REMAINING LAWS OP NATURE. 373 



of phenomena depend are so imperlectly acxcssible to our observation, 

 that we cannot ascertain, by a proper induction, their numerical laws; 

 or where the causes are so numerous, and intermixed in so complex a 

 manner with one another, that even sujiposinir their laws known, the 

 computation of the ag^gi-egate cftect transcends the powers of the cal- 

 culus as it is, or as it is ever likely to be ; or lastly, where the causes 

 themselves aro in a state of perpetual fluctuation, as in physiology, 

 and still more, if possible, in the social science. As M. Comte* well 

 obsei'ves, the mathematical solutions of physical questions become 

 progressively more difficult and more imperfect, in proportion as the 

 questions divest themselves of their abstract and hypothetical character, 

 and approach nearer to the degree of complication actually existing in 

 nature; insomuch that beyond the limits of astronomical phenomena, 

 and of those most nearly analogous to them, mathematical accuracy is 

 generally obtained " at the expense of the reality of the inquiry :" 

 while, even in astronomical questions, " notwithstanding the admirstble 

 simplicity of their mathematical elements, our fc^eble intelligence 

 becomes incapable of following out effectually the logical combinations 

 of the laws on which the phenomena are dependent, as soon as we 

 attempt to take into simultaneous consideration more than two or three 

 essential influences." Of this, the problem of the Three Bodies has 

 already been cited by us, more than once, as a rcmarkalile instance ; 

 the complete solution -of so comparatively simple a question having 

 vainly tried the skill of the most profound mathematicians. We may 

 conceive, then, how chimerical would be the hope that mathematical 

 principles could ever be advantageously applied to phenomena depen- 

 dent upon the mutual action of the innumerable minute particles of 

 bodies, as those of chemistry, and still more, of physiology ; and for 

 similar reasons those principles must be for ever inapplicable to the 

 still more complex inquiries, the subjects of which are phenomena of 

 society and government. 



The value of mathematical instruction as a preparation for those 

 more difficult investigations, consists in the applicability not of its 

 doctrines, but of its method. Mathematics will ever remain the most 

 perfect typo of the Deductive Method in general ; and the applications 

 of mathematics to the simpler branches of physics, furnish the only 

 school in which j)hilosophers can eff"ectually learn the most difficult and 

 important portion of their art, the employment of the laws of simpler 

 phenomena for explaining and predicting those of the more complex. 

 These grounds are quite sufficient for deeming mathematical training 

 an indispensable basis of real scientific education, and regarding, with 

 Plato, one who is dyew/xt'rpT/Tor, as wanting in one of the most essential 

 qualifications for the successful cultivation of the higher branches of 

 philosophy. 



* Court de Philosophic Positive, iii., 414-416. 



