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LOGARITHMS are a set of artificial numbers, and
a may be considered as the indices of a series of Geometrical proportionals, and are so related to the natural numbers, that the addition of Logarithms is equivalent to the multiplication of the corresponding numbers ; also, the subtraction of logarithms is the same as the division of the corresponding numbers; their difference being the logarithm of the quotient.
Here it may be observed, that common numbers are a series whose differences are equal; such as, 2, 4, 6, 8, 10, &c. where the common difference is 2, and are called a series in arithmefical progression.
Also a series of numbers whose ratios are equal, are called a series in Geometrical progression ; such as, 2, 4, 8, 16, 32, 64, &c. the common ratio being 2.
The following table will, in fome meafure, illustrate these general observations.
Note, Column A is a series in arithmetical progression; the other columns are in Geometrical progression, the common ratios being 2, 3, 4, 5, TO.
Now, let it be required to multiply 9 by 81, the product will
The terms in column A, corresponding to the factors, are 2 and 4; and which being added together, will give 6; over against 6 in column A, is 729, the product in column C.
Again-Let it be required to divide 78125 by 125, the quot will be 625. By the table it may be performed thus : Find the
numbers in column A, answering to 78125, the dividend, and to 125 the divisor (both in column E); subtract the lesser from the greater, and over-against their difference in column A is 625 the quotient in column E.
By extending the foregoing table, many operations, both in multiplication and division might be facilitated, provided the same numbers occur in the table; but as this seldom happens, the use of such a table will be confined to a few instances. In order, therefore, to extend its uility, we shall shew a method by which this inconveniency is removed.
There was a method formerly in use in making logarithms : The first inventors chose a set of numbers in arithmetical
progression, that should answer to a set of geometrical ones; (this is entirely arbitrary ;) and they chose the decuple geometrical progression as the most convenient, corresponding to the arithmetical series 1, 2, 3, 4, 5, 6, 7, &c., as the simplest, whose common difference is, 1. as follows:
Arith. progression, or log. 0, 1, 2, 3, 4.
10, 100, iooo, 10000.
Hence it appears, that the logarithm of 1 is o, of 10 is 1, of 100, is 2, &c.: but several numbers may be interposed between each of these; for, between 1 and 10 are 2, 3, 4, 5, 6, 7, 8, 9; to them also might indices be adapted, suited to cach term between i and ro, considered in geometrical progreslion. Likewise indices may be found to each teim interposed between any two terms whatever, in geometrical progresion.
It is plain, that the indices to all the numbers under 10 is less than 1; that is, they are so many decimal parts; likewise, that the indices of numbers between 10 and 100 are 1 of an integer, and so many decimal parts, and so on of numbers greater than 100.
The integral part is commonly called the index, and the decimal part the logarithm.
But since the above method is so intolerably laborious, the more learned mathematicians have thought of a more compendious one, by the mensuration of hyperbolic spaces, contained between the portions of an asymptote, and right lines perpendicular to it and the curve of an hyperbola ; but such computations depend on principles that require the higher parts of Geometry, and cannot, therefore, according to our plan, be introduced here.
We shall subjoin the process for obtaining the logarithm of 9, as derived from progression.