22 THE PHILOSOPHY OF BIOLOGY 



themselves do not in general give this ideal result ? 

 We might accumulate a great series of measurements 

 of the angles of our triangle, and we should then find 

 that these results would tend to group themselves 

 symmetrically round a certain value which would be 

 i8o°. Some of the results would be considerably 

 less than the ideal, and some of them would be con- 

 siderably more ; but these relatively great deviations 

 would be small in number and most of the results 

 would be a very little less than i8o° or a very little 

 more, and there would be as many which would be a 

 little less as those that were a little more. We should 

 have formed a " frequency distribution " ^ with its 

 " mode " at i8o°. 



But by " reasoning " about the " properties " 

 of these lines and triangles in plane two-dimensional 

 space, we should arrive at the conclusion that the 

 angles of a triangle were equal to i8o°, and neither 

 more nor less. We should then think of a straight 

 line as still a path along which we move in imagination, 

 and a path which still has some width. But we 

 imagine the width of the path to become less and less, 

 so that, even if we imagine ourselves to become thinner 

 and thinner, we should be unable to deviate either to 

 the right or left in moving along the path, because the 

 thinner we make ourselves the thinner becomes also 

 the path. We imagine our intuition of a deviation 

 to the right or left becoming keener and keener, so 

 that, no matter how small the deviation we should still 

 be able to appreciate it by the extra exertion which it 

 would involve. We think of a point as a little spot,, 

 and we think of ourselves as being very small indeed, 

 so that we can move about on this spot. But we can 

 reduce the area of the spot more and more, until it 



^ See appendix, p. 350. 



