58. By adopting a suitable unit of angles we are able to express the magnitudes of angles in numbers. If we suppose OC (Fig. 17) to turn about O from coincidence with OA until it makes one three hundred and sixtieth of a revolution, it generates an angle at O, which is taken as the unit for measuring angles. This unit is called a degree. The degree is subdivided into sixty equal parts called minutes, and the minute into sixty equal parts, called seconds. Degrees, minutes, and seconds are denoted by symbols. Thus, 5 degrees 13 minutes 12 seconds is written, 5° 13′ 12′′. A right angle is generated when OC has made one-fourth of a revolution and is an angle of 90°; a straight angle is generated when OC has made one-half of a revolution and is an angle of 180°; and the whole angular magnitude about O is generated when OC has made a complete revolution, and contains 360°. The natural angular unit is one complete revolution. But the adoption of this unit would require us to express the values of all angles by fractions. The advantage of using the degree as the unit consists in its convenient size, and in the fact that 360 is divisible by so many different integral numbers. METHOD OF SUPERPOSITION. 59. The test of the equality of two geometrical magnitudes is that they coincide throughout their whole extent. Thus, two straight lines are equal, if they can be so placed that the points at their extremities coincide. Two angles are equal, if they can be so placed that they coincide. In applying this test of equality, we assume that a line may be moved from one place to another without altering its length; that an angle may be taken up, turned over, and put down, without altering the difference in direction of its sides. This method enables us to compare magnitudes of the same kind. Suppose we have two angles, ABC and DEF. Let the side ED be placed on the side BA, so that the vertex E shall fall on B; then, if the side EF falls on BC, the angle DEF equals the angle ABC; if the side EF falls between BC and BA in the direction BG, the angle DEF is less than ABC; but if the side EF falls in the direction BH, the angle DEF is greater than ABC. A BC D This method enables us to add magnitudes of the same kind. Thus, if we have two straight lines AB and CD, by placing the point C Con B, and keeping CD in the A same direction with AB, we shall Α D B FIG. 19. have one continuous straight line AD equal to the sum of the lines AB and CD. Again if we have the angles ABC and DEF, by placing the vertex E on B and the side ED in the direction of BC, the angle DEF will take the position CBH, and the angles DEF and ABC will together equal the angle ABH. If the vertex E is placed on B, and the side ED on BA, the angle DEF will take the position ABF, and the angle FBC will be the difference between the angles ABC and DEF. SYMMETRY. 60. Two points are said to be symmetrical with respect to a P and P' are symmetrical with respect to C as a centre, if C bisects the straight line PP'. 61. Two points are said to be symmetrical with respect to a straight line, called the axis of symmetry, if this straight line bisects at right angles the straight line which joins them. Thus, P and P' are symmetrical with respect to XX' as an axis, if XX' bisects PP' at right angles. 62. Two figures are said to be symmetrical with respect to a centre or an axis if every point of one has a corresponding symmetrical point in the other. Thus, if every point in the figure A'B'C' has a symmetrical point in ABC, with respect to D as a centre, the figure A'B'C' is symmetrical to ABC with respect to D as a centre. 63. If every point in the figure A'B'C' has a symmetrical point in ABC, with respect to XX' as an axis, the figure A'B'C' is symmetrical to ABC with respect to XX' as an axis. X A P p' FIG. 23. B D B' B B FIG. 25. c' A X 66. A proof or demonstration is a course of reasoning by which the truth or falsity of any statement is logically established. 67. A theorem is a statement to be proved. 68. A theorem consists of two parts: the hypothesis, or that which is assumed; and the conclusion, or that which is asserted to follow from the hypothesis. 69. An axiom is a statement the truth of which is admitted without proof. 70. A construction is a graphical representation of a geometrical figure. 71. A problem is a question to be solved. 72. The solution of a problem consists of four parts: (1) The analysis, or course of thought by which the construction of the required figure is discovered. (2) The construction of the figure with the aid of ruler and compasses. (3) The proof that the figure satisfies all the given condi tions. (4) The discussion of the limitations, which often exist, within which the solution is possible. 73. A postulate is a construction admitted to be possible. 74. A proposition is a general term for either a theorem or a problem. 75. A corollary is a truth easily deduced from the proposition to which it is attached. 76. A scholium is a remark upon some particular feature of a proposition. 77. The converse of a theorem is formed by interchanging its hypothesis and conclusion. Thus, If A is equal to B, C is equal to D. (Direct.) If C is equal to D, A is equal to B. (Converse.) 78. The opposite of a proposition is formed by negativing its hypothesis and its conclusion. Thus, If A is equal to B, C' is equal to D. (Direct.) If A is not equal to B, C is not equal to D. (Opposite.) 79. The converse of a truth is not necessarily true. Thus, Every horse is a quadruped is a true proposition, but the converse, Every quadruped is a horse, is not true. 80. If a direct proposition and its converse are true, the opposite proposition is true; and if a direct proposition and its opposite are true, the converse proposition is true. 1. That a straight line can be drawn from any one point to any other point. 2. That a straight line can be produced to any distance, or can be terminated at any point. 3. That a circumference may be described about any point as a centre with a radius of given length. |