THE CONCEPTUAL WORLD 21 



or less than two right angles. But our Infusorian could 

 not imagine a triangle in which the sum of the angles 

 was not greater than two right angles, for all its figures 

 would be drawn on a convex surface. 



Our three-dimensional geometry depends, therefore, 

 on our modes of activity and the concepts with which 

 it operates ; points, straight lines, etc. are conceptual 

 limits to those modes of activity. We can imagine 

 a straight line only as a direction along which we can 

 move without deviating to the right or the left, or up 

 or down. But even if we draw such a line on paper 

 with a fine pencil the trace would still have some width, 

 and we can imagine ourselves small enough to be able 

 to deviate to the right or the left within the width of 

 the line drawn on the paper. We might make a very 

 small mark on the paper, but no matter how small 

 this mark is it would still have some magnitude ; 

 otherwise we should be unable to see it. If the straight 

 line had no width and the point no magnitude they 

 would have no perceptual existence. Our perceptual 

 triangles are not figures, the angles of which are 

 necessarily equal to two right angles. If we drive 

 three walking sticks into a field and then measure 

 the angles between them by means of a sextant we 

 shall find that the sum is nearly i8o°, but in general not 

 that amount. If we stick a darning needle into the 

 heads of each of the walking sticks and then remeasure 

 the angles by means of a theodolite we shall obtain 

 values which are nearer to that of two right angles, 

 but we should not, except by " accident," obtain 

 exactly this value. We do not, therefore, get the 

 " theoretical " result, and we say this is because of the 

 errors of our methods of observation ; but why do we 

 suppose that there is such a theoretical result from 

 which our observations deviate, if our observations 



