DEFINITIONS, AXIOMS, AND POSTULATES OF DEFINITIONS DEF. 1. A point has position, but it has no magnitude. DEF. 2. A line has position, and it has length, but neither breadth nor thickness. The extremities of a line are points, and the intersection of two lines is a point. DEF. 3. A surface has position, and it has length and breadth, but not thickness. The boundaries of a surface, and the intersection of two surfaces, are lines. DEF. 4. A solid has position, and it has length, breadth, and thick ness. The boundaries of a solid are surfaces. DEF. 5. A straight line is such that any part will, however placed, lie wholly on any other part, if its extremities are made to fall on that other part. DEF. 6. A plane surface, or plane, is a surface in which any two points being taken the straight line that joins them lies wholly in that surface. DEF. 7. When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle. The point is called the vertex, and the straight lines are called the arms, of the angle. A line drawn from the vertex and turning about the vertex in the plane of the angle from the position of When the angle contained by two lines is spoken of with- When the arms of an angle are in the same straight line, the conjugate angles are equal, and each is then said to be a straight angle. DEF. 8. When three straight lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles: and the angle between the extremes, through which a line would turn in passing from one extreme through the mean to the other extreme, is the sum of the two adjacent angles. DEF. 9. The bisector of an angle is the straight line that divides it into two equal angles. DEF. 10. When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle. DEF. 11. A perpendicular to a straight line is a straight line that makes a right angle with it. DEF. 12. An acute angle is that which is less than a right angle. DEF. 13. An obtuse angle is that which is greater than one right angle, but less than two right angles. DEF. 14. A reflex angle is a term sometimes used for a major conjugate angle. DEF. 15. When the sum of two angles is a right angle, each is called the complement of the other, or is said to be complementary to the other. DEF. 16. When the sum of two angles is two right angles, each is called the supplement of the other, or is said to be supplementary to the other. DEF. 17. The opposite angles made by two straight lines that intersect are called vertically opposite angles. DEF. 18. A plane figure is a portion of a plane surface inclosed by a line or lines. DEF. 19. Figures that may be made by superposition to coincide with one another are said to be identically equal; or they are said to be equal in all respects. DEF. 20. The area of a plane figure is the quantity of the plane surface inclosed by its boundary. DEF. 21. A plane rectilineal figure is a portion of a plane surface inclosed by straight lines. When there are more than three inclosing straight lines the figure is called a polygon. DEF. 22. A polygon is said to be convex when no one of its angles is reflex. DEF. 23. A polygon is said to be regular when it is equilateral and equiangular; that is, when its sides and angles are equal. DEF. 24. A diagonal is the straight line joining the vertices of any angles of a polygon which have not a common arm. DEF. 25. The perimeter of a rectilineal figure is the sum of its sides. DEF. 26. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so on. DEF. 27. A triangle is a figure contained by three straight lines. DEF. 28. Any side of a triangle may be called the base, and the opposite angular point is then called the vertex. DEF. 29. An isosceles triangle is that which has two sides equal; the angle contained by those sides is called the vertical angle, the third side the base. DEF. 30. A triangle which has one of its angles a right-angle is called a right-angled triangle. A triangle which has one of its angles an obtuse angle is called an obtuse-angled triangle. A triangle which has all its angles acute is called an acute-angled triangle. DEF. 31. The side of a right-angled triangle which is opposite to the right-angle is called the hypotenuse. DEF. 32. The perpendicular to a given straight line from a given point outside it is called the distance of the point from the straight line. DEF. 33. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet. DEF. 34. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to the lines or to one another. 2/1 3/4 6/5 8 Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6 interior angles; again 4 and 6, 3 and 5, are called alternate angles; lastly, I and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles. DEF. 35. A parallelogram is a quadrilateral whose opposite sides are parallel. DEF. 36. A trapezium (or trapezoid) is a quadrilateral that has only one pair of opposite sides parallel. DEF. 37. A parallelogram, one of whose angles is a right angle, is called a rectangle. DEF. 38. A rhombus is a parallelogram that has all its sides equal. DEF. 39. A square is a rectangle that has all its sides equal. straight line is the portion of the latter intercepted between DEF. 41. 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. This point is called the centre of the circle. DEF. 42. A radius of a circle is a straight line drawn from the centre to the circumference. DEF. 43. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference. DEF. 44. If any and every point on a line, part of a line, or group of lines (straight or curved), satisfies an assigned condition, and no other point does so, then that line, part of a line, or group of lines, is called the locus of the point satisfying that condition |