Algebra, or even in Arithmetic, if you pass beyond the simpler rules. And it has seemed to me, that no one can consider the admirable discipline which Geometry gives to the mind, its intimate connection with almost all the arts and occupations of men, and its essential concern in those perceptions, comparisons, judgments, calculations, and acts, which, from the very necessity of our nature, constitute the great staple of life, without feeling, that, if the door of the famed school of Plato bore upon the outside the inscription, "Let no one enter without a knowledge of Geometry," the door of every common school ought to bear upon the inside the inscription, "Let no one go forth without a knowledge of Geometry." Aside from all other advantages of the study, who can compute the vast difference which it makes in the dignity and the pleasure of life, whether we tread its path with an imperfect conception or a distinct view of the relations and properties of the material objects of grandeur, beauty, and use which are all around us, rising in the distance, skirting our pathway, shining over our heads, and blooming beneath our feet? Who would be blind, when he may see? II. Its method is that of suggestion, instead of dictation. Its object is to guide and assist the student in discovering for himself the truths of Geometry with their proofs, instead of making his work consist wholly in possessing himself of the precepts and reasonings of another, a work in which, all teachers have observed, the memory has often a larger share than the understanding. That which, more than any thing else, has robbed Geometry of its proper attractiveness, is that the mind has been made too passive in its study. With Americans at least, no study will ever be a favorite, which does not call into exercise the more active and independent powers of the mind. With what interest would Arithmetic or Algebra be studied, if every question were answered as soon as asked, or rather the fact were stated without a question's being even raised, and then this statement were followed in every instance by the whole operation written out in its minutiæ, and the scholar's work consisted merely in tracing out this operation, of which he already knows the result, and going over it again and again, till he can reproduce it (sometimes with scarce a thought that it has any meaning), when called up for recitation? III. It abounds in illustrative questions, both general and numerical. A truth has become in an especial sense our own, when we have learned to apply it. It is now no longer a stranger, but a member of the household. In these questions I have avoided high and complicated numbers, in order that the arithmetical computation might not divert the mind from the geometrical truth to be illustrated. One important addition to the plan of the work selected as a model will not fail to be observed. It is the full discussion of the elementary ideas of the science, and the definite statement of the results of its investigation in the form of distinct theorems and corollaries. Especial pains have been taken to render the definitions clear, and to conform them to the ideas actually existing in the mind, instead of making them, as they have too often been, disguised presentations of some axiom or theorem. That "a straight line is the shortest distance between two points," that " parallel lines will never meet," and that, "if any two points are taken in a plane, the straight line connecting those points will lie wholly in the plane," are all true enough, and may be readily shown; but, when proposed as definitions, let me ask the thinking man if they do not fail to express the essential idea which the mind has of the object defined, and do not substitute instead of this an elementary proposition in regard to it. I might go still farther, and ask if these definitions have not led in some cases to a species of unconscious sophistry, a reasoning at one time according to the real conception of the mind, and at another according to the arbitrary definition.* In regard to the theorems, no pains have been spared to make them clear, concise, and comprehensive. In some instances, they have been intentionally so worded as to admit a double application, and to include two propositions, one of which is the converse of the other. Mathematical signs have been extensively used, from the great relief which they give both to the hand, the eye, and the mind; and from the conviction that every thing which shortens the expression of truth, without sacrificing clearness, assists the mind in apprehending and in retaining it. The plan of the book has led to the adoption of two signs of an interrogative character (§ 13). To assist still farther the eye and the mind, each step in a demonstration is commonly printed in a separate line. It will be observed that the section-mark is sometimes prefixed rather for convenience of reference than to denote change of subject; and that the lessons in Part First which are marked A. are simply introductory to those which follow, and might be omitted without affecting the completeness of the work. For the greater simplicity of plan, the work is chiefly confined to Plane Rectilinear Geometry, and treats of this only so far as its laws can be investigated without introducing the doctrine of proportion. The field occupied is very nearly the same with that of the first and second books of Euclid. An index is given of the corresponding propositions both of Euclid and of Legendre. "This identity of direction in all its parts is that peculiar property of the straight line, which enters into every consideration of angles and parallels; and the neglect of which has been the cause of most of the embarrassment that has been felt in discussing the doctrine of parallel lines. It is hardly credible that the authors themselves, in using parallel lines in the various demonstrations in which they occur, usually think of them as not meeting. They contemplate them merely as having the same direction, and mentally derive their results from this property. This is certainly true of those who read their books."- PROFESSOR HAYWARD. The whole work is so constructed, that it is believed that the teacher, even if the science is entirely new to him, will find no difficulty in taking it up, and, with a little preparatory study, instructing a class in it. Some suggestions in regard to the best method of study and recitation immediately follow the Introduction. I cannot express strongly enough my obligations to my associate, Professor Chase, for his encouragement, valuable suggestions, and important assistance in the preparation of this work. He has conferred a great additional favor, in consenting to prefix to it an Introduction. I am also deeply indebted to my associate, Professor Young, and to Professor Peirce of the University at Cambridge. March 1, 1847. A. C. |