QUANTITATIVE INDUCTION. 113 



the variant is of the variable, but we obtain another func- 

 tion which, within the bounds of our observation, gives 

 nearly the same series of values. 



Discovery of Rational Formula. 



Let us now proceed to consider the modes in which 

 from numerical results we can establish the actual relation 

 between the quantity of the cause and that of the effect. 

 "What we want is a rational formula or function, which 

 may exhibit the reason or exact character and origin of 

 the law in question. There is no word more frequently 

 used by mathematicians than the vrord function, and yet 

 it is difficult to define its meaning with perfect accuracy. 

 Originally it meant performance or execution, being equi- 

 valent to the Greek Xeirovpyla or TeXeoyia. Mathematicians 

 at first used it to mean, any pov:er of a quantity, but 

 afterwards generalized it so as to include ' any quantit y 

 formed in any manner whatsoever from another quantity?/ 

 Any quantity, then, which depends upon and varies witli 

 another quantity may be called a function of it, and 

 either may be considered a function of the other. 



Given the quantities, we want the function of which 

 they are the values. It may first of all be pointed out 

 that simple inspection of the numbers cannot as a general 

 rule disclose the function. In an earlier part of this work 

 (vol. i. p. 142) I put before the reader certain numbers, 

 and requested him to point out the law which they obey, 

 and the same question will have to be asked in every 

 case of quantitative induction. There are perhaps three 

 methods, more or less distinct, by which we may hope to 

 obtain an answer : 



1 i ) By purely haphazard trial. 



(2) By noting the general character of the variation of 



g Lagrange, ' Lemons sur le Calcul des Fonctions,' 1806, p. 4. 

 VOL. II. I 



