xxi r.] QUANTITATIVE INDUCTION. 489 



there indeed any general method of inferring laws from 

 facts it would overturn my statement, but it must be 

 carefully observed that these empirical formulae do not 

 coincide with natural laws. They are only approximations 

 to the results of natural laws founded upon the general 

 principles of approximation. It has already been pointed 

 out that however complicated be the nature of a curve, 

 we may examine so small a portion of it, or we may ex- 

 amine it with such rude means of measurement, that its 

 divergence from an elliptic curve will not be apparent. 

 As a still ruder approximation a portion of a straight line 

 will always serve our purpose ; but if we need higher pre- 

 cision a curve of the third or fourth degree will almost 

 certainly be sufficient. Now empirical formulae really re- 

 present these approximate curves, but they give us no 

 information as to the precise nature of the curve itself to 

 which we are approximating. We do not learn what func- 

 tion the variant is of the variable, but we obtain another 

 function which, within the bounds of observation, gives 

 nearly the same values. 



Discovery of Rational Formulae. 



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 

 will exhibit the reason or exact nature and origin of the 

 law in question. There is no word more frequently used 

 by mathematicians than the word 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 \eirovpry la or reXeafjua. Mathematicians 

 at first used it to mean any power of a quantity, but 

 afterwards generalised it so as to include " any quantity 

 formed in any manner whatsoever from another quantity." l 

 Any quantity, then, which depends upon and varies with 

 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 



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



