512 Transactions. — Chemistry and Physics. 



sional space systems. The author of " Flatland " makes a 

 sphere pass in and out of two-dimensional space, and thereby 

 conveys the suggestion that higher space beings might simi- 

 larly visit our space and similarly disappear. In fact, the 

 idea has been seized upon as explaining many of the so- 

 called phenomena of Spiritism. But writers on this point 

 have not reckoned with the difficulty of insuring that the 

 higher space being should always offer the same three-dimen- 

 sional section on entering our space system. Even so simple 

 a figure as a cube might appear in a two-dimensional universe 

 as a point, a line, a triangle, quadrilateral, or five-, or six- 

 sided figures. In fact, under each of the four last headings 

 an infinite variety of forms might be offered. And the pos- 

 sible sections of a four-dimensional figure in space of three 

 axes offers, of course, a far greater variety of forms. It is not 

 conceivable that a being moving freely in space of four di- 

 mensions could present itself repeatedly to us in sections even 

 suggesting identity of form. 



There is one further objection which must be dealt with 

 in reference to both the above questions. The assumption is 

 usually made by writers on this subject, and has been tacitly 

 accepted in this paper, that a figure might be removed from a 

 plane and afterwards replaced in that plane ; and, by analogy, 

 that one of our solids might conceivably be lifted into four- 

 dimensional space and afterwards replaced in our system. 

 Now, either of these processes endues the body dealt with for 

 the time being with an existence in a system higher by one 

 dimension than that in which it was assumed to exist. Two- 

 dimensional beings, if such there be, are by the nature of 

 their limitations placed absolutely without the scope of our 

 cognisance, and we, of course, without the scope of theirs. 

 So too we, as long as we continue to be three-dimen- 

 sional beings, are absolutely cut off from such four-di- 

 mensional beings as there may be. The method of treatment 

 adopted by writers in endeavouring to place the conception of 

 the fourth dimension within reach of their readers consists in 

 developing figures from one space system to another. But 

 it is a fallacy to suppose that the matter occupying a figure 

 can be similarly dealt with. In fact, we have no knowledge 

 or conception of fourth-dimensional matter, any more than 

 of two-dimensional matter. It is this fallacy which vitiates 

 the application of the fourth dimension to reported spiritist 

 wonders, and the recognition of it restores confidence in 

 the old theory that two bodies cannot occupy the same 

 space. J. B. Stallo remarks* that " the analytical argument 

 in favour of the existence or possibility of transcendental 



* " The Concepts and Theories of Modern Physics," p. 269 



