Mr Meikle vn the Theory of JParailel Lines. 811 



tensive writers on tLe subject in our own times, may be reckoned ^ 

 Legendre in his ElcmentSy and in the Memoires de VAcadcmie, as after 

 noticed ; and Colonel T. Perronet Thompson, in his " Geometry without 

 Axioms," and in his more recent tract proposing a proof by help of the 

 logarithmic spiral. To the former of his treatises, Colonel Thompson 

 has appended critical notices of thirt}' of the more plausible methods 

 which have, at very different times, been given by other authors as 

 demonstrations. His criticisms, though neither always the fir&t which 

 have been made on these methods,* nor yet all new, are generally just» 

 and even fatal to them. He strongly objects, though rather metaphysi- 

 cally, to employing infinite quantities, as is done by M. Bertrand, who 

 reasons upon a numerous set of areas, each of which is of infinite mag- 

 nitude, and such that, while on one side it has no limit of any kind, it 

 is, on other sides, bounded by infinite lines stretching immeasurably 

 beyond the fixed stars. But, from other and more elementary consi- 

 derations, I shall afterwards briefly shew Bertrand's method to be a 

 complete failure. Colonel Thompson himself, however, does not seem 

 to have been aware that, in his own attempt at a sort of mechanical 

 demonstration, in which he supposes a straight line to preserve its 

 parallelism whilst *' travelling" laterally, no finite line could suflBce for 

 such " travelling line." Again, in calling in the aid of the logarithmic 

 spiral, he has assumed that triangles formed partly of unequally curved 

 arcs of that spiral, and partly of arcs of unequal circles, are equal to 

 one another, and identical with rectilinear triangles. This, like most of 

 the attempts on Euclid's axiom, evidently assumes far more than the 

 whole thing to be proved. Nay, supposing the curved sides of those 

 triangles were of the same lengths as if formed of straight lines, it would 

 still be necessary to prove, what is not even true, that their angles are 

 the same. But these are by no means the only objections to which his 

 demonstrations are liable. 



Much of the form of the present essay has been adopted with a view 

 to brevity ; various minute and commonplace details are omitted which 

 every tyro can readily supply, and occasionally some abbreviations and 

 modes of reasoning are adopted, which, though not very common at that 

 stage of geometry, are weU known, and Jiave no dependence xm w"hat is 

 to be proved. But independently of this, the question is here treated 

 in a very different manner from anything I had previously met with on 



* For instance, in the Philosophical Magazine for December 1822, 1 had greatly 

 anticipated him in refuting the 80th, or Mr Ivorj's, method; for I have there 

 shewn that it depends on the very liberal assumption that four straight lines maj 

 Always be made to meet round any given polygon, so as completely to inclose i:^ 

 let it have ever so maqy sides : whioh is far more -cou^plicated and -lesfi evidexkt 

 than Euclid's axiom. 



