24 SCIENTIFIC APPARATUS. 



Although this conception of the absolute continuity of arithme- 

 tical magnitude is of a very abstract character, it has exercised a 

 preponderating influence over scientific thought. That, on the one 

 hand, all natural phenomena take place by a continuous process, 

 and that they are all measurable quantitatively : that, on the other 

 hand, the law of any continuous process can be expressed by 

 an arithmetical formula, and the amount of any quantitative 

 measurement can be stated in arithmetical figures, are propositions 

 which are admitted by every one who understands them, and 

 which, indeed, are in some instances believed with a more 

 unlimited faith than is warranted by the evidence, strong as it is, 

 which can be brought in support of them. Nor is this all; for if 

 there be any one opinion concerning nature at the present time 

 universally accepted by scientific men, it is that the minutest as 

 well as the greatest phenomena are subject to a " reign of law." 

 And if we ask for the strongest reasons which can be given for 

 this belief, they may be summed up by saying that, so far as our 

 measurements are exact, and so far as our arithmetic has been 

 able to cope with the arithmetic of nature, we have uniformly 

 found our observations of continuous phenomena to be in strict 

 accordance with our deductions from the abstract science of 

 continuous number. 



We proceed to offer a few observations with reference to each 

 of the two branches of arithmetic that of discontinuous and that 

 of continuous quantity. The course of these remarks will 

 make it clear, why it is that a science of incalculable impor- 

 tance to other sciences, does not, nevertheless, make any consider- 

 able display of its pretensions in an exhibition of scientific appa- 

 ratus. 



(i.) The simple operations of counting, and of recording num- 

 bers counted, and of comparing them with one another, which 

 constitute the main business of practical arithmetic, have been 

 so facilitated by the two great inventions of the decimal system 

 of notation, and of logarithms, that, in many cases, but little 



