612 Transactions. — Miscellaneous, 



upon them builds up, by a process of deduction, all its proposi- 

 tions. In the physical sciences, on the other hand, whatever 

 concepts we start with, we find that our results have continually 

 to be qualified by bringmg in new concepts — so that even in our 

 theories a continual process of approximation is going on. 



(2.) We now come to the speculation that the actual proper- 

 ties of space may be different from those generally ascribed to it. 

 This really comes to the same thing as saying that there may 

 be points in space whose position we cannot consider by refer- 

 ence to om- Euclidian system of geometry of three dimensions. 

 If we show that by our geometry of three dimensions we could 

 consider the position of all possible points in space, then its 

 methods would suffice for the investigation of any possible form 

 of surface or solid ; the so-called geodesies, parallel straight 

 lines which meet, and uniplanar non-parallel straight lines 

 which do not meet — all of which are drawn on this wonderful 

 surface to which Mr. Frankland refers — could be brought to 

 reason by considering the corresponding lines on a similar 

 surface of manageable extent. For it must be possible, on the 

 assumption that three-dimension geometry is sufficient, to obtain 

 a surface of small extent similar to any finite surface whatever. 



By many of the transcendental geometers this objection is 

 met by the answer, (which is, I believe, the only possible one,) 

 that there may be four or more dimensions in space, not three 

 only, as is usually imagmed. Now, as Lotze points out, if there 

 be a fourth dimension in the strict sense of the term, it must be 

 of the same kind as the other three — length, breadth, and 

 thickness: otherwise, our use of the word "dimension" is a 

 misnomer ; so also is our use of the word " space." Time, 

 density, thermal capacity, etc., are all excluded from being 

 regarded as corresponding in any real sense to dimensions of space. 



We have three concepts of the methods of extension in 

 space, the three dimensions already referred to : the question is, 

 whether space can be such that we cannot completely examine 

 by reference to our three concepts the form and position of a 

 space which is finite. 



Let the position of any point of which we are cognizant m 

 three- dimension space be referred to three co-ordinate axes, 

 OX, Y, Z, which are mutually at right angles. All points 

 in our space can be so referred, and every point with any finite 

 and real co-ordinates whatever can have its position assigned to 

 it. Let the fourth dimension be referred to an axis V : V 

 must bear the same relation to X and O Y as Z does, that 

 is, it must be at right angles to each of them. (This follows 

 from the fact that the fourth dimension must be of the same 

 kind as the other three.) 



An imaginary being might have tlic same X and Y, but 

 might have V instead oi (J Z for his third axis. 



