46. Shew that in Euclid's figure (Euc. 11. 11.) four other lines, besides the given line, are divided in the required manner. 47. Enunciate Euc. vi. 31. What theorem of a previous book is included in this proposition? 48. What is the superior limit, as to magnitude, of the angle at the circumference in Euc. vi. 33? Shew that the proof may be extended by withdrawing the usually supposed restriction as to angular magnitude; and then deduce, as a corollary, the proposition respecting the magnitudes of angles in segments greater than, equal to, or less than a semicircle. 49. The sides of a triangle inscribed in a circle are a, b, c units respectively: find by Euc. VI. c, the radius of the circumscribing circle. 51. Shew independently that Euc. vI. D, is true when the quadrilateral figure is rectangular. 52. Shew that the rectangles contained by the opposite sides of a quadrilateral figure which does not admit of having a circle described about it, are together greater than the rectangle contained by the diagonals. 53. What different conditions may be stated as essential to the possibility of the inscription and circumscription of a circle in and about a quadrilateral figure? 54. Find two lines which shall be reciprocally proportional to two given lines. 55. Apply Euc. vi. 11, to find a series of six lines which shall be in continual proportion when the first two lines of the series are given. 56. What problem in the Sixth Book is immediately solved by Euc. 11. 14. P 57. Find a fourth proportional to three given similar triangles. 58. How many similar triangles are in the figure Euc. VI. B.? Name them according to their homologous sides. 59. State the proposition from which it is shewn that the side of an isosceles triangle inscribed in a circle is a mean proportional between the altitude of the triangle and the diameter of the circle. 60. How do you explain the case of Euc. VI. A., when the triangle is isosceles? 61. Shew by the aid of Euc. vI. 31, that a trapezium may be constructed equal and similar to any number of similar trapeziums. 62. If the sides about each of the angles of two triangles be proportional, the triangles are similar. Is this necessarily true in the case of quadrilateral figures? 63. Under certain circumstances the reciprocals of theorems are necessarily true. Illustrate this by examples from the Sixth Book of the Elements. 64. The perimeters of similar polygons are proportional to the homologous sides of the polygons. 65. What are the four forms in which Euclid's definition of Proportion is applied in the Fifth and Sixth Books of the Elements? Quote instances. 66. If Euclid had proved first that rectangles and parallelograms of the same altitude are to one another as their bases; and then deduced that triangles of the same altitude are to one another as their bases :-does it appear that any advantages would have arisen in this mode of treating the subject ? 67. Point out those propositions in the Sixth Book in which Euclid's definition of proportion is directly applied. 68. In what cases are triangles proved to be equal in Euclid, and in what cases are they proved to be similar? BOOK XI. DEFINITIONS. I. A SOLID is that which hath length, breadth, and thickness. II. That which bounds a solid is a superficies. III. A straight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane. IV. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane. V. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane. VI. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane. VII. Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another. VIII. Parallel planes are such as do not meet one another though produced. IX. A solid angle is that which is made by the meeting, in one point, of more than two plane angles, which are not in the same plane. X. Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude. XI. Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes. XII. A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet. XIII. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms. XIV. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved. XV.. The axis of a sphere is the fixed straight line about which the semicircle revolves. XVI. The center of a sphere is the same with that of the semicircle. XVII. The diameter of a sphere is any straight line which passes through the center, and is terminated both ways by the superficies of the sphere. XVIII. A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled; and if greater, an acute-angled cone. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. The base of a cone is the circle described by that side containing the right angle, which revolves. XXI. A cylinder is a solid figure described by the revolution of a rightangled parallelogram about one of its sides which remains fixed. XXII. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXIII. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals. XXV. A cube is a solid figure contained by six equal squares. XXVI. A tetrahedron is a solid figure contained by four equal and equilateral triangles. XXVII. An octahedron is a solid figure contained by eight equal and equilateral triangles. XXVIII. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. An icosahedron in a solid figure contained by twenty equal and equilateral triangles. Def. A. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. PROPOSITION I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it: and since the straight line AB is in the plane, it can be produced in that plane: let it be produced to D; and let any plane pass through the straight line AD, and be turned about it until it pass through the point C: and because the points B, Care in this plane, the straight line BC is in it: (1. def. 7.) therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB; (1. 11. Cor.) which is impossible. Therefore, one part, &c. Q. E.D. PROPOSITION II. THEOREM. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Let two straight lines AB, CD cut one another in E; then AB, CD shall be in one plane: and three straight lines EC, CB, BE, which meet one another, shall be in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced, if necessary, until it pass through the point C. Then, because the points E, C are in this plane, the straight line EC is in it: (1. def. 7.) for the same reason, the straight line BC is in the same: therefore the three straight lines EC, CB, BE are in one plane; but in the plane in which EC, EB are, in the same are CD, AB: (x1. 1.) therefore, AB, CD are in one plane. If two planes cut one another, their common section is a straight line. Let two planes AB, BC cut one another, and let the line DB be their common section. If it be not, from the point D to B, draw, in the plane AB, the straight line DEB, (post. 1.) and in the plane BC, the straight line DFB: then two straight lines DEB, DFB have the same extremities, therefore BD, the common section of the planes AB, BC, cannot Wherefore, if two planes, &c. Q. E.D. |