490 THE PRINCIPLES OF SCIENCE. [CHAP. 



they are the values. Simple inspection of the numbers 

 cannot as a general rule disclose the function. In an 

 earlier chapter (p. 124) 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 per- 

 haps three methods, more or less distinct, by which we 

 may hope to obtain an answer : 



(1) By purely haphazard trial. 



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

 the quantities, and trying by preference functions which 

 give a similar form of variation. 



(3) By deducing from previous knowledge the form of 

 the function which is most likely to suit. 



Having numerical results we are always at liberty 

 to invent any kind of mathematical formula we like, and 

 then try whether, by the suitable selection of values for 

 the unknown constant quantities, we can make it give the 

 required results. If ever we fall upon a formula which 

 does so, to a fair degree of approximation, there is a pre- 

 sumption in favour of its being the true function, although 

 there is no certainty whatever in the matter. In this way 

 I discovered a simple mathematical law which closely 

 agreed with the results of my experiments on muscular 

 exertion. This law was afterwards shown by Professor 

 Haughtoii to be the true rational law according to his 

 theory of muscular action. 1 



But the chance of succeeding in this manner is small. 

 The number of possible functions is infinite, and even the 

 number of comparatively simple functions is so large 

 that the probability of falling upon the correct one by 

 mere chance is very slight. Even when we obtain the 

 law it is by a deductive process, not by showing that the 

 numbers give the law, but that the law gives the numbers. 



In the second way, we may, by a survey of the 

 numbers, gain a general notion of the kind of law they 

 are likely to obey, and we may be much assisted in this 



1 Haughton, Principles of Animal Mechanics, 1873, pp. /Hi 450. 

 Jevons, Nature, 3oth of Juue, 1870, vol. ii. p. 158. See also the 

 experiments of Professor Nipher, of Washington University, St. 

 Louis, in American Journal of Science, vol. ix. p. 130, vol. x. p. I ; 

 Nature, vol. xi. pp. 256,276 



