1 04 Transactions. — Miscellaneous . 



Suppose then, two straight luies inlinitely 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 Avhen 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 fortu- 

 nate 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 exten- 

 sive ground of the finite. He enlarges the angle which two non-intersect- 

 ing 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 other- 

 wise than that he is in very palpable error. 



It appears to me that at any finite angle of convergence oi 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 con- 

 ceived 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 



