Review ofT. P. Thompsons Geometry without Axioms. 



73 



and called us to consider whether the early introduction of the sphere would 

 not tend to perfect the doctrine of the straight line. 



On his second main proposition, (!iis eleventh) we have to remark similar 

 defects as in the former. It has two cases, of which the former is doubtfully 

 proved : the latter we think is certainly not proved: for he does not show 

 that the points M, N, O, P may not be in one straight line ; and this is es- 

 sential to the proof, if we rightly understand it. Yet we confess it is very 

 hard ; so hard as to make us doubt every thing ; where indeed all the 

 hypotlieses are so monstrous, that the mind is bewildered in the midst 

 of so many false liglits. AVhen it is hard to say what is more manifestly 

 absurd than what, the reductio ad absiirdum is a most dangerous proof; 

 needing a perpetual eftbrt of most painful vigilance, even from practised 

 mathematicians. 



His third main proposition is satisfactory, though excessively tedious. 

 But in fact, there is no dilSculty at all in the doctrine of the plane, so soon 

 as Euclid I. 8, has been established j which is virtually Mr. T.'s eleventh 

 proposition, just noticed. For if a right angle be first defined, (which is 

 not difficult) we may generate a plane by supposing one leg of the angle 

 fixed while the other revolves round itj then the general property of the 

 plane is readily established by a method analogous to Eucl. XI. 4. 



We now feel ourselves bound to attempt to convey to our readers some 

 notion of his method of treating parallels, the more especially as other 

 reviewers decline the task, vphich is not an easy one. Anxious to do him 

 justice, we have diligently studied what he rightly, but funnily calls, "the 

 ])inch and nip" of the argument ; and we would fain put to him some vivd 

 voce questions, where the slippery materials appear to elude the grasp of 

 our forceps. We shall, without apology, throw his matter into the form 

 that strikes us as most intelligible to our readers. 



Conceive the equal angles R A B, 

 A B S in a given plane, which may 

 be taken so small as that A R, B S 

 meet towards R and S ; while if they 

 be riglit angles, we know that A R, 

 B S will meet neither way. The 

 question to be decided is, whether 

 they will certainly meet, if the angles 

 be ever so little less than right angles. 

 If possible, let it be otherwise, (for 

 precision, we might suppose the angle 

 the least possible, consistently with 

 the condition of the lines not meet- 

 ing.) Suppose the system to turn 

 about B S, till R A B imprints its 

 counterpart T C B on the plane, at the opposite side of B S. Similarly 

 No. I.— Vol. I. . 



