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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 Ör 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.

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Sect. Page 1. Decimal Fractions . . 11 2. Involution and Evolution . . . . . . . . . 22 3. Of Logarithms . . . . 28 4. Elements of Geometry . . . . . . . 40 Mathematical Instruments . . . . . . 64 5. Trigonometry . . . . 82 PART II. 1. The Chain . . . . . . 109 The Circumferentor . 121 The Theodolite . . . 125 The Semicircle . . . 128 The Plane Table . . ib. Mensuration of Angles by these Instruments . . . . . . . 131 The Protractor . . . ib.

2. Mensuration of Heights 137

Of Distances . 146

3. Mensuration of Areas 151

General Method . . . 177

Pennsylvania Method 187
Of computing the A-
rea of a Survey, ge-
ometrically consid-

A 3

Sect. ered and demonstrated . . . . . . . 4. Of Offsets . . . . 5.

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Method of Surveying by Intersections . . 6. Changing the Scale of Maps . . . . . . . . 7. Method of dividing Land Maritime Surveying . Levelling . . . . . . Promiscuous Questions . . . . . . . .

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LIST OF TABLES.

Logarithms of Numbers.
Sines and Tangents.
Traverse Table.
Natural Sines.

EXPLANATION

OF THE MATHEMATICAL CHARACTERS USED IN THIS WORK

+ signifies plus, or addition.
minus, or subtraction.
× or . “ multiplication.
— “ division.
“ proportion.
- “ equality.
v “ square root.
V * cube root, &c.
Ca “ difference 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 × 3, or 7.3, denotes that 7 is to be multiplied by 3. 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.

v3, or a’, denotes the square root of the number 3.

o/5, or 5°, denotes the cube root of the number 5.

7°, denotes that the number 7 is to be squared.

8°, denotes that the number 8 is to be cubed. Etcetera.

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THE

THEORY AND PRACTICE

OF

SURVEYING.

The 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 representation 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 determining the variation of the compass, in observing and delineating coasts and harbours, 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 treatise.

PART I.

Containing Decimal Fractions, Involution and Evolution, the Nature and Use of Logarithms, Geometry, and Plane Trigonometry.

SECTION I.

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 de#

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