Review of T. P. Thompson s Geometry ivithout Axloim. 75 



/G 



such a fallacy. We believe the tedium of Mr. T.'s proofs was the true 

 dissuasive with hiai, aud we acknowledge the power of such a sedative. 



The attempt to pro\e that a line cannot turn 

 a convex front towards its asymptote, leads 

 directly to the consideration of deflections and 

 deviations. Some of our readers may exercise 

 their wits in confirming or disproving the fol- 

 lowing : — If a point P describes a line A P B 

 which perpetually deflects from the tangent towards the same side, and 

 ART, R P S are tangents at A and P, it is ordinary to assume the < 

 S R T as the measure of the total deflection performed through the arc 

 A P : and this, by reasoning from first principles, without any reference to 

 the doctrine of parallel lines. Is this warrantable or unwarrantable ? 

 Now consider a similar case. Let A B C be 

 any triangle : prolong A B to D : I say, the < 

 GBD=<A+<C. ForletB A, B C, A C 

 be prolonged to E, F, G : and let any point P 

 describe the crooked path E A C F. It makes 

 at A the deviation GAD, and at C the devia- 

 IS A B D (.Jqji F C G ; and in the rest of its motion does 



not deviate at all. But in describing C F, it is manifest that it has deviated 

 from the straight line E A D by the angle F B D ; therefore,* since the 

 total deviation is equal to the sum of the partial deviations, the < F B D = 

 <GAB+<FCG; or <C B D= <C A B+ < A C B : which was to be 

 proved. 



We are sure Mr. T. would desire no mercy shown to any of his proofs, 

 but only fair play. Indeed he handles his antagonists very roughly ; (for 

 all his predecessors in the attempt to solve the problem of parallels appear 

 as his antagonists) : and to say the truth, we do not admire the decision 

 with which he puts his extinguisher (in his Appendix) on some attempts 

 which have much interest for us, as hopeful and admitting improvement. 

 We do not assent to his condemnation of M. Bertrand's proof. But we 

 especially have in view Legendre's analytical proof, — not that we approve 

 the introduction of algebraic considerations: — we speak of the fundamental 

 j)rinciple, which, to say the truth, Mr. T. seems to us not at all to under- 

 stand. Indeed he himself complains, that when they insist on the " angle 

 being a portion of a finite whole; and the straight line, of an infinite 

 whole ; — there is no reasonable or demonstrated connexion between the 



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>- 



* Playfair's proof assumes more than this ; viz. 

 that the total deviation in a self-rejoining line is 

 four right angles ; which seeras equivalent to as- 

 suming outright what he is proving, that the four 

 exterior angles of any figure together equal four 

 right angles. Yet it may furnish suggestions. 



