69. What proposition is the converse to the twelfth axiom of the First Book? What other two propositions are complementary to these? 70. If lines being produced ever so far do not meet; can they be otherwise than parallel? If so, under what circumstances? 71. Define adjacent angles, opposite angles, vertical angles, and alternate angles; and give examples from the First Book of Euclid. 72. Can you suggest anything to justify the assumption in the twelfth axiom upon which the proof of Euc. 1. 29, depends? 73. What objections have been urged against the definition and the doctrine of parallel straight lines as laid down by Euclid? Where does the difficulty originate? What other assumptions have been suggested, and for what reasons? 74. Assuming as an axiom that "two straight lines which cut one another, cannot both be parallel to the same straight line"; deduce Euclid's twelfth axiom as a corollary of Euc. 1. 29. 75. From Euc. 1. 27, shew that the distance between two parallel straight lines is constant. 76. If two straight lines be not parallel, shew that all straight lines falling on them, make alternate angles, which differ by the same angle. 77. Taking as the definition of parallel straight lines, that they are equally inclined to the same straight line towards the same parts; prove that "being produced ever so far both ways they do not meet." Prove also Euclid's axiom 12, by means of the same definition. 78. What is meant by exterior and interior angles? Point out examples. 79. Can the three angles of a triangle be proved equal to two right angles without producing a side of the triangle? 80. Shew how the corners of a triangular piece of paper may be turned down, so as to exhibit to the eye that the three angles of a triangle are equal to two right angles. 81. Explain the meaning of the term corollary. Enunciate the two corollaries appended to Euc. 1. 32, and give another proof of the first. What other corollaries may be deduced from this proposition? 82. Shew that the two lines which bisect the exterior and interior angles of a triangle, as well as those which bisect any two interior angles of a parallelogram, contain a right angle. 83. The opposite sides and angles of a parallelogram are equal to one another, and the diameters bisect it. State and prove the converse of this proposition. Also shew that a quadrilateral figure is a parallelogram, when its diagonals bisec each other: and when its diagonals divide it into four triangles, which are equal two and two, viz. those which have the same vertical angles. 84. In constructing a parallelogram, state what data render the problem deter minate or indeterminate. Exemplify each of the cases. 85. If two straight lines join the extremities of two parallel straight lines, but not towards the same parts, when are the joining lines equal, and when are they unequal? 86. If either diameter of a four-sided figure, divide it into two equal triangles, is the figure necessarily a parallelogram? Prove your answer. 87. Shew how to divide one of the parallelograms in Euc. 1. 35, by straight lines, so that the parts when properly arranged shall make up the other parallelogram. 88. Distinguish between equal triangles and equivalent triangles, and give examples from the First Book of Euclid. ། 89. What is meant by the locus of a point? Adduce instances of loci from the first Book of Euclid. 90. How is it shewn that equal triangles upon the same base or upon equal bases, have equal altitudes, whether they are situated on the same side or upon opposite sides of the same straight line? 91. In Euc. I. 37, 38, if the triangles are not towards the same parts, shew that the straight line joining the vertices of the triangles is bisected by the line containing the bases. 92. If the complements (fig. Euc. 1. 43) be squares, determine their relation to the whole parallelogram. 93. What is meant by a parallelogram being applied to a straight line ? 94. Is the construction of Euc. 1. 45, perfectly general? 95. Define a square without including superfluous conditions, and explain the mode of constructing a square upon a given straight line in conformity with such a definition. 96. The sum of the angles of a square is equal to four right angles. Is the converse true? If not, why ? 97. Conceiving a square to be a figure bounded by four equal straight lines not necessarily in the same plane, what condition respecting the angles is necessary to complete the definition ? 98. In Euclid 1. 47, why is it necessary to prove that one side of each square described upon each of the sides containing the right angle, should be in the same straight line with the other side of the triangle? 99. On what assumption is an analogy shewn to exist between the product of two equal numbers and the surface of a square? 100. Is the triangle whose sides are 3, 4, 5 right-angled, or not? 101. Can the side and diagonal of a square be represented simultaneously by any finite numbers? 102. By means of Euc. 1. 47, the square roots of the natural numbers, 1, 2, 3, 4, &c. may be represented by straight lines. 