64 Transactions. — Miscellaneous. 



course, at any determinable or conceivable distance " (p. 103). Tins 

 is beside the question. The true question is, whether they will 

 necessarily intersect if the angle is, for instance, one decillionth 

 of a degree. Those who regard the Euclidian geometry as abso- 

 lutely true, must hold that they will. Modern mathematicians, 

 on the other hand, say that we do not know whether they will or 

 not. Who can prove that they will ? Euclid frankly admitted 

 that he could not, by assuming the alleged fact as his twelfth 

 axiom. Since Euclid's time, scores of mathematicians have 

 tried to prove it, but all their attempted proofs are justly re- 

 garded by their fellow-mathematicians as simply inconclusive. 

 It cannot be proved. Experiment cannot prove it ; reasoning 

 has failed to prove it : our intuitions — if, as disciples of the 

 experiential school of philosophy, we believe they have been 

 produced by the experience of our ancestors through millions of 

 years in the portion of space passed through by our solar 

 system in that time — cannot be trusted as infallible, and, there- 

 fore, cannot prove it. Lastly, it will not be contended that any 

 supernatural revelation has been vouchsafed on this point. 



11. "None of the evidence of Lobatchewsky in favour of this 

 is given by Mr. Fraukland " (p. 104). It did not fall within 

 my province to give this evidence. It is to be found in 

 Lobatchewsky's works. The evidence is admitted, and has long 

 been admitted, to be conclusive by all mathematicians who have 

 studied the question. Also, I think I may fairly add that the 

 burden of proof lies with those who say that an intersection 

 must and will take place, not with those who say that it may or 

 may not take place. 



12. "It appears to me that at any finite angle of convergence 

 of C T) to A B they will intersect at some determinable part of 

 the line A B,for a finite angle can only mean an angle of such a 

 size that it can be measured or conceived of." Just so : it can 

 be measured by the ratio of a finite arc (subtended by the angle) 

 to the radius of the same circle. But this does not prove that 

 it must be measured by a portion of the straight line A B. 

 How, then, does it follow as a " necessary corollary " that 

 " there is a point along A B which the line P will pass through?" 

 (p. 104.) It will hardly be considered a proof to say that " It 

 seems that the completion of the ideal construction thus begun 

 demands this intersection " (p. 103). If this can be proved, 

 the most remarkable advance in geometry since the time of 

 Euclid himself will have been made. A whole literature has 

 grown up in the attempt to furnish this proof. Its growth has 

 been arrested by the discoveries of Lobatchewsky and Gauss, 

 and I feel very sure that the desired proof will never be forth- 

 coming. 



13. Mr. Frankland (p. 106, note) " gravely informs us here, 

 that the finishing point or goal for a geodesic line in process 



