MATHEMATICAL SCIENCES SEVENTEENTH CENT. 147 



perform arithmetical operations on all numbers with the greatest ease. 

 Nor will it be difficult for him to see that the proper figures for the 

 lower series could be obtained with any required degree of approxima- 

 tion by the rules of ordinary arithmetic. Suppose, for instance, the 

 ratio or multiplier of the upper series were not 2, but \/2, then the two 

 series would become 



i ; Y/2 : 2 : 2^/2 : 4 : 4\/2 : 8 : 8\/2 : 16 : etc. 

 0:0-5:1: 1-5 : 2 : 2-5 13: 3-5 : 4 : etc. 



The term 2 \/2 does not equal 3 ; but it is plain that by continuing 

 sufficiently far the process of finding mean proportionals between the 

 terms of the upper series, and writing under them the corresponding 

 arithmetical means of the terms in the lower series, we can obtain a 

 number as near 3 as we choose, and its corresponding number in the 

 lower series. From this last the arithmetical terms for the numbers 

 6, 9, 12, 1 8, etc., could at once be found, and so on with the rest. 

 The lower series of numbers thus found would constitute a system of 

 logarithms, so called from Xo7os, ratio, and apifyws, number. * Different 

 systems of logarithms are formed according to the law of the progres- 

 sions which may be selected. Thus in the common system of loga- 

 rithms, for good reasons the series run thus, if we write down only the 

 logarithms which are whole numbers : 



Numbers: i 10 100 1,000 10,000. 



Logarithms: 012 34. 



That is, the logarithm of 10 is one, that of 100 is two, etc. ; and it 

 will be observed that the logarithm is the index of that power of 10 

 which is equal to the numbers. Intermediate numbers have loga- 

 rithms which are mixed numbers. For example, the logarithm of 40 

 is i '602, and as such fractional numbers now enter also into the notion 

 of indices in algebra, we can say 4o=io 1 " 602 . The theory of indices 

 was, however, not understood in Napier's time, and therefore greater 

 were Napier's originality and merit in having conceived the idea, and 

 in having discovered the means of finding these numbers. 



The logarithms which first presented themselves to Napier's mind 

 were those we now term hyperbolic logarithms, and these were adapted 

 to trigonometrical calculations. It often occurred to him that the 

 system in which the logarithm of 10 is i, that of 100 2, etc., would be 

 more convenient, and just before his death, which happened in 1614, 

 he explained to Henry Briggs, the Gresham Professor of Mathematics, 

 the nature of his new plan. In 1618 Briggs published a table of the 

 logarithms of all the natural numbers up to 1,000, and in 1624 the 

 logarithms of all numbers from i to 20,000, and from 90,000 to 

 100,000, calculated to fourteen decimal places. Briggs was one of 



10 2 



