REMAINING LAWS OF NATURE. 



403 



question is the binomial theorem. 

 The formula (a + 6)* = a* + * a* " ^6 + 



~^ a'*~'b'- + &c., shows in what man- 

 1.3 ' 



ner the number which is fonned by 



multiplying a-tb into itself n times, 



might be formed without that process, 



directly from 0, 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, produce the 



same number. 



Besides these general theorems or 

 formulae, what remains in the alge- 

 braical calculus is the resolution of 

 e(juations. But the resolution of an 

 equation is also a theorem. If the 

 equation he X' + ax = b, the resolution 

 of this equation, viz. x = - ^a + 

 y/ ^ ar + b, IB a, general proposition 

 which may be regarded as an answer 

 to the question. If 6 is a certain 

 function of arand a, (namely, X' + ax,) 

 what function is x of 6 and a ? The 

 resolution of equati«)ns is, therefore, 

 a mere variety of the general pro- 

 blem 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 de- 

 monstrating an algebraical theorem, 

 or in resolving an equation, we travel 

 from the datum to the qucesitum by 

 pure ratiocination ; in wliich the only 

 premises introduced, besides the ori- 

 ginal hypotheses, are the fundamen- 

 tal 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 dedu- 

 cible from them, as, that the differ- 

 ences, products, &c., of equal num- 

 bers are equal. 



It would be inconsistent with the 

 scale of this work, and not necessary 

 to its design, to carry the analysis of 

 th*' truths and processes of algebra 

 any farther ; 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 philoso- 

 phical 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 calcuius ; and the speculations 

 of M. Comte, in his Cours de I'hilo- 

 sophie Positive, on the philosophy of 

 the higher branches of mathenjatics, 

 are among the many valuable gifts 

 for which philosophy is indebted to 

 that eminent thinker. 



§ 7. If the extreme generality, and 

 remoteness not so nmch from sense as 

 from the visual and tactual imagina- 

 tion, of the laws of number, renders 

 it a somewhat ditticult effort of ab- 

 straction to conceive those laws as 

 being in reality physical truths ob- 

 tained by observation ; the same diffi- 

 culty does not exist with regard to 

 the laws of extension. The facts of 

 which those laws are expressions are 

 of a kind peculiarly accessible to the 

 senses, and suggesting eminently dis- 

 tinct images to the fancy. That geo- 

 metry is a strictly physical science 

 would doubtless have been recognised 

 in all ages, had it not been for the 

 illusions produced by two circum- 

 stances. One of these is the charac- 

 teristic property, already noticed, of 

 the facts of geometry, that they may 

 be collected from our ideas or mental 

 pictures of objects as efifectually as 

 from the objects themselves. The 

 other is, the demonstrative character 

 of f>reometrical truths ; which was at 



