176 THE PRINCIPLES OF SCIENCE. [CHAP 



matics. The forms of algebraical expressions are deter- 

 mined by the principles oT combination, and Hindenburg 

 recognised this fact in his Combinatorial Analysis. The 

 greatest mathematicians have, during the last three cen- 

 turies, given their best powers to the treatment of this 

 subject ; it was the favourite study of Pascal ; it early 

 attracted the attention of Leibnitz, who wrote his curious 

 essay, De Arte Combinatoria, at twenty years of age ; James 

 Bernoulli, one of the very profoundest mathematicians, 

 devoted no small part of his life to the investigation of the 

 subject, as connected with that of Probability ; and in his 

 celebrated work, De Arte Conjectandi, he has so finely 

 described the importance of the doctrine of combinations, 

 that I need offer no excuse for quoting his remarks at full 

 length. 



" It is easy to perceive that the prodigious variety which 

 appears both in the works of nature and in the actions of 

 men, and which constitutes the greatest part of the beauty 

 of the universe, is owing to the multitude of different ways 

 in which its several parts are mixed with, or placed near, 

 each other. But, because the number of causes that concur 

 in producing a given event, or effect, is oftentimes so im- 

 mensely great, and the causes themselves are so different 

 one from another, that it is extremely difficult to reckon up 

 all the different ways in which they may be arranged or 

 combined together, it often happens that men, even of the 

 best understandings and greatest circumspection, are guilty 

 of that fault in reasoning which the writers on logic call 

 the insufficient or imperfect enumeration of parts or cases : 

 insomuch that I will venture to assert, that this is the 

 chief, and almost the only, source of the vast number of 

 erroneous opinions, and those too very often in matters 

 of great importance, which we are apt to form on all the 

 subjects we reflect upon, whether they relate to the know- 

 ledge of nature, or the merits and motives of human 

 actions. 



It must therefore be acknowledged, that that art which 

 affords a cure to this weakness, or defect, of our under- 

 standings, and teaches us so to enumerate all the possible 

 ways in which a given number of things may be mixed 

 and combined together, that we' may be certain that we 

 have not omitted any one arrangement of them that can 



