LOGARITHMS.

LOGARITHMS are a fet of artificial numbers, and

may be confidered as the indices of a series of Geometrical proportionals, and are fo related to the natural numbers, that the addition of Logarithms is equivalent to the multiplication of the correfponding numbers; also, the subtraction of logarithms is the fame as the divifion of the corresponding numbers; their difference being the logarithm of the quotient.

Here it may be observed, that common numbers are a series whofe differences are equal; such as, 2, 4, 6, 8, 10, &c. where the common difference is 2, and are called a feries in arithmefical progreffion.

Also a series of numbers whofe ratios are equal, are called a feries in Geometrical progreffion; fuch as, 2, 4, 8, 16, 32, 64, &c. the common ratio being 2.

The following table will, in fome meafure, illuftrate these general obfervations.

Note, Column A is a feries in arithmetical progreffion; 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 be 729.

The terms in column A, correfponding 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

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numbers in column A, anfwering to 78125, the dividend, and to 125 the divifor (both in column E); fubtract the leffer 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 divifion might be facilitated, provided the fame numbers occur in the table; but as this feldom happens, the use of fuch a table will be confined to a few inftances. In order, therefore, to extend its utility, we fhall fhew a method by which this inconveniency is removed.

There was a method formerly in ufe in making logarithms: The first inventors chofe a fet of numbers in arithmetical progreffion, that fhould answer to a fet of geometrical ones; (this is entirely arbitrary;) and they chose the decuple geometrical progreffion as the moft convenient, correfponding to the arithmetical feries 1, 2, 3, 4, 5, 6, 7, &c., as the fimpleft, whose common difference is, 1. as follows:

of

Hence it appears, that the logarithm of 1 is o, of 10 is 1, 100, is 2, &c. but several numbers may be interpofed between each of thefe; for, between 1 and 10 are 2, 3, 4, 5, 6, 7, 8, 9; to them alfo might indices be adapted, fuited to each term between 1 and ro, confidered in geometrical progreflion. Likewife indices may be found to each teim interpofed between any two terms whatever, in geometrical progression.

It is plain, that the indices to all the numbers under 10 is lefs than 1; that is, they are fo many decimal parts; likewife, that the indices of numbers between 10 and 100 are 1 of an integer, and fo many decimal parts, and fo on of numbers greater than 100.

The integral part is commonly called the index, and the decimal part the logarithm.

C 2

But

But fince the above method is fo intolerably laborious, the more learned mathematicians have thought of a more compendious one, by the menfuration of hyperbolic spaces, contained between the portions of an afymptote, and right lines perpendicular to it and the curve of an hyperbola; but fuch computations depend on principles that require the higher parts of Geometry, and cannot, therefore, according to our plan, be introduced here."

We shall fubjoin the process for obtaining the logarithm of 9, as derived from progreffion.

Geo.