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District of New York, ss.

BE IT REMEMBERED, that on the twenty-eighth day of March, in the thirty-fifth year of the Independence of the United States of America, Evert Duyckinck, of the said district, hath deposited in this office the title of a book, the right whereof he claims as proprietor, in the words following, to wit :

“ The Theory and Practice of Surveying ; containing all the lastructions requisite for the skilful practice of this Art. By Robert Gibson. Illustrated by Copper-Plates. The whole corrected, newly arranged, and greatly enlarged, with useful Selections, and a new set of accurate Mathematical Tables. By D. P. Adams, Teacher of the Mathematics."

In conformity to the act of the Congress of the United States, en. titled “ An act for the encouragement of learning, by securing the copies of maps, charts, and books to the authors and propietors of such copies, during the times therein mentioned.” And also to an act, entitled " An act, supplementary to an act, entitled an act for the encouragement of learning, by securing the copies of maps, charts, and books to the authors and proprietors of such copies, during the times therein mentioned, and extending the benefits thereof to the arts of designing, engraving, and etching historical and other prints."

CHARLES CLINTON, Clerk of the District of New-York

of the Mathematical Characters used in this work.

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+ signifies plus, or addition.

minus, or subtraction. X or

multiplication. division. proportion. equality. square root. cube root, &c. diff. between two numbers when it is not

known which is the greater'. Thus,

5 + 3, denotes that 3 is to be added to 5. 6- 2, denotes that 2 is to be taken from 6. 7 x 3, or 8.3, denotes that 7 is to be multiplied by S. 8 • 4, denotes that 8 is to be divided by 4. 2:3:: 4:6, shows that 2 is to 3 as 4 is to 6. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10: ✓ 3, or 3, denotes the square root of the number 3. ✓ 5, or 5j, denotes the cube root of the number 5. 72, denotes that the number 7 is to be squared. 83, denotes that the number 8 is to be cabed.

&c.

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THE

HE word Surveying, in the Mathematics,

signifies the art of measuring land, and of delineating its boundaries on a map.

The Surveyor, in the practice of this art, directs his attention, at first, to the tracing and measuring of lines; secondly, to the position of these lines in respect to each other, or the angles formed by them; thirdly, to the plan, or representa tion of the field, or tract, which he surveys; and fourthly, to the calculation of its area, or superficial content When this art is employed in observing and delineating Coasts and Harbours, in determining their variation of the Compass, their Latitude, Longitude and soundings, together with the bearings of their most remarkable places from each other, it is usually denominated Maritime-Surveying. This branch of Surveying, however, demands no other qualifications than those, which should be thoroughly acquired by every Land-Surveyor, who aspires to the character of an accomplished and skilful practitioner. Surveying, therefore, requires an intimate acquaintance with the several parts of the Mathematics, which are here inserted as an introduction to this trea

tise.

B

PART I.

Containing Decimal Fractions, Involution and Evo

lution, the Nature and Use of Logarithms, Geometry and Plane Trigonometry.

SECTION I.

DECIMAL FRACTIONS.

If we suppose unity or any one thing to be di, vided 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 written over the denominator, and these are separated by a small line thus }, or 8; the first of these is called three-fourths, and the latter five-eighths of an inch, yard, &c. or of whatever the whole thing originally consisted: the 4 and the 8 are the denominators, showing into how many equal parts the unit is divided ; and the three and the five are the numerators, showing how many of those parts are under consideration.

Fractions are expressed in two forms, that is, either vulgarly or decimally.

All fractions whose denominators do not consist of a cipher, or ciphers, set after unity, are called vulgar; and their denominators are always written under their numerators. The treatment of these, however, would be foreign to our present purpose. But fractions whose denominators consist of an unit prefixed to one or more ciphers, are called decimal fractions; the numerators of which are written without tlieir denominators, and are distinguished from integers by a point prefixed: thus io, ioo and 103%, in the decimal form, are expressed by .2.42.172.

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