INTRODUCTORY EXPLANATIONS. 9 



quantity of expansion measures the quantity of heat 

 which produces it. 



The exactness of mathematical reasoning depends 

 upon that of our knowledge of the circumstances em- 

 ployed. No theorem about triangles, for instance,, is 

 true of any approach to a triangle such as we make on 

 paper ; but only more and more nearly true, the more 

 nearly we make our lines lengths without breadths, and 

 straight. Similarly, we cannot apply any theory of pro- 

 babilities to the circumstances of life, with any greater 

 degree of exactness than the data will allow. But as in 

 geometry we invent exactness by supposing the utmost 

 limits of our conceptions attainable in practice, so in the 

 present case we begin by reasoning on circumstances de- 

 fined by ourselves, and require adherence to certain 

 axioms, as they are called, meaning propositions of the 

 highest order of evidence. 



AXIOM 1. Let it be granted that the impression of 

 probability is one which admits perceptibly of the 

 gradations of more and less, according to the circum- 

 stances under which an event is to happen. 

 AXIOM 2. Let it be granted that when one out of a 

 certain number of events must happen, and these 

 events are entirely independent of one another, the 

 probability of one or other of a certain number of 

 events happening must be made up of the probabi- 

 lities of the several events happening. For instance, 

 in the lottery of letters, in which there are 26 inde- 

 pendent possible events, the probability of drawing 

 either a, b, c, or d is made up of the probabilities 

 of drawing a, of drawing b, of drawing c, and of 

 drawing d, put together. 



The latter axiom may excite some discussion ; but we 

 must observe that it is the uniform practice of mankind 

 to act upon it, which is a sufficient justification ; for 

 what are we doing but endeavouring to represent that 

 which actually exists ? With regard to the value of each 

 chance, suppose that one of the letters is a prize of 26/., 

 and that the 26 letters have been bought. If I buy 



