ÆäÀÌÁö À̹ÌÁö
PDF
ePub

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

algebraic topics are not discussed, because they are not useful to the main object of the work, and it was especially desirable that its purpose should not be hindered by the making of a large book.

A few innovations in notation and nomenclature have been unavoidably introduced. The temptation to replace the terms complex quantity, imaginary quantity and real quantity by some such terms as gonion, orthogon and agon has been successfully resisted.

Partly in order to aid the student in obtaining a comparative view of the subject, partly in order to indicate in some detail the sources of information and give due credit to other writers, numerous foot-note references have been introduced.

I take great pleasure in acknowledging my obligations to Professor Haskell for valuable criticism and suggestion.

IRVING STRINGHAM.

UNIVERSITY Of California,
BERKELEY, July 1, 1893.

« ÀÌÀü°è¼Ó »