CONTENTS. INTRODUCTION, by Professor Chase Suggestions upon the best Method of Study Preliminary Definitions, Explanations, and Axioms. Direction, Linear, Superficial Straight Line, Plane, Broken and Curved Lines and Surfaces Signs, Common, Interrogative,- Use of Letters Comparison of Magnitudes Length, Area, Solid Content, Equal, Similar, and Identical 76 Measure of a Rectangle, Parallelogram, Triangle, Trapezoid Rectangles and Squares of Lines and their Parts, and of their INTRODUCTION. THE importance of an earlier and more general study of Geometry, the expediency of bringing it into our common schools, and the need of a text-book adapted to that purpose, were, more than seven years ago, subjects of frequent conversation between myself and the author of the following treatise. I was strongly urged by him to undertake the preparation of a book. But no satisfactory plan occurring to me the subject was dropped. Recently, however, Professor Crosby has again turned his attention to the subject, and, as no work had appeared which met our views, determined himself to make an attempt to supply the want which we had so long felt. The treatise now published is the result of this determination. I shall avail myself of the opportunity offered by the publication of this work, and of the space afforded me by the author, to suggest some reasons for the introduction of Geometry into common schools, and certain principles, which should direct in the preparation of a text-book for this purpose. The study of Geometry is admirably adapted to the powers of the youthful mind. Its ideas are elementary. The ideas of form and size are among the earliest which children acquire. Why should we not continue to direct their attention to ideas so early awakened? Why should we not extend their knowledge of subjects, towards which the mind so naturally turns, and, while thus encouraging the mind to activity by leading it in the very path which itself has chosen, secure, instead of confused notions, distinct and accurate ideas? This cannot be difficult. A child can distinguish between a straight line and a curve; can understand the nature of an angle, a triangle, a parallelogram, a square, or a circle; can compare magnitudes, and apprehend the relations of equality and inequality. In Geometry, moreover, if anywhere, one thing can be learned at a time, a principle of admitted importance in the education of the young, though practically far too little regarded. Again, Geometry is, in many respects, more elementary and less difficult than Arithmetic. It is true, that Arithmetic is studied in the child's school; Geometry, in the college. Arithmetic is universally regarded, and spoken of, as an easy study; Geometry is, by some at least, regarded as sufficiently difficult. But the difficulties of Geometry are present or recent difficulties; those of Arithmetic have been forgotten with the griefs of childhood. Arithmetic has been a subject of study from infancy. One of the earliest mental operations which the child learns is to count. Yet it is an operation of no little difficulty. The relations of different numbers are to be learned, and a name to be remembered for each particular number; in other words, for each combination of units. The number of the things counted is to be abstracted from the things themselves. Is not this at least as difficult as to distinguish varieties of form, or to learn the relations of different magnitudes? Is it not as easy to apprehend the distinction between a polygon with three angles, and one with four, as between the abstract numbers 3 and 4? Is it not as easy to see that two equal straight lines will coincide, as to see that 9 times 9 are 81? Or, to understand the definition of a straight line, as to appreciate the local value of figures? Is not, in short, the idea of extension as elementary as that of number? Will a child amuse himself by cutting a sheet of paper into some definite number of pieces, or by endeavouring to produce some particular shape? by counting the pieces, or by comparing the forms with each other, and with other known forms? "But why," it will be asked, “is the study of an elementary subject frequently so difficult to young men of more mature minds?" Partly from the very fact of the maturity of their minds, or, rather, of their mental habits. They have long thought and spoken loosely of the distinctions of form and magnitude, and have never given a moment's attention to the careful and discriminating consideration of such subjects. Is it strange, that, when young men with such habits undertake to acquire, in a short time, perfectly correct notions of so many objects of study, either entirely new, or seen in new relations, they should find difficulty? Here are serious obstacles to be encountered, aside from any intrinsic difficulty in the subject. First, the subject is new, and every thing connected with it strange. Then, at the age at which they usually commence the study, they cannot, as they might at an earlier period, afford time to stop on each principle as it occurs, and revolve it, and view it on all sides, till they become thoroughly acquainted with it, before looking at another. Principle after principle must be learned in rapid succession; and, if a single principle is passed before it is perfectly known, the mind instantly becomes confused, and the study difficult. Another obstacle is found in the very simplicity of the subject. The topics first considered are so simple, that young men frequently cannot be persuaded to dwell upon them, and give them thought enough to make them perfectly familiar. Thinking it beneath them to sit down to learn the definition of a straight line or an angle, they give but little attention to the first principles, and wait for something more worthy of their study, till they find themselves lost in difficulties resulting from insufficient acquaintance with those very principles which they deemed so insignificant. Now, at the age when we would have this study commenced, there is ample time to make every term and principle, as it occurs, perfectly familiar, and to illustrate and im |