CHAPTEE XXL 



THEORY OF APPROXIMATION. 



IN order that we may gain a true understanding of the 

 kind, degree, and value of the knowledge which we ac- 

 quire by experimental investigation, it is requisite that 

 we should be fully conscious of its approximate character. 

 We must learn to distinguish between what we can know 

 and cannot know between the questions which admit of 

 solution, and those which only seem to be solved. Many 

 persons may be misled by the expression exact science, 

 and may think that the knowledge acquired by scientific 

 methods admits of our reaching absolutely true laws, 

 exact to the last degree. There is even a prevailing 

 impression that when once mathematical formulae have 

 been successfully applied to a branch of science, this por- 

 tion of knowledge assumes a new nature, and admits of 

 reasoning of a higher character than those sciences which 

 are still unmathematical. 



The very satisfactory degree of accuracy attained in the 

 science of astronomy gives a certain plausibility to erro- 

 neous notions of this kind. Some persons no doubt con- 

 sider it to be proved that planets move in ellipses, in such 

 a manner that all Kepler's laws hold exactly true ; but 

 there is a double error in any such notions. In the first 

 place, Kepler's laws are not proved, if by proof we mean 

 certain demonstration of their exact truth. In the next 

 place, even assuming Kepler's laws to be exactly true in a 



