KII.] QUANTITATIVE INDUCTION. 46 



by simply laying off the quantities of heat at the mean tem- 

 peratures, namely 2^, and 7^, and so on. Lord Eayleigh 

 has shown that if we have drawn such an incorrect curve, 

 we can with little trouble correct it by a simple geo- 

 metrical process, and obtain to a close approximation the 

 tone ordinates instead of those denoting areas. 1 



Interpolation and Extrapolation. 



When we have by experiment obtained two or more 

 numerical results, and endeavour, without further experi- 

 ment, to calculate intermediate results, we are said to 

 interpolate. If we wish to assign by reasoning results 

 lying beyond the limits of experiment, we may be said, 

 using an expression of Sir George Airy, to extrapolate. 

 These two operations are the same in principle, but differ 

 in practicability. It is a matter of great scientific im- 

 portance to apprehend precisely how far we can practise 

 interpolation or extrapolation, and on what grounds we 

 proceed. 



In the first place, if the interpolation is to be more than 

 empirical, we must have not only the experimental results, 

 but the laws which they obey we must in fact go through 

 the complete process of scientific investigation. Having 

 discovered the laws of nature applying to the case, and 

 verified them by showing that they agree with the experi- 

 ments in question, we are then in a position to anticipate 

 the results of similar experiments. Our knowledge even 

 now is not certain, because we cannot completely prove 

 the truth of any assumed law, and we cannot possibly 

 exhaust- all the circumstances which may affect the result. 

 At the best then our interpolations will partake of the 

 want of certainty and precision attaching to all our know- 

 ledge of nature. Yet, having the supposed laws, our results 

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

 such a complete procedure is more than we commonly 

 mean by interpolation, which usually denotes some method 

 of estimating in a merely approximate manner the results 



1 J. W. Strutt, On a correction sometimes required in curves pro- 

 fessing to represent the connexion between two physical magnitudes. 

 Philosophical Magazine, 4th Series, vol. xlii. p. 441. 



