INDEX

, in Arithmetic, is the same with what is otherwise called the characteristic or the exponent of a logarithm; being that which shews of how many places the absolute or natural number belonging to the logarithm confifts, and of what nature it is, whether an integer or a fraction; the Index being less by 1 than the number of integer figures in the natural number, and is positive for integer or whole numbers, but negative in fractions, or in the denominator of a frac-| tion; and in decimals, the negative index is 1 more than the number of ciphers in the decimal, after the point, and before the first significant figure; or, still more generally, the Index shews how far the first figure of the natural number is distant from the place of units, either towards the left hand, as in whole numbers, or towards the right, as in decimals; these opposite cases being marked by the correspondent signs + and —, of opposite affections, the sign — being set over the Index, and not before it, because it is this Index only which is understood as negative, and not the decimal part of the logarithm. Thus, in this logarithm 2.4234097, the figures of whose natural number are 2651, the 2 is the Index, and being positive, it shews that the first figure of the number must be two places removed from the units place, or that there will be three places of integers, the number of these places being always 1 more than the Index; so that the natural number will be 265.1. But if the same Index be negative, thus ―2.4234097, it shews that the natural number is a decimal, and that the first signisicant figure of it is in the 2d place from units, or that there is one cipher at the beginning of the decimal,

Number.	Logarithm.
2651	3.4234097
265.1	2.4234097
26.51	1.4234097
2.651	0.4234097
.2651	―1.4234097
.02651	―2.4234097
.002651	―3.4234097

being 1 less than the negative Index; and consequently that the natural number of the logarithm in this case is .02651. Hence, by varying the natural number, with respect to the decimal places in it as in the former of the two columns here annexed, the Index of their logarithm will vary as in the 2d column.

Mr. Townly introduced a peculiar way of noting these Indices, when they become negative, or express decimal figures, which is now much in use, especially in the log. sines and tangents, &c, viz, by taking, instead of the true Index, its arithmetical complement to 10; so that, in this way, the logarithm ―2.4234097 is written 8.4234097.

For the addition and subtraction of Indices, see Logarithm.

Index of a Globe, is a little style fitted on to the north-pole, and turning round with it, pointing out the divisions of the hour-circle.

Index of a Quantity, in Arithmetic and Algebra, otherwise called the exponent, is the number that shews to what power it is understood to be raised: as in 10³, or a³, the figure 3 is the Index or exponent of the power, signifying that the root or quantity, 10 or a, is raised to the 3d power. See this fully treated under Exponent.