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Division of the Circumference ...
Definitions of the Trigonometrical Lines.
Measurement of Lines and Angles..
Measurement of a Horizontal Line..
Measurement of a Horizontal Angle with the Theodolite.
Measurement of a Vertical Angle.
Measurements with the Tape or Chain
Double Meridian Distances and Area ..
Laying Out and Dividing Land...
Definitions and General Notions
Traversing-Compass-Miner's Semicircle ..
Method of Traversing with the Theodolite....
Modes of Connecting with Surface Survey
Reducing the Traverse-Office Book....
Method of Plotting the Traverse on the Surface
ELEMENTS OF SURVEYING.
Β Ο Ο Κ Ι.
LOGARITHMS AND TRIGONOMETRY.
1. THE LOGARITHM of a number is the exponent of the power to which it is necessary to raise a fixed number, to produce the given number.
The fixed number is called the base of the system. Any positive number, except 1, may be taken as the base of a system. In the common system, the base is 10.
2. If we denote any positive number byn, and the corresponding exponent of 10 by P, shall have the exponential equation, 10P = n.
In this equation, p is, by definition, the logarithm of n, which may be expressed thus, p = log n.
3. From the definition of a logarithm, it follows that, the logarithm of any power of 10 is equal to the exponent of that power : hence, the formula, log (10)” = log n = p.
If a number is an exact power of 10, its logarithm is a whole number.
If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of a logarithm is called the characteristic; the decimal part is called the mantissa.
4. If, in Equation (3), we make p successively equal to 0, 1, 2, 3, &c.; and then equal to -1, -2, -3, &c., we may form the following
When a number lies between 1 and 10, its logarithm lies between 0 and 1; that is, it is equal to 0, plus a decimal; if a number lies between 10 and 100, its logarithm is equal to 1, plus a decimal; if between 100 and 1000, its logarithm is equal to 2, plus a decimal; and so on : hence, we have the following
RULE.--The characteristic of the logarithm of any whole number is positive, and numerically 1 less than the number of places of figures in the given number.
When a decimal fraction lies between .1 and 1, its logarithm lies between – 1 and 0, that is, it is equal to -1, plus a decimal; if a number lies between .01 and .1, its logarithm is equal to - 2, plus a decimal; if between .001 and .01, its
logarithm is equal to – 3, plus a decimal; and so on : hence,