THE FOURTH DIMENSION. 65 



situated in a four-dimensioned space as a plane might be situated in 

 the space familiar to us, secondly, that the spirits of the dead lived 

 in such a four-dimensioned space, thirdly, that these spirits could 

 accordingly act upon the world and, consequently, upon the human 

 beings resident in it, exactly as we three-dimensioned creatures can 

 produce effects upon things that are two-dimensioned ; for example, 

 such effects as that produced when we shatter a lamina of ice, and 

 so influence some possibly existing two-dimensioned /V<?-world. 



Since spiritualism, under the leadership of a Leipsic Professor, 

 ZQllner, thus proclaimed the existence of a four- dimensioned space, 

 this notion, which the mathematicians are thoroughly master of, 

 for in all their operations with it, though they have forsaken the 

 path of actual representability, they have never left that of the truth, 

 this notion has also passed into the heads of lay persons who have 

 used it as a catchword, ordinarily without having any clear idea of 

 what they or any one else mean by it. To clear up such ideas and 

 to correct the wrong impressions of cultured people who have not a 

 technical mathematical training, is the purpose of the following 

 pages. A similar elucidation was aimed at in the tracts which 

 Schlegel(Riemann, Berlin, 1888) and Cranz (Virchow-Holtzendorff's 

 Sammlung, Nos. 112 and 113) have published on the so-called fourth 

 dimension. Both treatises possess indubitable merits, but their 

 methods of presentation are in many respects too concise to give to 

 lay minds a profound comprehension of the subject. The author, 

 accordingly, has been able to add to the reflections which these ex- 

 cellent treatises offer, a great deal that appears to him necessary 

 for a thorough explanation in the minds of non-mathematicians of 

 the notion of the fourth dimension. 



II. 

 THE, CONCEPT OF DIMENSION. 



Many text-books of stereometry begin with the words : "Every 

 body has three dimensions, length, breadth, and thickness." If we 

 should ask the author of a book of this description to tell us the 

 length, breadth, and thickness of an apple, of a sponge, or of a cloud 

 of tobacco smoke, he would be somewhat perplexed and would prob- 



