PRINCIPLES OF NUMBER. 173 



certainty begins and ends with numerical determination. 

 Nothing is more certain and accurate than logical truth. 

 The laws of identity and difference are the tests of all 

 that is true and certain throughout the range of thought, 

 and mathematical reasoning is cogent only when it con 

 forms to these conditions, of which logic is the first 

 development. And if it be erroneous to suppose that all 

 certainty is mathematical, it is equally an error to imagine 

 that all which is mathematical is certain. Many processes 

 of mathematical reasoning are of most doubtful validity. 

 There are many points of mathematical doctrine which are 

 and must long remain matter of opinion ; for instance, the 

 best form of the definition and axiom concerning parallel 

 lines, or the true nature of a limit or a ratio of infinitesimal 

 quantities. In the use of symbolic reasoning questions 

 occur at every point on which the best mathematicians 

 may differ, as Bernouilli and Leibnitz differed irreconcile- 

 ably concerning the existence of the logarithms of ne 

 gative quantities a . In fact we no sooner leave the simple 

 logical conditions of number, than we find ourselves in 

 volved in a mazy and mysterious science of symbols. 



Mathematical science enjoys no monopoly, and not even 

 a supremacy in certainty of results. It is the boundless 

 extent and variety of quantitative questions that surprises 

 and delights the mathematical student. When simple 

 logic can give but a bare answer Yes or No, the algebraist 

 raises a score of subtle questions, and brings out a score 

 of curious results. The flower and the fruit, all that is 

 attractive and delightful, fall to the share of the mathe 

 matician, who too often despises the pure but necessary 

 stem from which all has arisen. But in no part of human 

 thought can a reasoner cast himself free from the prior 

 conditions of logical correctness. The mathematician is 



a Montucla, Histoire cles Mathematiques, vol. iii. p. 373. 



