REMAINING LAWS OF NATURK. 369 



proposition, which may be rou;arded a-s an ansvv(>r to the question, If i 

 is a certain function of x and a (nanielv x^-{-ax), wliat function is 

 a; of b and a? The resolution of ciiimtions is, therefore, a mere 

 variety of the, general ])rol)lem as above stated. The problem is — 

 Given a function, what function is it of some othei- function ? And, in 

 the resolution of an equation, the question is, to find what function of 

 one of its own functions the number itself is. 



Such as above described, is the aim and end of the calculus. As 

 for its processes, every one knows that they are simply deductive. In 

 demonstrating an algebraical theorem, or in resolving an equation, 

 we travel from the datum to the qucesitum by pure ratiocination ; in 

 which the only premisses introduced, besides the original hypotheses, 

 are the fundamental axioms already mentioned — that things equal to 

 the same thing are equal to one another, and that the sums of equal 

 things are equal. At each step in the demonstration or in the calcu- 

 hition we apply one or other of those truths, or truths deduced from 

 them, as, that the -differences, products, &c., of equal numbers are 

 equal. . . • 



It would be inconsistent with the scale of this work, and not neces- 

 sary to its design, to carry the analysis of the truths and processes of 

 algebra any further ; which is moreover the less needful, as the task 

 has been recently and thoroughly performed by other writers. Profes- 

 sor Peacock's Algebra, and ]N[r. Whewell's Doctrine of Limits, should 

 be studied by every one who desires to comprehend the evidence of 

 mathematical truths, and the meaning of the obscurer processes of the 

 calculus; while, even after mastering these treatises, the student will 

 have much to learn on the subject from M. Comte, of whose admirable 

 work one of the most admirable portions is that in which he may 

 truly be said to have created the philosophy of the higher mathe- 

 matics.* 



§ 7. If the extreme generality and remoteness, not so much from 

 sense as from the visual and tactual imagination, of the laws of number, 

 render it a somewhat difficult effort of abstraction to conceive those 

 laws as being in reality physical truths obtained by observation; the 

 same difficulty does not exist with regard to the iaws of extension. 

 The facts of which these laws are expressions, are of a kind peculiarly 

 accessible to the sense, and suggesting eminently distinct images to the 

 fancy. That geometry is a strictly physical science would doubtless 

 have been recognized in all ag<.^s, had it not been for the illusions pro- 

 duced by two causes. One of these is the characteristic property, 

 already noticed, of the facts of geometry, that they may be collected 

 from our ideas or mental pictures of objects as effectually as from the 

 objects themselves. The other is, the demonstrative character of 

 geometrical truths ; which was at one time supposed to constitute a 

 radical distinction between them and physical truths, the latter, as 

 resting on merely probable evidence, being deemed essentially uncer- 



* In the concluding pages of his Cows de Philosophie Positive, of which the final volume 

 has but recently appeared, M. Cointe announces the intention of hereafter producing a 

 special and systematic work on the Philosophy of Mathematics. All cninpetent judges 

 who are acquainted with what M. Comte has already accomplished in that great depart- 

 ment of the philosophy of the sciences, will look with the highest expectations to this 

 promised treatise. 



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