THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A POINT is that which hath no parts, or which hath no magnitude.* II. A line is length without breadth. 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 hath 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, * See Notes. 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 some'where upon one of those 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 ABC, or ČBA; '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 placed at that point; < as the angle at E.' 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 that 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. 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 only two sides equal. B A scalene triangle, is that which has three unequal sides. A right-angled triangle, is that which has a right angle. An obtuse-angled triangle, is that which has an obtuse angle. XXIX. A An acute-angled triangle, is that which has three acute angles. XXX. Of four-sided figures, a square is that which 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, is that which has all its sides equal, but its angles are not right angles. 00 XXXIII. A rhomboid, is that which has its opposite sides equal to one another, 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 which being produced ever so far both ways, do not meet. POSTULATES. I. 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. And that a circle may be described from any centre, at any distance from that centre. AXIOMS. I. THINGS which are equal to the same are equal to one another. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. If equals be taken from unequals, the remainders are unequal, VI. Things which are double of the same, are equal to one another. VII. Things which are halves of the same, are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. The whole is greater than its part. |