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We have not given demonstrations of the binomial theorem, nor made any investigations of logarithms, or the higher equations, for these subjects belong exclusively to the higher order of Algebra, and will be found very clear and full in the University Edition of Algebra by the same author.

In relation to great generalities, all books on the same science, are, in substance, much alike, yet, in the clearness and distinctness with which they present principles, they may be very different; and to arrive at perfection in this particular, is, and should be, the highest ambition of an author.

For peculiarities in this work, the teacher is respectfully referred to abbreviations generally in solving equations, to the philosophical uses made of equations in demonstrating principles-the formation of problems, and the manner of arriving at arithmetical rules, which may be found in various parts of this work.

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INTRODUCTION.

ALGEBRA is the science of computation by means of symbols. Letters of the alphabet are generally used to represent quantities or numbers, and conventional signs are employed to represent operations, and to abridge and generalize the reasoning in relation to propositions or problems.

We sometimes meet with persons who can readily solve quite difficult problems, and yet are not able to explain the steps in the process: they call their operations working in the head—and, indeed, their reasoning, properly written out, is Algebra; but not having a knowledge of the signs, and possessing no skill in writing out the thoughts of the mind, they do not know it is Algebra.

This natural adaptation of the mind, to solve problems without the aid of writing down the operation, is very essential to success in this science. But the mind can only go a very short distance, unaided by the pen; nor is it important that it should, for the aid given by that instrument is efficient and complete, secures the ground gone over, and leaves the mind free to advance indefinitely.

In a purely mental process, the mind must retain all the results thus far attained, and continue the reasoning onward at the same time. And this, carried to excess, breaks down the mind rather than strengthens it; and for this reason, a mere mental Algebra must be regarded as one of the ephemeral efforts of the times. But let no reader construe these sentiments into a disapproval of mental Algebra. Every Algebra, properly understood, is mental Algebra; for the mental process-the reasoning power-must precede every operation.

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