650 SCIENCE AND HYPOTHESIS 



will be less than two right angles." Thus, however far the conse- 

 quences of Lobatschewsky's hypotheses are carried, they will never 

 lead to a contradiction; in fact, if two of Lobatschewsky's theorems 

 were contradictory, the translations of these two theorems made by 

 the aid of our dictionary would be contradictory also. But these 

 translations are theorems of ordinary geometry, and no one doubts that 

 ordinary geometry is exempt from contradiction. Whence is the 

 certainty derived, and how far is it justified? That is a question 

 upon which I cannot enter here, but it is a very interesting question, 

 and I think not insoluble. Nothing, therefore, is left of the objection 

 I formulated above. But this is not all. Lobatschewsky's geometry 

 being susceptible of a concrete interpretation, ceases to be a useless 

 logical exercise, and may be applied. I have no time here to deal 

 with these applications, nor with what Herr Klein and myself have 

 done by using them in the integration of linear equations. Further, 

 this interpretation is not unique, and several dictionaries may be 

 constructed analogous to that above, which will enable us by a simple 

 translation to convert Lobatschewsky's theorems into the theorems of 

 ordinary geometry. 



Implicit Axioms. Are the axioms implicitly enunciated in our 

 text-books the only foundation of geometry? We may be assured of 

 the contrary when we see that, when they are abandoned one after 

 another, there are still left standing some propositions whicTi are com- 

 mon to the geometries of Euclid, Lobatschewsky, and Kiemamu 

 These propositions must be based on premisses that geometers admit 

 without enunciation. It is interesting to try and extract them from 

 the classical proofs. 



John Stuart Mill asserted 1 that every definition contains an axiom, 

 because by defining we implicitly affirm the existence of the object 

 defined. That is going rather too far. It is but rarely in mathematics 

 that a definition is given without following it up by the proof of 

 the existence of the object defined, and when this is not done it is 

 generally because the reader can easily supply it; and it must not be 

 forgotten that the word " existence " has not the same meaning when 

 it refers to a mathematical entity as when it refers to a material 

 object. 



A mathematical entity exists provided there is no contradiction 

 implied in its definition, eithe ' in itself, or with the propositions pre- 

 viously admitted. But if the observation of John Stuart Mill cannot 

 be applied to all definitions, it is none the less true for some of them. 

 A plane is sometimes defined in the following manner : The plane is 

 a surface such that the line which joins any two points upon it lies 

 wholly on that surface. Now, there is obviously a new axiom con- 

 cealed in this definition. It is true we might change it, and that 

 i Logic, c. viii., cf. Definitions, 5-6. TB. 



