134 DEVELOPMENT AND PURPOSE CHAP. 



ceive them modified in similar degree, similar results must 

 be arrived at so that (in terms of magnitude) if they are 

 increased equally or decreased equally or left unchanged, 

 the resulting terms will be equal. Different conditions, 

 conversely, must have different results. Thus the principal 

 axioms involved in calculation are applications of the 

 general principle that precisely similar syntheses of similar 

 elements yield similar results, while the specific geometrical 

 axioms appear to be generalised forms of certain simple 

 combinations. 



There remains a second condition of valid construction. 

 The elements that enter into combination must have no 

 effect on one another other than that contemplated in the 

 rule of combination itself. This condition can never be 

 affirmed a priori of physical things. Suppose that a body 

 A acted on by a force P undergoes a certain acceleration in 

 a certain direction. Suppose that when acted upon by Q 

 it undergoes a certain acceleration in a certain direction. 

 What will happen when it is acted upon by P and Q 

 together ? It may be that when we compare the resultant 

 movement with that which would arise from P (say) acting 

 alone the difference corresponds accurately in direction and 

 magnitude with the movement which would result from Q 

 acting alone. In that case it may be said that the action of 

 each force in the combination corresponds precisely to its 

 action apart from the combination. Conversely, if this 

 correspondence can be assumed it is possible by a geometri- 

 cal construction to determine the joint effect of P and Q 

 from a knowledge of their separate effects. Accordingly, 

 wherever this assumption can be experientially justified we 

 can apply such a construction in the prediction or explana- 

 tion of phenomena, and this is what is effected with success 

 in the whole realm of mechanics. But it may be also that P 

 and Q are so modified by the very fact of their co-presence 

 that there is no precise correspondence or no correspondence 

 at all between their action in combination and their action 

 apart from combination. In that case we cannot apply our 

 geometrical construction to infer from the separate to the 

 joint effect. We can do nothing without specific experience 

 of the effect of the combination itself. More generally, 



