Math. Also 7 erron. logorythm. [ad. mod.L. logarithm-us (Napier, 1614), f. Gr. λόγ-ος word, proportion, ratio + ἀριθμός number.
Napier does not explain his view of the literal meaning of logarithmus. It is commonly taken to mean ‘ratio-number,’ and as thus interpreted it is not inappropriate, though its fitness is not obvious without explanation. Perhaps, however, Napier may have used λόγος merely in the sense of ‘reckoning,’ ‘calculation’ (cf. LOGISTIC).]
One of a particular class of arithmetical functions, invented by John Napier of Merchiston (died 1617), and tabulated for use as a means of abridging calculation. The essential property of a system of logarithms is that the sum of the logarithms of any two or more numbers is the logarithm of their product. Hence the use of a table of logarithms enables a computer to substitute addition and subtraction for the more laborious operations of multiplication and division, and likewise multiplication and division for involution and evolution.
The word is now understood to refer only to systems in which the logarithm of any number ax is x, a being a constant which is called the base of the system. The logarithms (of sines) tabulated by Napier himself were not logarithms in this restricted sense, but were functions of what are now called the Napierian (also Neperian), hyperbolic, or natural logarithms, the base of which, denoted by the symbole e or e, is 2.71828+. This system is still in use for analytical investigations, but for common purposes the system used is that invented by Napier’s friend Henry Briggs (died 1630), the base of which is 10; the Briggsian or Briggian logarithms are also known as common or decimal logarithms. For binary, Gaussian logarithm, see the adjs. Logistic logarithms (see quot. 1795); also called proportional logarithms.
In mathematical notation ‘the logarithm of’ is expressed by the abbreviation ‘log’ prefixed to numeral figures or algebraical symbols. When necessary, the base of the system is indicated by adding an inferior figure: thus ‘log10 a’ means ‘the logarithm of a to the base 10.’
[1614. Napier (title), Mirifici Logarithmorum Canonis descriptio….]
1615–6. H. Briggs, in Ussher’s Lett. (1686), 36. Napper, Lord of Markinston, hath set my Head and Hands a Work, with his new and admirable Logarithms.
1616. E. Wright, trans. Napier’s Logarithmus, Ded. This new course of Logarithmes doth cleane take away all the difficultye that heretofore hath beene in mathematicall calculations.
1631. H. Briggs, Logarithm. Arithm., i. 1. The Logar, of 1 is 0. Ibid., 2. The Log. of proper fractions is Defective.
1632. B. Jonson, Magn. Lady, I. i. Sir Interest … will tell you instantly, by Logorythmes, The utmost profit of a stock imployed.
1706. W. Jones, Syn. Palmar. Matheseos, 173. Mr. Halley … has … drawn a very curious Method for Constructing Logarithms.
1795. Hutton, Math. Dict., s.v. Logarithms, Logistic Logarithms, are certain Logarithms of sexagesimal numbers or fractions, useful in astronomical calculations.
1827. Scott, Napoleon, VI. 80. Bonaparte said that his favourite work was a book of logarithms.
c. 1865. in Circ. Sci., I. 519/1. This advantage, which the base 10 has over any other, was first seen and applied by Briggs…; the logarithms are, therefore, sometimes called the ‘Briggian Logarithms.’
