REMAINING LAWS OF NATURE. 437 



the improvement of othei* sciences. Such principles are manifestly inap- 

 plicable, where the causes on which any class of phenomena depend are so 

 imperfectly accessible to our observation, that we can not ascertain, by a 

 proper induction, their numerical laws ; or where the causes are so numer- 

 ous, and intermixed in so complex a manner with one another, that even 

 supposing their laws known, the computation of the aggregate effect tran- 

 scends the powers of the calculus as it is, or is likely to be ; or, lastly, where 

 the causes themselves are in a state of perpetual fluctuation ; as in physiol- 

 ogy, and still more, if possible, in the social science. The mathematical so- 

 lutions of physical questions become progressively more difficult and im- 

 perfect, in proportion as the questions divest themselves of their abstract 

 and hypothetical character, and approach nearer to the degree of complica- 

 tion actually existing in nature ; insomuch that beyond the limits of astro- 

 nomical phenomena, and of those most nearly analogous to them, mathe- 

 matical accuracy is generally obtained " at the expense of the reality of the 

 inquiry :" while even in a.stronomical questions, " notwithstanding the ad- 

 mirable simplicity of their mathematical elements, our feeble 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 influ- 

 ences."* Of this, the problem of the Three Bodies has already been cited, 

 more than once, as a remarkable instance ; the complete solution of so com- 

 paratively simple a question having vainly tried the skill of the most pro- 

 found mathematicians. We may conceive, then, how chimerical would be 

 the hope that mathematical principles could be advantageously applied to 

 phenomena dependent on the mutual action of the innumerable minute par- 

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

 for similar reasons those„principles remain 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 type of the 

 Deductive Method in general ; and the applications of mathematics to the 

 deductive branches of physics, furnish the only school in which philosophers 

 can effectually learn the most difficult and important portion of their art, 

 the employment of the laws of simpler phenomena for explaining and pre- 

 dicting those of the more complex. These grounds are quite sufficient for 

 deeming mathematical training an indispensable basis of real scientific ed- 

 ucation, and regarding (according to the dictutn which an old but unau- 

 thentic tradition ascribes to Plato) one who is a-yewfie-priTOQ, as wanting in 

 one of the most essential qualifications for the successful cultivation of the 

 higher branches of philosophy. 



* Philosophie Positive, iii., 414-416. 



