A SEQUEL TO ELEMENTARY ALGEBRA FOR SCHOOLS BY H. S. HALL, M.A., = FORMERLY SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE, AND S. R. KNIGHT, B.A., FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE, LATE ASSISTANT-MASTER AT MARLBOROUGH COLLEGE. FOURTH EDITION. London: MACMILLAN AND CO. AND NEW YORK. 1891 [The Right of Translation is reserved.] LIBRARY OF THE LLAND STANFORD JUNIOR A 3104 First Printed 1887. Second Edition with corrections 1888. PREFACE. THE present work is intended as a sequel to our Elementary Algebra for Schools. The first few chapters are devoted to a fuller discussion of Ratio, Proportion, Variation, and the Progressions, which in the former work were treated in an elementary manner; and we have here introduced theorems and examples which are unsuitable for a first course of reading. From this point the work covers ground for the most part new to the student, and enters upon subjects of special importance: these we have endeavoured to treat minutely and thoroughly, discussing both bookwork and examples with that fulness which we have always found necessary in our experience as teachers. It has been our aim to discuss all the essential parts as completely as possible within the limits of a single. volume, but in a few of the later chapters it has been impossible to find room for more than an introductory sketch; in all such cases our object has been to map out a suitable first course of reading, referring the student to special treatises for fuller information. In the chapter on Permutations and Combinations we are much indebted to the Rev. W. A. Whitworth for permission to make use of some of the proofs given in his Choice and Chance. For many years we have used these proofs in our own teaching, and we are convinced that this part of Algebra is made far more intelligible to the beginner by a system of common sense reasoning from first principles than by the proofs usually found in algebraical text-books. The discussion of Convergency and Divergency of Series always presents great difficulty to the student on his first reading. The inherent difficulties of the subject are no doubt considerable, and these are increased by the place it has ordinarily occupied, and by the somewhat inadequate treatment it has hitherto received. Accordingly we have placed this section somewhat later than is usual; much thought has been bestowed on its general arrangement, and on the selection of suitable examples to illustrate the text; and we have endeavoured to make it more interesting and intelligible by previously introducing a short chapter on Limiting Values and Vanishing Fractions. In the chapter on Summation of Series we have laid much stress on the " Method of Differences" and its wide and important applications. The basis of this method is a wellknown formula in the Calculus of Finite Differences, which in the absence of a purely algebraical proof can hardly be considered admissible in a treatise on Algebra. The proof of the Finite Difference formula which we have given in Arts. 395, 396, we believe to be new and original, and the development of the Difference Method from this formula has enabled us to introduce many interesting types of series which have hitherto been relegated to a much later stage in the student's reading. We have received able and material assistance in the chapter on Probability from the Rev. T. C. Simmons of Christ's College, Brecon, and our warmest thanks are due to him, both for his aid in criticising and improving the text, and for placing at our disposal several interesting and original problems. It is hardly possible to read any modern treatise on Analytical Conics or Solid Geometry without some know |