146 HISTORY OF SCIENCE. 



perfect from the hands of its author, whereas most other inventions and 

 discoveries have originally presented themselves in a more or less crude 

 form, and have advanced to perfect and practical shapes by the labour 

 of those who successively improved upon the work of the originators. 

 It has been truly remarked that the subsequent advance of science has 

 found no better method of facilitating computations than logarithms, 

 and has only extended the field of their application. 



It would, of course, be impossible to explain fully to the non-mathe- 

 matical reader all the properties and uses of logarithms, but perhaps 

 enough can be stated in a few sentences to give him some idea of the 

 nature of the invention and of some of its uses, although we necessarily 

 adopt only the simplest form which will illustrate the principle. Let 

 him write down, first, a series of numbers which form an arithmetical 

 progression that is, each term is formed by the addition of a constant, 

 say i, to the preceding term; and over the several terms of the series 

 so formed, let him write the terms of a geometrical series that is, one 

 in which each term is formed by multiplying the preceding term by a 

 constant number. Supposing, for simplicity, that the multiplier is 2, 

 he will then have the two progressions thus : 



i 2 4 8 16 32 64 128 256 

 012345678 



These series should be carried out by the reader to a greater number 

 of terms for the sake of illustration. It will be found that many arith- 

 metical problems regarding the numbers in the upper series can be 

 solved with the greatest ease by very simple operations with the cor- 

 responding terms of the lower series. Examples will make this clear. 

 Multiplication and division To find the product of 32 and 4: 5+2 

 = 7, and above 7 is 128, the product required. To divide 256 by 32 : 

 8 5=3, and over 3 is 8, the quotient required. Involution and evolu- 

 tion To find the square of 16 : 4x2=8, and over 8 is the required 

 square. To extract the square root of 64 : 6-5-2 = 3, and over 3 is 8, the 

 root required. To find the cube of 256 : 8 x 3 = 24, and over 24 in the 

 lower series would be found the required cube, if the progressions are 

 carried out. What is the fourth root of 256? 8-7-4=2, and over 2 is 

 4, the required fourth root. If the reader attempts a few problems of 

 this kind with an extended series, he will soon be convinced that all 

 calculations into which the numbers of the upper series enter will be 

 greatly facilitated by thus using the lower series ; for multiplication then 

 becomes addition ; division, subtraction ; involution, a single multipli- 

 cation ; evolution, a simple division. He will further see that if the 

 gaps in the upper series were filled up, so that he had 3 between 2 

 and 4; 5, 6, 7 between 4 and 8, etc., and the arithmetical series were also 

 filled in with its corresponding numbers ; it would then be possible to 



