This method enables us to compare magnitudes o kind. Suppose we have two angles, ABC and D the side ED be placed on the side BA, so that the shall fall on B; then, if the side EF falls on BC DEF equals the angle ABC; if the side EF fal BC and BA in the direction BG, the angle DEFi ABC; but if the side EF falls in the direction BH DEF is greater than ABC. A This method enables us to add magnitudes of the 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 BC D B FIG. 19 have one continuous straight line AD equal to t the lines AB and CD. Again if we have the angles ABC and DEF, the vertex E on B and the side ED in the direction angle DEF will take the position CBH, and the an and ABC will together equal the angle ABH. If the vertex E is placed on B, and the side ED c angle DEF will take the position ABF, and the at will be the difference between the angles ABC and e same Let -tex E - angle etween ss than e angle e kind. D sum of 1 place BC, the 3 DEF BA, the 60. Two points are said to be symmetrical with respect to a third point, called the centre of sym metry, if this third point bisects the P C + FIG 22 P 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 FIG. 23. B D B' B B' C A -X 66. A proof or demonstration is a course of re which the truth or falsity of any statement is established. 67. A theorem is a statement to be proved. 68. A theorem consists of two parts: the hyp that which is assumed; and the conclusion, or tha asserted to follow from the hypothesis. 69. An axiom is a statement the truth of which without proof. 70. A construction is a graphical representation metrical figure. 71. A problem is a question to be solved. 72. The solution of a problem consists of four pa (1) The analysis, or course of thought by whi struction of the required figure is discovered; (2) The construction of the figure with the aid of compasses; (3) The proof that the figure satisfies all the gi 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 stating the negative of 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 1. Things which are equal to the same thing are each other. 2. If equals are added to equals the sums are equ 3. If equals are taken from equals the remai equal. 4. If equals are added to unequals the sums ar and the greater sum is obtained from the greater m 5. If equals are taken from unequals the rema unequal, and the greater remainder is obtained greater magnitude. 6. Things which are double the same thing, or eq are equal to each other. 7. Things which are halves of the same thing, things, are equal to each other. 8. The whole is greater than any of its parts. 9. The whole is equal to all its parts taken toget |