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I have endeavoured, in the present volume, 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.
It is true that the same views of the relations of the principles of Arithmetical and Symbolical Algebra formed the basis of my first publication on Algebra in 1830: but not only was the nature of the dependance of Symbolical upon Arithmetical Algebra very imperfectly developed in that work, but no sufficient attempt was made to reduce its principles and their applications to a complete and regular system, all whose parts were connected with each other : they have consequently been sometimes controverted upon grounds more or less erroneous; and notwithstanding a very general acknowledgment of their theoretical authority, they have hitherto exercised very little influence upon the views of elementary writers on Algebra.
It may likewise be very reasonably contended that the reduction of such principles, as those which I have ventured
to put forward, to an elementary form, in which they may be fully understood by an ordinary student, is the only practical and decisive test, I will not say of their correctness, but of their value: for we are very apt to conclude that the most difficult theories and researches which have become familiar to us from long study and contemplation, may be made equally clear and intelligible to others as well as to ourselves : and though I will not say that I feel perfectly secure that I may not have been, in some degree, under the influence of this very common source of self-deception and error, yet I have adopted the only course which was open to me, in order to bring this question to an issue, by embodying my own views in an elementary work, and by suppressing as much as possible any original or other researches, which might be considered likely to interfere with its complete and systematic developement.
It is from the relations of space that Symbolical Algebra derives its largest range of interpretations, as well as the chief sources of its power of dealing with those branches of science and natural philosophy which are essentially connected with them : it is for this reason that I have endeavoured to associate Algebra with Geometry thoughout the whole course of its developement, beginning with the geometrical interpretation of the signs + and - when considered with reference to each other, and advancing to that of the various other signs which are symbolized by the roots of 1: we are thus enabled, in the present volume, to bring the Geometry of Position, embracing the whole theory of lines considered both in relative position and magnitude and the properties of rectilineal figures, under the dominion of Algebra : in a subsequent volume this application will be further extended to the Geometry of Situation, (where lines are considered in
their absolute as well as in their relative position with respect to each other), and also to the theory of curves.
The theory of the roots of 1 is so important, not merely with reference to the signs of affection which they symbolize, but likewise in the exposition of the general theory of equations and in all the higher branches of Symbolical Algebra, that I have thought it expedient to give it with unusual fulness and detail : such roots may be considered as forming the connecting link between Arithmetical and Symbolical Algebra, without whose aid the two sciences could be very imperfectly separated from each other.
I have not entered further into the general theory of equations than was necessary to enable me to exhibit the theory of their general solution, as far it can be carried by existing methods, reserving the more complete exposition of their properties and of the methods employed for their numerical solution, to a subsequent volume.
The plan which I have adopted necessarily brings Trigonometry, or to speak more properly, Goniometry, within the compass of the present volume, not merely as forming the most essential element in the application of Algebra to the Geometry of Position, but as intimately connected with many important analytical theories.