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sines ar.d tangents. The Traverse Table commonly used in this country furnishes the latitude and departure to every quarter degree of the quadrant, for distances from 1 to 100, and occupies ninety pages. The accompanying table occupies but six pages, and yields ten times greater precision.

The Table of Meridional Parts extends to tenths of a mile, and great care has been taken to insure its accuracy. For this purpose, I have compared all the similar tables within my reach, and among them have found two which appeared to have been computed independently. Between them there were detected 674 discrepancies in the final figures. These cases were all recomputed, and 78 errors were detected in the Jest copy compared. It is probable that the numbers in this table are not in every instance true to the nearest tenth of a mile; but it is believed that the remaining errors are few in number, as well as minute. This table is confidently pronounced more accurate than any similar one with which I have been able to compare it.

The Table of Corrections to Middle Latitude was computed entirely anew. The corresponding table in common use, which was originally computed by Workman, contains more than four hundred errors, several of them amounting to two minutes.

On the whole, it is believed that the accompanying tables will be found more convenient to the computer than any tables of six decimal places hitherto published in this country; and that they will be pronounced sufficiently extensive for all purposes of academic and collegi ate instruction, as well as for practical mechanics and surveyors.

EXPLANATION OF THE TABLES.

TABLE OF LOGARITHMS OF NUMBERS, pp. 1-20.

LOGARITHMS are numbers contrived to diminish the labor of Multiplica. tion and Division by substituting in their stead Addition and Subtrac tion. All numbers are regarded as powers of some one number, which is called the base of the system; and the exponent of that power of the base which is equal to a given number, is called the logarithm of that

number.

The base of the common system of logarithms (called, from their inventor, Briggs' logarithms) is the number 10. Hence all numbers are to be regarded as powers of 10. Thus, since

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whence it appears that, in Briggs' system, the logarithm of every number between 1 and 10 is some number between 0 and 1, i. e., is a proper fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e., is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e., is 2 plus a fraction, and so on.

The preceding principles may be extended to fractions by means of negative exponents. Thus, since

10'=0.1, -1. is the logarithm of 0.1 102=0.01,

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in Briggs' system;

-2 66

0.01

66

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Hence it appears that the logarithm of every number between 1 and 0.1 is some number between 0 and -1, or may be represented by -1 plus a fraction; the logarithm of every number between 0.1 and .01 is

a fraction; the logarithm of every number between .01 and .001 is some number between -2 and -3, or is equal to -3 plus a fraction, and so on.

The logarithms of most numbers, therefore, consist of an integer and a fraction. The integral part is called the characteristic, and may be known from the following

RULE.

The characteristic of the logarithm of a number greater than unity, is one less than the number of integral figures in the given number.

Thus the logarithm of 297 is 2 plus a fraction; that is, the characteristic of the logarithm of 297 is 2, which is one less than the number of integral figures. The characteristic of the logarithm of 5673.29 is 3: that of 73254.1 is 4, &c.

The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which its first significant figure is removed from the place of units.

Thus the logarithm of .0046 is -3 plus a fraction; that is, the characteristic of the logarithm is 3, the first significant figure 4 being removed three places from units.

The accompanying table contains the logarithms of all numbers from 1 to 10,000 carried to 6 decimal places.

To find the Logarithm of any Number between 1 and 100.

Look on the first page of the table, along the column of numbers under N, for the given number, and against it, in the next column, will be found the logarithm, with its characteristic. Thus,

opposite 13 is 1.113943, which is the logarithm of 13;

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65 is 1.812913,

66

66

65.

To find the Logarithm of any Number consisting of three Figures. Look on one of the pages from 2 to 20, along the left-hand column marked N, for the given number, and against it, in the column headed 0, will be found the decimal part of its logarithm. To this the characteristic must be prefixed, according to the rule already given. Thus the logarithm of 347 will be found, from page 8, to be 2.540329;

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As the first two figures of the decimal are the same for several successive numbers in the table, they are not repeated for each logarithm separately, but are left to be supplied. Thus the decimal part of the logarithm of 339 is .530200. The first two figures of the decimal remain the same up to 347; they are therefore omitted in the table, and are to

be supplied.

To find the Logarithm of any Number consisting of four Figures.

and the fourth figure at the head of one of the other columns. Opposite to the first three figures, and in the column under the fourth figure, will be found four figures of the logarithm, to which two figures from the column headed O are to be prefixed, as in the former case. The characteristic must be supplied by the usual rule. Thus

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In several of the columns headed 1, 2, 3, &c., small dots are found in the place of figures. This is to show that the two figures which are to be prefixed from the first column have changed, and they are to be taken from the horizontal line directly below. The place of the dots is to be supplied with ciphers. Thus

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The two leading figures from the column 0 must also be taken from the horizontal line below, if any dots have been passed over on the same horizontal line. Thus

the logarithm of 1628 is 3.211654.

To find the Logarithm of any Number containing more than four

Figures.

By inspecting the table, we shall find that within certain limits the log. arithms are nearly proportional to their corresponding numbers. Thus the logarithm of 7250 is 3.860338;

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Here the difference between the successive logarithms, called the tabular difference, is constantly 60, corresponding to a difference of unity in the natural numbers. If, then, we suppose the logarithms to be proportional to their corresponding numbers (as they are nearly,, a difference of 0.1 in the numbers should correspond to a difference of 6 in the logarithms; a difference of 0.2 in the numbers should correspond to a difference of 12 in the logarithms, &c. Hence

the logarithm of 7250.1 must be 3.860344;

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In order to facilitate the computation, the tabular difference is inserted on page 16 in the column headed D, and the proportional part for the fifth figure of the natural number is given at the bottom of the page. Thus, when the tabular difference is 60, the corrections for .1, .2, .3, &c., are seen to be 6, 12, 18, &c.

If the given number was 72501, the characteristic of its logarithm

If it were required to find the correction for a sixth figure in the nat ural number, it is readily obtained from the Proportional Parts in the table. Thus, if the correction for .5 is 30, the correction for .05 is obviously 3.

As the differences change rapidly in ne first part of the table, it was found inconvenient to give the proportional parts for each tabular difference; accordingly, for the first seven pages they are only given for the even differences, but the proportional parts for the odd differences will be readily found by inspection.

Required the logarithm of 452789.

The logarithm of 452700 is 5.655810.

The tabular difference is 96.

Accordingly, the correction for the fifth figure, 8, is 77, and for the sixth figure, 9, is 8.6, or 9 nearly. Adding these corrections to the number before found, we obtain 5.655896.

The preceding logarithms do not pretend to be perfectly exact, bu only the nearest numbers having but six decimal places. Accordingly when the fraction which is omitted exceeds half a unit in the sixth deci mal place, the last figure must be increased by unity.

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To find the Logarithm of a Decimal Fraction.

The decimal part of the logarithm of any number is the same as that of the number multiplied or divided by 10, 100, 1000, &c. Hence, for a decimal fraction, we find the logarithm as if the figures were integers. and prefix the characteristic according to the usual rule.

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