268 tNDUCTION. 



another, witli a force directly as their mass and inversely as the square 

 of the distance, all the resources of the calculus have not hitherto 

 sufficed to obtain anything more than an approximate general solution. 

 In a case a little more complex, but still one of the simplest w^hich 

 arise in practice, thajt of the motion of a projectile, the causes which 

 affect tlie velocity and range (for example) of a cannon-ball may be all 

 known and estimated ; the force of the gunpowder, the angle of eleva- 

 tion, the density of ..the air, the strength and direction of the sound; 

 but it is one of the most difficult of all mathematical problems to 

 combine all these, so as to determine the effect resulting from their 

 collective action. i 



Besides the theorems of nvimber, those of geometry also come in as 

 premisses, where the effects take place in space, and involve motion 

 and' extension, as in mechanics, optics, acoustics, astronomy. But 

 when the complication increases, and the effects are under the influ- 

 ence of so many and such shifting causes as to give no room either for 

 fijXed numbers, or for straight lines and regular curves, as -in the case 

 of .physiological, to say nothing of mental and social phenomena, the 

 laws of number and extension are applicable, if at all, only on that 

 large scale on which precision of details becomes unimportant ; and 

 although these laws play a conspicuous part in the most striking 

 examples of the investigation of nature by the Deductive Method, as 

 for example in. the Newtonian theory of the celestial motions, they are 

 by, no means an indispensable part of every such process. All that is 

 essential in it is the ratiocination from a general law to a particular 

 case, that is, the determination, by means of the particular circum- 

 stances of that case, what result is required in that instance to fulfill the 

 law. Thus, in the Torricellian experiment, if the fact that air had 

 weight had been previously kno^yn, it would have been easy, without 

 any numerical data, to deduce from the general law of equilibrium, 

 that the .mercury W^ould stand in the tube at such a'height that the 

 column of mercury would exactly balance a column of the atmosphere 

 of equal diameter; because otherwise, eqiiilibrium would not exist. 



-By such ratiocinations from the separate laws of the causes, we may, 

 to a certain extent, succeed in answering either of the following ques- 

 tions : Given a certain combination of causes, what effect will follow] 

 and, What combinatioTn of causes, if it existed, would produce a given 

 effect ] Tti the one case, we determine the effect to be exj>ected in 

 any complex circumstances of which the different elements are known : 

 in the other case we learn, according to what law — -under what ante- 

 cedent conditions — a given complex effect will recur. 



§ 3. But (it may here be asked) are not the same arguments by 

 which the -methods of direct observation and experiment were set 

 aside as illusory when applied to the laws of complex phenomena, 

 applicable with equal force against the Method of Deduction 1 When 

 in every single instance a multitude, often an unknown multitude, of 

 ag'encies, are clashing and combining, what security have we that in 

 our computation a priori we have taken all these into our reckoning ? 

 How many must we not generally be ignorant of? Among those 

 which we know, how probable that some have been overlooked ; and 

 even were all included, how vain the pretence of summing up the 

 effects of many causes, unless we know accurately the numerical law 



