PRACTICAL ARITHMETIC, UNITING THE INDUCTIVE WITH THE SYNTHETIC MODE OF INSTRUCTION. FOR SCHOOLS AND ACADEMIES. BY JAMES B. THOMSON, LL. D. AUTHOR OF MENTAL ARITHMETIC; SLATE AND BLACK-BOARD EXERCISES; LEGENDRE'S GEOMETRY, ETC. ~ NEW STEREOTYPE EDITION, REVISED AND ENLARGED. NEW YORK: IVISON, PIIINNEY, BLAKEMAN & CO., 1864. DAY & THOMSON'S MATHEMATICAL SERIES, FOR SCHOOLS AND ACADEMIES. I ARITHMETICAL TABLES. For Primary Schools. Revised and Enlarged. II. MENTAL ARITHMETIC, or First Lessons in Numbers. For Children. Revised and Enlarged. III RUDIMENTS OF ARITHMETIC, or Exercises for the Slate and Black board. For beginners. Revised and Enlarged. IV. EXERCISES IN ARITHMETICAL ANALYSIS, or Higher Mental Arithmetic. Being a Sequel to Thomson's First Lessons in Numbers. For Advanced Classes. V. PRACTICAL ARITHMETIC-Uniting the Inductive with the Synthetic mode of Instruction; also illustrating the Principles of CANCELLATION. Revised and Enlarged. VI. KEY TO PRACTICAL ARITHMETIC. Revised and Enlarged VIII. KEY TO HIGHER ARITHMETIC. For Teachers. X. KEY TO THOMSON'S DAY'S ALGEBRA. For Teachers. XII. ELEMENTS OF TRIGONOMETRY, MENSURATION, AND LOGARITHMS. XIII. ELEMENTS OF SURVEYING. Entered, according to Act of Congress, in the year 1853, In the Clerk's Office of the Southern District of New York. STEREOTYPED BY THOMAS B. SMITH, 216 WILLIAM STREET, N. Y. Ir has been well said, that "whoever shortens the rosto knowledge, lengthens life." The value of a knowledge of Arithmetic is too generally appreciated to require comment. When properly studied, two important ends are attained, viz: discipline of mind, and facility in the application of numbers to business calculations. Neither of these results can be secured, unless the pupil thoroughly understands the principle of every operation he performs. There is no uncertainty in the conclusions of mathematics; there should be no guess-work in its operations. What then is the cause of so much groping and fruitless effort in this department of education? Why this aimless, mechanical" ciphering," that is so prevalent in our schools? Many of these evils, it is believed, arise from the practice of requiring beginners to solve problems above their comprehension, and to learn abstract rules without analysing their principles, or explaining the reasons upon which they are based. Taking his slate and pencil, the pupil sits down to the solution of his problem, but soon finds himself involved in an impenetrable maze. He anxiously asks for light, and is directed "to learn the rule." He does this to the letter, but his mind is still in the dark. By puzzling and repeated trials, he at length finds that certain multiplications and divisions produce the answer in the book; but so far as the reasons of the process, and the principles of the rule are concerned, he is totally ignorant. It needs no arguments to show that this course is calculated to dampen the ardor of a child, and make him a mechanical M290003 cipherer. To require a pupil to learn and understand a rule before he is permitted to see its principles illustrated by simple practical examples, places him in the condition of the boy, whose mother charged him never to go into the water till he had learned to swim. These embarrassments are believed to be as unnecessary, as they are deleterious. The present work was undertaken, with the hope of contributing something towards their removal. Its plan is the following: 1. The definitions are designed to be simple, brief, and comprehensive. If they are not simple, children can not understand them; if long, it is difficult to remember them; and if not comprehensive, they are not worth remembering. 2. The pupil is led to a knowledge of the rule by induction, a process by which he is taught to reason from particular examples to general principles. To this end the examples at the commencement of the rules are practical, and are adapted to illustrate the particular principles under consideration. Every teacher can bear testimony, that children reason upon practical questions with far greater facility and accuracy, than they do upon abstract numbers. 3. The separate principles being analyzed and understood, the general rule is then deduced, and arranged in a convenient manner for reference and review; thus combining the advantages both of the inductive and synthetic modes of instruction. 4. The rules, as far as possible, are constructed in such a manner as to suggest the principles upon which they depend; and the reasons for the various operations are carefully given. 5. The work abounds in examples for practice, which are drawn from the various departments of business and science, and are calculated to call into exercise the different principles of the rules, to wake up thought, and to prepare the learner for the active duties of life. Their arrangement is gradual and progressive. At first, the numbers are small and refer to objects with which the pupil is acquainted, in order that he may clearly understand the nature of the question and the reason for every step in its solution. As he becomes familiar with the operation and the principles of the rule, the numbers are larg and the combinations more complicated and difficult. 6. Mental exercises are frequently interspersed through the work, which, if properly attended to, are among the best means to arrest and prevent habits of mechanical ciphering. 7. In the arrangement of subjects, it has been a cardinal point to follow the natural order of the science. No principle is used in the explanation of another, until it has itself been demonstrated or explained. Common fractions, therefore, are placed immediately after division, for two reasons. First, they arise from division, and are inseparably connected with it. Second, in Reduction, Compound Addition, &c., it is frequently necessary to use fractions; consequently fractions must be explained before the Compound rules can be understood. For the same reason, Federal Money, which is based upon the decimal notation, is placed after Decimal Fractions. Interest, Insurance, Commission, &c., are also placed after Percentage, upon whose principles they are based. 8. In preparing the tables of Weights and Measures, particular pains have been taken to ascertain those that are in present use in our country, and to give the legal standards as adopted by the General Government, in 1834.* 9. The subject of Analysis is deemed so essential to a thorough knowledge of arithmetic and business calculations, that an entire section is devoted to its development and application. The principles of Cancellation are carefully explained, and its important applications pointed out, in their proper places. The Square and Cube Roots are illustrated by geometrical figures and cubical blocks. 10. The work contains much valuable information respecting business transactions and matters of science, not found in other school Arithmetics. * Nearly twelve years had elapsed after the Government adopted a uniform standard of Weights and Measures, before the publication of the first edition of this work; and yet not a single arithmetic, so far as we know, had given these standards to the public, or even intimated that any thing had been done upon the subject. In the year 1836, Congress directed the Secretary of the Treasury to cause to be delivered to the Governor of each State in the Union, or to such per son as he should appoint, a complete set of all the Weights and Measures adopte as standards, for the use of the States respectively; to the end that a unifor standard might be established throughout the United States. Since that, many of the States have adopted the same, and it is to be hoped that every State of the Union will promptly unite in the accomplishment of an object so conducive both to individual and public good. |