12 ROYAL SOCIETY OF CANADA 



analogies which he believes to hold good between the conditions of exist- 

 ence in spaces of lower dimensions. 



But arguments in mathematics, drawn from analogy, even where 

 the analogy is unquestionable, are not alwaj^s trustworthy. Of this I 

 shall adduce one single example out of many : 



It is well known that a quadi'atic equation has a reducing linear, or 

 can by a substitution be reduced to a linear. So also a cubic equation 

 has a reducing quadratic, and a quartic equation has a reducing cubic. 

 Analogy would tell us then that the quintic equation should have a 

 reducing quartic. But all attempts at finding such a quartic have failed, 

 and Abel has shown that no such quartic exists. Here then analogy 

 fails us, and in a case where we would particularly expect it to hold. 



What then can we say of arguments resting upon doubtful analogies 

 between the abstract and the concrete, and between the real and the 

 inconceivable, and where there is, and can be, nothing in our conceptions 

 to support such arguments. 



In Clifford's worm we are asked to conceive an intelligent being re- 

 duced to a segment of a line, and in his fish, a similar being reduced to a 

 portion of a surface, and from imaginary experiences of these hypotheti- 

 cal beings we are to rise, by a very doubtful analogy, to space of four 

 dimensions, to the properties of such a space, and to the experiences of 

 creatures or intelligences inhabiting it. 



Let us see what all this means. 



It is very convenient to speak of a plane as being space of two 

 dimensions, and of the straight line as space of one dimension. But, 

 considering the matter a little more in detail, we see that a surface is the 

 boundary or limit belonging to an object of three dimensions, and sepa- 

 rating this object from adjacent ones, or from the space which lies with- 

 out it. A surface considered in these relations, and as the limits or 

 bounds of a material object is real enough ; but separate it from these 

 relations and it becomes a mere abstraction. And similar remarks might 

 be made in regard to the line. How then can we imagine an intelligent 

 being as inhabiting an abstraction ? For so long as we fail to consider 

 the surface in relation to the third dimension, that is what it comes to. 

 And the fish, which has no relation to the third dimension, is an abstrac- 

 tion also. And yet we are asked to find an analogy between the life and 

 experiences of an abstraction and those of real creatures who lead a real 

 life in space of three dimensions. And from this analogy we are to draw 

 conclusions with respect to relations existing between the real space 

 which we inhabit and the inconceivable unknown called space of four 

 dimensions. For my own part I must confess that I cannot see any 

 analogy upon which an argument can be built. 



Again, it is argued that a four-dimensional figure may possibly be 

 projective into a figure of thi'ee dimensions, just as one of thi'ee dimen- 



