FOR

SECONDARY SCHOOLS

BY

ERNEST L. THURSTON

ASSISTANT SUPERINTENDENT OF PUBLIC SCHOOLS, DISTRICT OF COLUMBIA

STANFORD LIBRARY

New York

THE MACMILLAN COMPANY

LONDON: MACMILLAN & CO., LTD.

COPYRIGHT, 1913,

BY THE MACMILLAN COMPANY.

Set up and electrotyped. Published March, 1913.

179207

C

Norwood Press:

Berwick & Smith Co., Norwood, Mass., U.S.A.

PREFACE.

The use and applications of number vary with the changes and development of our industrial, commercial and social relations. The arithmetic of to-day is not the arithmetic of ten years ago. The fundamental principles and processes do not alter, but new applications constantly develop and older applications lessen in usefulness. In this sense, arithmetic is a living subject which deals with living phenomena.

In these days arithmetic is commonly used not only for the necessary routine computations incident to our private, business, or scientific affairs, but its processes are called upon to study facts through number and to interpret these facts-to put number into things rather than to take number from them. Moreover, the modern use of number often takes the form of a language, as in the construction or reading of numerical illustrations, and statements or statistics which are designed to impress upon others certain facts or conditions. Thus modern arithmetic may be said to have a use as a tool, tool, i. e., for routine computation, an interpretive use, and a language

use.

But accuracy and facility in the use of most working instruments presuppose a knowledge of the fundamental parts of the instrument, of the interrelation of these parts, of their working principles, and of the classes of material or work to which they may be applied. Thus, facility in the three "uses" of the arithmetic instrument is founded, primarily, on a logical working knowledge of the fundamental operations and principles. This is the more necessary because the arithmetic problems of actual life do not have a text-book form of statement. If formulated at all, they may offer no direct clue to a particular subdivision of the subject. Often the problem must first be formulated, at least mentally, the power of

selection being exercised to choose from a group of number facts those essential to the computation desired.

This book is an attempt to construct the arithmetic instrument logically, in its simplest form, as a branch of mathematics; then to apply this tool to general computation, and to the study, preparation and interpretation of varied number material. Involved in this is the purpose to develop arithmetic as a language of business, or as a means of interpretation and study of business, and economic conditions; to acquire the capacity properly to present numerical facts by tabulations or graphs; to cultivate clearness of thought and expression, and an appreciation of the value of order and system in applied number work.

In the preparation of the subject matter of this book, no attempt has been made to draw from other text-books, although much may be found here that is in harmony with them. Material has been gathered only from living and reliable sources. The material has been selected with the purpose of emphasizing how arithmetic is applied for a definite object. The problem exercises are not intended, as is so often the case, as a means of conveying to the mind indigestible information on most known subjects. The author believes that the book will be found none the less rich in content because it aims at the digestion of facts, rather than at surfeiting with them.

Much thought has been given to the problem work and, as one result, considerable variety will be noticed in form of statement. Thus the old time "problem in question form" is used, also the incomplete statement, the statistical table, the business form, the written article, the memorandum, etc. Place is also given to series or related problems, composite problems, and "central" problems, to be viewed from different standpoints. Exercises in the preparation of original problems are included for the purpose of cultivating powers of selection

and statement, and of distinguishing between essentials and non-essentials. Mental development, and growth in power of analysis, must necessarily follow the doing of the work that these exercises require.

A method of development has been followed, in condensed and simplified form, and rules have been reduced to an absolute minimum. It has been assumed, however, that the pupil has had considerable experience in grade arithmetic, before undertaking the study of this book. For this reason illustrative examples are frequently given only in outline. To give facility in the fundamental operations and main processes, a considerable quantity of abstract work has been introduced. In this line, special emphasis has been laid on the fundamental processes as applied to whole numbers, decimals and fractions. Short methods have been emphasized for paper work, and methods of checking have been introduced or suggested. It is felt that every important computation should be checked.

Oral exercises, planned with the utmost care to serve as models, have been given throughout the book, but they should be supplemented by further work along the lines that they suggest. The exercises should vary in form, being, at different times, sight exercises, exercises by dictation, and exercises based upon some fundamental expressions that are placed upon the board, and around which individual examples are built. Many types of exercises will be found illustrated in the following pages.

In all oral work, it should be remembered that facility will come (1) from constantly varying the simple number combinations, or "ringing the changes" on fundamentals, and (2) from fitting the examples to the individual pupil, giving him something that he can do and that will develop him in the doing. Attention to these requirements will bring a rich return in the increased capacity of the pupils.

The author wishes to acknowledge his indebtedness to Mr.