QUANTITATIVE INDUCTION. 121 



results will be as sure and accurate as any we can attain to. 

 But such a complete procedure is more than we generally 

 mean by interpolation, which generally denotes the em- 

 ployment of some general method of estimating in a 

 merely approximate and probable manner the results 

 which might have been expected independently of any 

 complete theoretical investigation. 



Regarded in this light, interpolation is in reality an 

 indeterminate problem. From given values of a function 

 it is impossible to determine that function ; for we can 

 always invent an infinite number of functions which would 

 give those values if we are not restricted by any other 

 conditions, just as through a given series of points we can 

 always draw an infinite number of curves, if we may di- 

 verge between or beyond the points into bends and cusps 

 as we think fit . In any process of interpolation we must 

 in fact be guided more or less by a priori considerations ; 

 we must know, for instance, whether or not periodical 

 fluctuations are to be expected, and we must be guided 

 accordingly in the choice of mathematical formulae. Sup- 

 posing, for the present, that the phenomenon is non- 

 periodic, we next proceed to assume that the function 

 can be expressed in a limited series of the powers of the 

 variable. The number of powers which can be included 

 depends upon the number of experimental results avail- 

 able, and must be at least one less than this number. By 

 processes of calculation, which have been already alluded to 

 in the section on empirical formulas, we can then calculate 

 the coefficients of the powers, and obtain an empirical 

 formula which will give the required intermediate results. 

 In reality, then, we return to the methods treated under 

 the head of approximation and empirical formulas; and 

 interpolation, as commonly understood, consists in assum- 



Herschel, ' Appendix to Translation of Lacroix' Differential Calculus,' 

 P.- 55i. 



