198 THE PRINCIPLES OF SCIENCE. 



much confidence in the accuracy of his result . b It required 

 the skill of James Bernouilli to decide the number of 

 transpositions to be 3312, under the condition that the 

 sense and metre of the verse shall be perfectly preserved. 

 In approaching the consideration of the great Inductive 

 problem, it is very necessary that we should acquire correct 

 notions as to the comparative number of combinations 

 which may exist under different circumstances. The 

 doctrine of combinations is that part of mathematical 

 science which applies numerical calculation to determine 

 the number of combinations under various conditions. 

 It is a part of the science which really lies at the base 

 not only of other sciences, but of other branches of mathe- 

 matical science. The forms of algebraical expressions are 

 determined by the principles of combination, and Hinden- 

 burg recognised this fact in his Combinatorial Analysis. 

 The greatest mathematicians have, during the last three 

 centuries, 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 Bernouilli, 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 Con- 

 jectandi, he has so finely described the importance of 

 the doctrine of combinations, that I need offer no excuses 

 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 



b Wallis, 'Of Combinations,' &c., p. 119. 



