LOGARITHMS, a series of numbers having a certain relation to the series of natural numbers, by means of which. many arithmetical operations are made comparatively easy. The nature of the relation will be understood by considering two simple series such as the following, one proceeding from unity in geometrical progression, the other from 0 in arithmetical progression: Geometrical series-1, 2, 4, 8, 16, 32, 64, 128, 256, 512, etc.
Arithmetical series-0, 1, 2, 3, 4, 5, 6, 7, 8, 9, etc.
Here the ratio of the geometrical series is 2, and any term in the arithmetical series expresses how often 2 has been multiplied into 1 to produce the corresponding terra of the geometrical series; thus, in proceeding from 1 to 32, there have been 5 steps or multiplications by the ratio 2; in other words, the ratio of 32 to 1 is compounded five thnes of the ratio of 2 to 1. It was this conception of the relation that led to giving the name of logarithms to the arithrnetical series, the word logarithm (Gr. logo& arith mos) meaning " the number of the ratios." As to the use that may be made of suclt series, it will be observed that the sum of any two logarithms (as we shall now call the lower series) is the logarithm of their product; c.g., 9 (=- 3 -I- 6) is the logarithm of 512 (= 8 X 64). Similarly, the difference of any two logarithms is the logarithm of the quo tient of the numbers; a, multiple of any logarithm is the logarithm of the corresponding number raised to the power of the multiple; e.g., 8 (= 4 X 2) is the logarithm of 256 (= 162); and a submultiple of a logarithm is the logarithm of the corresponding root of its number. In this way, with complete tables of numbers and their corresponding loga rithms, addition is made to take the place of multiplication, subtraction of division, multiplication of involution, and division of evolution.
In order to make the series above given of practical use, it would be necessary to complete them by interpolating a set of means between the several terms, as will be explained below. We have chosen 2 as the fundamental ratio or base, as being most convenient for illustration; but any other number (integral or fractional) might be taken; and every different base, or radix, gives a different system of logarithms. The system now in use has 10 for its base; in other words, 10 is the number whose logarithm is 1.
The idea of making use of series in this way would seem to have been known to Archimedes and Euclid, without, however, resulting in any practical scheme; but by the end of the 16th c., trigonometrical operations had become so complicated that thh wits of several mathematicians were at work to devise means of shortening them. The
real invention of logarithms is now universally ascribed to John Napier (q.v.), baron of Merchistoun, who in 1614 printed his Canon Hirabilis Logarithmorum. His tables only give logarithms of sines, cosines, and the other functions of angles; they also labor under the three defects of being sometimes + and sometimes -, of decreasing as the corre sponding natural numbers increase, and of having for their radix (the number of which 1 the logarithm is 1) the number which is the sum of 1+1 + etc.
1.2 1.2.3These defects were, however, soon remedied: John Speidell, in 1619, amended the tables in such a manner that the logarithms became all positive, and increased along with their corresponding natural numbers. He also, in the sixth edition of his work (1624), constructed a table of Napier's logarithtns for the integer numbers, 1, 2, 3, etc., up to 1000. with th,,,i. differences and arithmetical complements, besides other improvements. Speidell's table:, are now known as hyperbolic logarithms. But the greatest improvement was made in 1615 by prof. Henry Briggs (q.v.), of London, who substituted for Napier's inconvenient " radix' the number 10, and succeeded before his death in calculating the logarithms of 30,000 natural numbers to the new radix. Briggs's exertions were ably seconded; and before 1628 the logarithms of all the natural numbers up to 100,000 had been computed. Computers have since chiefly occupied themselves rather in repeatedly revising the tables already calculated than in extending them.
Construction of Tables.-The following is the simplest method of constructing a table of logarithms on Briggs's system, The log.. of 10 = 1. ; the log. of 100 (which is twice compounded of 10) = 2. ; the log. of 1000 = 3., etc.; and the logarithms of all powers of 10 can be found in the same manner. The intermediate logarithms arc found by con tinually computing geometric means between two numbers, one greater and the other less than the number required. Thus, to find the log,. of 5, take the geometric mean between 1 and 10, or 3.162..., the correTonding, arithmetic mean (the log. of 1 being 0, and that of 10 being 1.) being -5; the geometric mean between 8.162... and 10, or 5.623_, corresponds to the arithmetic mean between .5 and 1., or -75; the geometric seven pliices (180), issued in a more Cow:0110A form, with improvements, by MesarS. W, & R. Chambers; the most accurate of all, however, are supposed to be those which Mr. Babbage produced with the aid of his ingenious calculating-machine.