An Elementary Treatise on Algebra, Theoretical and Practical

has, consequently, at length acquired such extent and importance, as to have assumed the form of a distinct department of analysis. The discussion of this subject is therefore reserved for a separate volume, now at press, which will form a supplement to the present treatise.

The following brief enumeration of the principal topics discussed in this work is extracted, with slight modifications, from the preface to the former edition.

Chapter I. contains the Preliminary Rules of the science, in which the fundamental principles of operation are explained and illustrated.

Chap. II. is on Simple Equations, and commences with some propositions preparatory to entering upon the solution of an equation, which operation they are intended to render more easy and inviting. Then follow the several methods of solving simple equations involving one, two, and three or more unknown quantities; each of these methods being illustrated separately, not only by algebraical examples, but also by practical questions; a mode rather different from that usually adopted, but which appears to be preferable, as it affords the student an early opportunity of applying the principles that he has acquired to useful and interesting inquiries, an exercise which is generally found to be peculiarly pleasing and encouraging.

Chap. III. treats of Ratio, Proportion, and Progression, both arithmetical and geometrical; and, although the general formulas are fewer in number than those given in most books on this subject, yet it is shown that they are amply sufficient for every variety of case, and that therefore it would be superfluous to extend their number.

Chap. IV. is on Quadratics, and on Imaginary Quantities. This chapter is of a more difficult nature than either of the preceding, and proportionate pains have been taken to render the modes of operation clear and intelligible; the solutions to some of the more difficult examples, which are given at length, will be of service to the student in cases of a similar nature, and will manifest to him how much a little judgment and ingenuity on his part will add to the elegance of his operation. The article on Imaginary Quantities, with which this chapter concludes, will be found to contain some observations tending to remove the obscurity in which the subject is usually enveloped.

Chap. V. contains the general investigation of the Binomial Theorem. The demonstration of this celebrated Theorem in a manner adapted to elementary instruction, has always been considered as an object greatly to be desired, and many attempts have accordingly been made by different mathematicians for this purpose; all, however, that have yet appeared have been objected to, either on account of unwarrantable assumptions, at the outset, which have consequently weakened the evidence, and rendered the demonstration incomplete, or because of a too tiresome and obscure method

of reasoning, which has been incomprehensible to a learner. The demonstration given in this chapter is, I believe, different from any that has been previously offered, and appears to be more simple and satisfactory than any which I have had an opportunity of seeing. In the practical application of this theorem to the expansion of a binomial, it is always best to separate the case in which the exponent is integral, from that in which it is fractional, because, in the former instance, the process by the general formula is unnecessarily long and troublesome; a different method of proceeding is therefore usually pointed out; but it is rather singular that it has been applied only when the exponent is a positive integer: as, however, it is equally applicable when the exponent is a negative integer, it is here extended to that case.

Chap. VI. explains the nature and construction of Logarithms, and shows their importance in their application to several useful inquiries relating to interest, annuities, &c.

Chap. VII. is devoted to Series; and a new method for the summation of infinite series is given, which it is thought will be found to be more direct and easy than those generally used in elementary works. Several interesting subjects connected with series will be found in this chapter.

Chap. VIII. is on Indeterminate Equations of the first degree. In this chapter also some improvements will be found. The rule given at page 234 for solving an indeterminate equation involving two unknown quantities, is more direct and concise than the usual method, and equally simple.

Chap. IX. contains the principles of the Diophantine Analysis, or of indeterminate equations above the first degree, and concludes with a collection of diophantine questions; several of which are solved, in order to exhibit to the student the artifices which are sometimes to be employed in this part of the subject. This chapter has little claim to novelty, except as far as relates to the introduction of some new questions, and to the new solutions given to others.

From the above outline an idea may be formed of the nature and pretensions of the work here submitted to the judgment of an impartial public; and if, upon examination, it shall be found that I have at all succeeded in my endeavours to lessen the labours of the student, it will afford me the nighest satisfaction.

ROYAL COLLEGE, Belfast;

August 19th, 1834.

J. R. YOUNG.

ADVERTISEMENT.

THE American publishers of Professor Young's Elementary Treatise on Algebra, with the desire of rendering the work as useful and correct as possible, have been induced to submit it to the perusal of a competent mathematician. At his suggestion, they have retained the Theory of Equations, which formed a part of the first London edition, and which they have judged to be a necessary supplement to the present volume. From the care taken by the reviser, and the number of corrections made, they are confident that exceedingly few, if any errors, could have escaped notice, and they accordingly offer the work thus improved to the attention of the public.

PHILADELPHIA, April, 1838.

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