DEMONSTRATION, AND NECESSARY TRUTHS. 169 



to me psychologically incorrect. The points, lines, circles, and squares 

 whicJi. any one has in his mind, are (I apprehend) simply copies of the 

 points, lines, circles, and squares which he has known in his experience. 

 Our idea of a point, I apprehend to be simply our idea of the minimum 

 vislbile, the smallest portion of surface which we can see. A line, as de- 

 fined by geometers, is wholly inconceivable. We can reason about a line 

 as if it had no breadth ; because we have a power, which is the foundation 

 of all the control we can exercise over the operations of our minds ; the 

 jDOWpr, when a perception is present to our senses, or a conception to our 

 intellects, of attending to a part only of that perception or conception, in- 

 stead of the whole. But we can not conceive a line without breadth -, we 

 can form no mental picture of such a line : all the lines which we have in 

 our minds are lines possessing breadth. If any one doubts this, we may 

 refer him to his own 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 possible, 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 can not be supposed to be conversant about nonentities ; nothing 

 remains but to consider geometry as conversant w'ith 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, «&■ gener- 

 alizations, 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 impor- 

 tance in practice will be incurred by feigning it to be exactly true. 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 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 aberration; just as we also take in proposi- 

 tions 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 ex- 

 cept 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 : according- 

 ly, 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 w^e there- 

 fore conceive, or have an idea of, the object, denuded of its other propei*- 

 ties. 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 which are material to our purpose, <ind in regard 

 to which we design to consider them. 



The peculiar accuracy, supposed to be characteristic of the first princi- 



