LOGARITHMS are series of artificial numbers, so arranged with reference to a set of natural numbers that the addition of the logarithms shall correspond with the multiplication of the natural numbers belonging to them ; and sub traction of logarithms answers for division ; while involution, or the raising of powers, is performed by the multiplication of logarithms ; and evolution, or the extraction of roots, by the division of logarithms.
To illustrate this, let us take— For Natural Numbers the Geometrical Series .. f And for their Logarithms t the Arithmetical Series From this it appears that the log. of 1 is 0, that of 10 is 1, of 100 is 2, &c.; that the log. of any number below 10 is a fraction, above 10 and under 100 is 1; with a fraction, and between 1000 and 100, is 2 with a fraction, and so on. Hence it is evident that the portion of a log. which constitutes the whole number, and is denominated the INDEX, is always one less than the numbers of figures for which it is the log. This general rule is so easy of application, that the Indexes of Logarithms are never printed in the tables, but left to be supplied by the operator.
The rule for determining the Index descends as well as ascends, and applies with equal facility to numbers below and above unity ; but when applied to numbers below unity, it must be distinguished by a negative sign thus— To furnish the means of illustrating this important subject by a few examples, and to give the reader an opportunity of working cases by logarithms when the numbers to be operated upon are not very large, we subjoin It is not our province, in this 'brief article, to explain the use of the larger Logarithmic Tables, as whoeversuch have of course their author's own explanations, and therefore the illustrative examples are selected to suit the table here given.
Here the first factor, 388, being a whole number, consisting of three figures, has for its index 2; and the second, 22•5, having but two figures, without the decimal part, has for its index I. To these are subjoined the decimal portions of the logarithms taken from the Table, and the sum of the two being found in the Table opposite to 828, which would be the answer were the index 2; but as the index is 3, the answer must be made to consist of four figures, which is done by supplying to the right of the figures a cipher, making the answer, as above, 8280.
inw;:te;:;;i7 fine grain, capable of being polished, and so durable, as to be scarcely susceptible of decay. Its predominant colour is red, tinged with orange, yellow, and black. It yields its colour both to spirituous and watery menstrua. Alcohol extracts it more readily and copiously than water. The colour of its dye is a fine red, inclined a little to violet or purple, which left to itself, becomes yellowish, pur ple, and at length black. Acids turn it yellow, alkalies deepen the colour, and give it a purple or violet hue. A blue colour is obtained from logwood, by verdigris with it in the dye bath. The great consumption of logwood is for s, to which it gives a lustre and velvety cast ; it is also extensively used as a red, purple, or black dye to beech, and various white woods.