MENTAL ARITHMETIC; COMBINING A COMPLETE SYSTEM OF RAPID COMPUTATIONS, WITH CORRECT LOGIC OF THE SOLUTIONS OF PROBLEMS, BY JOHN H. FRENCH, LL.D. Mental Arithmetic is the Logic of the Common School. NEW YORK: HARPER & BROTHERS, PUBLISHERS, FRANKLIN SQUARE. 1871. HARVARD COLLEGE LIBRARY BY EXCHANGE MAY 18 1938. PUBLISHERS' NOTICE. FRENCH'S ARITHMETICS. This Series consists of Five Books, viz.: I. FIRST LESSONS IN NUMBERS. II. ELEMENTARY ARITHMETIC. III. MENTAL ARITHMETIC. IV. COMMON SCHOOL ARITHMETIC. V. ACADEMIC ARITHMETIC. (In preparation.) The Publishers present this Series of Text-Books to American Teachers, fully believing that they contain many new and valuable features that will especially commend them to the practical wants of the age. The plan for the Series, and for each book embraced in it, was fully matured before any one of the Series was completed; and as it is based upon true philosophical principles, there is a harmony, a fitness, and a real progressiveness in the books that are not found in any other Series of Arithmetics published. Entered according to Act of Congress, in the year 1870, by In the Office of the Librarian of Congress at Washington. PREFACE. PREVIOUS to the introduction of the study of Mental Arithmetic into American schools, pupils were seldom required to give reasons for processes of computation; and few teachers could be found, who could give any reason for an arithmetical process other than "The rule says so." The publication of Warren Colburn's First Lessons, in 1826, marked the beginning of an era of progress in the art of teaching. The use of that book turned the attention of teachers to two facts, viz: 1st. That for every arithmetical process there is a reason; and 2d. That it is quite as important that pupils learn the why as the how. The general principle that reformers go to extremes, found no exception in the case of the introduction of the study of Mental Arithmetic into our schools. The chief aim of some teachers of this subject seems to have been, to see how much a child could be made to say, in solving a problem, rather than to cultivate his power to combine numbers, and his ability to explain processes and, state reasons clearly, concisely, and understandingly. By this class of teaders, much talking has been regarded as indispensable to good scholarship in this subject. But, within the past ten years, many prominent educators in different sections of the country have had sufficient independence to question the value of the discipline resulting from lengthy solutions, recited from memo rized formulæ, and to test the comparative value of these with concise processes, which appeal constantly to the reason and understanding of the learners. It is scarcely needless to remark that, wherever these tests have been made, they have resulted in favor of concise methods of solution. The objects for which children should study Mental Arithmetic are two, namely: First. To acquire accuracy and rapidity in combinations; and Second. To acquire the power to reason correctly. The attainment of this second object will give them the ability to combine a process with a reason, and to frame logical statements adapted to the solution of problems, which shall be clear, concise, and correct. In the preparation of this work these two objects have been kept constantly in view. The attention of teachers and parents is especially invited to the following general plan and distinctive features of the work: General Plan.—The book is divided into eight chapters, the first one of which is devoted to combinations in integers in which no result exceeds 100, and is called a First Course in Integers; the second chapter is a Second Course in Integers, and embraces combinations in which the results do not exceed 1,000; the third chapter is devoted to United States Money; the fourth to Compound Numbers; the fifth to Fractions; the sixth to Converse Operations in the fundamental rules, and in the reductions of Compound Numbers and Fractions; the seventh to the five general cases of Percentage, and their special applications to Insurance Commission, Profit and Loss, Stocks, Taxes, Interest, and Discount; and the eighth to Miscellaneous Review Problems. The First Course in Integers is designed especially for children who have had no previous instruction in Mental Arithmetic. Beginning at the Second Course in Integers, page 57, the order of subjects is the same as that in the Common School Arithmetic; and, from this place to the end of the book, the two works can be used together, the same subject in the two books being studied at the same time. This arrangement has received the approval of many of the best teachers in the land. Drills on Combinations.-The Addition, Subtraction, Multiplication, and Division tables are omitted. Thorough drills in the exercises given on pages 15, 17 to 22, 27, 30 to 35, 41, 51, and 52 will make pupils accurate and rapid in every possible combination of numbers. Problems. These are all new, and being prepared from material gathered from the various departments of actual business life, are, in numerous instances, the medium of instruction in the usages of business, and the uses of business terms and expressions. Illustrations.-The cuts and diagrams, all of which were designed and engraved especially for this work, possess the superior artistic merit which has been conceded to the illustrations in the other works of this Series. Manual. A manual occupies the last ten pages of the work. The first three pages of this Manual contain hints and suggestions to teachers, references to which are made in the body of the work; and the last seven pages present methods of solving the different classes of problems found in the book, and are embraced together under the head Methods. All methods for the solution of problems being omitted from the body of the work, are here placed together, and are referred to after the problems to which they apply. These Methods are intended as models, to be varied at the discretion of the teacher. They are logical and progressive, while they avoid all useless verbiage and repetition. The arrangement of the subjects, the manner in which they are presented, and the methods for the solution of problems, |