74 Review ofT. P. Tliompson's Geometry without Axioms. 



we may produce the systems T C D U, U D E V, V E F W, W F G X. . 

 ^c.,ad 2;j/?w.— all counterparts of the first. It follows also that C T never 

 meets B S, nor does D U meet C T, nor E V meet D U, &c.. . so that a for- 

 tiori, no line in the series as F W or G X can meet A R. This puts the 

 absurdity of the original supposition in a striking light before the eyes : but 

 this is not to give reasons for the thing. Let now M N be perpendicular 

 to A M. It is then easy to take the distance A M so short, that M N may 

 meet the crooked path ; but if A M be great, it is not so obvious that this 

 will happen. In reality we know it will not j for we know that that path 

 will bend round and cross A R — but this is excluded by hypothesis— so 

 that the argument is a dilemma ; either the path shall cross A R, and then 

 all we want to prove is conceded ; else, however distant M may be, I say, 

 M N shall meet the path. Here is the pinch and nip ! " For else, let A M 

 be the least distance such that M N does not meet it. Then A G, M N 

 are infinite lines asymptotic to each other ; of which the latter is straight 

 and the former is convex towards the latter, which is absurd." Thus we 

 have ventured to supply our author's reasoning. It seems to us, that 

 the effect of his labours is to reduce the twelfth axiom of Euclid to 

 another much more obvious, and which perhaps may be proved, but which 

 nevertheless he has not proved ; that " a crooked line cannot turn its con- 

 vexity towards a straight line which is asymptotic to it." If this be granted 

 him, he proves that any tessera A B /3 a [cut off from A R and B S, A a= 

 B /3, and join a /3] which has the angles at A and B acute, cannot have the 

 angles at a and ji right ; which leads by easy steps to Euclid's twelfth 

 axiom. We conceive he is still engaged with the second part of the di- 

 lemma. Either the crooked path meets A R, and so the twelfth axiom is 

 granted outright : else he proves this and that about the tessera, whence 

 ultimately the twelfth axiom is still made to follow. If we be right, Mr. 

 T. ought not to have isolated his propositions as he has done. 



And this strikes us as a sufficient reply to the startling objection of an 

 ingenious reviewer,* that for any disproof which Mr. T. has offered, the 

 point D might coincide with A, E with B, F with C, &c for all con- 

 ceivable values of the angle and length. We are not surprised that Mr. T. 

 did not foresee so extraordinary a thought, and we give credit to our con- 

 temporary for the inventiveness of it. Doubtless it would vitiate all Mr. 

 T.'s after-reasoning : but it would amount to a direct concession of all that 

 he is seeking to prove ; viz. that the path must meet A R a second time. 

 The objection however was just, as levelled against a proposition, which 

 ought not to stand (we conceive) as a positive truth, but as on a hypothesis 

 still kept up till the chain of argument is complete. The reviewer, though 

 acute and clear, had evidently a strong prepossession against the possibility 

 of Mr. T.'s success ; and was deterred from reading further, when he met 



* Quarterly Journal of Education, No. XIII. 



