DEMONSTRATION, AND NECESSARY TRUTHS. 253 



experience. I much question if any one who fancies that he 

 can conceive what is called a mathematical line, thinks so 

 from the evidence of his consciousness : I suspect it is rather 

 because he supposes that unless such a conception were possi- 

 ble, mathematics could not exist as a science : a supposition 

 which there will be no difficulty in showing to be entirely 

 groundless. 



Since, then, neither in nature, nor in the human mind, do 

 there exist any objects exactly corresponding to the definitions 

 of geometry, while yet that science cannot be supposed to be 

 conversant about non-entities ; nothing remains but to consider 

 geometry as conversant with such lines, angles, and figures, as 

 really exist ; and the definitions, as they are called, must be 

 regarded as some of our first and most obvious generalizations 

 concerning those natural objects. The correctness of those 

 generalizations, as generalizations, is without a flaw : the 

 equality of all the radii of a circle is true of all circles, so far 

 as it is true of any one : but it is not exactly true of any 

 circle ; it is only nearly true ; so nearly that no error of any 

 importance in practice will be incurred by feigning it to be 

 exactly true. When we have occasion to extend these in- 

 ductions, or their consequences, to cases in which the error 

 would be appreciable to lines of perceptible breadth or 

 thickness, 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 aberra- 

 tion ; just as we also take in propositions relating to the 

 physical or chemical properties of the material, if those 

 properties happen to introduce any modification 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 properties, 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 



