THE FOURTH DIMENSION. tg 



will then be independent of one another. To reach the same result 

 in the case of six atoms we must assume a five-dimensional space ; 

 and so on. 



Now, if the independence of all the possible distances between 

 the atoms of a molecule is absolutely required by theoretical chem- 

 ical research, the science is really compelled, if it deals with mole- 

 cules of more than four atoms, to make use of the idea of a space of 

 more than three dimensions. This idea is, in this case, simply an 

 instrument of research, just as are, also, the ideas of molecules and 

 atoms means designed to embrace in a perspicuous and systematic 

 form the phenomena of chemistry and to discover the conditions 

 under which new phenomena can be evoked. Whether a four- 

 dimensioned space really exists is a question whose insolubility 

 cannot prevent research from making use of the idea, exactly as 

 chemistry has not been prevented from making use of the notion 

 of atom, although no one really knows whether the things we call 

 atoms exist or not. 



v. 



REFUTATION OF THE ARGUMENTS ADDUCED TO PROVE THE 



EXISTENCE OF A FOUR DIMENSIONED SPACE INCLUSIVE 



OF THE VISIBLE WORLD. 



The considerations of the preceding section will have convinced 

 the cultured non-mathematician of the service which the theory of 

 multi-dimensioned spaces has done, and bids fair to do, for geo- 

 metrical research. In addition thereto is the consideration that 

 every extension of one branch of mathematical science is a constant 

 source of beneficial and helpful influence to the other branches. The 

 knowledge, however, that mathematicians can employ the notion of 

 four-dimensioned space with good results in their researches, would 

 never have been sufficient to procure it its present popularity ; for 

 every man of intelligence has now heard of it, and, in jest or in 

 earnest, often speaks of it. The knowledge of a four-dimensioned 

 space did not reach the ears of cultured non-mathematicians until 

 the consequences which the spiritualists fancied it v/as permissible 

 to draw from this mathematical notion were publicly known. But 

 it is a tremendous step from the four-dimensioned space of the 



