants. DEFINITION, SU 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 part of the Mathematics, which must be known before we can treat of it, wherefore we shall begin with DECIMAL FRACTION S. 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 The |