212 REASONING. 



its probative force. But he imagined this to be a peculiarity 

 belonging to axioms ; and argued from it, that axioms are not 

 the foundations or first principles of geometry, from which all 

 the other truths of the science are synthetically deduced (as 

 the laws of motion and of the composition of forces in dyna 

 mics, the equal mobility of fluids in hydrostatics, the laws of 

 reflection and refraction in optics, are the first principles of 

 those sciences) , but are merely necessary assumptions, self- 

 evident indeed, and the denial of which would annihilate all 

 demonstration, but from which, as premises, 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 well where they lie hid, and not 

 seeing that this is equally true in every other case of generali 

 zation, 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 confined 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 proposition 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 definition of a circle 

 in general; but it is only necessary that the assumption be 

 granted in the case of the particular circle supposed. From 

 this, which is not a general but a singular proposition, com 

 bined with other propositions of a similar kind, some of which 

 when generalized are called definitions, and others axioms, we 

 prove that a certain conclusion is true, not of all circles, but 



