« ΠροηγούμενηΣυνέχεια »
The work, the first volume of which is now offered to the public, was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.
I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system.
In the preface to my former Treatise I have given a
general exposition of my reasons for distinguishing arith
metical from symbolical algebra, and of my views of the just relations which their principles bear to each other, though I did not then consider it necessary to separate the exposition of one science altogether from the other. A more matured consideration of the subject, however, has convinced me of the expediency of this separation; for it is extremely difficult, when the two sciences are treated simultaneously, to keep their principles and results apart from each other, and to obviate the confusion, obscurity
and false reasoning which thence arises: a short stateme of the distinct and proper provinces of these two scienc will make this difficulty sufficiently manifest. In arithmetical algebra, we consider symbols as repr senting numbers, and the operations to which they a. submitted as included in the same definitions (whether e. pressed or understood) as in common arithmetic: the sig + and – denote the operations of addition and subtractic in their ordinary meaning only, and those operations an considered as impossible in all cases where the symbo subjected to them possess values which would render ther so, in case they were replaced by digital numbers: thus i expressions, such as a + b, we must suppose a and b to b quantities of the same kind: in others, like a - b, we mu suppose a greater than b, and therefore homogeneous wit
it: in products and quotients, like ab and o, We mu
suppose the multiplier and divisor to be abstract numbers all results whatsoever, including negative quantities, whic are not strictly deducible as legitimate conclusions from the definitions of the several operations, must be rejecte as impossible, or as foreign to the science. Numerical fractions, which have not a common deno minator, are not homogeneous, and are incapable of addition and subtraction in arithmetic, and therefore in arithmetica algebra; and the multiplication and division of a numbe. or fraction by a fraction is only admissible in arithmetic, and therefore in arithmetical algebra, in virtue of a con. vention which assumes the permanence of forms', which constitutes the great and fundamental principle of sym. bolical algebra; but by thus trenching upon the province of another and more comprehensive science, we are enabled to give an extension to our notion of number, which greatly enlarges the province of arithmetic and arithmetical algebra. Without the aid of such an extension, the sciences of arithmetical and symbolical algebra must have long since been separated from each other.
* See Art. 135, 136, 137, 138, 139, 140.
Again, the generalizations of arithmetical algebra are generalizations of reasoning, and not of form. Thus we conclude that (a + b)” is equivalent to a” + 2 ab + bo, for all homogeneous values of a and b, inasmuch as it may be easily shewn that their specific magnitudes, if expressible by numbers, (using the term in its largest sense,) cannot in any way affect the formation of the result: but it is only for values of the symbol a which are not less than b, that the product (a + b) (a – b) can be assumed to be equivalent to a”— b”; for in no other case can this product be formed consistently with the arithmetical definition of the operation of multiplication. Again, the product of a” and a” can be shewn to be equivalent to a”", when m and n are integral and abstract numbers, and in no other case ; for no other values are recognized in the definition of the power of a symbol, and therefore in no other case can we appeal to it in determining the form of the product: the series for (a + b)" is also a necessary result of the same definition, and subject therefore to the same limitations: and we conclude generally that all the necessary results of arithmetical algebra must be rigorously restricted to those conditions of value and representation which the definitions require.
But though the rules for performing the operations of symbolical algebra must embody the limitations which the definitions impose, yet it is very difficult, in innumerable cases, to discover the impossibility of the operation or the inadmissibility of the result, before the operation is performed or the result is obtained: thus if we are required to subtract a + b from a, we do not attempt an operation . which is as obviously impossible in arithmetical algebra, as it would be, in common arithmetic, to subtract 3 from 2:
Put if it was required to subtract from 7a + 5b, the several
subtrahends a + 3b, 3a – 2b and 3a + 7b, we should probably proceed without hesitation to apply the general rule of subtraction, which would give us
a result which indicates that the final operation, to which the application of the rule conducts us, is impossible: we are perpetually encountering, in arithmetical algebra, as it were unconsciously and unawares', similar examples of operations which cannot be performed or of results which cannot be recognized, consistently with the definitions upon which that science is founded. Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed: as far as those relations are admissible, therefore, in the latter science, they are in every respect the same: in both sciences also we change the signs of the terms of the subtrahend, and ther proceed as in addition: but it is in the former science only that we form and recognize the result, whatever it may be without any reference to its consistency with the definition upon which those rules in arithmetical algebra are founded we are thus enabled to subtract a + b as well as a — b from a, obtaining by the unrestricted rule a — (a + b) = a – a – b = — b in one case, and a – (a – b) = a – a + b = + b in the other: the same observation extends to the rule for performing all the other operations of arithmetic: algebra and to the results to which they lead; adoptin also the results themselves, whether common to arithmetics algebra or not, whether negative or positive, as equally th subjects of the fundamental and other operations. It is this adoption of the rules of the operations , arithmetical algebra as the rules for performing the oper tions which bear the same names in symbolical algebr which secures the absolute identity of the results in th two sciences as far as they exist in common : or in oth, words, all the results of arithmetical algebra which a deduced by the application of its rules, and which a general in form, though particular in value, are resul
Art. 31, 380, 389, Ex. 5. 410, Ex. 7.