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THE POPULAR SCIENCE MONTHLY.— SUPPLEMENT. 



that in these very eases our science has not out- 

 stepped its own legitimate range, but that even 

 art and literature have unconsciously employed 

 methods similar in principle. The three methods 

 in question are — 1. That of imaginary quantities ; 

 2. That of manifold space ; and, 3. That of geom- 

 etry 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 nomen- 

 clature, 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 Ire- 

 land we are indebted for that marvelous calculus 

 of quaternions, which is only now beginning to 

 be fully understood, and which has not yet re- 

 ceived all the applications of which it is doubt- 

 less capable. And even although this calculus 

 be not coextensive with another (the Ausdeh- 

 nungslehre of Grassmann), 1 which almost simul- 

 taneously germinated on the Continent, nor with 

 ideas more recently developed in America 

 (Peirce's "Linear Associative Algebras"); 8 yet 

 it must always hold its position as an original 

 discovery and as a representative of one of the 

 two great groups of generalized 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 recollect 

 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 sub- 

 mitted to the present and ever-ready counsel of 

 Sir YV. R. Hamilton himself. 



To fix our ideas, consider the measurement of 

 a line, or the reckoning of time, or the perform- 

 ance of any mathematical operation. A line 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 



1 Grunert's " Arcbiv," vol. vi., p. 337 ; also separate 

 work, Berlin, 1862. 



- " Linear Associative Algebra," by Benjamin 

 Peirce, Washington City, 1870. 



it a second time, we shall find that we have come 

 back to the original direction of measurement or 

 reckoning, or to the original kind of operation. 



Suppose, however, that at some stage of a 

 calculation our formulae indicate an alteration in 

 the mode of measurement such that if the altera- 

 tion be repeated, a condition of things not the 

 same as but the reverse of the original will be 

 produced. Or suppose that, at a certain stage, 

 our transformations indicate that time is to be 

 reckoned in some manner different from future or 

 past, but still in a way having definite algebraical 

 connection with time which is gone and time 

 which is to come. 1 It is clear that in actual ex- 

 perience there is no process to which such meas- 

 urements correspond. Time has no meaning ex- 

 cept as future or past ; and the present is but the 

 meeting-point of the two. Or, once more, sup- 

 pose that we are gravely told that all circles pass 

 through the same two imaginary points at an in- 

 finite distance, and that every line drawn through 

 one of these points is perpendicular to itself. On 

 hearing the statement we shall probably whisper, 

 with a smile or a sigh, that we hope it is not true, 

 but that in any case it is a long way off, and per- 

 haps, after all, it does not very much signify. If, 

 however, we are not satisfied to dismiss the ques- 

 tion on these terms, the mathematician himself 

 must admit that we have here reached a definite 

 point of issue. Our science must either give a 

 rational account of the dilemma, or yield the posi- 

 tion as no longer tenable. 



Special modes of explaining this anomalous 

 state of things have occurred to mathematicians. 

 But, omitting details as unsuited to the present 

 occasion, it will, I think, be sufficient to point out 

 in general terms that a solution of the difficulty 

 is to be found in the fact that the formulae which 

 give rise to these results are more comprehensive 

 than the signification which has been given to 

 them ; and when we pass out of the condition of 



1 Sir W. Thomson, Cambridge Mathematical Jour- 

 nal, vol. Hi., p 174 ; Jevons's " Principles of Sci- 

 ence," vol. ii., p. 438. But an explanation of the dif- 

 ficulty seems to me to be found in the fact that the 

 problem, as stated, is one of the conduction of heat, 

 and that the " impossibility" which attaches itself to 

 the expression for the " time " merely means that 

 previous to a certain epoch the conditions which pave 

 rise to the phenomena were not those of conduction, 

 but those of some other action of heat. If, therefore, 

 we desire to comprise the phenomena of the earlier as 

 well as of the later period in one problem, we must 

 find some more general statement, viz., that of phys- 

 ical conditions which at the critical epoch will issue 

 in a case 6f conduction. I think that Prof. Clifford 

 has somewhere given a similar explanation. 



