Frankland, — The Non-EucUtUan Geometry Vindicated. 59 



paragraph, and the number of the page from which the quota- 

 tion is made is indicated. It seemed to me that in this way 

 only could a searching and exhaustive refutation of his argu- 

 ments be given. 



1. What is meant by the assertion that "the axioms of geo- 

 metry may be only approximately true "? (p. 100) It means that 

 the actual physical constitution of the space in which we live may 

 be different from the space treated of in works on solid geometry, 

 but that it must be so nearly the same that we cannot detect the 

 difference by the most delicate experimental methods at our 

 command. 



2. " The author then adverts to ' the existence ' of a particular 

 manifoldness, which has been treated by Professor Clifford in 

 a lecture on the postulates of space" (p. 101). I mean it exists 

 in the sense of being logically constructive, not in the sense that 

 any surface in the space in which we live possesses such pro- 

 perties. It may be that planes (or flattest surfaces, if the expres- 

 sion be preferred,) in the space in which we live possess the 

 properties of this "manifoldness." We cannot know whether 

 they do or not. If they do, at any rate their total areas must 

 be immensely large. 



Perhaps it may be said that any absurd scheme of pseudo- 

 geometry is " logically constructible." But this is not the case. 

 It is not possible, for instance, to construct a scheme of geometry 

 in which two shortest lines enclose a space (all shortest lines 

 being supposed congruent), and in which the three angles of a 

 triangle are always less than two right angles. Such a scheme 

 would be logically self-contradictory. For it is logically involved 

 in the assertion that two shortest lines may meet twice, assuming 

 all Euclid's other axioms "to be true, that the three angles of a 

 triangle are always greater than two right angles. They cannot, 

 under such circumstances be either equal to 180° or less than 180°. 



3. " Then he describes how this space is analytically 

 conceived, with the object of putting us in a position to 

 apprehend certain discoveries of his own, which relate to its 

 very singular properties" (p. 101). The manifold" I described 

 in my paper is not a space. It is a manifold of two dimensions, 

 not of three. It may be described as an unimaginable but 

 logically constructible surface. 



4. It is not accurate to say that Professor Clifford " imputes 

 finiteness " to the universe or to space. He says, in common 

 with most living mathematicians who have studied this question, 

 that space may be finite — not that it is finite. Its possible 

 finiteness is spoken of, not in the sense of its having a boundary, 

 which would be unmeaning, but as implying that space may 

 return into itself, so to speak, just as the surface of a sphere and 



* This term is now generally used instead of the more cumbrous 



" manifoldness." 



