148 



REASONING. 



will perhaps be said that the assump- 

 tion does not extend to the actual, 

 but only to the possible existence of 

 such things. I answer that, accord- 

 ing to any test we have of possibility, 

 they are not even possible. Their 

 existence, so far as we can form any 

 judgment, would seem to be incon- 

 sistent with the physical constitution 

 of our planet at least, if not of the 

 universe. To get rid of this diflfi- 

 culty, and at the same time to save 

 the credit of the supposed system of 

 necessary truth, it is customary to say 

 that the points, lines, circles, and 

 squares which are the subject of geo- 

 metry, exist in our conceptions merely, 

 and are part of our minds ; which 

 minds, by working on their own 

 materials, construct an A priori 

 science, the evidence of which is 

 purely mental, and has nothing what- 

 ever to do with outward experience. 

 By howsoever high authorities this 

 doctrine may have been sanctioned, it 

 appears to me psychologically incor- 

 rect. The points, lines, circles, and 

 squares which 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 

 visibUe, the smallest portion of sur- 

 face which we can see. A line as 

 defined by geometers is wholly incon- 

 ceivable. 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 

 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, in- 

 stead 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 

 Oe can conceive what is called a mathe- 



matical line, thinks so from the evi< 

 dence 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 cannot be supposed to be 

 conversant about non-entities ; no- 

 thing remains but to consider geo- 

 metry 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 generalisations con- 

 cerning those natural objects. The 

 correctness of those generalisations, 

 as generalisations, is without a flaw : 

 the equality of all the radii of a circle 

 is true of all circles, so far as it ia 

 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 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 conclu- 

 sions 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 modification into the result ; 

 which they easily may, even with re- 

 spect to figure and magnitude, as in 

 the case, for instance, of expansion 

 by heat. So long, however, as there 

 exists no practical necessity for at- 

 tending to any of the properties of 

 the object except its geometrical pro- 

 perties, or to any of the natural irre- 

 gularities in those, it is convenient to 

 neglect the consideration of the other 



