DESIGN FOR A BRAIN 19/13 



in which the right-hand side contains no functions of t except those 

 whose fluxions are given on the left. 



19/13. A simple example of a system which is regular but not 

 state-determined is given by the following apparatus. A table 

 top is altered so that instead of being flat, it undulates irregularly 

 but gently like a putting-green (Figure 19/13/1). Looking down 



Figure 19/13/1. 



on it from above, we can mark across it a grid of lines to act 

 as co-ordinates. If we place a ball at any point and then release 

 it, the ball will roll, and by marking its position at, say, every 

 one-tenth second we can determine the lines of behaviour of the 

 two-variable system provided by the two co-ordinates. 



If the table is well made, the lines of behaviour will be accur- 

 ately reproducible and the system will be regular. Yet the 

 experimenter, if he knew nothing of forces, gravity, or momenta, 

 would find this two- variable system unsatisfactory. He would 

 establish that the ball, started at A, always went to A' \ and 

 started at B it always went to B'. He would find its behaviour at 

 C difficult to explain. And if he tried to clarify the situation by 

 starting the ball at C itself, he would find it went to D ! He would 

 say that he could make nothing of the system ; for although each 

 line of behaviour is accurately reproducible, the different lines 

 of behaviour have no simple relation to one another. He will, 

 therefore reject this two-variable system and will not rest till 

 he has discovered, either for himself or by following Newton, 



248 



