1. Uniplanar Algebra, Vol. 1: Being of a Propaedeutic to the Higher Mathematical Analysis (1893).
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.
The labours 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 Propaedeutic to the Higher Analysis, endeavours to place concisely in connected sequence the argument required for its solution.
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 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 formulae 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 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.
University of California
Berkeley, July 1, 1893.
1.2. Review by: Paul Saurel.
The School Review 2 (4) (1894), 241-242.
If any one expects to find in this little book a text-book on algebra like, except in name, to most text-books on that subject he will be disappointed. The book is not a beginner's book; it is elementary only in so far as it begins at the beginning. Starting with the theory of proportion as stated by Euclid, the author builds upon this the algebra of real quantities and establishes the laws of combination of such quantities by simple geometrical constructions. After devoting a chapter to the definition and discussion of the circular and hyperbolic functions he takes up the algebra of complex quantities. By means of Argand's mode of representation he shows that the laws which were established for real apply as well to complex quantities. At the end of this chapter he states briefly and clearly the characteristics of a logically complete algebra, and incidentally points out that an algebra which "admits evolution and the logarithmic process, but precludes the imaginary and the complex quantity is logically only the fraction of an algebra." ... We think it safe to say that both the teacher and the student of mathematics will find the book eminently pleasing and stimulating.
The transition from the traditional algebra of many of our secondary schools to the reconstructed algebra of the best American colleges is more abrupt than is necessary or creditable. This lack of articulation between the work of the schools and the colleges emphasizes the need of a fuller and more thorough course in elementary algebra than is furnished by the text-books now most commonly used. It is with the hope of supplying this new demand that an American edition of Charles Smith's Elementary Algebra is published; a work whose excellencies, as represented in former editions, have been recognized by able critics on both sides of the Atlantic.
In the rearrangement of the work and in its adaptation to American schools many changes have been made, too many to be noted in a short preface, and a considerable amount of new subject-matter has been introduced. The following are innovations of some importance:
Chapter I, consisting of a series of introductory lessons, is wholly new, and Chapter XIII is partly new and partly transferred from Chapter XXVIII of the second edition. Horner's synthetic division is made prominent in the chapter on division, an early introduction to quadratic equations finds its appropriate place in the chapter on factoring, the binomial theorem for positive integral exponents is demonstrated by elementary methods in the chapter on powers and roots, and the chapter on surds has been enlarged by a short discussion of complex numbers. Some new collections of examples have been introduced, and several of the older lists have been extended.
As thus reconstructed, the book constitutes a rounded course in what may be called the newer elementary algebra, and includes the subject-matter specified by nearly all American colleges as the requirement for admission. It will prove especially helpful to students preparing for such colleges as are using Mr Smith's Treatise on Algebra for advanced work.
Those who are familiar with the second edition will notice that the chapters on permutations and combinations, on logarithms and exponentials, on the binomial theorem for negative and fractional indices, on scales of notation, and on cube root have been omitted. ...
I am indebted to my colleagues of the mathematical of the University of California for valuable suggestions, but especially to Professor Haskell and Dr Hengstler for contributions to subject-matter and for reading many of the proof-sheets.
Special thanks are due to Mr Smith, for allowing free scope for this revision.
University of California, May, 1894.
2.2. Review by: S L Howe.
The School Review 3 (4) (1895), 240-243.
Among the many excellencies of this American edition, are the introductory lessons by Professor Stringham. These are intended to form a natural bridge between arithmetic and algebra. Factoring is made prominent, and is considered fundamental in solving equations of higher degree. The explanations are, for the most part, clear and abundant, though not always direct and concise. Subtraction and division are made easy by regarding these processes as the reverse of addition and multiplication respectively.