REMAINING LAWS OF NATURE. 171 



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 #, produce 

 the same number. 



Besides these general theorems or 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 x 2 + ax = 6, the reso- 

 lution of this equation, viz., # = % a +_^/ j a 2 + &, 

 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? + ax,) what function is x of b 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 qu&situm by pure ratio- 

 cination; 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 calculation we apply one or 

 other of these truths, or truths deduced from them, 

 as, that the differences, 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 



