THE DEDUCTIVE METHOD. 513 



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 

 towards 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 solu 

 tion 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 extension, as in mechanics, optics, acous 

 tics, astronomy. But when the complication increases, and 

 the effects are under the influence of so many and such shift 

 ing causes as to give no room either for fixed numbers, or for 

 straight lines and regular curves, (as in the case of physio 

 logical, 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. 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 indis 

 pensable 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 fulfil the 

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

 has weight had been previously 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 

 VOL. i. 33 



