INTRODUCTORY EXPLANATIONS. lS 



force directed towards the central body, which, for dif- 

 ferent distances, changes inversely as the square of the 

 distances." This is well known to be a fundamental 

 part of the system of astronomy which has enabled one 

 century to do more towards correct prediction of the 

 state of the heavens than the twenty centuries which 

 preceded it; and yet the apparatus of mathematics 

 which is required to establish this result, which is of the 

 nature of a principle, is enormous. But in the present 

 subject we shall establish all our principles without the 

 aid of any more mathematics than is contained in arith- 

 metic ; and when we draw upon the science, it shall be 

 for nothing but abbreviation of long processes. The 

 principle upon which mathematical abbreviation fre- 

 quently proceeds is this : that where the calculation of 

 a few results materially aids the production of a great 

 many more, it is advisable to calculate a multitude of 

 results, to arrange them in convenient tables of reference, 

 and to publish them ; so that by means of one person 

 taking a little more trouble than would otherwise fall to 

 his share, all others may be saved labour altogether. 

 Mathematical tables are frequently nothing but the re- 

 sult of labour performed once for all ; but it also some- 

 times happens, that the principle on which the labour 

 is performed can be exemplifie.d by a familiar case of it. 

 "We shall take that of logarithms as an instance. 



Every table of logarithms is an extensive table of com- 

 pound interest. Not to embarrass ourselves with* frac- 

 tions, let us take a table of cent, per cent, compound 

 interest. We have then the following amounts of II. in 

 1, 2, 3, &c. years: 



