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ADVERTISEMENT TO THE FOURTH EDITION.

In preparing this edition for the press, several alterations have been made, which, it is believed, will be found to be real improvements. A number of new Problems has been introduced, and a more methodical arrangement of the whole has been adopted. Instead of three different rules for calculating the content of a Survey, one general rule, including these, is now given. It may be further added, that the rules for solving several of the problems in Division of Land, have been considerably simplified. The Mathematical Tables have been stereotyped, after carefully revising them and comparing them with the most correct European Editions. THE AUTHOR.

ADVERTISEMENT TO THE FOURTEENTH EDITION

To meet the wants of the Student of Civil Engineering, this edition has been enlarged by the addition of several chapters, in which the Theodolite and Levelling Instrument are described, the methods of adjusting and using them are given, and the principles and practice of Levelling and Topography are explained and illustrated.-The whole has been carefully revised and the few typographical errors existing in former editions have been corrected.

J. G.

A

TREATISE ON SURVEYING.

OF LOGARITHMS

LOGARITHMS are a series of numbers so contrived, that by them the work of multiplication is performed by addition, and that of division by subtraction.

If a series of numbers in arithmetical progression be placed as indices, or exponants, to a series of numbers in geometrical progression, the sum or difference of any two of the former, will answer to the product or quotient of the two corresponding terms of the latter. Thus,

0. 1. 2. 3. 4. 5. 6. 1. 2. 4. 8. 16. 32. 64.

Now 2+3=5.

And 4×8=32.

7. &c. arith. series, or indices. 128. &c. geom. series.

also 7-3-4.

and 128-8-16.

Therefore the arithmetical series, or indices, have the same properties as logarithms; and these properties hold true, whatever may be the ratio of the geometrical series.

There may, therefore, be as many different systems of logarithms, as there can be taken different geometrical series, having unity for the first term. But the most

convenient system is that in which the ratio of the geometrical series is 10; and this is the one in common use. Thus,

0. 1. 2. 3.

4.

5. &c. indices or logar. 1. 10. 100. 1000. 10000. 100000. &c. natural numbers

In this system the log. of 1. is 0,* the log. of 10 is 1. the log. of 100, is 2, &c. Hence it is plain that the log. of any number between 1 and 10, will be expressed by a decimal, the log. of any number between 10, and 100, by 1 and a decimal, the log. of any number between 100 and 1000, by 2 and a decimal, &c.

The numbers, 0, 1, 2, 3, &c. that stand before the decimal part of logarithms, are called indices and are always less by unity, than the number of figures in the integral part of the corresponding natural number.

The index of the logarithm of a number, consisting in whole, or in part of integers, is affirmative; but if the number be a decimal, the index is negative, and is mark ed by a negative sign (-) placed either before or above it. If the first significant figure of the decimal be adjacent to the decimal point, the index is, -1, or 1; if there be one cipher between them, the index is-2, or 2; if there be two ciphers between them, the index is ―3 or 3, &c.

The decimal parts of the logarithms of numbers, consisting of the same figures and in the same order, are the same, whether the number be integral, fractional, or mixed. This is illustrated as follows:

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The method of finding logarithms in the tables, and of multiplying, dividing, &c. by them, is contained in the following problems.

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To find the Logarithm of a given number.

If the given number consists of one or two figures only, find it in the column marked No. in the first page of the table, and against it, in the next column, marked log. is the logarithm. Thus the log. of 7 will be found 0.84510, and the log. of 85 will be found 1.92942.

But if the given number be either wholly or in part decimal, the index must be changed accordingly. Observing that the index must always be one less than the number of figures in the integral part of the given number; also, when the given number is wholly a decimal, the index is negative, and must be one more than the number of the ciphers between the decimal point and first significant figure on the left hand. Thus the log. of .7 is -1.84510, and the log. of .0085 is -3.92942.

If the given number consists of three figures, find it in one of the other pages of the table, in the column marked

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