PREFACE. In this edition, it is endeavoured, by a new, yet extremely simple, typographical arrangement, to render the text of Euclid more perspicuous than when printed in the ordinary manner, and to make it, as it were, its own interpreter, so as to obviate, as much as possible, the necessity of books of questions, notes, and explanations. The leading feature of the work is, that the principal steps in every demonstration have the conclusions numbered, and printed in separate lines with a different type from the premises; thus presenting to the eye the substance of the whole in the form of a synopsis which, it is hoped, the teacher will find of no small advantage in the important business of examination. This plan will, it is believed, give considerable aid to the student, in enabling him not merely to acquire a clear perception of the subject as he proceeds, but to fix in his memory, with comparatively little trouble, the order in which the lines, angles, &c., composing the diagrams, are to be considered in the course of demonstration, thereby preventing the confused notion of the whole, by which learners are so commonly embarrassed at the commencement of their geometrical studies. EUCLID'S ELEMENTS OF GEOMETRY. Book I. DEFINITIONS. I. A point is that which has no parts, or which has no magnitude. A line is length without breadth. II. III. The extremities of a line are points. IV. A straight line is that which lies evenly between its extreme points. V. A superficies is that which has only length and breadth. VI. The extremities of a superficies are lines. VII. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. VIII. "A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction." IX. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. N.B. "When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these two is somewhere upon one of these straight lines, and the other upon the other line. Thus the angle which is contained by the straight lines AB, CB, is named the angle ÃBC, or CBA; that which is contained by AB, BD, is named the angle ABD, or DBÁ; and that which is contained by DB, CB, is called the angle DBC, or CBD; but, if there be only one angle at a point, it may be expressed by a letter placed at that point, as the angle at E." X. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it. XI. An obtuse angle is that which is greater than a right angle. XII. An acute angle is that which is less than a right angle. XIII. is the extremity of any thing.” XIV. A figure is that which is inclosed by one or more boundaries. XV. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. XVI. And this point is called the centre of the circle. XVII. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A segment of a circle is the figure contained by a straight line, and the circumference it cuts off." XX. Rectilineal figures are those which are contained by straight lines. XXI. Trilateral figures, or triangles, by three straight lines. XXII. Quadrilateral, by four straight lines. XXIII. Multilateral figures, or polygons, by more than four straight lines. XXIV. Of three-sided figures, an equilateral triangle is that which has three equal sides. |