LOGARITHMS. Numbers so contrived and adapteu to other numbers, that the sums and differenas of the former shall correspond to and show the products and 9uotients of the latter, or more properly a eines of numbxs in arithmetical progression answering to ano ther series in geometrical progression, thus, 0, 1, 2, &c. Indices ar Logarithms. , 2, 4 8, 16, 32, 4:c. Geomet. Progression. Or 0, 1, 2, 3, 4, 6, ex. Indices or Logs. 1, 3, 9, 27, 81, 243, ac. Genoa& Prog. Or, 0, I, 2, 3, 4, &c. Indices or Log.
1, 10, 100, 1000, 10,000, &c. Geom. Prog. Where it is evident that the same numbers in arithmetical progression, which are the logarithms or indices, serve equally fir any geometrical series, consequently there may be an endless' variety of systems of logarithms to the same common numbers, by only chang ing the second term, 2, 3, 10, Ac. of the geo metrical series. If any two indices be added taiether„ , their sum wiM be equal to the product of the mu terms in geometrical progression with which those indices correspond, thus, 2 and 3 added together are equal to 5, and the numbers 4 and 8, corresponding with three indices, being multiplied together are equal to 32, which is the number answering to the index 5. So if any index be subtracted from
another, the difference will be the index of that number, which is equal to the quotient of the two terms to which those indices belong, thus the index 6-4=2, then 64 divided by 16, the terms corresponding to these two indices leaves the quotient 4, which answers to the index 2. Logarithms being the exponents of ratios are on that account called indices, thus the logarithm 2 is the exponent or index of the several numbers in the geometrical series over which it stands, as or the square of 2 equal to 4 in the first genes, 32 or square of 3, that is 9, in the second series, and 102 or the square of 10, that is 100, in the third series ; so likewise 3 is the index or exponent for the cube numbers 8, 27, 1000, &c. over which it stands.