Suppose then, two straight lines infinitely long, joining each other at an angle infinitely small, and the remarkable consequence follows, that at no conceivable length along these lines would they be apart any conceivable distance; and still the “analytical conception” (to use Mr. Frankland's term) is a valid one, that at some point they widen out to such an extent that a line joining their free ends is infinitely long. But then to our further embarrassment we have in this way upon our hands, or rather upon our minds, a triangle infinitely large, knowing full well the while, that aught which has a shape, must ever finite be. Thus are we again led to conclusions which are self-contradictory, and we learn thereby that geometry is not likely to be advanced or served by us when we go out of our proper beat to soar in the regions of the infinite. But whatever may be your views in regard to this aspect of the question, which I have thus so superficially and hastily treated, it is perhaps a fortunate thing for my continued sanity, that for his contention Lobatchewsky does not use arguments based upon the properties of lines which converge at angles infinitely small; possibly seeing, as I think we have, that this gives him nothing, he takes us on to the more solid if less extensive ground of the finite. He enlarges the angle which two non-intersecting infinitely extended straight lines in the same plane may make with each other, to a finite one. None of the evidence of Lobatchewsky in favour of this is given by Mr. Frankland, but simply the bare supposition itself. We cannot, therefore, examine the position fairly to Lobatchewsky, but being unaided by his arguments, I feel it impossible to conceive otherwise than that he is in very palpable error. It appears to me that at any finite angle of convergence of C D to A B they will intersect at some determinable part of the line A B, for a finite angle can only mean an angle of such a size that it can be measured or conceived of, or its value numerically assigned. To hold it to be otherwise is really to hold that an angle finitely large is infinitely small, which either is a contradiction, or these qualifying terms are divested of all meaning. This granted, it then follows as a necessary corollary that there is a point along A B which the line P will pass through, and a point, too, capable of being exactly determined. It appears, then, that here Lobatchewsky, in trying to secure something tangible in support of his idea, has overshot the mark, and so entangled himself and his disciples in a fallacy. If this is so, can we wonder that, starting in this way, Lobatchewsky gets, as Mr. Frankland says, “very curious results.” Triangles, the sum of whose internal angles is less than 180°; triangles which get out at their