PREFACE. THIS book, Part I., a work on Mental Arithmetic, is designed to be introductory to Part II., a work on Written Arithmetic, in which the science of Arithmetic is more fully developed, and the system rendered more complete. Part I. differs in many respects from any work of the kind now in use. It is peculiar in its commencement and mode of procedure. It commences where the child commences, and proceeds as the child proceeds, -falls in with his own mode of arriving at truth,-discovers to him his process,-cultivates his desire to know the cause, and furnishes him with language properly to express his thoughts. In this way we aid the child to think for himself, excite his interest, and secure his future progress. Hence a series of introductory exercises are given, by which the child is made familiar with his own mental operations, the mystery of his own thoughts is revealed, and he learns to express in the language of mathematical demonstration, those truths which have ordinarily been charged over to memory, or been considered intuitive. But where does the child commence, and how does he proceed? What is the natural order in which his thoughts are suggested? Obviously, his first idea is of unity. An object passes before his eye, and the idea of a thing is suggested. Another object passes, and the idea of succession, of another thing, follows. Contrast and comparison succeed, and the idea of individuality (or the distinctive traits of character peculiar to each) is the result. With the foregoing exercises and thoughts, the mind of the child is occupied until the power of language is acquired, when it learns to count and apply names to the succession of things which passes before it. A single formula is given in this work, by which the mind, through the eye, may be aided in comprehending those few elementary truths, which are the foundation of all mathemati cal calculations. Only one is given, because by it all these truths may be illustrated; and when they are clearly perceived, no more objects should be presented to the eye. Since the study of mathematics is eminently calculated to improve the reflective faculties, it is important to remove from the sight every object that would interrupt or lessen the labor of reflection. Having illustrated by the formula a few simple truths, remove it from before the pupil, and teach him to reflect upon these truths, and show him how to make use of the knowledge he has already gained, in acquiring more. A familiarity with the simple and earliest processes of the mind in search of truth, will not be deemed unimportant by those who require the same of the more advanced pupil in all analytic and synthetic processes. Still there will be objectors to the mode here adopted. Some will object to its novelty, some to its simplicity, and others to its in tricateness. The first objection has already been done away by the practices of the present age. To the second it may be replied, that it is nature's own mode, and therefore it is the best; and this sufficiently meets the third objection. But it may be added, that if it be found difficult, this fact indicates that the child is not sufficiently advanced to commence the study of Mental Arithmetic; for no child should commence the study of any science, until he is able to comprehend its simplest ele ments. But at a much earlier age than many are aware, children employ more difficult mental processes than are necessary in the commencement of the study of the science of numbers. Even when they can command but few words, and arrange these but imperfectly in sentential order, their reasonings give birth to feelings and sentiments, most of which it is true, like early fruit, quickly perish, but some, indeed, live, and are fresh even in life's winter; and not unfrequently, too, is the eye of childhood moist ened with tears, and the heart saddened by the workings of the mind within. A single instance will serve to illustrate. The younger brother of a little boy, two and a half years of age, sickened and died. The mother was also sick. One day by himself alone, the little boy was seen to burst into tears, and to weep bitterly. Being asked why he wept, he replied, Mother will die. He was asked why he thought his mother would die. He replied, Brother was sick, and died: mother is sick, and she will die. The experience of the child fully established this proposition. He had observed sickness to terminate in death, and had never known it otherwise. His reasoning was logical; it was natural that he should rely upon his conclusion with implicit confidence. He had taken into consideration the observation of his whole life, and a man of threescore years and ten could do no more. But he failed in one point; he did not know that a single fact was not enough to establish a principle. This child was not remarkable for early mental development. The case is a fair sample of the reasonings of children at an early age. The importance of adapting our teachings to the laws of mind, of seeking to excite thoughts in their natural order,-of making every new idea the result of a previous one, is too ob vious to need discussion. TO TEACHERS, Teachers will be careful to adopt and carry out the mode of instructing here proposed, unless they can substitute a better. In all questions involving analysis and synthesis, the process of the solution should be insisted on. The teacher can adopt his own mode of recitation, whether to read the question himself, or require the pupil to read it. In all cases, he should require the pupil to be so familiar with his lesson, as to be able to recite rapidly and correctly. It is no unimportant part of education to be able to bring our mental powers to act instantly and efficiently upon a given subject. Throughout the work, after each question, the process is generally required. This is to remind both teacher and scholar of |