ADDRESS. 17 



l'ects, to prove that hers is not after all such a far-off region, nor so unde- 

 cipherable an alphabet, and to show that eveu at unlikely spots we may 

 trace under-currents of thought which having issued from a common source 

 fertilise 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 litera- 

 ture have unconsciously 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 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 counterpart 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 applications of which it is doubtless capable. And even 

 although this calculus be not coextensive with another which almost simul- 

 taneously germinated on the Continent, nor with ideas more recently 

 developed in America ; yet it must always hold its position as an original 

 discovery, and as a representative of one of the two great groups of gene- 

 ralised algebras (viz., those the squares of whose units are respectively 

 negative unity and zero), the common origin of which must still be 

 marked on our intellectual map as an unknown region. Well do I re- 

 collect how in its early days we used to handle the method as a magician's 

 page might try to wield his master's wand, trembling as it were between 

 hope and fear, and hardly knowing whether to trust our own results 

 until they had been submitted to the present and ever-ready counsel of 

 Sir W. R. Hamilton himself. 



To fix our ideas, consider the measurement of a line, or the reckoning 

 of time, or the performance of any mathematical operation. A Hue may 

 be measured in one direction or in the opposite ; time may be reckoned 

 forward or backward ; an operation may be performed or be reversed, it 

 may be done or may be undone ; and if having once reversed any of these 

 processes we reverse it a second time, wc shall find that we have come 

 1878. B 



