that each of its terms is the logarithm or index of each correfponding term of the geometrical feries as formerly.

In this manner a table of logarithms is constructed:

14. A geometrical mean proportional is found between two numbers [17. 6.] by extracting the square root of the product of the two numbers propofed. Thus, a geometrical mean proportional between 4 and 8 is the nearest fquare root of 32.

An arithmetical mean proportional between two numbers is [1. of this] half the fum of the two propofed numbers: fo the half of 5 is an arithmetical mean proportional between 2 and 3.

15. The imperfections of a table of logarithms are, that an exact mean propors tional cannot always be found between each two of the terms of the feries, and that the difference between each two terms of the geometrical feries cannot be exactly unit. The firft of thefe imperfections is remedied by dividing each term into a great number of decimal places; and the lait, by interting fo many terms, as that their difference may not much exceed unit.

16. A geometrical 1, 10, 100, 1000, &c. feries beginning at u- o, I, 2, 3, &c. nit, and increasing in a tenfold proportion; and an arithmetical feries beginning at nothing, and increafing by unit, as in the margin, are found moft convenient for conftructing a table of logarithms.

17. Hence the logarithm of 1 is o, the logarithm of 10 is 1, that of 100 is 2, and that of 1000 is 3, &c.; therefore the logarithm of each intermediate term between I and 10 is a proper fraction; the logarithm of each term between 10 and 100 is unit with a proper fraction, &c.

18. It is manifeft that the logarithms of each of the intermediate terms between I and 10, 10 and oo, &c. of the geometrical feries, cannot [art. 14.] be found exactly; fuch as the logarithm of 2, of 3, of 4, &c.; also that of 11, of 12, of 13, &c.; but to find them to fuch a degree of exactness as anfwers any ule, each term is to be divided into many decimal parts, as follows.

1.0000000, 10.0000000, 100.0000000, &c. 0.00000000, 1.00000000, 2.00000000, &c.

19. In order to find the logarithm of each whole number between 1 and 10,

.

viz. the logarithm of 2, of 3, &c. as many geometrical mean terms are to be found between 1 and 10, as there are whole numbers between them; and these mean terms must be fuch, that the difference between any one of them, and the next preceding or following, is very near unit. Arithmetical mean terms are to be found between 0 and 1, each of them anfwering to each of the geometrical mean terms. In like manner, geometrical mean terms are to be found between 10 and 100, and as many arithmetical mean terms between 1 and 2, &c.

20. To find the logarithm of 9.

Between 1 and 10, find a geometrical mean proportional; that is, [art. 14.] the nearest square root of 10.0000000, viz. 3.1622777.

Between 0.0ooooooo, which is the logarithm of 1.0000000, and 1.00000000, which is the logarithm of 10.0000000, find an arithmetical mean proportional; that is, [art. 14.] the half of 1.00000000, viz. 0.50000000; this laft number is the logarithm of the geometrical mean 3.1622777. But because 3.1622777 is much less than 9.0000000, another geometrical mean must

must be found between 3.1622777 and 10.0000000, viz. 5.6234132; and then an arithmetical mean between the former arithmetical mean 0.50000000 and 1.00000000, viz. 0.75000000. This last number is the logarithm of 5.6234132, but is ftill far from the logarithm of 9.0000000; because 5.6234132 is much less than 9.0000000; and therefore another geometrical mean must be found between 5.6234132 and 10.0000000 with its correfponding logarithm, and fo on, till a geometrical mean is found, whose difference from 9.0000000 may be neglected. An arithmetical mean is to be found as often; and the arithmetical mean, correfponding to the geometrical mean that comes nearest to 9.0000000, is the logarithm of 9.

21. It will be found, on trial, that the fifth geometrical mean, found as above, is 9.3057204, which is which is greater than 9.0000000; therefore the next geometrical mean must be found between 9.3057204 and the next preceding geometrical mean 8.6596432. The geometrical mean between these laft is 8.9768713, which is lefs than 9.0000000. The next following

is to be found between 9.3057204 and 8.9768713, which will be greater than 9.0000000; and the next again is to be found between this laft one and the next preceding; and fo on.

22. No fewer than twenty-fix geometrical mean terms, and as many arithmetical, are neceffary, in order to find the logarithm of 9, by this method, tolerably exact, aş may be seen by the following table.

The