DEMONSTRATION, AND NECESSARY TRUTHS. 281 



matter the elements of all our experience and all our knowledge, surely these are 

 the truths which, if experience suggest to us any truths at all, it ought to suggest 

 most readily, clearly, and unceasingly. If it were a truth, universal and neces- 

 sary, that a net is spread over the whole surface of every planetary globe, we 

 should not travel far on our own without getting entangled in its meshes, and 

 making the necessity of some means of extrication an axiom of locomotion. . . . 

 There is, therefore, nothing paradoxical, but the reverse, in our being led by 

 observation to a recognition of such truths, as general propositions, coextensive 

 at least with all human experience. That they pervade all the objects of expe- 

 rience, must ensure their continual suggestion by experience ; that they are 

 true, must ensure that consistency of suggestion, that iteration of uncontra- 

 dicted assertion, which commands implicit assent, and removes all occasion of 

 exception ; that they are simple, and admit of no misunderstanding, must 

 secure their admission by every mind." 



"A truth, necessary and universal, relative to any object of our knowledge, 

 must verify itself in every instance where that object is before our contemplation, 

 and if at the same time it be simple and intelligible, its verification must be 

 obvious. The sentiment of such a truth cannot, therefore, but be present to our 

 minds whenever that object is contemplated, and must therefore make a part of the 

 mental picture or idea of that object which we may on any occasion summon before 

 our imagination. . . . All propositions, therefore, become not only untrue but 

 inconceivable, if ... axioms be violated in their enunciation." 



Another eminent mathematician had previously sanctioned by his authority 

 the doctrine of the origin of geometrical axioms in experience. * ' Geometry 

 is thus founded likewise on observation ; but of a kind so familiar and obvious, 

 that the primary notions which it furnishes might seem intuitive." Sir John 

 Leslie, quoted by Sir William Hamilton, Discourses, &c. p. 272. 



