114 THE PRINCIPLES OF SCIENCE. 



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 certain numerical results we are always at 

 perfect 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 presumption in favour of its being the true 

 function, although there is no certainty whatever in the 

 matter. In this way I happened to discover a simple 

 mathematical law which closely agreed with the results 

 of certain experiments on muscular exertion. This law 

 was afterwards shown by Professor Haughton to be the 

 true rational law according to his theory of muscular 

 action h . 



But the chance of succeeding in this manner is usually 

 very small. The number of possible functions is certainly 

 infinite, and even the number of comparatively simple 

 functions is so very large that the probability of falling 

 upon the correct one by mere chance is very slight. Let 

 the reader observe that even when we can thus 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 place, 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 

 process by drawing them out in the form of a curve, as 

 wi]l be presently considered. We can in this way ascer- 

 tain with some probability whether the curve is likely to 



h Haughton, 'Principles of Animal Mechanics,' 1873, pp. 444-450. 

 Nature, 3oth of June, 1870, vol. ii. p. 158. 



