THE DEDUCTIVE METHOD. 329 



only are the most advanced truths of mathematics often required to enable 

 us to compute an effect, the numerical law of which we already know ; but, 

 even by the aid of those most advanced truths, we can go but a little way. 

 In so simple a case as the common problem of three bodies gravitating to- 

 ward one another, with 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 any general solution, but an approximate one. In a case 

 a little more complex, but still one of the simplest which arise in practice, 

 that of the motion of a projectile, the causes which affect the velocity and 

 range (for example) of a cannon-ball may be all known and estimated : the 

 force of the gunpowder, the angle of elevation, the density of the air, the 

 strength and direction of the wind; but it is one of the most difficult of 

 mathematical problems to combine all these, so as to determine the effect 

 resulting from their collective action. 



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

 premises, where the effects take place in space, and involve motion and ex- 

 tension, as in mechanics, optics, acoustics, asti'onomy. But when the com- 

 plication increases, and the effects are under the influence of so many and 

 such shifting causes as to give no room either for fixed 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 exten- 

 sion are applicable, if at all, only on that large scale on which precision of 

 details becomes unimportant. 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 reasoning from a general law to a particular 

 case, that is, determining by means of the particular circumstances of that 

 case, what result is required in that instance to fulfill the law. Thus in 

 the Torricellian experiment, if the fact that air has weight had been pre- 

 viously known, it would have been easy, without any numerical data, to 

 deduce from the general law of equilibrium, that the mercury would stand 

 in the tube at such a height that the column of mercury would exactly bal- 

 ance a column of the atmosphere of equal diameter; because, otherwise, 

 equilibrium 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 questions : 

 Given a certain combination of causes, what effect will follow ? and, What 

 combination of causes, if it existed, would produce a given effect ? In 

 the one case, we determine the effect to be expected in any complex cir- 

 cumstances of which the different elements are known : in the other case 

 we learn, according to what law — under what antecedent conditions — a 

 given complex effect will occur. 



§ 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 illuso- 

 ry when applied to the laws of complex phenomena, applicable with equal 

 force against the Method of Deduction ? When in every single instance a 

 multitude, often an unknown multitude, of agencies, are clashing and com- 

 bining, 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 pretense of 



