[Dupuisl PRESIDENTIAL ADDRESS 13 



sions is projective into one of two, etc. Or, in other terms, just as the 

 whole of phine geometrj- may be looked upon us a projection of solid 

 geometry, or geometry of three dimensions upon a plane of projection, 

 so geometry of three dimensions might be considered to be the projection 

 of four-dimensional geometry upon what we would have to call a tri- 

 dimensional space of projection, whatever such a jihrase may mean. 



Of course there is more analogy in this argument than in the pre- 

 ceding one, and yet, whatever it may render possible, it proves nothing 

 whatever. 



A line, which is of one dimension, can be projected into a point, 

 which is of no dimensions, and thus into a geometrical figure one dimen- 

 sion lower, only when the line to be projected is normal to the plane of 

 projection. And a plane, which is of two dimensions, can be projected 

 into a line, which is of one dimension, and therefore one dimension lower 

 than the original, only when the plane to be projected is normal to the 

 plane of projection. 



But in these very cases it is impossible to know, from anything in 

 the nature of the projection itself, whether the original was of higher, or 

 of the same dimensions as the projection. So also a figure of three dimen- 

 sions gives rise, when projected, to a figure of two dimensions. But 

 nothing in the natiire of the figure thus produced can indicate whether 

 the original was of three dimensions or of only two. And were it not 

 that we know beforehand the character of the original, it would not be 

 legitimate to infer, from anything presented to us in the projection, that 

 the original must be of three dimensions. 



Reasoning then from analogy, all that we are justified in saying is, 

 that if there be such a thing as a four-dimensional space, our solid figures 

 may possibly be projections from figures in that space, although we fail 

 to conceive how such a projection could beefiected. But w^eare certainly 

 not justified in assuming that there is a four-dimensional space, unless we 

 can first know something about the nature of a figure in such space. 



And then again, if we can reason by analogy from a three-dimen- 

 sional space to a four-dimensional space, we can rise by the same means 

 from a four to a five-dimensional space, and so on by induction to a space 

 of infinite dimensions. For if we admit the legitimacy of the method 

 there can be no stopping point, except a merely arbitrary one. But it 

 has been shown that even in cases where we have the strictest analogy, 

 this method of argument may fail. Then in the j^resent case where the 

 analogy is certainl}^ not very close, it is highly probable that at some 

 point it will fail, and it is just as reasonable, and more so, to suppose that 

 it fails in rising from three to four dimensions, as to suppose that it will 

 fail at some higher point. 



It is said, again, that the mathematician frequently works upon the 

 assumption of a four-dimensional space, as when he employs four co-ordin- 



