W. Skey. — On Franktand^ s Pajjer on Two Dimensions. 103 



Here, then, are two propositions : first, that the lines A B, C D, though 

 infinitely long, may lay angularly to each other without making an inter- 

 section ; and, second, that this angularity may be such as to be of a finite 

 value. Now as much is made to turn upon the supposed truth of these 

 propositions it may be expedient (notwithstanding what I have already said) 

 that I should make a few remarks upon this matter also. 



You will hardly fail to note the very easy manner in which lines 

 infinitely long are spoken of by the proposition ; in effect it says : — Take two 

 such lines, manipulate them in the manner described, and a certain result 

 follows, — ^just as if this were as sure and tangible an operation, and one not so 

 very dissimilar besides, as that of preparing puddings by a recipe out of 

 some standard cookery book. Surely the mere taking of any line stamps 

 that line as a line of but finite length. However this is, one of these lines 

 is to pass through a point outside the other line, but nothing is said as to 

 the distance away from this line at which the point is to be placed. As we 

 have been started into infinities, it is open to us to place it at an infinite 

 distance away or not ; but if, in a friendly spirit towards Lobatchewsky, we 

 place it where his result seems the more likely to be secured, that is at an 

 infinite distance away from A B, the proposition becomes simply a truism, 

 and by its wide significance defeats the end desired ; for in this way any 

 number of lines may strike through the point P, and at all angles to A B — 

 each of which may be infinitely extended without intersecting this line. It 

 is seen then that this proposition, as it stands, requires amending by an 

 addition thereto which shall restrict us to the placing of the point P at 

 a finite distance away from A B. Thus checked, we have only now 

 to ascertain whether or no any line inclining to A B may be extended 

 infinitely through P as now placed, without intersecting this companion 

 line. 



It is very difficult for me to work, or even suppose I am working, with 

 lines of such unwieldy length as these we are set to improvise for our 

 geometrical constructions, but it appears to me that even if the angle of 

 convergence is infinitely small the lines would intersect, but not, of course, 

 at any determinable or conceivable distance. It seems that the comple- 

 tion of the ideal construction begun demands this intersection ; and yet, on 

 the other hand, I cannot but allow that to realize an intersection at all is to 

 reduce the lines themselves to finite proportions. Clearly, then, dealing in 

 this way with infinities places us on the " horns of a dilemma." 



But that the attempt at operating in this way with infinitely long lines 

 is clearly futile if not absurd, is perhaps better manifested by conceiving, or 

 rather trying to conceive, of the exact converse of the proposition in ques- 

 tion. 



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