103. Prove the 47th Prop. of Book 1. by describing the squares on the sides towards the hypotenuse, and shewing that they are divided by the sides of the square on the hypotenuse into segments which may be so placed as to cover exactly that square. 104. If Euclid II. 2, be assumed, enunciate the form in which Euc. 1. 47 may be expressed. 105. Classify all the properties of triangles and parallelograms, proved in the First Book of Euclid. 106. Mention any propositions in Book 1. which are included in more general ones which follow. 107. Beginning with the forty-seventh proposition of the First Book of Euclid's Elements, trace backwards how many of the propositions of the book are necessary to the proof. 108. How are converse propositions generally proved? Do you know of any exception to this general rule? 109. Taking solidity as a fundamental idea of Geometry, how would you define a superficies, a line, a point? 110. What general classification may be made of the Propositions contained in the First Book of Euclid? , BOOK II, DEFINITIONS. ` I. EVERY right-angled parallelogram is called a rectangle, and is said to be contained by any two of the straight lines which contain one of the right angles. II. In every parallelogram, any of the parallelograms about a diameter together with the two complements, is called a gnomon. "Thus the parallelogram HG together with the complements AF, FC, is the gnomon, which is more briefly expressed by the letters AGK, or EHC, which are at the opposite angles of the parallelograms which make the gnomon.' PROPOSITION I. THEOREM. If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines, is equal to the rectangles_contained by the undivided line, and the several parts of the divided line. Let A and BC be two straight lines; and let BC be divided into any parts BD, DE, EC, in the points D, E. Then the rectangle contained by the straight lines A and BC, shall be equal to the rectangle contained by A and BD, together with that contained by A and DE, and that contained by A and EC. Through G draw GH parallel to BC, (í. 31.) and through D, E, C, draw DK, EL, CH parallel to BG, meeting GH in K, L, H. Then the rectangle BH is equal to the rectangles BK, DL, EH. And BH is contained by A and BC, for it is contained by GB, BC, of which GB is equal to A: for it is contained by GB, BD, of which GB is equal to 4: because DK, that is, BG, (1. 34.) is equal to A; and in like manner the rectangle EH is contained by A, EC: therefore the rectangle contained by A, BC, is equal to the several rectangles contained by A, BD, and by A, DE, and by A, EC. Wherefore, if there be two straight lines, &c. Q.E.D. PROPOSITION II. THEOREM. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square on the whole line. Let the straight line AB be divided into any two parts in the point C. Then the rectangle contained by AB, BC, together with that contained by AB, AC, shall be equal to the square on AB. Upon AB describe the square ADEB, (1. 46.) and through C draw CF parallel to AD or BE, (I. 31.) meeting DE in F. Then AE is equal to the rectangles AF, CE. And AE is the square on AB; and AF is the rectangle contained by BA, AC; for BE is equal to AB : therefore the rectangle contained by AB, AC, together with the Q. E.D. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square on the aforesaid part. Let the straight line AB be divided into any two parts in the point C. Then the rectangle AB, BC, shall be equal to the rectangle AC, CB, together with the square on BC. Upon BC describe the square CDEB, (1. 46.) and produce ED to F, for it is contained by AB, BĚ, of which BE is equal to BC: therefore the rectangle AB, BC, is equal to the rectangle AC, CB, together with the square on BC. If therefore a straight line be divided, &c. Q.E.D. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the parts. Let the straight line AB be divided into any two parts in C. Then the square on AB shall be equal to the squares on AC, and CB, together with twice the rectangle contained by AC, CB. Upon AB describe the square ADEB, (1. 46.) join BD, through draw CGF parallel to AD or BE, (1. 31.) meeting BD in G and DE in F; and through G draw HGK parallel to AB or DE, meeting AD in H, and BE in K; Then, because CF is parallel to AD and BD falls upon them, therefore the exterior angle BGC is equal to the interior and opposite. ́angle BDA; (1. 29.) but the angle BDA is equal to the angle DBA, (1. 5.) because BA is equal to AD, being sides of a square; wherefore the angle BGC is equal to the angle DBA or GBC; and therefore the side BC is equal to the side CG; (1. 6.) but BC is equal also to GK, and CG to BK; (1. 34.) wherefore the figure CGKB is equilateral. It is likewise rectangular; for, since CG is parallel to BK, and BC meets them, therefore the angles KBC, BCG are equal to two right angles; (1. 29.) but the angle KBC is a right angle; (def. 30. constr.) wherefore BCG is a right angle: and therefore also the angles CGK, GKB, opposite to these, are right angles; (1. 34.) |