78 Review of T. P. Thompsons Geometrif without .4<rioms. 



ever, we were chiefly startled by the inconsistency of a passage which 

 seems to show that he regards Geometry, as a science whose elements may 

 possibly depend on the planetary system. He asks how we could tell ^ 

 priori that "Mature," instead of making the angles of a triangle equal to 

 two right angles, had not made their sum depend on the relation of one of 

 their sides to some grand modulus e.risting in nature, suppose, the radius 

 of the earth's orbit. If this be for one moment even conceivable, all Geo- 

 metry must be based on experimental laws, and nothing then can be so 

 unreasonable as Mr. T.'s aversion to appeal to the senses. We trace 

 however the same indistinctness of apprehension, when he complains of 

 Euclid's definition of a straight line, as being "an identical proposition." 

 Had he said, " the definition itself needs an explanation," we should as- 

 sent. But his tone seems to show, that he considers identical propositions 

 as nugatory. All propositions in pure science are as truly identical as 

 the binomial theorem and the multiplication table : yet they are not there- 

 fore nugatory. Either geometrical propositions are such ; or Geometry is a 

 mixed science, and must be based ultimately on sense. AVe can find no 

 intermediate view. 



"We further feel that Mr. T. inadequately appreciates the force of the 

 objection ; that " a proof is too difficult j" and again, that "the parts of a 

 subject are straggling." He says : " It may be a great irregularity, that 

 nature should not have framed the elements of Geometry, so as to present 

 a concinnous whole j" but " we ought not to quarrel with the dispensation." 

 If indeed our object were to ascertain the certainty of a practical truth, of 

 course we should be thankful to have atit/ demonstration : and where the 

 truth is such as no man is aware of till it is formally proved, we do not 

 complain if the proof be difficult. But when the mind has a direct and 

 distinct perception of a truth without proof at all, it cannot be the natural 

 method to lead us through many tedious and intricate ways to arrive at it. 

 Let an intelligent person unversed in Mathematics be asked "whether 

 spheres, if very large, might touch in more points than one : and probably 

 his fi.-st conception will be that they may ; because he unawares appeals 

 to experience, by trying to conceive how it would be to his eje. But 

 when reminded that it is owing to the roughness of the surface and the 

 compressibility of the material, that they come into contact more exactly : 

 lie soon convinces himself that perfect spheres would touch but in one 

 point. It is then clear, first, that the truth is not intuitive ; next, that it 

 is not learned by experience ; thirdly, that it is not gained by wading 

 through a whole book about the triangle and circle, nor yet through 

 Mr. T.'s long proof. The mind does certainly find some short cut, con- 

 necting the definition of perfect sphericity with the property alleged : and 

 by analysis of the vulgar reasoning, the philosophic geometer should en- 

 deavour to arrive at the natural method. By this not only would the study 

 be made more pleasant and satisfactory, but more light would be thrown 



