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Of the Mathematical Characters used in this work.


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+ signifies plus, or addition.

minus, or subtraction. xor.,. multiplication. X

division. proportion.


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 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. N 3, or 3$, denotes the square root of the number 3. À 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 cubed.






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 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 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


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PART 1. Containing Decimal Fractions, Involution and Evo

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



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 written over the denominator, and these are separated by a small line thus, ori; 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 their denominators, and are distinguished from integers by a point prefixed : thus is, 4 and 17en, in the decimal


1766 form, are expressed by .2.42,172.

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003 1000

The denominators of such fractions consisting always of an unit, prefixed to as many ciphers as there are places of figures in the numerators, it follows, that any number of ciphers put after those numerators, will neither increase nor lessen, their value : for is, and are all of the same value, and will stand in the decimal form thus .3 .30 .300"; but a cipher, or ciphers prefixed to those numerators lessen their value in a tenfold proportion : for io, and Toot, which in the decimal form we denote by .3 .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 number of figures, in common arithmetic is enlarged, and becomes ten, or an hundred, or a thousand times greater, by placing one or two, or three ciphers after it ; .so in decimal arithmetic, the value of any figure, or number of figures, decreases, and becomes ten, or a hundred, or a thousand times less, while the denomination of it increases, and becomes so many times greater, by prefixing oné, or two, or three ciphers to it: and that any number of ciphers, before an integer, or after a decimal fraction, has no effect in changing their values.

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ADDITION OF DECIMALS. Write the numbers under each other according to the value or denomination of their places ; which position will bring all the Decimal points into a column, or vertical line, by themselves. Then, beginning at the right hand column of figures, add in the same manner as in whole numbers, and put the decimal point, in the sum directly beneath the other points.

EXAMPLES. Add 4.7832 3.2543 7.8251 6.03 2.857 and 3,251 together. Place them thus,



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