THE FOURTH DIMENSION. Jl 



point-aggregate, and around each centre there may be additionally 

 conceived a one-dimensional totality of spheres, the radii of which 

 can be expressed by every numerical magnitude from zero to infinity. 

 Further, if we imagine a measuring-stick of invariable length to as- 

 sume every conceivable position in space, the positions so obtained 

 will constitute a five-dimensional aggregate. For, in the first place, 

 one of the extremities of the measuring stick may be conceived to 

 assume a position at every point of space, and this determines for 

 one extremity alone of the stick a three-dimensional totality of po- 

 sitions ; and secondly, as we have seen above, there proceeds from 

 every such position of this extremity a two-dimensional totality of 

 directions, and by conceiving the measuring-stick to be placed 

 lengthwise in every one of these directions we shall obtain all the 

 conceivable positions which the second extremity can assume, and 

 consequently, the dimensions must be 3 plus 2 or 5. Finally, to 

 find out how many dimensions the totality of all the possible posi- 

 tions of a square, invariable in magnitude, possesses, we first give 

 one of its corners all conceivable positions in space, and we thus 

 obtain three dimensions. One definite point in space now being 

 fixed for the position of one corner of the square, we imagine drawn 

 through this point all possible lines, and on each we lay off the 

 length of the side of the square and thus obtain two additional di- 

 mensions. Through the point obtained for the position of the sec- 

 ond corner of the square we must now conceive all the possible di- 

 rections drawn that are perpendicular to the line thus fixed, and 

 we must lay off once more on each of these directions the side of 

 the square. By this last determination the dimensions are only in- 

 creased by one, for only one one- dimensional totality of perpen- 

 dicular directions is possible to one straight line in one of its points. 

 Three corners of the square are now fixed and therewith the posi- 

 tion of the fourth also is uniquely determined. Accordingly, the 

 totality of all equal squares which differ from one another only by 

 their position in space, constitutes a manifoldness of six dimen- 

 sions. 



