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Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the … - George Peacock
" "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... all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of <math>a^{m}</math> and <math>a^{n}</math>, which is <math>a^{m+n}</math> when <math>m</math> and <math>n</math> are whole numbers, and therefore general in form though particular in value, will be their product likewise when <math>m</math> and <math>n</math> are general in value as well as in form: the series for <math>(a+b)^{n}</math>, determined by the principles of arithmetical algebra, when <math>n</math> is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for <math>(a+b)^n</math>, when <math>n</math> is general both in form and value.
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About George Peacock
George Peacock (April 9, 1791 – November 8, 1858) was an English mathematician and author of books on mathematics and a biography of Thomas Young. He became a deacon, then priest, in the Church of England, and later, Vicar of Wymewold and Dean of Ely cathedral, Cambridgeshire. He was also professor of astronomy at the University of Cambridge.
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Additional quotes by George Peacock
In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs <math>+</math> and <math>-</math> denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as <math>a + b</math> we must suppose <math>a</math> and <math>b</math> to be quantities of the same kind; in others, like <math>a - b</math>, we must suppose <math>a</math> greater than <math>b</math> and therefore homogeneous with it; in products and quotients, like <math>ab</math> and <math>\frac{a}{b}</math> we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.
I have endeavoured... to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained.
I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science.
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