The conception of this surface is, in the author's own words, arrived at as follows:—“To obtain the simplest case of such manifoldness [i.e., surface] we must suppose that the point towards which two geodesic lines converge is separated from their starting point not by half but by the entire length of a geodesic line, or what amounts to the same thing, that it coincides with the starting point.”* Sorely Mr. Frankland must take a positive delight in tormenting us with paradoxes. 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! 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 ultimate converging point that identical position whence it started; indeed, as it travels along, it may properly 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 converges there, is only got by simultaneously producing 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 surfaces; and, for the purpose these latter-day geometricians have in view, it is necessary that they shall enclose but one. This idea, or rather proposition, 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 connected, 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.† “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 be very great as compared with any magnitude which we can bring under our observation.”—(Frankland, l. c., p. 278.) With this knowledge it is manifest that the analytical conception of two geodesic lines refusing to intersect each other more than