Of the Measurement and Calculation of Lines and Angles, CHAPTER III. INTRODUCTION, CHAPTER I. OF LOGARITHMS. 1. LOGARITIIMS of numbers are the indices that denote the different powers to which a given number must be raised to produce those numbers. 2. If a be the given number, whose indices and powers are to be considered, then a++ being put equal to n, a, the given number, or root, is called the base of the system of logarithms, n the number whose logarithm is considered, and Ex, the logarithm of that number. 3. Any number, except 1, may be taken for the base of a system of logarithms. In the system in general use, the base is 10; and this system affords the greatest facilities in calculations, because 10 is the base of the common numeration, both in whole numbers and decimal fractions. 4. Taking a+r=n, we have, +x=log. n; and putting aty= m, gives, +y=log. m. If the equations ar=n, and av=m, be multiplied together, member by member, we have, a' Xa= nxm, or ax+y=nXm. In this expression, x+y is the logarithm of nxm (2); from which we conclude, that the sum of the logarithms of any two numbers, is equal to the logarithm of |