256 REASONING. 



the equal mobility of fluids in hydrostatics, the laws 

 of reflection and refraction in optics, are the first prin- 

 ciples of those sciences) ; but are merely necessary 

 assumptions, self-evident indeed, and the denial of 

 which would annihilate all demonstration, but from 

 which, as premisses, nothing can be demonstrated. In 

 the present, as in many other instances, this thoughtful 

 and elegant writer has perceived an important truth, 

 but only by halves. Finding, in the case of geometrical 

 axioms, that general names have not any talismanic 

 virtue for conjuring new truths out of the pit of 

 darkness, and not seeing that this is equally true in 

 every other case of generalization, he contended that 

 axioms are in their nature barren of consequences, and 

 that the really fruitful truths, the real first principles 

 of geometry, are the definitions ; that the definition, 

 for example, of the circle is to the properties of the 

 circle, what the laws of equilibrium and of the pressure 

 of the atmosphere are to the rise of the mercury in 

 the Torricellian tube. Yet all that he had asserted 

 respecting the function to which the axioms are con- 

 fined in the demonstrations of geometry, holds equally 

 true of the definitions. Every demonstration in 

 Euclid might be carried on without them. This is 

 apparent from the ordinary process of proving a pro- 

 position of geometry by means of a diagram. What 

 assumption, in fact, do we set out from, to demonstrate 

 by a diagram any of the properties of the circle ? 

 Not that in all circles the radii are equal, but only 

 that they are so in the circle ABC. As our warrant 

 for assuming this, we appeal, it is true, to the defini- 

 tion of a circle in general ; but it is only necessary 

 that you should grant the assumption in the case of 

 the particular circle supposed. From this, which is 

 not a general but a singular proposition, combined 



