EUCLID'S ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS. I. A POINT is that which has no parts, or which has no magnitude. II. A line is length without breadth. III. The extremities of lines are points. IV. A right 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 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 each other in a plane which meet together, but are not in the same straight line. 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, either 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 right 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 right lines, and the other upon the other line. Thus the angle which is contained by the right lines AB, CB, is named the angle ABC, or CBA; that which is contained by AB, DB, is named the angle ABD, or DBA; 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 the letter at that point; as the angle at E. X. When a straight line standing on another straight line makes the adjacent angles equal to each other, each of these 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. A term or boundary 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. The centre of a semicircle is the same with that of the circle. 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. XXV. An isosceles triangle is that which has two sides equal. XXVI. A scalene triangle is that which has three unequal sides. XXVII. A right-angled triangle is that which has a right angle. XXVIII. An obtuse-angled triangle is that which has an obtuse angle. XXIX. An acute-angled triangle is that which has three acute angles. XXX. Of quadrilateral or four-sided figures, a square has all its sides equal and all its angles right angles. XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal. XXXII. A rhombus has all its sides equal, but its angles are not right angles. XXXIII. A rhomboid has its opposite sides equal to each other, but all its sides are not equal, nor its angles right angles. XXXIV. All other four-sided figures besides these, are called Trapeziums. XXXV. Parallel straight lines are such as are in the same plane, and whic being produced ever so far both ways, do not meet. A. A parallelogram is a four-sided figure, of which the opposite sides are parallel: and the diameter or the diagonal is the straight line joining two of its opposite angles. POSTULATES. LET it be granted that a straight line may be drawn from any one point to any other point. II. That a terminated straight line may be produced to any length in a straight line. III. And that a circle may be described from any centre at any distance from that centre. AXIOMS. THINGS which are equal to the same thing are equal to one another. II. If equals be added to equals, the wholes are equal. |