Most teachers will agree that at the present time the work of mathematical specialists in schools is heavily handicapped by the absence of suitable text-books. There have been such radical changes in method and outlook that it has become necessary to treat large sections of some of the standard books merely as (moderately) convenient collections of examples and to supply the bookwork in the form of notes; especially is this true of Algebra, Trigonometry, and the Calculus.
Dividing lines between these subjects tend nowadays to be obliterated. Methods of the Calculus are freely used in courses of Algebra and Trigonometry, while matter which used to find a place in the Algebra text-book is now included more conveniently elsewhere. Perhaps the most important example of this re-arrangement is the treatment of the logarithmic function. For many years past leading mathematicians have advocated a definition which transfers the chapter on the theory of logarithms from the Algebra to the Calculus text-book, and makes it the basis from which the exponential function is discussed, thus reversing the order commonly followed. The authors are convinced by their own experience that this is the best mode of approach. On general principles it would seem desirable also to follow the same order for the complex variable, but unfortunately in practice this point of view appears to be too difficult for school work. By tradition the theory of the exponential and logarithmic functions of a complex variable is included in books on Advanced Trigonometry and this is a very reasonable arrangement ; it seems equally desirable to include also the theory of the corresponding functions of a real variable instead of relegating it to the Calculus book.
The interest and value of advanced trigonometry lies in regarding it as an introduction to modern analysis. The methods by which results are obtained are often more important - that is, educationally more valuable - than the results themselves. The character of the treatment in this book is shaped and controlled by that idea. Thus the methods for expanding functions in series focus attention on "remainders" and "limits"; the methods for factorizing functions turn on establishing possible forms and then using the fundamental factor-theorem; the discussion of complex numbers emphasises the fact that complex numbers are just as "real" as real numbers, etc. For the same reason no apology need be offered for the prevalence, in this book, of "inequalities." Their importance in higher mathematics can hardly be exaggerated, and they are invaluable too in elementary work. The "useful inequalities" of Chapter IV will, it is believed, be found fully worthy of their name.
The authors are planning text-books parallel to the present volume on Advanced Algebra and Calculus, written from a similar point of view. In all these subjects, it must be admitted, there are certain difficulties which the average student will never face, but which are all-important for the real mathematician; these include, for example, the purely arithmetical treatment of real number, limits, continuity, convergence, mean-value theorems, the analysis of area, length of a curve, etc. The authors propose to deal with these matters in a book which is cited as a "companion volume on Analysis," limiting the treatment, however, to what seems suitable for specialist work at schools. Although planned, no part of this book is yet written. The theory of Infinite Products has been left for this companion volume; it is not so easy to provide a satisfactory ab initio treatment for products as it is for series and the alternative of taking for granted everything that really matters is undesirable. Happily also Infinite Products are of small value in elementary work and they are not required for most examinations. See. however, pp. 223, 240.
As a text-book on Trigonometry, this volume is a continuation of Durell and Wright's Elementary Trigonometry, and Chapter I should be regarded partly as a revision course. The sole object of Chapter XIV is to give opportunity for practice in mechanical manipulation to those who require it. The course really closes with Chapter XIII, which deals with a difficult subject and one which should be done carefully if it is done at all.
A Key is published, for the convenience of teachers, in which solutions are given in considerable detail, and in some cases alternative methods of solution are supplied, so that to some extent the Key forms a supplementary teaching manual.
The authors gratefully acknowledge help with the proofs received from Mr J C Manisty, whose numerous criticisms and suggestions have enabled them to effect many improvements.
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