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at unlikely spots we may trace under-currents of thought 

 which having issued from a common source fertilize 

 alike the mathematical and the non-mathematical world. 



Having this in view, I propose to make the subject 

 of special remark some processes peculiar to modern 

 Mathematics ; and, partly with the object of incidentally 

 removing some current misapprehensions, I have selected 

 for examination three methods in respect of which 

 Mathematicians are often thought to have exceeded all 

 reasonable limits of speculation, and to have adopted 

 for unknown purposes an unknown tongue. And it will 

 be my endeavour to show not only that in these very 

 cases our Science has not outstepped its own legitimate 

 range, but that even Art and Literature have uncon- 

 sciously employed methods similar in principle. The 

 three methods in question are, first, that of Imaginary 

 Quantities ; secondly, that of Manifold Space ; and 

 thirdly, that of Geometry not according to Euclid. 



First it is objected that, abandoning the more cautious Imao-inaries. 

 methods of ancient Mathematicians, we have admitted 

 into our formulas quantities which by our own showing, 

 and even in our own nomenclature, are imaginary or 

 impossible ; nay, more, that out of them we have formed 

 a variety of new algebras to which there is no counter- 

 part whatever in reality ; but from which we claim to 

 arrive at possible and certain results. 



On this head it is in Dublin, if anywhere, that I may 

 be permitted to speak. For to the fertile imagination 

 of the late Astronomer Royal for Ireland we are 

 indebted for that marvellous calculus of Quaternions, 

 which is only now beginning to be fully understood, 

 and which has not yet received all the applica- 

 tions of which it is doubtless capable. And even 

 although this calculus be not coextensive with another 



