ARITHMETIC AND ITS APPLICATIONS DESIGNED AS A TEXT BOOK FOR COMMON SCHOOLS, HIGH SCHOOLS, AND ACADEMIES. BY DANA P. COLBURN, PRINCIPAL OF THE RHODE ISLAND STATE NORMAL SCHOOL, PHILADELPHIA: H. COW PERTHWAIT & CO. C56 Phi Delta Kappa (Lambela dun.) to Education Entered, according to Act of Congress, in the Year 1855, by DANA P. COLBURN, In the Clerk's Office of the District Court of the District of Rhode Island. PRINTED BY SMITH & PETERS. LBC PR_E FAC Е. THE principles involved in Arithmetic are few, the methods of applying them many. To be a perfect master of the subject, a person must possess, -- 1. A knowledge of the nature and use of numbers, with the methods of representing and expressing them. 2. A knowledge of the nature and use of the various numerical operations, with the methods of indicating and of performing them. 3. Such mental training and cultivation of the reasoning powers as shall enable him to understand the conditions of any given problem, and to determine from them what operations are necessary to its solution. The first of these includes every thing belonging to Notation and Numeration. The second includes the operations of Addition, Subtraction, Multiplication, and Division; to which some would add, as a separate operation, the Comparison of Numbers, as in Fractions and Ratio. The third requires a power of grasping the various conditions of a problem, of tracing their relations to each other, and of finding from them what operations must be performed, and what new relations determined, to obtain the result required. They are, however, mutually dependent, so that no person (iii) can master one without learning much of the others. Count ing is but addition; and to understand the nature and use of the number two, we must know that it equals 1 and 1, two ones, or two times 1; that it is 1 more than 1; that if 1 be taken from it 1 will be left, and so on with the other numbers things which require a knowledge of numerical operations, and also a power of tracing and appreciating relations. The operations and exercises included in the first two points are eminently adapted to give quickness of thought and rapidity of mental action. They are to Arithmetic what a knowledge of the nature and power of letters, and of their combination into words, is to reading. The processes included in them may be called the mechanical processes of Arithmetic, and by practice may and should be made so familiar that the moment a number or a combination is suggested, the mind can appreciate it, and determine the result. The third requires and imparts a power of investigation, of tracing out the relations of cause and effect, and habits of accuracy both in thought and expression. To secure these results, it is necessary that the pupil should be taught in the simplest as well as in the most complicated problems to reason for himself; to trace fully and clearly the connection between the conditions of a problem and the steps taken in its solution; to state not only what he does, but why he does it, and indicate the precise character of the result obtained by each step. Finally, he must learn to grasp the whole mechanical process before performing any part of it, so that he may know before writing a figure just what additions, subtractions, multiplications, divisions, and comparisons he has to make, and be assured that if made correctly they will lead to the true result. Such a course as this is usually taken in works on Oral Arithmetic. In studying them the scholar is thrown on his own resources; is compelled to learn principles; to follow out rigid reasoning processes and connected trains of thought; to examine and know for himself the necessity and the reason for each step taken, and for each operation performed. The result is, that the study gives strength, vigor, and healthful discipline to the mind, and becomes an almost invaluable part of the educating process. Why should not the same result follow a similar course in Written Arithmetic? Aside from the writing of numbers, there is no difference in the principles involved, in the reasoning processes demanded, or in the operations required. In the preparation of this work, the author has kept these things in view. He has endeavored to present the subject of Arithmetic as it lies in his own mind, and without any effort either to follow or t deviate from the course pursued by other writers. He has aimed to arrange the work in such a way as to lead those who may study it to understand the principles which lie at the foundation of the science, to learn to reason upon them, apply them, and to trace out their connections, relations, and combinations. He has given very full explanations and illustrations, especially of the fundamental operations; he has endeavored every where to state principles rather than rules; to throw the pupil constantly on his own resources, and force him to investigate and think for himself. He has omitted some subjects usually found in school arithmetics, because they do not belong legitimately to the subject of Arithmetic, because they are of theoretical rather than of practical importance, or because they require neither special explanation nor peculiar exercise of the mind. He has differed from other authors of school arithmetics in giving algebraic rather than geometrical explanations of the principles involved in Square and Cube Roots. In this way he has been able to give more rigid demonstrations, and more a |