Mr. W. H. Walenn on Negative and Fractional Unitates. 37 



Association Meeting of 1869 at Exeter, Professor J. J. Sylvester 

 laid much stress upon the employment of inductive philosophy 

 in mathematics. He said that he was aware that many who had 

 not gone deeply into the principles of mathematical science be- 

 lieved that inductive philosophy, or the method of evolving new 

 truths by induction, was reserved for the experimental sciences, 

 and that the methods of investigation in mathematical science 

 might all be classified as deductive. He went on to say that 

 this opinion is not a correct one, and that many valuable results 

 are obtained in mathematical science by induction, or reasoning 

 from particulars to generals, which could not otherwise be ob- 

 tained so easily. Although making a distinction between ma- 

 thematical induction and the induction used in natural philo- 

 sophy, De Morgan, in his article in the ' Penny Cyclopaedia 3 on 

 this subject, states that an instance of mathematical induction 

 occurs in every equation of differences and in every recurring 

 series. Taking the definition of induction as given by Dr. 

 Whateley, namely, " a kind of argument which infers respecting 

 a whole class what has been ascertained respecting one or more 

 individuals of that class," it will be evident to any experimenter 

 in chemical or physical science who is also acquainted with the 

 use of induction in mathematical science, that mathematical in- 

 duction is of a higher and more perfect kind than the induction 

 used in the physical sciences, especially when it assumes the form 

 of successive induction as De Morgan calls it, and as it is em- 

 ployed in recurring series. 



It is this high class of reasoning, which is involved in the con- 

 struction of series that recur according to a given law, that makes 

 the use of recurring series so valuable in imitation. Neverthe- 

 less, in addition to the policy of investigating every general 

 result by the use of a more general application of inductive phi- 

 losophy, namely by the calculation or examination of particular 

 examples (and so, having by one train of reasoning obtained a 

 general result, proving or illustrating the use of that general 

 result by particular applications), it is especially necessary to 

 study examples of every possible type in unitation. It is more- 

 over necessary to have other means than the principles of recur- 

 ring series, and of verification by examples, to ascertain the 

 truth of certain results, in consequence of the number of anoma- 

 lies that present themselves for explanation. 



These anomalies occur not only at the ordinary singular values 

 that the theory of numbers affords, such as and 1, but the base 

 of the system of .imitation — take 9 for instance — together with 

 the submultiples of that base, namely six and three, also furnish 

 singular or anomalous values. 



In respect of operations, as might be expected^ inverse opera- 



