REMAINING LAWS OF NATURE. 153 



formation from a certain function of x, produce the same 

 number. 



Besides these general theorems of formulae, what remains 

 in the algebraical calculus is the resolution of equations. But 

 the resolution of an equation is also a theorem. If the 

 equation be # 2 + ax = b, the resolution of this equation, viz. 

 x = a + ,Jia* + b, is a general proposition, which may 

 be regarded as an answer to the question, If & is a certain 

 function of x and a, (namely a? 2 + ax), what function is x of & 

 and a ? The resolution of equations is, therefore, a mere 

 variety of the general problem as above stated. The problem 

 is Given a function, what function is it of some other 

 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 

 quasitum by pure ratiocination ; in which the only premises 

 introduced, besides the original hypotheses, are the funda- 

 mental 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 calculation, we apply one or other of these truths, or 

 truths deducible 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 necessary to its design, to carry the analysis of the truths 

 and processes of algebra any farther; which is also the 

 less needful, as the task has been, to a very great extent, per- 

 formed by other writers. Peacock's Algebra, and Dr. Whewell's 

 Doctrine of Limits, are full of instruction on the subject. The 

 profound treatises of a truly philosophical mathematician, 

 Professor De Morgan, 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; 

 and the speculations of M. Comte, in his Cours de Philosophic 



