120 THE PRINCIPLES OF SCIENCE. 



geometrical process, and obtain to a very close approxi- 

 mation the true ordinates instead of those denoting 

 areas". 



Interpolation and Extrapolation. 



When we have by experiment obtained two or more 

 numerical results, and endeavour, without further resort 

 to experiment, to infer and 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 to a certain 

 extent the same in principle, but differ in practicability. 

 It is a matter of great scientific importance to appre- 

 hend precisely how far we can interpolate or extend 

 experimental results by extrapolation, and on what 

 grounds we proceed. 



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

 than empirical and speculative, we must have not only 

 the experimental results, but the laws which they obey 

 we must in fact go through the complete process of scien- 

 tific investigation. Having discovered the laws of nature 

 applying to the case, and verified them by showing that 

 they agree with the experiments in question, we are then 

 in a fair position to anticipate the results of any 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 cir- 

 cumstances which may more or less affect the result. 

 Even at the best then our interpolations will partake of 

 the want of certainty and precision attaching to all our 

 knowledge of nature. Yet having the supposed laws, our 



n 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. 



