од

UNIVERSITY

ALGEBRA.

COMPRISING

1.-A COMPENDIOUS, YET COMPLETE AND THOROUGH COURSE IN ELEMENTARY

ALGEBRA, AND

II. AN ADVANCED COURSE IN ALGEBRA, SUFFICIENTLY EXTENDED TO MEET THE
WANTS OF OUR UNIVERSITIES, COLLEGES, AND SCHOOLS OF SCIENCE.

BY

EDWARD OLNEY,

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN, AND AUTHOR OF
A SERIES OF MATHEMATICS.

NEW YORK:

SHELDON & COMPANY,

677 BROADWAY.

1873.

PREFACE.

THE Author's COMPLETE SCHOOL ALGEBRA was written to meet the wants of our Common and High Schools and Academies, and to afford adequate preparation for entering our best Colleges, Schools of Science, and Universities.

The present volume is designed for use in these advanced courses of training. Thus, while it is thought that the former affords as extended a course in Algebra as is expedient for the preparatory schools, it is believed that this will be found to contain all that these higher schools require.

It was deemed necessary to make the work a complete treatise, including the Elements, for purposes of reference, and for reviews, and also in consideration of the fact that our higher institutions have various standards of requirement for admission. In fact, there are few students of Higher Algebra who do not find it necessary to have the Elements at hand for occasional consultation.

This Elementary portion is embraced in the first 150 pages, and contains all the definitions, principles, rules, and demonstrations of the COMPLETE SCHOOL ALGEBRA, with an abundant collection of New Examples; but from it all elementary illustrations, explanations, solutions, and suggestions, are omitted. The whole is so arranged as to secure readiness of reference and convenience of review by somewhat mature students.

The subjects treated in PART III., which constitutes the Advanced Course proper, will be best seen by turning to the Table of Contents. In this place the author wishes merely to call attention to a few of the distinguishing features of this Part.

1. The conception of Function and Variable is introduced at once, and is made familiar by such use of it as mathematicians are constantly making. No one needs to be told that this conception lies at the foundation of all higher algebraic discussion; yet, strangely enough, the very terms are scarcely to be found in our common text-books, and the practical use of the conception is totally wanting.

2. The first chapter in the Advanced Course is given to an elementary and practical exposition of the Infinitesimal Analysis. The author knows from his own experience, and from that of many others, that this subject presents no peculiar difficulties to ordinary minds; and everybody knows that it is only by this analysis that the development of functions, as in the Binomial Formula, Logarithmic Series, etc., the general relation of function and variable, the evolution of many of the principles requisite in solving the Higher Equations, and many other subjects, are ever treated by mathematicians, except when they attempt to make Algebras. No mathematician thinks of using the clumsy and antiquated processes by which we have been accustomed to teach our pupils in algebra to demonstrate the Binomial Formula, produce the Logarithmic Series, deduce the law of derived polynomials, examine the relative rate of change of a function and its variable, etc., except when he is teaching the tyro. Why not, then, dismiss forever these processes, and let the pupil enter at once upon those elegant and productive methods of thinking which he will ever after use ?

3. By the introduction of a short chapter on Loci of Equations, which any one can read even without a knowledge of Elementary Geometry, and which in itself is always interesting to the pupil, and of fundamental use in the subsequent course, all the more abstruse principles of the Theory of Equations are illustrated, and the student is thus enabled to see the truth, as well as to demonstrate it abstractly. How great an advantage this is, no experienced teacher needs to be told.

4. In the treatment of the Higher Equations, while some things have been discarded which everybody knows to be worthless, but which have in some way found a place in our text-books, a far more full and clear discussion of practical principles and methods is given, than is found in any of the treatises in common use.

5. The important but difficult subject of the Discussion of Equations has been reserved till late in the course, for several reasons. Thus, when the pupil reaches this topic, he has become familiar with most of the principles to be applied, and has become sufficiently imbued with the spirit of the algebraic analysis to be enabled to grasp it. To discuss an equation independently and well, is a high mathematical accomplishment, and should not be expected of the tyro. It is nothing else than to think in mathematical formulæ, and hence is one of the later products of mathematical study. It is hoped that the position assigned to this subject in the course, and the manner of treating it, will insure better results than we have hitherto been able to obtain.

6. In the selection of Subjects to be Presented, constant regard has been had

to the demands of the subsequent mathematical course. This has led to the omission of a number of theorems and methods, which, though well enough in themselves as mere matter of theory, find no practical application in a subsequent course, however extended; and has, at the same time, led to the introduction of not a few things which the advanced student always finds occasion to use, but for which he searches his Algebra in vain, if he has at hand nothing but our common American text-books.

7. In Method of Treatment the following principles have been kept constantly in mind: 1. That the view presented be in line with the mathematical thinking of to-day. 2. That everything be rigidly demonstrated and amply and clearly illustrated. 3. When long experience has shown that the majority of good students have difficulty in comprehending a subject, special pains should be taken to elucidate it. 4. No principle is thoroughly learned by a pupil until he can apply it; and nothing so fixes principles in the mind as the use of them. Hence an unusually large number of examples has been introduced. 5. It is often necessary to multiply examples in order to meet the requirements of the class-room.

8. Answers.-The answers to examples are not generally annexed to them in the text. There are, however, two editions of the volume, one with the answers at the end, and the other without any answers, except an occasional one in the body of the book.

9. Finally, the Order of Topics is such that a student requiring a less extended course than the entire volume presents, can stop at any point, and feel assured that what he has studied is of more elementary importance than what follows. Thus students who do not desire to study the Higher Equations can conclude their course with the first chapter of Part III.; and a course which includes the first three chapters of this part will be found as extended as most of our Academies, and perhaps many of our Colleges, will find expedient.

Such works as those of SERRET, CIRODDE, COMBEROUSSE, WOOD, HYMERS, HIND, TODHUNTER, YOUNG, and most of our American treatises, have been at hand during the preparation of the entire volume. To WHITWORTH'S charming little treatise on Choice and Chance, the author is indebted for a number of examples in the last section.

The quick eye and cultivated taste of my friend, Mr. W. W. BEMAN, A.M., Instructor of Mathematics in the University, have done me excellent service in reading the proof-sheets, and have, I trust, given the work a degree of typographical accuracy not usually found in first issues of such treatises.