412 



Mr. J. P. Nichol on Parallel Lines. 



wire lies, we have 



e=E.i(l--e-/"); 



and on the side towards the end, 



e=E/^|-(l-e-''0 + e-'''T. 



Fig. 3. 

 If e and s be made the 

 rectangular coordinates 

 of a point, then for a 

 certain value of t we 

 obtain a line of the 

 form shown in fig. 3; 

 when t does not exceed 

 a moderate multiple of 

 T, the line has the form 

 shown in fig. 4; the 

 more t increases, the 

 more nearly does the ^ 

 figure approximate to the straight line, fig. 5. 

 Fig. 5. 



Fig. 4. 



LV. On Parallel Lines. By J. P. Nichol, Esq. 

 To the Editor's of the Philosophical Magazine and Journal. 



Observatory, Glasgow, 

 Gentlemen, May 6, 1857. 



THE frequent appearance of late in your publication, of 

 memoirs concerning the doctrine of parallel lines, assuring 

 me that the subject continues to possess an interest, I have ven- 

 tured to request insertion for the enclosed article, which has 

 just been printed in a ' Cyclopaedia of the Physical Sciences,^ 

 edited by myself. It contains views not touched by your corre- 

 spondents. I shall not say that those views do, in my own 

 opinion, exclude the applicability of other modes of solving the 

 difficulty, or that they are in themselves free fi'om objection. 

 Nevertheless, I think that principles are indicated there, which 

 must command a prominent place in any successful attempt to 

 regenerate our elementary geometry. It is certainly out of rule 



