578 VIII. ON THE STRUCTURE OF MATHEMATICS
toward zero and the aggregate of such points of division represents (in the rough only) a continuous line.
It is important that the reader should become thoroughly acquainted with the above simple considerations as they will be very useful in any line of endeavour. Here we already have learned how, somehow, to translate discontinuous jumps into 'continuous' smooth entities. Because of the structure of our nervous system we 'feel' 'continuity', yet we can analyse it into a smaller or larger number of definite jumps, according to our needs. The secret of this process lies in assigning an increasing number of jumps, which as they become vanishingly small, or tend to zero, as we say, cease to be felt as jumps and are felt as a 'continuous' motion, or change, or growth or anything of this sort.
An excellent example is given by the motion pictures. When we look at them we see a very good representation of life with all its continuity of transitions between joy and sorrow. If we look at an arrested film we find a definite number of static pictures, each differing from the next by a measurable difference or jump, and the joy or sorrow which moved us so in the play of the actors on the moving film, becomes a static manifold of static pictures each differing measurably from its neighbour by a slightly more or less accentuated grimace. If we increase the number of pictures in a unit of 'time' by using a faster camera and then release this film at the ordinary speed, we get what is called slow motion pictures with which we are all familiar. In them we notice a much greater smoothness of movements which in life are jerky, as, for instance, the movements of a running horse. They appear smooth and non-jerky, the horse looks as if it were swimming. Indeed we do swim no less than fishes, except that our medium; namely, air, is less dense than water, and so our movements have to be more energetic to overcome gravitation. The above example is indeed the best analogy in existence of the working of our nervous system and of the difference between orders of abstractions. Let us imagine that some one wants to study some event as presented by the moving picture camera. What would he do? He would first see the picture, in its moving, dynamic form, and later he would arrest the movement and devote himself to the contemplation of the static extensional manifold, or series, of the static pictures of the film. It should be noticed that the differences between the static pictures are finite, definite and measurable.
The power of analysis which we humans possess in our higher order abstractions is due precisely to the fact that they are static and so we can take our 'time' to investigate, analyse , . The lower order abstractions, such as our looking at the moving picture, are shifting and non-permanent and thus evade any serious analysis. On the level of looking at the moving film, we get a general feeling of the events, with a very imperfect memory of what we have seen, coloured to a large extent by our moods and other 'emotional' or organic states. We are on the shifting level of lower order abstractions, 'feelings', 'motions', and 'emotions'. The first lower centres do the best they can in a given case but the value of their results is highly doubtful, as they are not especially reliable. Now the higher order abstractions are produced by the