69.- Name all the triangles in the accompanying figure. 70. Name the additional triangles that I would be formed if AD were joined. 71. Name by three letters all the angles opposite to BC; to BE; to CE. 72. Name all the sides that are opposite to angle A; to angle D. 73. Name all the angles in the figure that are called exterior angles of the B Ꭰ triangle BEC; of the triangle AEB; of the triangle CED. 74. ABCD is a quadrilateral. Name it in seven other ways. 75. If the diagonals AC, BD be A 76. Name the two angles opposite to the diagonal AC. D through which the diagonal AC passes. BD 80. Could a square, with propriety, be called a rhombus ? 81. Could a rhombus be called a square? 82. Could a rectangle be called a parallelogram? 83. Could a parallelogram be called a rectangle? C 84. Would it be a sufficient definition of a parallelogram to say that it is a figure whose opposite sides are parallel? Why? 85. Could a parallelogram or a rectangle be called a trapezium? 86. Could a trapezium be called a parallelogram or a rectangle? 87. What is a diagonal of a quadrilateral, and how many diagonals has a quadrilateral? 88. How many sides has a polygon ? 89. Which postulate allows us to join two points? 92. In what sense is the word 'circle' used in the third postulate? 93. What are the only instruments that may be used in elementary plane geometry? Under what restrictions are they to be used? 94. What is an axiom? Give an example of one. 95. State Euclid's axiom about magnitudes which coincide. 96. Would it be correct to say, magnitudes which fill the same space, instead of magnitudes which coincide? Illustrate your answer by reference to straight lines, and angles. 97. What is Euclid's axiom about right angles? 98. What is the axiom about parallels ? 99. Would it be correct to say, two straight lines which pass through the same point cannot be both parallel to the same straight line? 100. Could two straight lines which do not pass through the same point be both parallel to a third straight line? EXPLANATION OF TERMS. Propositions are divided into two classes, theorems and problems. A theorem is a truth that requires to be proved by means of other truths already known. The truths already known are either axioms or theorems. A problem is a construction which is to be made by means of certain instruments. The instruments allowed to be used are (see the remarks on the postulates) the ruler and the compasses. A corollary is a truth which is (more or less) easily inferred from a proposition. In the statement of a theorem there are two parts, the hypothesis and the conclusion. Thus, in the theorem, 'If two sides of a triangle be equal, the angles opposite to them shall be equal,' the part, 'if two sides of a triangle be equal,' is the hypothesis, or that which is assumed; the other part, 'the angles opposite to them shall be equal,' is the conclusion, or that which is inferred from the hypothesis. The converse of a theorem is derived from the theorem by interchanging the hypothesis and the conclusion. Thus, the converse of the theorem mentioned above is, 'If in a triangle the angles opposite two sides be equal, the sides shall be equal.' When the hypothesis of a theorem consists of several hypotheses, there may be more than one converse to the theorem. In proving propositions, recourse is sometimes had to the following method. The proposition is supposed not to be true, and the con sequences of this supposition are then examined, till at length a result is reached which is impossible or absurd. It is therefore inferred that the proposition must be true. Such a method of proof is called an indirect demonstration, or sometimes a reductio ad absurdum (a reducing to the absurd). SYMBOLS AND ABBREVIATIONS. +, read plus, is the sign of addition, and signifies that the magnitudes between which it is placed are to be added together. read minus, is the sign of subtraction, and signifies that the magnitude written after it is to be subtracted from the magnitude written before it. read difference, is sometimes used instead of minus, when it is not known which of the two magnitudes before and after it is the greater. = is the sign of equality, and signifies that the magnitudes between which it is placed are equal to each other. It is used here as an abbreviation for 'is equal to,' 'are equal to,' 'be equal to,' and 'equal to.' ▲ stands for 'perpendicular to,' or 'is perpendicular to.' 'parallel to,' or 'is parallel to.' 'angle.' 'triangle.' 'parallelogram.' 'circle.' 'circumference.' 'therefore.' This symbol turned upside down ('.'), which is sometimes used for 'because' or 'since,' I have not introduced, partly because some writers use it for‘therefore,’ and partly because it is easily confounded with the other. AB2 stands for 'the square described on AB.' AB BC stands for 'the rectangle contained by AB and BC!' AB stands for 'the ratio compounded of the ratios of A to B and B to C. A: B: = C: D stands for the proportion 'A is to B as C is to D.' The small letters a, b, c, m, n, p, &c. stand for numbers. App. stands for 'appendix.' In the references given at the right-hand side of the page (Euclid gives no references), the Roman numerals indicate the number of the book, the Arabic numerals the number of the proposition. Thus, I. 47 means the forty-seventh proposition of the first book. In the figures to certain of the theorems, it will be seen that some lines are thick, and some dotted. The thick lines are those which are given, the dotted lines are those which are drawn in order to prove the theorem. [In a few figures this arrangement has been neglected to attain another object.] In the figures to certain of the problems, some lines are thick, some thin, and some dotted. The thick lines are those which are given, the thin lines are those which are drawn in order to effect the construction, and the dotted lines are those which are necessary for the proof that the construction is correct. In the figures which illustrate definitions, the lines are almost invariably thin. To describe an equilateral triangle on a given straight line. D Let AB be the given straight line : it is required to describe an equilateral triangle on AB. With centre A and radius AB, describe O BCD. Post. 3 Post. 3 Post. 1 ABC shall be an equilateral triangle. I. Def. 16 I. Def. 16 BC. I. Ax. 1 I. Def. 23 For AB = AC, being radii of the ○ BCD; and AB BC, being radii of the O ACE; AC = = AB, AC, BC are all equal, and ABC is an equilateral triangle. DEDUCTIONS. 1. If the two circles intersect also at F, and AF, BF be joined, prove that ABF is an equilateral triangle. 2. Show how to find a point which is equidistant from two given points. 3. Show how to make a rhombus having one of its diagonals equal to a given straight line. 4. Show how to make a rhombus having each of its sides equal to a given straight line. |