Moreover, however little we reflect upon the certainty of the truths intimately connected with experience, such as are arithmetical and geometrical truths, we cannot fail to perceive that the confidence with which we build upon them is not founded upon induction, but that we assent to them independently of any particular fact, and consider their truth as absolutely necessary, although we cannot verify it by the touchstone of experience. 145. The verification of ideas by facts is in many cases impossible, because the weakness of our perception and of our senses, and the coarseness of the instruments we use, fail to render us certain that the facts correspond exactly to the ideas. It is sometimes absolutely impossible to establish this proof, since geometrical truth supposes conditions such as cannot be realized in practice. 146. Let us apply these observations to the simplest truths of geometry. Certainly no one will doubt the solidity of the proof called superposition: that is to say, if one of two lines, or surfaces, be placed upon the other, and they exactly correspond, they will be equal. This truth cannot depend upon experience: first, because experience is limited to a certain number of cases, whereas the proposition is general. To say that one serves for all is to say that there is a general principal, independent of experience, since, without recognizing an intrinsic necessity in this truth, the universal could in no other way be deduced from the particular. Secondly, because even where experience avails, it is impossible for us to make it exact, since superposition made in the most delicate manner imaginable, can never attain to geometrical exactness, which repudiates the minutest difference in any point. It is an elementary theorem, that the three angles of a triangle are equal to two right angles. This truth does not rest upon experience: first, because the universal cannot be deduced from the particular; secondly, because, however delicate be the instruments for measuring angles, they cannot measure them with geometrical exactness; thirdly, because geometry supposes conditions which we cannot realize in practice; lines have no thickness, and the vertices of angles are indivisible points. 147. If general principles depended upon experience they would cease to be general, and would be limited to a certain number of cases. Neither would their enunciation be absolute, even for the cases already observed; for it would of necessity be reduced to what had been observed, that is to say, to a little more or less, but never be perfect exactness. Consequently we could not assert that the three angles of every triangle are equal to two right angles; all that we could say would be, that so far as our experience goes, we have observed that in all triangles the three angles are very nearly equal to two right angles.