1. A point is that which hath no parts, or 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 hath only length and breadth. 6. The extremities of a 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 one another in a plane, which meet together, but are not in the same direction. 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. 10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a right angle; and the straight line which stands on the other is 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 anything. 14. A figure is that which is enclosed by one or more boundaries. 15. A circle is a plane figure contained (or bounded) 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. 18. A semicircle is the figure contained by a diameter and the part of the circumference it cuts off. 19. A segment of a circle is the figure contained by a straight line and the part of the circumference it cuts off. 20. Rectilineal figures are those which are contained by straight lines. 21. Trilateral figures, or triangles, by three straight lines. 22. Quadrilateral, by four straight lines. 23. Multilateral figures, or polygons, by more than four straight lines. 24. Of three-sided figures, an equilateral triangle is that which has three equal sides. 25. An isosceles triangle is that which has only two sides equal. 26. A scalene triangle is that which has three unequal sides. A 27. A right-angled triangle is that which has a right angle. 28. An obtuse-angled triangle is that which has an obtuse angle. 29. An acute-angled triangle is that which has three acute angles. 30. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. 31. An oblong is that which has all its angles right angles, but not all its sides equal. 32. A rhombus is that which has all its sides equal, but its angles are not right angles. 33. A rhomboid is that which has its opposite sides equal to one another, but all its sides not equal, nor its angles right angles. 34. All other four-sided figures besides these, are called Trapeziums. 35. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways do not meet. 36. A parallelogram is a four-sided figure of which the opposite sides are parallel; and the diagonal is the straight line joining the vertices of two opposite angles. POSTULATES. 1. Let it be granted, that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any centre, at any distance from that centre. AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double of the same are equal to one another. 7. Things which are halves of the same are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. 9. The whole is greater than its part. 10. Two straight lines cannot enclose a space. 11. All right angles are equal to one another. 12. If a straight line meets two straight lines so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles. BOOK II. 1. Every right-angled parallelogram is called a rectangle, and is said to be contained by any two of the straight lines which form one of its right angles. 2. In every parallelogram, any of the parallelograms about a diagonal, together with its complements, is called a gnomon. BOOK III. 1. Equal circles are those of which the diameters are equal, or those from the centres of which the straight lines drawn to the circumference are equal. 2. A straight line is said to touch a circle when it meets the circumference, and being produced does not cut the circle. 3. Circles are said to touch one another when their circumferences meet in a point, but do not cut one another. 4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal. 5. And the straight line which has the greater perpendicular drawn to it, is said to be further from the centre. 6. The angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment to the extremity of the straight line which is the base of the segment. 7. An angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle. 8. A sector of a circle is the figure contained by two straight lines drawn from the centre, and the arc, or part of the circumference between them. 9. Similar segments of circles are those in which the angles are equal, or which contain equal angles. |