Mr. J. P. Nichol on Parallel Lines. 413 



to ask you to accept an extract from a published work : but, iu 

 the present case, the vohime is only just published ; and as it is 

 of a miscellaneous nature, the brief article of which a copy is 

 now sent, may easily escape the notice of those of your corre- 

 spondents who take a critical interest in the very curious subject 

 under discussion. 



I am. Gentlemen, 



Yours truly, 



J. P. Nichol. 



Parallel Lines. 



" The characteristic of two lines in the same plane, to which 

 the name of parallel is given in geometry, is simply this, — that 

 although produced ever so. far either way they will never meet. 

 The theory of these lines continues a stain on our elementary 

 science. It is easy to prove, that if certain conditions are ful- 

 filled when two lines cut a transverse line, these two will never 

 meet, or must be parallel; but to establish the converse, — to 

 prove, viz. that if the two lines are parallel, these same conditions 

 are lawfully predicable, — has hitherto defied the logic of all geo- 

 meters — a fact certainly most remarkable in this purely deduct- 

 ive science : nevertheless, its causes do not appear remote. The 

 existence of such a defect unquestionably argues some oversight 

 in the list of geometrical axioms, — the oversight of the nature of 

 some of our primal perceptions regarding magnitude : but it 

 does not follow that the missing axiom has immediate relation 

 to parallels, or that it ought to enable us to resolve directly the 

 specific proposition at which the acknowledged difficulty first 

 appears. On the very contrary, it may be asserted with abun- 

 dant confidence, that nothing but failure could attend the effort 

 — originated by Euclid, and since his time all but universally 

 followed — to supply the deficiency by new postulates or axioms 

 regarding parallel lines. To prove that under certain conditions 

 two lines will never meet, or what is the same thing, that no 

 triangle can be formed in such circumstances, involves no con- 

 ception with which we cannot readily cope ; but to deduce the 

 properties of two lines postulated as parallels, involves a direct 

 dealing with the positive idea of infinity — a task utterly beyond 

 the reach of our faculties. Whether we have a p)ositive idea of the 

 Infinite, is a question concerning which the profoundest meta- 

 physicians have differed and continue to differ; but, considera-. 

 tion of the origin and formation of language, suffices of itself to 

 leave no doubt, that we cannot speak of Infinity otherwise than as 

 a negation, and therefore that no positive axiom can be laid down 

 respecting it. 



