DEMONSTRATION, AND NECESSARY TRUTHS, 309 



the lines to any distance we please ; but we cannot 

 follow them to infinity : for aught our senses can 

 testify, they may, immediately beyond the farthest 

 point to which we have traced them, begin to approach, 

 and at last meet. Unless, therefore, we had some 

 other proof of the impossibility than observation 

 affords us, we should have no ground for believing 

 the axiom at all. 



To these arguments, which I trust I cannot be 

 accused of understating, a satisfactory answer will, I 

 conceive, be found, if we advert to one of the charac- 

 teristic properties of geometrical forms their capacity 

 of being painted in the imagination with a distinctness 

 equal to reality: in other words, the exact resemblance 

 of our ideas of form to the sensations which suggest 

 them. This, in the first place, enables us to make 

 (at least with a little practice) mental pictures of all 

 possible combinations of lines and angles, which 

 resemble the realities quite as well as any which we 

 could make upon paper ; and in the next place, makes 

 those pictures just as fit subjects of geometrical 

 experimentation as the realities themselves ; inasmuch 

 as pictures, if sufficiently accurate, exhibit of course 

 all the properties which would be manifested by the 

 realities at one given instant, and on simple inspec- 

 tion : and in geometry we are concerned only with 

 such properties, and not with that which pictures 

 could not exhibit, the mutual action of bodies one 

 upon another. The foundations of geometry would 

 therefore be laid in direct experience, even if the 

 experiments (which in this case consist merely in 

 attentive contemplation) were practised solely upon 

 what we call our ideas, that is, upon the diagrams in 

 our minds, and not upon outward objects. For in all 

 systems of experimentation we take some objects to 



