THE CONCEPTUAL WORLD 27 



There is nothing at all about motion here. All 

 that we have done is to measure lengths. We have 

 made a kind of counterpoint, ^-points against Y-points, 

 but we have not even made a curve. We connect the 

 points A, B, C, D, E, etc., by means of short, straight 

 lines, and then we may connect together these short 

 lines, and, if we plot a number of intermediate points 

 between those that we have already obtained and join 

 these, the points may be so close together that they 

 may seem to be indistinguishable from a curve. Yet, 

 no matter how numerous they may be, they can never 

 be connected together so as to form a curve ; we there- 

 fore draw a curved 

 line freehand through 

 them, and at once, 

 in so doing, we aban- 

 don our intellectual 

 methods, for our curve 

 depends on our intui- 

 tion of continuously 

 changing direction. 



But if we think about it we shall find that we can 

 form no clear intellectual notion of continuity and we 

 can only measure the curvature of a line at a point in 

 the line by drawing a tangent to the curve at this 

 point, and then by measuring the slope of the tangent. 

 The curve itself we obviously leave out of considera- 

 tion. 



We cannot conceive of the point moving along the 

 locus OD. We can think of it only as at the places 

 0, A, B, C, D, E, etc., but we must neglect the intervals 

 OA, AB, BC, CD, DE, and so on, or we can divide them 

 into smaller intervals by supposing the point to have 

 occupied the positions /, g, i, j, between the points 

 A and B, for instance. Yet, no matter how many these 



