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III. A TABLE of LOGARITHMS for NUMBERS ; and

IV. A TABLE of LOGARITHMIC or ARTIFICIAL SINES, TANGENT'S and SECANTS.

Explunation of the Table of Logarithms for Numbers. LOGARITHMS are Numbers in Arithmetical Progression, corresponding to other Numbers in Geometrical Proportion. As, 0. 1. 2.

3.

4. Logarithms.
1.
10.

100. 1000. 10000. Numbers. The Logarithm for any Number less than 10 is a certain number of Decimals ; for any Number between 10 and 100 it is 1 with Decimals ; for any Number between 100 and 1000 it is 2 with Decimals, &c. The whole Number in Logarithms, or the Number which stands at the Left hand of the Decimal Point is called the Index; and is always a Unit less than the places of figures in the whole Number for which it is the Logarithm : Thus, The Log. of 6543 is 3.81578 654.3

2.81578 65.43

1.81578 6.543

0.81578 'The Log. of a Decimal Fraction is the same as that of an Integer, only the Index is negative ; and is distinguished from an absolute one by placing a Point or a negative Sign before it : Thus, The Log. of 0.6543 is .9.81578 or - 1.81578 0.06543

.8.81578 or - - 2.81578 By the following Table the Log. of any Number, containing three places of figures, whether whole Numbers, mixed Numbers or Decimals, may be found true at once.

Look for the two first figures in the Left or Right hand Column, markéd No. and for the third figure on the Top of the Påge ; against the two first figures and under the third will be the Loga: rithm.

EXAMPLES.

Required the Logarithm for 346 Look for 34 in the Column marked No. and for 6 on the Top of the Page, under which and against 34 you find 53908 to which prefix 2 for the Index, because the Number consists of three places of figures.

In the same way the Log. for 28.3 will be found to be 1.45179 And the Log. for 3.23 to be 0.50920

To find the Number corresponding to any Logarithm. Look in the Table till you find the given Log. without regarding the Index ; the Number standing against it in the Column marked No. together with the figure on the Top, form the corresponding Number; whether whole, mixed or Decimals, will be determined by the Index. If you cannot find the exact Log, take the nearest to it.

If the Log. of any Number between 10 and 100, with two places of Decimals, be required, take the nearest number of tenths, which will be sufficiently exact for common practice. But, if great accuracy be desired, work by Natura] Sines, in the manner pointed out in 'Trigonometry, and in the Introduction to tbe Table of Natural Sines. Or,

The Log. of any Number containing more than three places of figures, may be found by the Table in this Book, 19 follows :

tract that from the next greater Log. contained in the Table ; multiply the difference by the remaining figure or figures in the given Number, and from the Product cut off as many figures from the Right hand as remain in the given Number ; add the figure or figures standing at the Left hand to the Log. of the three first figures, and the Sum will be the Log. required, to which prefix the proper Index.

EXAMPLES. 1. Required the Logarithm of 7624 Log. of 763

.88252 762

.88195

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Required Log 3.88217 Note. This is also the Log. of 762.4 or 76.24, &c. varying the

Index according to the preceding directions. 2. Required the Logarithm of 541.25 Log. of 542

.73400 541

.73320

Difference

80 Remaining figures of the given Numb. 25

400 160

20.00 Log. of 541

.73320

Required Log. 2.73340 To find the nearest Number corresponding to any Logarithm for more than three places of figures.

Find the Log. next less than the given one, and take the difference between that and the given one ; also take the difference between the next greater and the next less Log. than the given one; divide the former difference by the latter, according to the Rule in Division of Decimals ; add the Quotient to the number answering to the Log. next less than the given one, and you will have the required Number; whether a whole or a mixed Number will be determined by the Index.

EXAMPLES. 1. Required the Number to the Logarithm 3.88218 Given Log. .88218 Next greater Log.

.88252 Next less .88195 Next less

.88195

Difference

23

Difference

57

+57)23.0(4

228

2

The Number to the Log. next less than the given one is 7620 because the Index is 3 ; to this add 4 and it makes 7624 the required Number.

2. Required the Number to the Logarithm 2.73340
Given Log.
.73340 Next greater Log

.73400 Next less .73320 Next less

.73320

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The Number to the Log. next less than the given one is 541, to this add the figures in the preceding Quotient, which are known to be Decimals from the Index of the given Log. and the required Number will be 541.25

The addition and subtraction of Logarithms answers the same purpose as the multiplication and division of their corresponding Numbers : That is, the Log. of any two Numbers being added, their Sum will be the Log. of the Product of those Numbers ; and the Log. of one Number being subtracted from the Log. of another Number, the Remainder will be the Log. of the Quotient of one of those Numbers divided by the other. Again, the Log. of any Number being doubled will produce the Log. of the Square of that Number; and one half the Log. of any Number is the Log. of the Square Root of that Number.

To perform Addition or Subtraction by Logarithms. The following Theorems for adding and subtracting by Logarithms were invented by Mr. EBENEZER R. WHITE of DANBUNY, and by him communicated to the Compiler. Though in common cases, they may not be particularly useful, yet in the solution of many Mathematical Questions they will greatly abridge the numerical operation. They are therefore here inserted. Let a = greater } number to be added or subtracted.

b = lesser Then +1xb=a+b And i-lxb= a_b These Theorems may be expressed in words as follows : From the Log. of the greater number subtract the Log. of the lesser, and find the number corresponding to the Remainder : Then, if the original numbers are to be added together, add i to the number last found ; but if they are to be subtracted, subtract 1 from it; and the Log of the number thus increased or diminished added to the Log. of the lesser original number, will give the Log.

Of the TABLE OF LOGARITHMIC or ARTIFICIAL SINEs, Tangent's and SECANTS.

76 find the Logarithmic Sine, &c. for any number of Degrees and Minutes, within the Compass of the Table.

If the Degrees be less than 45, look for them at the Top of the Columns, and under Sine, Tangent or Secant, whichever is wanted, and for the Minutes at the Left hand ; but if more than 45, look for the Degrees at the Bottom over Sine, &c. and for the Minutes at the Right hand ; under or over the Degrees and against the Minutes will be the required Log. Sine, &c.

To find the Degrees and Minutes corresponding to a given Logarithmic Sine, &c.

Look in the proper Column for the nearest Log. to the given one ; and the Degrees and Minutes standing over or under and against it, are those required. Note. When the Log. Sine, &c. for more than 90° is required,

subtract the given number of Degrees from 180° and make

use of the Remainder. It will be observed that this Table is calculated only for every 5 Minutes. This was thought sufficient for Surveyors, as few Compasses will take a Course to greater exactness. If however a Question is to be solved where greater accuracy is required, work hy Natural Sines. Or,

The Log. Sine, &c. for any Minute may be found as follows :

Look in the Table for the Log, of the nearest number of Minutes greater than the given one, and from this subtract the next less Log. contained in the Table : Then say, As 5 Minutes, Is to this difference ; So is the excess of the given Minutes above 5, 10, 15, 20, 25, &c; To a fourth number, which add to the Log. of the Minutes next less than the given number, and the Sum will be the Log. required.

EXAMPLE. Required the Logarithmic Sine of 34° 23' Sine of 34° 25'

9.75221 34 20

9.75128

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To find the nearest Minutes corresponding to a given Logarithmie. Sine, &c.

Look in the Table, in the proper Column, for the Log. next less than the given one, and take the difference between that and the given one ; also take the difference between the next greater and the next less Log. than the given one : Then say, As the latter difference ; Is to 5 Minutes ; So is the former difference ; To the number of Minutes to be added to the Minutes of the Log.

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