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IRVING STRINGHAM, Ph. D.
Professor of Mathematics in the University of California


SAN FRANCISCO

THE BERKELEY PRESS

1893

Engineering
Library

COPYRIGHT, 1893

BY

IRVING STRINGHAM

QA153
577

Engineerin
Library

Typography and Presswork by C. A. MURDOCK & CO., San Francisco

PREFACE.

FROM the beginning, with rare exceptions,* a singular logical incompleteness has characterized our text-books in elementary algebra. By tradition algebra early became a mere technical device for turning out practical results, by careless reasoning inaccuracies crept into the explanation of its principles and, through compilers, are still perpetuated as current literature. Thus, instead of becoming a classic, like the geometry handed down to us from the Greeks, in the form of Euclid's Elements, algebra has become a collection of processes practically exemplified and of principles inadequately explained.

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The labors of the mathematicians of the nineteenth century Argand, Gauss, Cauchy, Grassmann, Peirce, Cayley, Sylvester, Kronecker, Weierstrass, G. Cantor, Dedekind and others**-have rendered unjustifiable the longer continuance of this unsatisfactory state of algebraic science. We now know what an algebra is, and the problem of its systematic unfolding into organic form is a definite and achievable one. The short treatise here presented, as the first part of a Propedeutic to the Higher Analysis, endeavors to place concisely in connected sequence the argument required for its solution.

*Notably, in English, Chrystal's Algebra, 2 vols., Edinburgh, 1886, 1889. On the continent of Europe the deficiency has been compensated mainly in works on the Higher Analysis.

**The literature through which algebra has been rehabilitated during the present century is extensive. See Stolz: Allgemeine Arithmetik, Leipzig, 1885, for many valuable references.

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The first three chapters were made public, substantially in their present form, in a course of University Extension Lectures in San Francisco during the winter of 1891-92, a synopsis of which was issued from the University Press in October, 1891. At the close of these lectures the manuscript of the complete work: was prepared for the press; but unavoidable obstacles prevented its immediate publication and a consequent delay of somewhat more than a year has intervened. This delay, however, has made possible a revision of the original sketch and some additions to its subject-matter.

The logical grounding of algebra may be attained by either of two methods, the one essentially arithmetical, the other geometrical. I have chosen the geometrical form of presentation and development, partly because of its simpler elegance, partly because this way lies the shortest path for the student who knows only the elements of geometry and algebra as taught in our schools and requires mathematical study only for its disciplinary value. The choice of method, therefore, is not to be interpreted to mean that the writer underestimates the value and the importance to the special mathematical student of the Number-System.* This system, however, has no appropriate place in the plan here presented.

The point of departure is Euclid's doctrine of proportion, and the point of view is the one that Euclid himself, could he have anticipated the modern results of mathematical science, would naturally have taken. It is interesting to note that of logical necessity the development falls mainly into the historical order. For convenience of reference the fundamental propositions of proportion are enunciated and proved in an Introduction, in which I have followed the

*Fine: The Number-System of Algebra, Boston, 1891.

method recommended by the Association for the Improvement of Geometrical Teaching, and published in its Syllabus of Plane Geometry. Except a few additions and omissions, the enunciations and numbering in Sections. B and C of this Introduction are those of Hall and Stevens' admirable Text-Book of Euclid's Elements, Book V; and in Section D those of the Syllabus of Plane Geometry, Book IV, Section 2. The proofs vary in unessential particulars from those of the two texts named.

The subject-matter and treatment are such as to constitute, for the student already familiar with the elements of algebra and trigonometry, a rapid review of the underlying principles of those subjects, including in its most géneral aspects the algebra of complex quantities. All the fundamental formulæ of the circular and hyperbolic functions are concisely given. The chapter on Cyclometry furnishes, presumptively, a useful generalization of the circular and hyperbolic functions.

The generalized definition of a logarithm (Art. 68) and the classification of logarithmic systems,* first made public, outside of the mathematical lecture-room, in a paper read before the New York Mathematical Society in October, 1891, and subsequently published in the American Journal of Mathematics, are here reproduced in the revised form suggested by Professor Haskell. A chapter on Graphical Transformations, giving the orthomorphosis of the exponential and cyclic functions, appropriately concludes this part of the subject.

Many incidental problems are suggested in the form of Agenda, useful to the student for exemplification and practice. But on the other hand, many elementary

* American Journal of Mathematics, Vol. XIV, pp. 187-194, and Bulletin of the New York Mathematical Society Vol. II, pp. 164-170.

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