From the Preface.This book is designed to teach the principles of Arithmetical Science as far as they are involved in the elementary processes, and to secure a reasonable degree of accuracy and rapidity in expressing numbers and computing results.The pupils for whose use this book is intended cannot be expected, at the outset, to explain processes, give definitions, or assign reasons, though their notions of all of them may be quite correct; much less should they be expected to grasp the principles of a science by committing to memory statements which are the deductions and generalizations of persons entirely familiar with the subjects treated. And, therefore, this work has been prepared upon the inductive and objective methods, so that the pupil may obtain his knowledge of arithmetic from actual work with numbers, rather than from the stereotyped statements of the book.The true method of awakening in the child a clear idea of numbers is through a perception of objects, or by pictures or other representations of them. After he has become familiar with things and their names, the name will signify to him all that he needs to know about the objects so far as number is concerned; and it is no longer necessary that the objects should be present before him. The necessity, too, of associating the same numbers with such a variety of things, causes him at length to cease to regard them in connection with any particular thing, or as anything more than abstractions.In harmony with these laws of the development and growth of mind, this work presents at first the idea of number connected with visual representations of objects; afterward, when the pupils become familiar with the objects, concrete examples are given without illustration, and these are immediately followed by exercises upon abstract numbers, so that the pupil may become familiar with the various combinations that may be formed.The first lessons are devoted to instruction in counting, reading, and writing numbers up to twenty. Immediately following these, are lessons upon Addition, in which abundant practice is given in combining numbers. The fundamental processes of Subtraction, Multiplication and Division, are treated upon a plan similar to that exemplified in Addition, including besides, numerous examples combining two or more of the previous processes. The lessons on Fractions are so simple and yet so thorough, that the pupil is enabled to read, write, add, subtract, multiply, and divide fractions when the processes are not very complex. The work on Written Arithmetic, which follows, is believed to be simple, thorough, and practical; and the brief and practical treatment of the various classes of denominate numbers in common use, makes the work quite comprehensive within its scope.With the hope that the features of this book will commend it to all intelligent persons, and that instruction in the rudiments of arithmetic may be rendered more interesting and effective through its agency, it is confidently offered to the public.