EUCLID'S ELEMENTS OF GEOMETRY. BOOK I. DEFINITIONS. 1. A POINT is that which has no parts, or which has no magnitude. 2. A line is length without breadth. 3. The extremities of a line are points. 4. A straight line is that which lies evenly between its extreme points. 5. A superficies is that which has only length and breadth. 6. The extremities of superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 8. 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. A 9. 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 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 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. 10. 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. 11. An obtuse angle is that which is greater than a right angle. 12. An acute angle is that which is less than a right angle. 13. A term or boundary is the extremity of any thing. 14. A figure is that which is inclosed by one or more boundaries. 15. 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. 16. And this point is called the centre of the circle. 17. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference. 9. 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 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 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 anglé 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. 10. 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. 11. An obtuse angle is that which is greater than a right angle. |