REMAINING LAWS OF NATURE. 433 



tion F will be of any function of that number. For example, a binomial 

 a+b is a function of its two parts a and b, and the parts are, in their 

 turn, functions of a-\-b: now (a+J)" is a certain function of the binomial; 

 what function will this be of a and b, tiie two parts? The answer to this 



question is the binomial theorem. The formula (« + 6)"=a"+- a"~'J + 



n.n — 1 



,n-2 



6'+, etc., shows in Avhat manner the number which is formed by 



multiplying a-\-b into itself n times, might be formed without that process, 

 directly from a, b, and n. And of this nature are all the theorems of the 

 science of number. They assert the identity of the result of different 

 modes of formation. They affirm that some mode of formation from x, 

 and some mode of formation from a certain function of x, pi'oduce the 

 same number. 



Besides these general theorems or formulae,- what remains in the algebra- 

 ical calculus is the resolution of equations. I3ut the resolution of an equa- 

 tion is also a theorem. If the equation be x^-\-ox=b, the resolution of this 

 eqnuiion, \iz., x=z —^ adz -x/^ a^ -\-b, is a general j^roposition, which may be 

 regarded as an answer to the question, If J is a certain function of x and a 

 (namely tc^ + a.^), what function is a; of 6 and a? The resolution of equa- 

 tions 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 func- 

 tion? 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 demon- 

 strating an algebraical theorem, or in resolving an equation, we travel from 

 the datum to the qiccesitmn by pure ratiocination ; in which the only prem- 

 ises introduced, besides the original hypotheses, ai'e the fundamental ax- 

 ioms 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, 

 etc., of equal numbers are equal. 



It would be inconsistent with the scale of this work, nnd not necessary 

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

 any further; which is also the less needful, as the task has been, to a very 

 great extent, performed 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 proc- 

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

 Philosophie Positive, on the philosophy of the higher branches of mathe- 

 matics, are among the many valuable gifts for which philosophy is indebted 

 to that eminent thinker. 



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

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

 it a somewhat difficult effort of abstraction to conceive those laws as being 

 in reality physical truths obtained by observation ; the same difficulty does 

 aot exist with regard to the laws of extension. The facts of which those 



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