13. A term or boundary is the extremity of any thing. 14. A figure is that which is enclosed 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. [A radius of a circle is a straight line drawn from the centre to the circumference.] 18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 19. A segment of a circle is the figure contained by a straight line and the circumference which 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 figures 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: 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 are 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. [Note. The terms oblong and rhomboid are not often used. Practically the following definitions are used. Any four-sided figure is called a quadrilateral. A line joining two opposite angles of a quadrilateral is called a diagonal. A quadrilateral which has its opposite sides parallel is called a parallelogram. The words square and rhombus are used in the sense defined by Euclid; and the word rectangle is used instead of the word oblong. Some writers propose to restrict the word trapezium to a quadrilateral which has two of its sides parallel; and it would certainly be convenient if this restriction were universally adopted.] Let it be granted, POSTULATES. 1. 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 thing are equal to one another. 7. Things which are halves of the same thing 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 lmes cannot enclose a space. 11. All right angles are equal to one another. 12. If a straight line meet 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 on that side on which are the angles which are less than two right angles. PROPOSITION 1. PROBLEM. To describe an equilateral triangle on a given finite straight line. Let AB be the given straight line: it is required to describe an equilateral triangle on AB. From the centre A, at the distance AB, describe the circle BCD. [Postulate 3. From the centre B, at the distance BA, describe the circle ACE. [Postulate 3. From the point C, at which the circles cut one another, draw the straight lines CA and CB to the points A and B. [Post. 1. ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal to AB. [Definition 15. And because the point B is the centre of the circle ACE, BC is equal to BA. [Definition 15. But it has been shewn that CA is equal to AB; therefore CA and CB are each of them equal to AB. But things which are equal to the same thing are equal to one another. Therefore CA is equal to CB. Therefore CA, AB, BC are equal to one another. Wherefore the triangle ABC is equilateral, and it is described on the given straight line AB. [Axiom 1. [Def. 24. Q.E.F. |