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.... ELEMENTS OF SURVEYING. Β Ο Ο Κ Ι. I. LOGARITHMS AND TRIGONOMETRY. SECTION I. LOGARITHMS. 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. we 2. If we denote any positive number byn, and the corresponding exponent of 10 by P, shall have the exponential equation, 10P = n. (1.) In this equation, p is, by definition, the logarithm of n, which may be expressed thus, p = log n. . (2.) . 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. (3.) 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, the following |
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