38. A SEMI-CIRCLE is the half of a circle, terminated by a diameter and the semi-circumference. Thus, in fig. 27, the diameter a b divides the circle into two semi-circles. 39. A SEGMENT of a circle is a portion cut off by a chord, and the part of the circumference intercepted by the chord. Thus, ab c, figures 28 and 29, are segments; and fig. 30, though a semi-circle, is still a segment, terminated by the diameter, instead of a lesser chord. 40. A SECTOR of a circle is the portion contained by two radii and the intercepted part of the circumference. Thus, a bc, fig. 31, is the sector of a circle. 41. The QUADRANT of a circle is a sector contained by two radii, at a right-angle with each other, and the intercepted part of the circumference; as, ab c, in fig. 32. 42. An ARC of a circle is any portion of its circumference. 43. The ALTITUDE of a figure is a straight line drawn from the vertical angle, perpendicular to the opposite side, or to the opposite side produced or continued. Thus, CD, fig. 33, is the altitude of the triangle ABC, drawn from the vertical angle C to the opposite side AB produced to D. 44. NOTATION. When several angles unite at a point, each angle is indicated by three letters, the middle letter denoting the angular point, and the others the sides containing that angle. Thus, in fig. 34, ABC, ABD, ABE, the middle letter B indicates the angular point: in the first, AB, BC, the two sides; in the second, AB, BD, the sides; and, in the third, AB, BE, the sides. 45. EXPLANATION OF TERMS. An AXIOM is a self-evident truth. A THEOREM is a truth which becomes evident by a process of reasoning called a demonstration. A PROBLEM is a thing required to be done, or a question proposed for solution. A LEMMA is a truth premised to facilitate either the demonstration of a theorem, or the solution of a Problem. A PROPOSITION is the common name of a Theorem or Problem. A COROLLARY is a consequence or deduction which follows from a Proposition. A SCHOLIUM is an explanatory remark upon one or more preceding Proposition or Propositions. An HYPOTHESIS is a supposition made either in the enunciation of a Proposition, or in the course of a demonstration. 46. AXIOMS. 1. Things which are equal to the same thing, or things, are equal to one another. 2. If equals be added to equals, the wholes will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the wholes will be unequal. 5. If equals be taken from unequals, the remainders will be 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. Only one straight line can be drawn from one point to another. 11. Two straight lines cannot be drawn through the same point, parallel to the same straight line, without coinciding with one another. 47. POSTULATES, OR DEMANDS. 1. Let it be granted that a straight line may be drawn from any one point 2. That a terminated straight line may be produced, or continued, to any length. 3. That a circle may be described from any centre, and at any distance from that centre, or with any radius. THEOREMS. THEOREM 1. 48. Any straight line, CD, which meets another straight line, AB, makes with it two adjacent angles, ACD, BCD; which, taken together, are equal to two right angles.* B At the point C, let the straight line CE be drawn, perpendicular to AB. The angle ACD is the sum of the angles ACE and ECD; therefore ACD+DCB shall be the sum of the three angles ACE, ECD, DCB, (Axiom 2, page 15). Now the A angle ACE is a right angle, and the sum of the angles ECD, DCB, make a right angle; therefore the sum of the two angles ACD, BCD, is equal to two right angles. 49. COROLLARY.-If one of the angles ACD, BCD, is a right angle, the other is, also, a right angle. THEOREM 2. 50. If the sum of two adjacent angles, ACD, DCB, be equal to two right angles, the exterior sides form one continued straight line. For, if CB is not the continuation of AC, let CE be its continuation; then the sum of the angles ACD, DCE, is equal to two right angles, (theorem 1,) but, by hypothesis, the sum of the angles ACD, DCB, is equal to two right angles; therefore the two angles ACD, DCE, is equal to the two angles The signs used in Algebraic Notation are explained hereafter; but, as several may previously occur, it may here be noticed that + (plus) signifies more, or one quantity or thing added to another: The sign - (minus) signifies less, or one quantity subtracted from another: = means is, or are, equal to x into, or multiplied by: divided by, as 30-3 or 10. ACD, DCB (Ax. 1, page 15); and, taking from each of these equals the angle ACD, there will remain the angle DCE, equal to DCB, a part equal to the whole, which is impossible (Ax. 9, page 15). THEOREM 3. 51. If two straight lines, AB, DE, cut one another, the opposite angles shall be equal to one another. X B For, since DE is a straight line, the sum of the two angles ACD, a ACE, is equal to two right angles (theorem 1); and, because AB is a straight line, the sum of the angles ACE, ECB, is equal to two right angles (theorem 1); therefore the sum of the angles ACD, ACE, is equal to the sum of the angles ACE, ECB; and, taking away from each the common angle ACE, there will remain the angle ACD, equal to the vertical opposite angle ECB. THEOREM 4. 52. Two straight lines, which have two common points, coincide entirely throughout their whole extent. Let A and B be the two common points; in the first place, the two lines can make but one from A to B, (Ax. 10, p. 15). If it were possible that they could separate, let C be the point B At the point C suppose CF to be drawn, perpendicular to AC; then, because ACD is, by hypothesis, a straight line, the angle FCD is a right angle; (Def. art. 9;) in like manner, because ACE is supposed to be a straight line, the angle FCE is a right angle; therefore the angles FCD, FCE, are equal; but this is impossible (Ax. 9, page 15); therefore the two straight lines, which have two common points, A and B, cannot separate, but must form one continued line. THEOREM 5. 53. Two triangles are equal when, in the one, an angle and the two sides which contain it are equal, in the other, to an angle and the two sides which contain it. Let the angle A be equal to the angle D, the side AB equal to DE, and the side AC equal to DF; then the triangles ABC, DEF, shall be equal. Suppose the triangle ABC to be placed upon the tri- B D angle DEF, so that AB may be upon DE; then, because the angles A and D are equal, AC will fall upon DF; and, because AB is equal to DE, and AC equal to DF, the point B will coincide with E, and C with F: therefore, the base BC will coincide with the base EF (theorem 4); and, since the sides of the triangles coincide, the other two angles must also coincide; that is, they must be equal to each other. THEOREM 6. 54. Two triangles are equal when a side and two adjacent angles of the one are respectively equal to a side and two adjacent angles of the other. Let the side BC be equal to the side EF, the angle B equal to the angle E, and the angle C equal to the angle F, the triangles shall be equal. For, suppose the triangle ABC to be placed upon the triangle DEF, so that their bases, BC and EF, may coincide; then, because the angles B and E are equal, the straight line BA will fall upon ED; and, because the angles C and F are equal, the straight line CA will fall upon FD: therefore, the three sides of one triangle will coincide with the three sides of the other; and, consequently, the triangles themselves will be equal: and, since therefore the sides of the triangles coincide, the corresponding angles will be equal. |