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Example. Find the continued product of 3.902, 597.16, and 0.0314728.

Log. 3.902 .
Log. 597.16
Log. 0.0314728

Operation. 0.591287 2.776091 2.497936

.

1.865314

log. 73.3354, the product.

Here the 2 cancels the + 2, and the 1 carried from the decis mal part is set down.

DIVISION BY MEANS OF LOGARITHMS.

8. Find the logarithms of the dividend and the divisor, and subtract the latter from the former; then find the number corresponding to the resulting logarithm, and it will be the quotient required.

Estample 1. Divide 24163 by 4567.

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Here 1 taken from - gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic way.

9. The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of the arithmetical complement.

is the result obtained by subtracting it from 10: it may be written out by commencing at the left hand, and subtracting each figure from 9 until the last significant figure is reached, which must be taken from 10. Thus 8.130456 is the arithmet. ical complement of 1.869544.

To divide one number by another by means of the arithmetical complement, find the logarithm of the dividend and the arithmetical complement of the logarithm of the divisor; add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required.

Example. Multiply 358884 by 5672, and divide the product by 89721.

Operation.
Log. 358884

5.554954
Log. 5672

3.753736 (a.c.) Log. 89721

5.047106

4.355796 = log. 22688, the result. The operation of subtracting 10 is performed mentally.

TO RAISE A NUMBER TO ANY POWER BY MEANS OF LOGA.

RITHMS.

10. Find the logarithm of the number, and multiply it by the exponent of the power; then find the number corresponding to the resulting logarithm, and it will be the power required.

Example. Find the 5th power of 9.

Log. 9 ..

Operation. 0.954243

5

4.771215=log. 59049, the power.

TO EXTRACT ROOTS BY MEANS OF LOGARITHMS.

11. Find the logarithm of the number, and divide it by the mdex of the root; then find the number corresponding to the

Example. Find the cube root of 4,096.

Operation. Log. 4,096, 3.612360; one-third of this is 1.204120, to which the corresponding number is 16, which is the root sought.

12. When the characteristic is negative, and not divisible by the index, add to it the smallest negative number that will make it divisible, and then prefix the same number, with a plus sign, to the mantissa.

Example Find the 4th root of .00000081. The logarithm of this number is 7.908485, which is equal to 3 + 1.908485, and one-fourth of this is 2.477121; the number corresponding to this logarithm is .03: hence .03 is the root required.

13. Five-figure logarithms are sufficiently accurate for ordi. nary railroad field-work. The tables in this book may therefore, as a rule, be used without interpolation.

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PLANE TRIGONOMETRY.

III:

DEFINITIONS.

1. Plane Trigonometry treats of the solution of plane tri. angles.

In every plane triangle there are six parts, – three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solution of the triangle.

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