102 Transactions, — Miscellaneous. 



do, and as I have every reason to do, from my knoAvledge of his mathe- 

 matical attahameuts, that the arguments of the geometers he cites therein 

 are correctly rendered and are their best. 



To commence then, as to the Euclidian axiom, which Lobatchewsky 

 assumes should be incorrect — namely, the twelfth — that relating to parallel 

 straight lines, an equivalent form of which the author gives as being the 

 one " now generally employed in works on geometry" — it runs thus : "It 

 is impossible to draw more than one straight line parallel to a given straight 

 line through a given point outside it." But, observe, that it is not this 

 equivalent which Lobatchewsky is supposed to use in his attempt at demon- 

 strating the truth of his assumption, but an equivalent of the above-given 

 equivalent ; and so, as we have to deal with an equivalent thus twice re- 

 moved, we must be doubly careful to see whether or not equivalence is here 

 maintained. 



As given by Mr. Frankland, this supposed equivalent is as follows : " If 

 we take a fixed straight line, A B, prolonged infinitely in both directions, 

 and a fixed point, P, outside it ; then, if a second 

 straight line " — (say, C D) — " also infinitely prolonged 

 m both directions be made to rotate about P, there 

 is oiily one position in which it will not intersect the 

 line A P." 



Now, I contend that this is not what it purports to be — an equivalent of 

 the Euclidian axiom set before us ; and, I think it can plainly be seen that 

 parallelism, as a quality of such lines to be sought for or maintained, is 

 given up ; for, allowing the reverse of the proposition to be true — allowing 

 that the second straight line may be made to occupy more than one position 

 relative to the line A B without intersecting it, still it will be parallel to it 

 in one position only. 



In reality the substituted proposition which we have here, does not 

 even touch the original in its essential part ; it is, in fact, an independent 

 one, exchiding the idea of parallelism as either a necessity or even a deside- 

 ratum, and merely affirming something which, whether true or otherwise, 

 has nothing to do with the matter now before us. 



To any one who will examine the subject, all this must, I think, appear 

 so palpable, that I may leave it now and turn to the discussion of what 

 Lobatchewsky would make of this pseudo-equivalent. 



Taking up Mr. Frankland at this point, we have him rendermg the 

 master thus : " Now Lobatchewsky made the supposition that this axiom " 

 (meaning, of course, the equivalent in question) " should be untrue, and 

 that there should be a finite angle through which the rotating line might be 

 turned without ever intersecting the fixed straight line A P." 



