LOGARITHM

From Agepedia

LOGARITHM (from fc"? Greek ^ps, proportion, and a^fyfo, number). * The logarithms of numbers are the exponent*' of the different powers to which a con stant number must be raised, in order tt be equal to those numbers ; the princi pies, therefore, which apply to exponents in general, apply to logarithms." To con stitute a logarithm, it is necessary that tht exponent should refer to a system or se ries. These exponents, therefore, consti tute a series of numbers in arithmeticsproportion, corresponding to as many others in geometrical proportion. Take, foi instance, the series 10l = 10; 102 = 100; 103 = 1000 ; 104 = 10,000 : then we have the logarithm of 10=1 ; logarithm, 100=2 ; logarithm, 1000 = 3 ; logarithm, 10,000 = 4, &c. Perhaps the definition of a logarithm may be more scientifically expressed thus: Logarithmic a mathematical term for a number by which the magnitude of a certain numerical ratio is expressed in reference to a fundamental ratio. The value of a ratio becomes known to us by the comparison of two numbers, and is expressed by a number called the quotient of the ratio ; for instance, 12:4 is expressed by 3, or 18:9 by 2; 3 and 2 being called the quotients of the two proportions, 12:4 and 18:9. If we now imagine a series of proportions, which have all the same value Or quotient, as, for instance, 1 to 3, 3 to 9, 9 to 27, 27 to 81, &c. (in which 9 and 3, 27 and 9, 81 and 27, are in the same ratio as 3 and 1)? and if we at the same time adopt the ratio 3 to 1, as the fundamental ratio (or the unit of these ratios), then 9 to 1 is the double of this ratio, 27 to 1 the triple, 81 to I the quadruple, and so on. The numbers 1, 2, 3, 4, which indicate the value of such ratios, in respect to the fundamental ratio, are called logarithms. If, therefore, in this case, 1 is the logarithm of 3, 2 must be the logarithm of 9, 3 of 27, 4 of 81, &c. If we adopt, however, the ratio of 4:1 as the fundamental one, and hence 1 as the logarithm of 4, then 2 would be the logarithm of 16, 3 of 64, &c. The, logarithms of the numbers which lie between, must be fractions, and are to be calculated and put in a table. A table of logarithms, made according to an assumed basis or fundamental ratio, of all numbers to a certain limit, is called a logarithmic system. The most common, at present, is that of Briggs, in which the fundamental basis is 10 to 1; hence I is the logarithm of 10, 2 of 100, 3 of 1000, 4 of 10*000, &c. It is evident that all logarithms of numbers between 1 and 10, must be mor" man U, yet less tnan 1, 1. e. a traction; thus the logarithm of 6 is 0.7781513. In the same way, the logarithms of the numbers between 10 and 100 must be more than 1, but less than 2, &c.; thus the logarithm of 95 is = 1.9777236. All logarithms of the numbers between 0, 10, 100, 1000, &c.., are arranged in tables, the use of which, particularly in calculations with large numbers, is very great. The process is simple and easy. If there are numbers to be multiplied, we only have to add the logarithms; if the numbers are to be divided, the logarithms are merely to be subtracted; if numbers are to be raised to powers, their logarithms are multiplied; if roots are to be extracted, the logarithms are merely to be divided by the exponent of the root. In a table of logarithms, the integer figure is called the index or characteristic. The decimals are called, by the Germans and Italians, the mantissa. In general, the logarithms of the system in which 1 indicates 10, are called common or Briggs's logarithms. The properties of logarithms, and some of their uses, were taken notice of by Stiefel or Stifelius, a German clergyman, who wrote as early as 1530; but the use of them in trigonometry was discovered by John Napier, a Scotch baron, and made known by him in a work published at Edinburgh, in 1614. Logarithmic tables are of great value, not only to mathematicians, but to all who have to make calculations with large numbers. The best logarithmical tables are those of Vega (q. v.) and of Callet. The former are calculated with ten decimals.* Logarithms are of incalculable importance in trigonometry and in astronomy. Vega's edition of Vlacq's tables contains a trigonometrical table of the common logarithms of the radius or log. sin. tot. = 10.0000000, which gives the logarithms of sines, arcs, cosines, tangents, and cotangents for each second of the two first and two last degrees, and for each ten seconds of the rest of the quadrant. Under Napier's direction, B. Ursinius first gave the logarithm of the sines of the angles from 10 to 10 seconds, the logarithm of the tangents, which are the differences of the logarithms of each sine and cosine, together with the natural sine for a radius of 100,000,000 parts. Kepler turned his attention particularly upon the invention of Napier, and gave a new theory and* Logarithmic and Trigonometric Tables have lately been published by F. R. Hassler (NewYork, 1830); and Mathematical Tables, comprising Logarithms of Numbers, &c. (Boston, 1830). The English Tables are too numerous to mention. new tables. xSriggs was also conspicuous in the construction of tables. Mercator shows a new way for calculating the logarithms easily and accurately. Newton, Leibnitz, Halley, Euler,, L'Huillier, and others, perfected the system much, by applying to it the binomial theorem and differential calculus. The names of Vlacq, Sherwin, Gardiner, Hutton, Taylor, Callet, and others, deserve to be honorably mentioned. The edition of Vlacq, within a few years, by Vega, is particularly valuable. During the French revolution, wThen all measures were founded on the decimal division, new tables of the trigonometrical lines and their logarithms became necessary. The director of the bureau du catastre, M. Prony, was ordered, by government, to have tables calculated, which were to be not only extremely accurate, but to exceed all other tables in magnitude. This colossal work, for which the first mathematicians supplied the formulas and the methods for using the differences in the calculations, was executed, but the depreciation of the paper money prevented its publication. The tables would have occupied 1200 folio pages. (Notices sur les grandes Tables Logarithmiques et Trigonometriques, calcules au Bureau du Catastre a Paris, an IX.)