Franklan'd. — The Xon-Euclidifin Geometry Vindicated. 65 



of construction is to be the length of such a line away Loin the 

 starting point of that line. The two points are to be apart, yet 

 coincide!'' Where is the contradiction? In the manifold I 

 describe, as on the surface of a sphere, a geodesic starting 

 from any point leads back eventually to that point. So far, my 

 manifold and the surface of a sphere resemble one another. 

 '] he difference is this : If two persons on the surface of a sphere 

 (say the earth) were to start from the same place, and travel 

 along geodesic lines, they would cross each other's paths at a 

 half-way house (on the other side of the sphere], and then again 

 at the starting point. But on the manifold I have investigated 

 they would, after travelling a certain distance, get back to the 

 starting point, but without ever having crossed each other's paths 

 in the meanwhile. On a Euclidian plane, on the other hand, 

 they would obviously never either cross each other's paths or 

 get back to the starting point at all. 



14. " Geodesic lines, then, proceeding from some common 

 point of a surface, are to diverge somehow from the polar of that 

 point" (p. 106). I do not know what Mr. Skey means by the 

 " polar of that point," unless, indeed, it be the opposite point. 

 If so, I reply that in my manifold, which for the future we may 

 for convenience call the " finite plane,"* a point has not one 

 opposite only (like a point on a sphere), but a whole row of 

 opposite points : that is to say, an opposite line. The geodesic 

 lines proceeding from a common point cut this " opposite line" 

 (which I have called the polar) in separate points, each of which 

 is equally " opposite" to the common centre of radiation. 



15. " He is assuming a uniformly curved surface of immense 

 size" (p. 106). By no means. The manifold may be of any 

 size, large or small. Its total area may be less than the 

 decillionth part of a square inch — yet it will have its complete 

 and thoroughly self-consistent, though, I admit, quite un- 

 imaginable, geometry. What I do say is that, if any surface 

 constructible in the space in which ice live possesses the properties 

 of a " finite plane," then that surface must be of immense size, 

 for we can prove by experiment that no closed surface of 

 moderate area constructible in our space does possess these 

 properties. 



16. " It is manifest that the analytical conception of two geo- 

 desic lines refusing to intersect each other more than once, and 

 so enclosing but one space, is founded upon Lobatchewsky's con- 

 ception of what parallel straight lines are capable of" (p. 106). 

 This is not so. It is founded on just the opposite conception. 

 Lobatchewsky's conception is that of two geodesic lines which, 

 even though converging at first, do not ultimately intersect ; 

 mine is that of two geodesic lines which ultimately intersect, 



* The manifold in question possesses the same properties as the " plane 

 at infinity," well known to students of solid geometry. 



