DEMONSTRATION, AND NECESSARY TRUTHS. 149 



copies of the points, lines, circles, and squares which he has known in 

 his experience. A line as defined by geometers is wholly inconceiva- 

 ble. 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 exer- 

 cise over the operations of our minds ; the power, 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, instead of the whole. 

 But we cannot 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 be- 

 cause he supposes that unless such a conception were possible, mathe- 

 matics 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 ex- 

 ist any objects exactly coiTesponding 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 obvi- 

 ous 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 occa- 

 sion 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 abeiTation; just as we also take in 

 propositions relating to the physical or chemical propeities of the ma- 

 terial, if those properties happen to introduce any modification into the 

 result, which tliey easily may, even with respect to figure and magni- 

 tude, as in the case, for instance, of expansion by heat. So long, how- 

 ever, 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 con- 

 sideration of the other properties and of the iri-egularities, and to rea- 

 son 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 

 eiTor to suppose, because we resolve to confine our attention to a cer- 

 tain 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 characteristic of the first 

 principles of geometry, thus appears to be fictitious. The assertions 



