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XXXI. — On the Limits of our Knowledge respecting the Theory of Parallels. 



Bj Professor Kelland. 



(Read December 21, 1863.) 



The subject of this paper is a very old one, and may to many appear to be 

 sufficiently worn ; but I venture to hope, that there are some to whom a glimpse 

 of the successive approaches of the human mind towards the right understanding 

 of a question of pure logic, may have an interest, — even although the problem 

 solved be an abstract one, and the conclusion a negative conclusion, having little 

 practical application. Like the kindred problem of the quadrature of the circle, 

 or the metaphysical problem of "Knowing and Being," the theory of parallels 

 has been attacked in various directions, and although it is true that no one ever 

 reached the goal he aimed at, yet it is not the less certain that great and posi- 

 tive results have followed in the history of human attainment. If no other lesson 

 has been learnt, this at least may have been : that in reasoning it is necessary to 

 look warily around and abroad at every step, seeing that admissions, the most 

 obviously inadmissible, or evasions the most palpable, have foiled generations 

 of thinkers, whilst those who have detected their errors have fallen into others 

 of an equally destructive character. 



It is not my intention to give an account of the successive failures of different 

 geometers in their attempts on this problem . That has already been done by 

 Colonel Thompson, in his " Geometry without Axioms." My object will be rather 

 to show what has been successfully accomplished, and by going over in a positive 

 form the ground which is forbidden to those who attack the problem directly, to 

 indicate as clearly as I can the limits within which future research may be 

 confined. 



I say I am going over the negative limits of the discussion of the problem in a 

 positive form. What I mean by this statement is, that I am about to start with 

 the assumption, as though it were an axiom, of that very problem the incorrect- 

 ness of which it is the object of future geometers to demonstrate, and by a purely 

 logical process, to ascertain whither those false premises (if they be false) shall 

 lead me. I am much mistaken if those who give themselves the trouble to 

 examine the argument do not find it both interesting and instructive. So far as 

 I know, it has never yet been developed in this country, although the circumstance 

 that for the last seventeen years I have made it the repeated subject of Lectures 

 and Essays in my Class may possibly take from it some of the appearance of 



VOL. XXIII. PART III. 6 B 



