THE FOURTH DIMENSION. 



77 



of great value. So, also, in multiplication the numbers which are 

 combined, that is, the factors, may be changed about in any way, 

 and thus the introduction of involution is of value. But in involution 

 the base and the exponent cannot be interchanged, and consequently 

 the introduction of any higher operation is almost valueless. 



But with the introduction of the idea of pointraggregates of 

 multiple dimensions the case is wholly different. The innovation 

 in question has proved itself to be not only of great importance to 

 research, but the progress of science has irresistibly forced investi- 

 gators to the introduction of this idea, as we shall now set foryi in 

 detail. 



In the first place, algebra, especially the algebraical theory of 

 systems of eqations, derives much advantage from the notion -of 

 multi-dimensioned spaces. If we have only three unknown quanti- 

 ties, x, y, z, the algebraical questions which arise from the possible 

 problems of this class admit, as we have above seen, of geometrical 

 representation to the eye. Owing to this possibility of geometrical 

 representation, some certain simple geometrical ideas like "mov- 

 ing," "lying in," "intersecting," and so forth, may be translated 

 into algebraical events. Now, no reason exists why algebra should 

 stop at three variable quantities ; it must in fact take into considera- 

 tion any number of. variable quantities. 



For purposes of brevity and greater evidentness, therefore, it is 

 quite natural to employ geometrical forms of speech in the consid- 

 eration of more than three variables. But when we do this, we as- 

 sume, perhaps without really intending to do so, the idea of a space 

 of more than three dimensions. If we have four variable quantities, 

 x, y, z, u, we arrive, by conceiving attributed to each of these four 

 quantities every possible numerical magnitude, at a four-dimen- 

 sioned manifoldness of numerical quantities, which we may just as 

 well regard as a four-dimensioned aggregate of points. Two equa- 

 tions which exist oh this supposition between x, y, z, and u, define 

 two three-dimensioned aggregates of points, which intersect, as we 

 may briefly say, in a two-dimensioned aggregate of points, that is, 

 in a surface ; and so on. In a somewhat different manner the de- 

 termination of the contents of a square or a cube by the involution 



