70 THE FOURTH DIMENSION. 



in a systematic scheme all these isolated events, facilitates our men- 

 tal control of the phenomena of nature, and enables us to produce 

 these phenomena at will. But it must not be forgotten in such re- 

 flexions that the ether itself is even a greater problem for man, and 

 that the ether-hypothesis does not solve the difficulties of phenom- 

 ena, but only puts them in a unitary conceptual shape. Notwith- 

 standing all this, physicists have never had the least hesitation in 

 employing the ether as a means of investigation. And as little do 

 reasons exist why the mathematicians should hesitate to investigate 

 the properties of a four-dimensioned point-aggregate, with the view 

 of acquiring thus a convenient means of research. 



rv. 



THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT- 

 AGGREGATES OF SERVICE TO RESEARCH. 



From the concession that the mathematician has the right to 

 define and investigate the properties of point-aggregates of more 

 than three dimensions, it does not necessarily follow that the intro- 

 duction of an idea of this description is of value to science. Thus, 

 for example, in arithmetic, the introduction of operations which 

 spring from involution, as involution and its two inverse operations 

 proceed from multiplication, is undoubtedly permitted. Just as for 

 "a times a times a " we write the abbreviated symbol "# 8 ," (which 

 we read, a to the third power,) and investigate in detail the opera- 

 tion of involution thus defined, so we might also introduce some 

 shorthand symbol for "a to the th power to the th power" and thus 

 reach an operation of the fourth degree, which would regard a as 

 a passive number and the number 3, or any higher number, as the 

 active number, that is, as the number which indicates how often a 

 is taken as the base of a power whose exponent may be a, or "a to, 

 the th ," or "a to the th 'to the a* power." 



But the introduction of such an operation of the fourth degree 

 has proved itself to be of no especial value to mathematics. And 

 the reason is that in the operation of involution the law of commuta- 

 tion does not hold good. In addition, the numbers to be added may 

 be interchanged and the introduction of multiplication is therefore 



