106 Trmuactions. — Miscellaneoils. 



The conception of this surface is, in the author's own words, arrived at as 

 follows :■ — " To ohtain the simplest case of such manifoldness [?'.<?., surface] 

 we nmst suppose that the point towards which two geodesic lines converge is 

 separated from their starting point not by half hut by the entire length of a 

 geodesic line, or what amounts to the same thing, that it coincides with the 

 starting point."* Now it appears to me that Mr. Frankland is unduly 

 cautious here, in stating as a supposition that which is a fact ; for it is 

 certain that any point which is describing a geodesic line, has for its ulti- 

 mate converging point that identical position whence it started ; indeed, as 

 it travels along, it may projperly be considered to converge every part of its 

 road in succession, this, however, in a subordinate manner; but that part of 

 a geodesic line which happens to be intersected by another geodesic line, is 

 no more a point of convergence for that line than any other part along it. 

 The idea of two principal converging points to every such line seems a false 

 one. On extending a single line of this kind we are not at all impressed with 

 the idea that it converges to a sort of half-way house on its route ; the idea 

 of a convergence there, is only got by simultaneously producmg two such 

 lines or more. That a geodesic line, then, converges to its own starting 

 point, admits of no supposition, being a fact ; but this is not all that is 

 wanted. Two such lines, as heretofore known, enclose two spaces or sur- 

 faces ; and, for the purpose these latter-day geometricians have in view, it 

 is necessary that they shall enclose but one. This idea, or rather proposi- 

 tion, is conveyed to us in rather a queer manner, considering what it 

 involves and clashes with, viz., (in the retrospective sentence which follows 

 thus), — "It is true that we are utterly unable to figure to ourselves a 

 surface in which two geodesic lines shall only have one point of intersection, 

 and yet shall enclose space." Geodesic lines, then, proceeding from some 

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

 point ; but, at this part, Mr. Frankland, otherwise so full, lucid, and con- 

 nected, is singularly curt and, to me at least, hardly intelligible, so that it 

 was not until I got nearly through his paper that I found what he omits 

 to inform us of here, — that he is assuming a uniformly curved surface of 

 immense size.j With this knowledge it is manifest that the analytical con- 

 ception of two geodesic lines refusing to intersect each other more than 



* Surely Mr. Frankland must take a pogitive delight in tormenting us with para- 

 doxes. He gravely informs us here that the finishing point or goal for a geodesic line 

 in process of construction is to be the length of such line away from the starting point of 

 that line. The two points are to be apart, yet coincide ! 



t " And on this ground it has been argued that the Universe may in reality be of finite 

 extent, and that each of its geodesic lines may return into itself, provided only that its 

 total magnitude he very great as compared loith any magnitude ivhich loe can bring under 

 our observation." — (Frankland, I. c, p. 278.) 



