80 POPULAR SCIENCE MONTHLY 



agreement, the aggregate of distinct series 2 will still form a physical 

 continuum and the number of dimensions will be less but still very- 

 great. 



To each of these series 2 corresponds a point of space ; to two series 

 2 and 2' thus correspond two points M and M'. The means we have 

 hitherto used enable us to recognize that M and M' are not distinct in 

 two cases : (1) if % is identical with %' ; (2) if 2' = 2 -f- s + s '> s an( l s ' 

 being inverses one of the other. If in all the other cases we should 

 regard M and W as distinct, the manifold of points would have as 

 many dimensions as the aggregate of distinct series 2, that is, much 

 more than three. 



For those who already know geometry, the following explanation 

 would be easily comprehensible. Among the imaginable series of mus- 

 cular sensations, there are those which correspond to series of move- 

 ments where the finger does not budge. I say that if one does not 

 consider as distinct the series 2 and ~%-\- a, where the series a corre- 

 sponds to movements where the finger does not budge, the aggregate 

 of series will constitute a continuum of three dimensions, but that if 

 one regards as distinct two series 2 and 2' unless 2' = 2 + s -\-s', s and 

 s' being inverses, the aggregate of series will constitute a continuum of 

 more than three dimensions. 



In fact, let there be in space a surface A, on this surface a line B, 

 on this line a point M . Let C be the aggregate of all series 2. Let 

 Ci be the aggregate of all the series 2, such that at the end of cor- 

 responding movements the finger is found upon the surface A, and C 2 

 or C 3 the aggregate of series 2 such that at the end the finger is found 

 on B, or at M. It is clear, first that C x will constitute a cut which will 

 divide C , that C 2 will be a cut which will divide C 1} and C 3 a cut which 

 will divide C 2 . Thence it results, in accordance with our definitions, 

 that if C 3 is a continuum of n dimensions, C will be a physical con- 

 tinuum of n -f- 3 dimensions. 



Therefore, let 2 and 2' -f- a be two series forming part of C 3 ; for 

 both, at the end of the movements, the finger is found at M ; thence 

 results that at the beginning and at the end of the series a, the finger is 

 at the same point M . This series a is therefore one of those which 

 correspond to movements where the finger does not budge. If 2 and 

 2 -f- o- are not regarded as distinct, all the series of C 3 blend into one ; 

 therefore C 3 will have dimension, and C will have 3, as I wished to 

 prove. If, on the contrary, I do not regard 2 and 2 + cr as blending 

 (unless o- = s -f- s', s and s' being inverses), it is clear that C 3 will con- 

 tain a great number of series of distinct sensations; because, without 

 the finger budging, the body may take a multitude of different atti- 

 tudes. Then C 3 will form a continuum and C Q will have more than 

 three dimensions, and this also I wished to prove. 



