154 REASONING. 



arguments. These are reducible to two, which I shali endeavor to 

 state as clearly and a^ forcibly, as possible. 



§ 5. In the first place it is said, that if ovir assent to the proposition 

 that two straight lines cannot inclose a space, were derived from the 

 senses, we could only be convinced of its truth by actual trial, that is, 

 by seeing or feeling the straight liijes ; whereas in fact it is seen to be 

 ti-ue by. merely thinking of them. That a stone thrown into water 

 goes to the bottom,'may be perceived by our senses, but mere think- 

 ing of a stone thrown into the water will never lead us to that conclu- 

 sion : not so, however, with the axioms relating to sti'aight lines : if I 

 could be made to conceive what a straight line is, without having seen 

 one, I should at once recognize that two such lines cannot inclose a 

 space. Intuition is "imaginary looking;"* but experience must be 

 real looking : if we see a property of straight lines to be true by merely 

 fancying ourselves to be looking at them, the ground of our belief can- 

 not be the senses, or experience ; it must be something mental. 



To this argument it might be added in the case of this particular 

 axiom (for the assertion would not be true of all axioms), that the evi- 

 dence of it from actual ocular inspection, is not only unnecessary, but 

 unattainable. What says the axiom 1 That two straight lines cannot 

 inclose a space ; that after having once intersected, if they are pro- 

 longed to infinity they do not meet, but continue to diverge fi-om one 

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

 observation 1 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 furthest 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 under- 

 stating, a satisfactory answer will, I conceive, be found, if we advert 

 to one of the characteristic properties of geometrical forms — their 

 capacity of being painted in the imagination with a distinctness equal 

 to reality : in other wordsi, 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 accu- 

 rate, 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 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 talvO some 

 objects to serve, as representatives of aU whjich resemble them ; and in 



* Whewell's Philosophy of the Inductive Sciences, i., 130. 



