262 . REASONING. 



How can this, in any single case, be proved by actual obser 

 vation ? We may follow 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 im 

 possibility 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 characteristic properties of geome 

 trical forms their capacity of being painted in the imagina 

 tion 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 on paper ; and in 

 the next place, make those pictures just as fit subjects of 

 geometrical experimentation as the realities themselves ; inas 

 much 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 inspection : 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 there 

 fore 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 

 serve as representatives of all which resemble them ; and in 

 the present case the conditions which qualify a real object to 

 be the representative of its class, are completely fulfilled by an 

 object existing only in our fancy. Without denying, therefore, 

 the possibility of satisfying ourselves that two straight lines 

 cannot inclose a space, by merely thinking of straight lines 

 without actually looking at them ; I contend, that we do not 



