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



shortly to be assumed as in substantive existence : for we have afterwards 

 a new process to secure ourselves from fallacy. Thus, though we may, if 

 we please, use axioms to fix and ascertain the idea of straightness, it will 

 remain to be proved that there is between every two points one line and 

 one only conceivable, to have the property of straightness. 



But when a geometer professes to define all his terms, he has not the 

 shadow of a pretence to put forth axioms. If the definitions are sound, 

 they are infallibly adequate to prove all truths essentially connected with 

 them. If the definitions be inadequate, — descriptions rather than defini- 

 tions, — it should be avowed. Indeed the words Equal and Quantity are 

 not ordinarily defined by geometers ; hence the axioms which relate to 

 these terms are bearable enough : (though they belong to the earlier theory 

 of Quantity and Number, on which all Mathematics is based ; and not 

 specially to Geometry :) but those axioms which involve the terms line, 

 angle, straight, &c. are nothing but unwarrantable assumptions, as all 

 these have been (ill or well) defined. 



It is easy to see the analogy between axioms in pure science and laws 

 in mixed science. Tlie only reason why a treatise on Hydrostatics needs 

 to be based on the experimental laws of fluids, is because the writer can- 

 not or will not give a definition oi fluidity. If he does give a definition, he 

 turns his treatise into pure science} but it is possible then that he is 

 amusing himself with writing on that which does not exist. A metaphy- 

 sician might yet arise among us, who should not unplausibly maintain, that 

 Geometry, though it treat not of substantial bodies, yet as it treats of 

 space, which is a something with which we have acquaintance through the 

 senses, ought philosophically to be regarded as one of the mised sciences. 

 Such a one would not be inconsistent in making simple experiments the 

 basis of his reasoning : and his justification would be in this, that the 

 words space or solidity are undefined or illdefined : a circumstance which 

 seems most remarkably to escape notice. 



We are led to the following observations by the perusal of the little 

 book which is named at the head of this article. The number of past 

 fruitless attempts to get rid of the twelfth axiom of Euclid, ought not to 

 make us look with contempt on new efforts, until some one has shown that 

 it cannot be proved. But our present author, in his fourth edition, which 

 we have before us, has net confined himself to the much vexed question of 

 parallelism : he has spent much labour on the definitions of the straight 

 line and plane. And what gives new interest to his treatise is, that he 

 has set out on an entirely original method, making the discussion of certain 

 properties of the sphere precede the definition of the straight line and 

 plane. This makes the book peculiarly worthy of notice, whatever judg- 

 ment be formed of it : and we propose to set before our readers some 

 account of his object and method, before making our own remarks on the 

 execution. 



