ELEMENTARY THEORY of EQUATIONSby
by LEONARD EUGENE DICKSON, Ph.D. (Professor of Mathematics in the University of Chicago)
London: CHAPMAN & HALL, Limited 1914
In the Preface Dickson describes the contents of the book, as well as pointing out his reasons to present material in the way that he chooses. We give a version of the Preface below:
The longer an engineer has been separated from his alma mater, the fewer mathematical formulas he uses and the more he relies upon tables and, when the latter fail, upon graphical methods. Although graphical methods have the advantage of being ocular, they frequently suffer from the fact that only what is seen is sensed. But this defect is due to the kind of graphics used. With the aid of the scientific art of graphing presented in Chapter I, one may not merely make better graphs in less time but actually draw correct negative conclusions from a graph so made, and therefore sense more than one sees. For instance, one may be sure that a given cubic equation has only the one real root seen in the graph, if the bend points lie on the same side of the x-axis.
Emphasis is here placed upon Newton's method of solving numerical equations, both from the graphical and the numerical standpoint. One of several advantages (well recognized in Europe) of Newton's method over Horner's is that it applies as well to non-algebraic as to algebraic equations.
In this elementary book, the author has of course omitted the difficult Galois theory of algebraic equations (certain texts on which are very erroneous) and has merely illustrated the subject of invariants by a few examples.
It is surprising that the theorems of Descartes, Budan, and Sturm, on the real roots of an equation, are often stated inaccurately. Nor are the texts in English on this subject more fortunate on the score of correct proofs; for these reasons, care has been taken in selecting the books to which the reader is referred in the present text.
The material is here so arranged that, before an important general theorem is stated, the reader has had concrete illustrations and often also special cases. The exercises are so placed that a reasonably elegant and brief solution may be expected, without resort to tedious multiplications and similar manual labour. Very few of the five hundred exercises are of the same nature.
Complex numbers are introduced in a logical and satisfying manner. The treatment of roots of unity is concrete, in contrast to the usual abstract method.
Attention is paid to scientific computation, both as to control of the limit of error and as to securing maximum accuracy with minimum labour.
An easy introduction to determinants and their application to the solution of systems of linear equations is afforded by Chapter XI, which is independent of the earlier chapters. Here and there are given brief, but clear, outlooks upon various topics of decided intrinsic and historical interest, - thus putting real meat upon the dry bones of the subject.
To provide for a very brief course, certain sections, aggregating over fifty pages, are marked by a dagger for omission. However, in compensation for the somewhat more advanced character of these sections, they are treated in greater detail.
In addition to the large number of illustrative problems solved in the text, there are five hundred very carefully selected and graded exercises, distributed into seventy sets. As only sixty of these exercises (falling into seventeen sets) are marked with a dagger, there remains an ample number of exercises for the briefer course.
The author is greatly indebted to his colleagues Professors A C Lunn and E J Wilczynski for most valuable suggestions made after reading the initial manuscript of the book. Useful advice was given by Professor G A Miller, who read part of the galley proofs. A most thorough reading of both the galley and page proofs was very generously made by Dr A J Kempner, whose scientific comments and very practical suggestions have led to a marked improvement of the book. Moreover, the galleys were read critically by Professor D R Curtiss, who gave the author the benefit not merely of his wide knowledge of the subject but also of his keen critical ability. The author sends forth the book thus emended with less fear of future critics, and with the hope that it will prove as stimulating and useful as these five friends have been generous of their aid.
CHICAGO, February, 1914.
This book occupies a middle ground in difficulty, being too advanced for the average freshman, but still of an elementary character, suitable for a second course in the theory of equations. It is such a book as may be read with profit by any one who wants an exact statement and rigorous proof of the elementary theorems - not involving group-theory or invariants - concerning algebraic equations; a work of value to all teachers of algebra, whether elementary or advanced. In particular every teacher of algebra should read the proof of the fundamental theorem of algebra and the work on graphing; while every teacher of geometry, should read the proofs given in Chap. VIII relating to the trisection of an angle duplication of the cube, and construction of regular polygons of 7, 9, and 17 sides. An exact treatment of these topics cannot but be of aid to anyone interested in elementary mathematics.
The early introduction of graphing and the use of derivatives in finding "bend points" enable the writer at the beginning of the book to give a discussion of the discriminant of the cubic x3 - 3px + q = 0, while the work throughout the book is rendered clear by the use of graphs. In the first chapter we are given also the graphical solution of a quadratic. Complex numbers are next introduced in a soul-satisfying way. This chapter should be read by everyone who thinks that complex numbers are "imaginary" and that we gain nothing by their use except to make certain equations have roots.
"The fundamental theorem of algebra" is also treated in a satisfactory way, the graphical proof being clear and elementary. ...
The theorem that an integral root of an equation with integral coefficients divides the constant term might well be supplemented by the similar theorem that if an equation with integral coefficients has a fractional root
The treatment of symmetric functions is unusually complete and careful.
The reviewer is glad to see in an accessible place a treatment given to the problems of trisecting an angle and duplicating a cube. These puzzle students and often teachers, partly because the problem is not clearly understood, and partly because there is so obviously a solution; and yet their impossibility may readily be made plausible to a student familiar with coordinate geometry and is here rigorously proved in an elementary way. We are also shown why there can be no construction, in the Euclidean sense, of regular polygons of 7 and 9 sides.
The usual theorems for the isolation of the roots are given in Chap. IX, as well as some theorems that are not found in most textbooks. The use of elementary calculus allows a clear treatment and a complete solution of the problem, "given an equation to locate its real roots," while the methods of Chap. X show how to compute them. Besides Horner's well-known method for the numerical computation of roots, Newton's is given and emphasized as one that is effective for non-algebraic as well as for algebraic equations; and Gräffe's little known but very ingenious scheme of solution by forming equations whose roots are powers of the roots of the given equation, and Lagrange's solution by continued fractions are also explained.
In Chap. XI determinants receive a clear natural treatment, while the subject of resultants and discriminants is carefully and rigorously discussed in the closing chapter.
On the whole in this book there is much to praise and little with which to find fault.