238 INDUCTIOiV. 



perpendicular, wliicli in ordinary circumstances lasts but a few minutes, 

 was prolonged in Borda's experiments to more than thirty hours, by 

 diminishing as much as possible the friction at the point of susjjension, 

 and by making the body oscillate in a Space exhausted as nearly as 

 possible of its air. There could therefore be no hesitation in assign- 

 ing the whole of the retardation of motion to tbe influence of the 

 obstacles ; and since, after subducting this retardation from the total 

 phenomenon, th© remainder was an uniform velocity, the result was 

 the proposition known as the first law of motion. 



There is also another characteristic uncertainty affecting the infer- 

 ence that the law of variation which the quantities observe within our 

 limits* of observation, will hold beyond those limits. There is of 

 course, in the first instance, the possibility that beyond the limits, and 

 in circumstances, therefore of which we have no direct experience, 

 some counteracting cause might develop itself; either a new agent, or 

 a new property of the agents concerned, which lies dormant in the 

 circumstances we are^ able to observe. This is an element of uncer- 

 tainty which enters largely into all our predictions of effects'; but it is 

 Jiot peculiarly applicable to the Method of Concomitc^nt Variations. 

 The uncertainty, however, of which I am about to speak, is character- 

 istic of that method ; especially in the cases in which the extreme 

 limits, of our observation are very narrow, in comparison with the 

 possible variations in the quantities of the phenomena. Any one ^^■ho 

 has the slightest acquaintance with mathematics, is aware that very 

 different laws of variation may produce numerical results which differ 

 but slightly from one another within narrow limits ; and it is often 

 only when the absolute amounts of variation are considerable, that the 

 difference between the results given by one law and by another, be- 

 comes appreciable. When, therefore, such variations in the quantity 

 of the antecedents as we have the means of observing, are but small in 

 comparison with the total qua:ntities, there is much danger lest we 

 should mistake the numerical law, and be led quite to miscalculate the 

 variations which would take place beyond the limits; a miscalculation 

 which would vitiate any conclusion respecting the dependence of the effect 

 upon the cause, whidi could be founded upon those variations. Exam- 

 ples are not wanting of such mistakes. "The formulae," says Sir John 

 Herschel,* " which have been empirically deduced for the elasticity of 

 steam (till very recently), and those for the resistance of fluids, and 

 other similar subjects," when relied on beyond the limits of the obsei-- 

 vations from which they were deduced, " have almost invariably failed 

 to support the theoretical structures which have been erected on them." 



Under this uncertainty j the conclusion we may draw froin the con- 

 comitant variations of a and A, to the existence of an invariable and 

 exclusive connexion between them, or to the permanency of the same 

 numerical relation .between their variations when the quantities are 

 much gi-eater or smaller than those whiqh we have had the means of 

 observing, cannot be considered to rest upon a complete induction. 

 All that in such a case can be regarded as proved on the subject of 

 causation, is that there is some connexion between the two phenomena ; 

 that A, or something which can influence A, must be one of the causes 

 which collectively determine a. We may, however, feel assured that 



* Discourse on the Sludy of Natural Philosophy, p. 179, 



