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[Entered, according to the Act of Congress, in the year one thousand eight hundred and thirty-two, by J. & J. Harper, in the Clerk's Office of the District Court of the United States for the Southern District of New-York.]

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PREFA CE.

THE word Geometry imports no more than to measure the earth, or to measure the 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, from the river Nile's destroying and confounding all their landmarks by its annual inundations, which laid them under the necessity 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and measures to enable them to distinguish and adjust the limits of their respective grounds when the waters were withdrawn.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 Moses 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 posterity.

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, the rest have met with much greater improvements from the pens of 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.

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:

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section the first (Part the First) you-have Decimal Fractions. The second section contains Involution and Evolution. The third section contains the nature and power of Logarithms, with their application, and 'the method of computing them.

The fourth section contains geometrical definitions, theorems, and problems, with the description and use of the sector, Gunter's scale, and other mathematical drawing instruments used by surveyors.

The fifth section contains-Plane Trigonometry, rightangled and oblique, with a variety of rules and practical examples.

The first section (Part the Second) 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 several instruments used in surveying, with their respective uses.

The second section contains the mensuration of heights and distances, with a great variety of problems and practical examples.

The third section contains the mensuration of areas, or the various methods of calculating the superficial contents of any field; also several new rules and problems, with practical examples, and various methods of finding the areas of maps from their geometrical construction; two of which, more concise than the rest, were first published in this work. Also, it 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; to these is added the method of calculating the area of a survey, by having the meridian pass through the east or west point of the survey, with the method of discovering these points from the fieldnotes, and the method of correcting the errors by the pen, when the survey does not close: also another new method for calculating the area, by having a parallel of latitude pass through the north or south point of the survey. The whole geometrically considered and demonstrated.*.

* The remaining part of the Author's Preface I have altered according

The fourth section contains the nature of offsets, and the method of casting them up by the pen.

The fifth section contains the method of finding the areas by intersections.

The sixth section shows how to enlarge or diminish a map, or to reduce a map from one scale to another; also the manner of uniting separate maps of lands which join each other into one map of any assigned size.

The seventh section contains the method of dividing land, or of taking off or enclosing any given quantity. Section the eighth treats of surveying harbours, shoals, sands, &c.

Section the ninth treats of levelling, adapted to the surveying of roads and hilly ground, with promiscuous questions.

Section the first (Part the Third) contains the astronomical methods of finding the latitude, variation of the compass, &c., with a description of the instruments used in these operations.

Section the second contains a description of the instruments requisite in astronomical observations.

Section the third shows how to find the variation, of the compass; with a description of the azimuth compass, and its use.

In this edition is introduced a new set of accurate Mathematical Tables.

Truth calls upon me to acknowledge, that the methods. of calculation herein set forth got their rise from those of the late Thomas Burgh, Esq.,* who first discovered a universal method for determining the areas of right-lined figures, and for which he obtained a reward of twenty thousand pounds sterling from the Irish Parliament. 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.

* This method, with very little alteration and improvement, in this country, is usually called the Pennsylvania Method of Calculation.-ED.