88 THE FOURTH DIMENSION. 



number of points lie in every ray, are called configurations. Other 

 configurations may, of course, be produced, by taking a different 

 number of points and by assuming that the points taken lie in a 

 space of different or even higher dimensions. The author of this 

 article was the first to draw attention to configurations derived from 

 spaces of higher dimensions. As we see, then, the notion of a space 

 of more than three dimensions has performed an important ser- 

 vice in the investigations of common plane geometry. 



In conclusion, I should like to add a remark which Cranz makes 

 regarding the application of the idea of multi-dimensioned space 

 to theoretical chemistry. (See the treatise before cited.) In chem- 

 istry, the molecules of a compound body are said to consist of the 

 atoms of the elements which are contained in the body, and these are 

 supposed to be situated at certain distances from one another, and 

 to be held in their relative positions by certain forces. At first, the 

 centres of the atoms were conceived to lie in one and the same 

 plane- But Wislicenus was led by researches in paralactic acid 

 to explain the differences of isomeric molecules of the same struc- 

 tural formulae by the different positions of the atoms in space. (Com- 

 pare La chimie dans I'espace by van't Hoff, 1875, preface by J. 

 Wislicenus). In fact four points can always be so arranged in space 

 that every two of them may have any distance from each other; 

 and the change of one of the six distances does not necessarily in- 

 volve the alteration of any other. 



But suppose our molecule consists of five atoms? Four of these 

 may be so placed that the distance between any two of them can be 

 made what we please. But it is no longer possible to give the fifth 

 atom a position such that each of the four distances by which it is 

 separated from the other atoms may be what we please. On the 

 contrary, the fourth distance is dependent on the three remaining 

 distances ; for the space of experience has only three dimensions. 

 If, therefore, I have a molecule which consists of five ato'ms I can- 

 not alter the distance between two of them without at least altering 

 some second distance. But if we imagine the centres of the atoms 

 placed in a four-dimensioned space, this can be done ; all the ten 

 distances which may be conceived to exist between the five points 



