DEMONSTRATION, AND NECESSARY TRUTHS. 299 



When we have occasion to extend these inductions, or 

 their consequences, to cases in which the error would 

 be appreciable to lines of perceptible breadth or thick- 

 ness, parallels which deviate sensibly from equidistance, 

 and the like we correct our conclusions, by combining 

 with them a fresh set of propositions relating to the 

 aberration; just as we also take in propositions relating 

 to the physical or chemical properties of the material, 

 if those properties happen to introduce any modifica- 

 tion into the result, which they easily may, even with 

 respect to figure and magnitude, as in the case, for 

 instance, of expansion by heat. So long, however, as 

 there exists no practical necessity for attending to any 

 of the properties of the object except its geometrical pro- 

 perties, or to any of the natural irregularities in those, 

 it is convenient to neglect the consideration of the other 

 properties and of the irregularities, and to reason as if 

 these did not exist : accordingly, we formally announce, 

 in the definitions, that we intend to proceed on this 

 plan. But it is an error to suppose, because we resolve 

 to confine our attention to a certain number of the 

 properties of an object, that we therefore conceive, 

 or have an idea of, the object, denuded of its other 

 properties. We are thinking, all the time, of precisely 

 such objects as we have seen and touched, and with 

 all the properties which naturally belong to them ; 

 but, for scientific convenience, we feign them to be 

 divested of all properties, except those in regard to 

 which we design to consider them. 



The peculiar accuracy, supposed to be charac- 

 teristic of the first principles of geometry, thus appears 

 to be fictitious. The assertions on which the reason- 

 ings of the science are founded, do not, any more 

 than in other sciences, exactly correspond with the 

 fact ; but we suppose that they do so, for the sake of 



