522 Transactions. — Chemistry and Physics. 



is, supposing all the error attributable to the values of Y — 

 i.e., where values of X may be taken as accurate for the pur- 

 poses of reasoning, as we always suppose). 



6. It is obvious, however, that systematic, or what we 

 may call instrumental, error must be eliminated or it will 

 infallibly render any reasoning wrong which is based on the 

 results, provided, of course, that the error be sensible in 

 amount. 



7. While the graph forms a very complete representation 

 of the observed facts, and indicates interpolation in the case 

 of datum observations, and in the hands of a person of clear 

 insight may often be the means of reasoning which may not 

 be practicable or even possible by the more formal means of 

 algebraic symbols, yet it is clearly necessary to find, if pos- 

 sible, some formula or function of X which will stand for the 

 graph as well as may be. There are many reasons for this, 

 the chief theoretic one being the enormous developing-power 

 of the algebraic calculus. 



8. In the preceding we have considered the graph as the 

 most natural mode of recording phenomena of variation, but 

 we may have occasionally inferential reasons for believing 

 that the phenomenon should follow some particular function 

 of X more or less completely, and it is necessary to examine 

 the rationale of the functions in various cases. 



(a.) A function may be logically applicable to a pheno- 

 menon. For instance, formulae which state the results of 

 definition, or those which state such inferences as that the 

 angles in a plane triangle are 180°, may be regarded as truly 

 applicable. Even this class may be subject to systematic 

 instrumental error. 



(b.) Functions in which there are strong inferential 

 grounds for the belief that they express the substantial truth. 

 For instance, formulae deduced from Newton's laws of mo- 

 tion may be expected to apply closely to the motion of the 

 major objects of the solar system ; but experiments of an 

 accuracy greater than those upon which such laws were 

 founded may always be apt to demonstrate that the functions 

 are not strictly applicable to any given phenomenon, and that 

 there are systematic residual causes which should be taken 

 into account. 



(c.) Functions which have some inferential foundation, 

 but the substantial applicability of which it is worth while to 

 question and examine. 



(cZ.) Functions whose foundation is largely hypothetical. 

 This class we may term " empirical." 



(c.) Functions which have no foundation except, perhaps, 

 certain notions of continuity in rates of change, and so on. 

 This class we may term " arbitrary." 



