PROGRESSIVE EFFECTS. 363 



sis be considered as a number of causes exactly similar, successively intro- 

 duced, and producing by their combination the sum of the effects which 

 they would severally produce if they acted singly. The progressive rust- 

 ing of the iron is in strictness the sum of the effects of many particles of 

 air acting in succession upon corresponding particles of iron. The con- 

 tinued action of the earth upon a falling body is equivalent to a series of 

 forces, applied in successive instants, each tending to produce a certain con- 

 stant quantity of motion ; and the motion at each instant is the sum of the 

 effects of the new force applied at the preceding instant, and the motion 

 ali-eady acquired. In each instant a fresh effect, of which gravity is the 

 proximate cause, is added to the effect of which it was the remote cause ; 

 or (to express the same thing in another manner), the effect produced by 

 the earth's influence at the instant last elapsed is added to the sum of the 

 effects of which the remote causes were the influences exerted by the earth 

 at all the previous instants since the motion began. The case, therefore, 

 comes under the principle of a concurrence of causes producing an effect 

 equal to the sum of their separate effects. But as the causes come into 

 play not all at once, but successively, and as the effect at each instant is the 

 sum of the effects of those causes only which have come into action up to 

 that instant, the result assumes the form of an ascending series ; a succes- 

 sion of sums, each greater than that which preceded it ; and we have thus 

 a progressive effect from the continued action of a cause. 



Since the continuance of the cause influences the effect only by adding 

 to its quantity, and since the addition takes place according to a fixed law 

 (equal quantities in equal times), the result is capable of being computed 

 on mathematical principles. In fact, this case, being that of infinitesimal 

 increments, is precisely the case which the differential calculus was invent- 

 ed to meet. The questions, what effect will result from the continual ad- 

 dition of a given cause to itself, and what amount of the cause, being con- 

 tinually added to itself, will produce a given amount of the effect, are evi- 

 dently mathematical questions, and to be treated, therefore, deductively. 

 If, as we have seen, cases of the Composition of Causes are seldom adapt- 

 ed for any other than deductive investigation, this is especially true in the 

 case now examined, the continual composition of a cause with its own pre- 

 vious effects ; since such a case is peculiarly amenable to the deductive 

 method, while the undistinguishable manner in which the effects are blend- 

 ed with one another and with the causes, must make the treatment of such 

 an instance experimentally still more chimerical than in any other case. 



§ 2. We shall next advert to a rather more intricate operation of the 

 same principle, namely, when the cause does not merely continue in action, 

 but undergoes, during the same time, a progressive change in those of its 

 circumstances which contribute to determine the effect. In this case, as 

 in the former, the total effect goes on accumulating by the continual addi- 

 tion of a fresh effect to that already produced, but it is no longer by the 

 addition of equal quantities in equal times; the quantities added are un- 

 equal, and even the quality may now be different. If the change in the 

 state of the permanent cause be progressive, the effect will go through, a 

 double series of changes, arising partly from the accumulated action of the 

 cause, and partly from the changes in its action. The effect is still a pro- 

 gressive effect, produced, however, not by the mere continuance of a cause, 

 but by its continuance and its progressiveness combined. 



A familiar example is afforded by the increase of the temperature as 



