542 INDUCTION. 



powder, the angle of elevation, the density of the air, 

 the strength and direction of the sound ; hut 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. 



Besides the theorems of number, those of geo- 

 metry 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 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 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 pro- 

 cess. 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 fulfil the law. Thus, in the Torricellian 

 experiment, if the fact that air had 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 balance a column of the atmosphere of 

 equal diameter ; because otherwise, equilibrium would 

 not exist. 



