· A MENTAL ARITHMETIC, ON THE INDUCTIVE PLAN; BEING AN ADVANCED INTELLECTUAL COURSE, DESIGNED FOR COMMON SCHOOLS AND ACADEMIES. BY BENJAMIN GREENLEAF, A.M., AUTHOR OF "NATIONAL ARITHMETIC," ETC. IMPROVED ELECTROTYPE EDITION. BOSTON: PUBLISHED BY ROBERT S. DAVIS & CO. NEW YORK: MASON, BAKER, & PRATT, 142 & 144 GRAND STREET. PHILADELPHIA: J. A. BANCROFT AND COMPANY. CHICAGO: S. C. GRIGGS AND COMPANY. ST. LOUIS HENDRICKS & CHITTENDEN. EducT 118,72,438 ист GREENLEAF'S NEW COMPREHENSIVE SERIES. GREENLEAF'S NEW PRIMARY ARITHMETIC. UNIFORMITY ARITHMETICAL SERIES GREENLEAF'S NEW PRIMARY ARITHMETIC. Each book in the series complete in itself. Entered according to Act of Congress, in the year 1851, by in the Clerk's Office of the District Court for the District of Massachusetts. Entered according to Act of Congress, in the year 1857, by in the Clerk's Office of the District Court for the District of Massachusetts. Entered according to Act of Congress, in the year 1863, by in the Clerk's Office of the District Court for the District of Massachusetts. PRINTED BY H. O. HOUGHTON AND COMPANY. MARVARD COLLEGE LIBRARY. FROM THE GIFT OF CHARLES HERGERT THURBER PREFACE. THE great benefits to be derived from the study of Mental Arithmetic have, at length, become universally admitted. This appreciation of the science has given rise to an urgent demand for improved methods of teaching it. The mechanical way of arranging a set of examples to a model, which should give the pupil a solution of the whole, with little or no mental effort, is no longer approved. The object of this book is, therefore, to furnish a graded course of lessons, fully up to the most approved standards of instruction. It has been the constant aim of the author, in its prepa ration, to unfold inductively the science of numbers in such a series of progressive intellectual exercises as should awaken latent thought, encourage originality, give activity to invention, and develop the power of discriminating justly, reasoning exactly, and applying readily results to practical purposes. Forms of analysis have been introduced throughout the work, as a guide to the learner, but in connection with such examples as shall, nevertheless, give proper scope to his reasoning powers. It will be noticed, as a valuable original feature of this work, that two forms of analysis are given in the first part of the book, one full, and the other abbreviated. In the notes, aid is furnished more by hints and suggestions than by full and formal solutions, which, if too numerous, might discourage sufficiently persevering effort, and the all-important habit of self-reliance. Percentage and Interest receive full attention, and are treated, for the most part, in an original manner. The advanced exercises in the fundamental processes of the science, given in the Appendix, constitute another feature peculiar to this work. These will be found not only useful as an intellectual drill, but also exceedingly valuable for preparing the learner to dispense with written operations in business life, to a far greater extent than has hitherto been deemed practicable. The latter part of the Appendix, although quite brief, is intended to give the plan of the work, with additional models of analysis, and suggestions valuable alike to pupil and teacher. SUGGESTIONS TO TEACHERS. THE extent to which the book can be dispensed with by the class, in recitation, should be determined by the nature of the lesson and the attainments of the pupil. When the book is not used, each question should be repeated by the pupil after the teacher, and the required solution should always be given promptly. A full form of analysis should be insisted upon at first, but when it has become familiar, a more abbreviated one may be allowed. No form of solution should be permitted to pass, unless it is neatly expressed, and is entirely accurate. In general, when a pupil has thoroughly mastered the first fifty pages of this book, he may advantageously enter upon the study of the Common School Arithmetic, or the New Practical Arithmetic of Greenleaf's Series, and continue the intellectual course in connection with the written. Classes in higher Arithmetic, and even in Algebra, may often be benefited by a review of the more difficult exercises of this book, in connection with those branches. MENTAL ARITHMETIC. LESSON I. 1. John had 1 peach, and his father gave him 1 more; how many peaches did he then have ? SOLUTION. Since John had 1 peach, and his father gave him 1 more, he then had 1 peach and 1 peach, which are 2 peaches: 2. Susan has 2 books, and Mary has 1 book; how many books have they both? 3. If you had 2 cherries, and I should give you 2 more, how many cherries would you then have? 4. Lucy found 2 pins, and Sarah found 3 pins; how many did they both find? 5. If you should recite 2 lessons to-day, and 4 more to-morrow, how many would you recite in all? 6. A lemon cost 2 cents, and an orange cost 5 cents; how many cents did both cost? 7. Gave for a pencil 2 cents, and for some paper 6 cents; what was the cost of both? 8. On one bush there are 2 roses, and on another there are 7 roses; how many on both bushes? 9. 2 boys and 8 boys are how many boys? 10. A farmer sold a lamb for 2 dollars, and a calf for 9 dollars; how many dollars did he get for both? 11. Alfred caught 3 birds, and Jason caught 1 bird; how many birds did they both catch? 12. James has 3 marbles, and Charles has 2 marbles; how many marbles have they both ? |