INTRODUCTION. THE science of Geometry is the development of a system of truths in reference to lines, angles, surfaces, and solids. Plane Geometry treats of lines, surfaces, and angles situated in a plane. The truths of Geometry are deduced one from another, in a regular concatenation of propositions, the foundation or starting-point being certain first principles called axioms, or self-evident truths, and definitions derived from the most obvious properties of figures. This being the case, it is only necessary that the student appeal to the dictates of his own mind, in order to see whether the various propositions be true; he is therefore not under the necessity of taking anything on the testimony of another, or even of appealing to experiment, as in the case of the physical sciences, but can see clearly for himself at every step whether what is affirmed be true or not. The only arbitrary thing in Geometry is the names given to the various figures, and these are derived by the ordinary methods of forming words. Thus, a figure having three sides is called a triangle or a trilateral figure, from its having three angles or three sides; a figure having four sides is called a quadrilateral figure; and, in like manner, a figure having many angles or sides, is called a polygon or a multilateral figure. Since every proposition in Geometry is used as an axiom ever after it has been demonstrated, it is evident that the progress of the student can only be secured by his learning perfectly every single proposition as he proceeds; and if he has patience to do so for some time, he will acquire a habit which will be of very great advantage to him in after-life, not only in the study of Mathematics, but in everything else; for he will never afterwards be satisfied with any assertion, whether made by himself or others, without knowing distinctly the grounds on which it rests. Thus, in learning Geometry in a proper manner, he will at the same time be laying a sure foundation for excellence in any branch of study that he may afterwards undertake. Besides, the facts thus acquired are the key, as it were, to all the exact sciences; for science is only rendered exact by bringing it under the domain of Mathematics. Thus, without a knowledge of Geometry, it is impossible to learn Mechanical Philosophy, Surveying, Navigation, or Astronomy. The propositions of Geometry are each composed of four distinct parts-namely, the General Enunciation, the Particular Enunciation, the Construction, and the Demonstration. In this edition, these parts have been so separated as to make it evident at a glance where the one ends and the other begins, so that the pupil may always know exactly with what part he is engaged, and be enabled to make himself master of the one before he proceeds to the other. The General Enunciation is that statement which stands in plain language at the top; the Particular Enunciation is the same repeated with reference to the diagram; next comes the Construction, except when the proposition is of a kind that requires none; and lastly, Demonstration. As the force of the reasoning can only be appreciated by having a clear conception of the previous parts, the pupil should master each part in its order. |