Review of T. P. Thompson s Geometry without Axioms. 79 



on the processes followed by tl»e mind. With such views, we much depre- 

 cate the artificial system and unbending style, pursued in most geometrical 

 treatises. Indeed we suspect that Mr. T. practically mistakes formality 

 for rigour; and thinks that his proofs are more cogent, because he puts 

 in premises which in ordinary reasoning we suppress. If a man reason 

 thus : " Good statesmen are valuable to a country : Lord Liverpool is a 

 good statesman ; therefore he is valuable to the country :" do we think his 

 argument the sounder for the aflected formality ?* It may be requisite now 

 and then ; but is disgusting in the extreme, and painfully perplexing, if 

 much carried out. The moment we quit Geometry, we gi\'e up the innu- 

 merable formalities of etiquette therein insisted on, (and to which Mr. T. 

 seems much attached,) nor is it conceived that we lose in rigour. In factf 

 all illustration, so that it be marked off as illustration, tends to rigour, by 

 giving more vivid apprehensions : while it relieves the tedium of dry rea- 

 soning. But as regards the order of a geometrical treatise, nothing, we 

 apprehend, need be said of this a priori. Let only the natural methods of 

 proof be well analyzed, and some luminous order will assuredly result ; 

 though it may be of a kind that we did not anticipate. But we do think 

 that the appearance of complete disorder, is a most serious presumption 

 against a geometrical compendium : the more especially since the value of 

 mathematical study is quite as much in training the mind to a habit of 

 clearness, as in the bare apprehension of what is good and what is bad 

 logic ; — nay, we should say, much more. It is a valuable habit, to discri- 

 minate quickly what is easier, what is more dii3ficult ; what is best proved 

 from what; in what order truths must be set before the mind: — but a 

 geometrical work which is just not fallacious, may train us to the very 

 reverse of this. 



And this leads us to express our conviction, that no patching of Euclid 

 will make his Elements a work worthy of the eulogiums that have been 

 lavished on it. We do not complain that it is ill adapted as an introduc- 

 tion to modern Mathematics, (though that be quite true) ; nor that he has 

 various petty flaws ; nay, nor that he has fundamental defects ; (for we are 

 supposing them all mended up :) but the treatise will still remain disordered, 

 crude, and formal. We will take this opportunity of throwing together 

 reasons, why all pretence of making his work the text book should be 

 abandoned. 



The natural connexion of subjects is broken, first, by the strange intro- 

 duction of ratios in the fifth book ; secondly, by interweaving with his 

 theorems problems of mere construction. Wc shall speak of each sepa- 



• Our illustration is pei'haps too strong. We only mean to say, that liis i)roofs 

 would gain clearness and pleasantness, and lose notiiing in rigour, by being less 

 diffuse. 



t Mr. T.'s Scliolia are very good ; and his cxi)lanations of many terms would be 

 serviceable to a beginner, though not to a person capable of reading Ms book. 



