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answering to the next less log., will give the number sought.

Let it be required to find the number answering to the log. 2.26589.

Next log. greater
Next log. less

2.26600 2.26576

24, first difference.

Given log. 2.26589

Next less 2.26576, log. of 184.4

13, second difference.

As 24 13 10 : 5, the fifth figure.

:

Hence the number required to five places of figures is 184.45.

When the number answering to the above log. is required to six places of figures, say, as 24:13::100:53. Hence the number required to six places is 184.453.

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Note. When the logs. next less and next greater, fall in the latter part of the tables, where the differences are very small, the number answering to the given log. cannot be always depended on to more than five places of figures.

In treating of logs. having negative indices, we might have mentioned that the negative sign is put over the index, in order to distinguish it from the decimal part found in the tables, which is always positive: so -2 +69897, which is the log. of .05, is written -2.69897.

It

may be necessary to observe further, that on some occasions it is convenient to reduce the whole expression to a negative form, which is done by making the characteristic less by 1, and taking the arithmetical

complement of the decimal part of the log., that is, beginning at the left hand, subtract each figure from 9, except the last significant figure, which is subtracted from 10, then shall the remainders express the log. wholly negative. Thus the log. of .05, which is -2.69897, is expressed -1.30103, which is all negative.

Sometimes also it is thought more convenient to express such log. entirely as positive, by only joining to the tabular decimal the complement of the index to 10; and in this way the above log. is expressed by 8.69897, which is only increasing the index in the scale by 10.

MULTIPLICATION BY LOGARITHMS.

Multiplication is performed by adding the logs. of the factors together, and finding the natural, number corresponding to the sum.

1. What is the product of 26 by 74?

26 log. 1.41497

74 log. 1.86923

1924 log. 3.28420

2. What is the product of .0054 by .95?

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If we choose to avoid negative indices, each of the factors, when decimals, may be multiplied by 10,

100, &c. so as to make the product whole or mixed numbers; then having added the logs. of those factors together, the natural number corresponding to the sum must be divided by 10, 100, &c. according as the factors were increased.

Taking the last example :

.0054× 1000 5.4 log. 0.73239

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Then 51.3÷10000=00513, the product as before.

DIVISION BY LOGARITHMS.

This is performed by subtracting the log. of the divisor from the log. of the dividend, and finding the natural number corresponding to the remainder.

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TO EXTRACT THE SQUARE ROOT BY LOGARITHMS.

Divide the log. of the given number by 2, and the natural number corresponding to the quotient, will be the root required.

1. What is the square root of 4096 ?

4096 log. 3.61236; then 3.61236÷2-1.80618, the number corresponding to which is 64, the root required.

To extract the cube root, we divide the log. of the number by 3.

To raise any number to the second power:

Multiply the log. of the number by 2, and the natural number corresponding to the product, will be the square or second power of the number.

1. What is the square of 64?

64 log. 1.80618

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To find the natural sine or cosine of an arc, also the log., sine, tangent, secant, &c.

If the given degrees be less than 45°, look for them at the top of the table, and for the minutes in the left

hand column, marked M, against which, in the column. marked at the top with the proposed name, viz. sine, cosine, &c., stands the sine, cosine, &c. required.

When the degrees are more than 45°, look for them at the bottom of the table, and for the minutes, if any, in the right hand column.

The name of this, viz. sine, cosine, &c. must be taken at the bottom of the table.

The natural sines, &c. must be looked for in the tables marked natural sines, &c. at the top; and the log. sines, &c. in the table marked log. sines, cosines, &c. at the top.

The natural sine of 39° 42′ is .63877
The natural cosine of 39° 42′ is .76940
Logarithmic sine of 39° 42′ is 9.80534
Logarithmic cosine of 39° 42′ is 9.88615

What are the natural sine and cosine of 73° 27'?
Natural sine of 73° 27' is .95857

Natural cosine of 73° 27' is .28485

When the sine, tangent, &c. are required to any number of degrees above 90° :

Subtract those degrees from 180°, and find the natural sine, &c. or log. sine, &c. of the remainder, as before.

Required the log. sine, cosine, tangent, &c. of 137° 29'.

180°
137° 29'

42° 31′ sine=9.82982, cosine=

51. tangent 9.96231, cotangent=10.03769.

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