_Calculation of Logarithms._--The name logarithm is derived from the words [Greek: logon arithmos], the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or _ratiunculae_, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or _ratiuncula_, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a^{301,030} to 1, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a^{301,030}, 3 will be nearly equal to a^{477,121}, and so on. Napier's original work, the _Descriptio Canonis_ of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fluxions. An account of the processes by which Napier constructed his table was given in the _Constructio Canonis_ of 1619. These methods apply, however, specially to Napier's own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the _Arithmetica Logarithmica_, although some of the latter are the same in principle as the processes described in an appendix to the _Constructio_. The processes used by Briggs are explained by him in the preface to the _Arithmetica Logarithmica_ (1624). His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows. He first formed the table of numbers and their logarithms:-- Numbers. Logarithms. 10 1 3.162277... 0.5 1.778279... 0.25 1.333521... 0.125 1.154781... 0.0625