П 39. Scholium. By other more expeditious methods the value of π has been found accurately to two hundred and fifty places of decimals. For practical purposes it is sufficiently accurate to call 3.14159. PRACTICAL QUESTIONS. 1. What is the circumference of a circle whose radius is 10 feet? 2. What is the diameter of a circle whose circumference is 57 rods? 3. What is the area of a circle whose radius is 40 feet? 4. What is the area of a circle whose circumference is 18 inches? 5. What is the circumference of a circle whose area is 116 square feet? 6. The radii of two concentric circles are 40 and 54 feet; what is the area of the space bounded by their circumferences? 7. A has a circular lot of land whose diameter is 95 rods, and B a similar lot whose area is 750 square rods; compare these lots. 8. What is the difference between the perimeters of two lots of land each containing an acre, if one is a square and the other a circle? 9. What is the area of a square inscribed in a circle whose area is a square metre? 10. What is the area of a regular hexagon inscribed in a circle whose area is 567 square feet. 11. If a rope an inch in diameter will support 1,000 pounds, what must be the diameter of a rope of like material to support 4,000 pounds? 12. If a pipe an inch in diameter will fill a cistern in 25 minutes, how long will it take a pipe 5 inches in diameter ? 13. If a pipe an inch in diameter will empty a cistern in an hour, how long will it take this pipe to empty the cistern if there is another pipe one third of an inch in diameter through which the fluid runs in? Ans. 67 minutes. 14. If a pipe 3 inches in diameter will empty a cistern in 3 hours, how long will it take the pipe to empty the cistern if there are 3 other pipes each an inch in diameter through which the fluid runs in. Ans. 4 hours. EXERCISES. The following Theorems, depending for their demonstration upon those already demonstrated, are introduced as exercises for the pupil. In some of them references are made to the propositions upon which the demonstration depends. They are not connected with the propositions in the following books, and can be omitted if thought best. 40. Every diameter bisects the circle and the circumference. 41. A straight line can meet the circumference of a circle in only two points. (4.) (I. 51.) 42. The diameter is greater than any other chord of the circle. 43. In the same or equal circles, when the sum of the arcs is less than a circumference, the greater arc is subtended by the greater chord; and, conversely, the greater chord is subtended by the greater arc. Draw A C (21.) (I. 47.) What is the case when the sum of the arcs is greater than a circumference? D B C 44. Equal chords are equally distant from the centre; and of two unequal chords the greater is nearer the centre. 45. The shortest and the longest line that can be drawn from any point to a given circumference lies on the line that passes from the point to the centre of the circle. 46. Two parallels cutting the circumference of a circle intercept equal arcs. 47. A straight line perpendicular to a diameter at its extremity is a tangent to the circumference. Draw CB. (I. 51.) 48. The lines joining the extremities of two diameters are parallel. B 49. If the extremities of two chords are joined, the triangles thus formed are similar. 50. If two circumferences cut each other, the chord which joins their points of intersection is bisected at right angles by the line joining their centres. (17.) 51. If two circumferences touch each other, their centres and point of contact are in the same straight line, perpendicular to the tangent at the point of contact. (47.) 52. The distance between the centres of two circles whose circumferences cut one another, is less than the sum, but greater than the difference, of their radii. 53. Every angle inscribed in a segment greater than a semicircle is acute; and every angle inscribed in a segment less than a semicircle is obtuse. (21.) 54. The angle made by a tangent and a chord is measured by half the included arc. Draw the diameter A B. (47.) (21.) 55. The angle formed by two chords cut- 4 ting each other within the circle is measured by half the sum of the intercepted arcs. Join BC (in lower figure). (21.) 56. By moving the point of intersection of the two chords, show that (14) and (21) can be deduced from (55). 57. The segments of two chords cutting each other within a circle are reciprocally proportional. Join AD, BC. (21.) (II. 20.) E B B 58. The opposite angles of a quadrilateral inscribed in a circle are supplementary. (21.) 59. A quadrilateral whose opposite angles are supplementary, and no other, can be inscribed in a circle. 60. Circles are as the squares of their radii, or diameters, or circumferences. (32.) 61. The area of a sector is equal to half the product of its arc by the radius of the circle. (31.) 62. Show how to find the area of a segment of a circle. 63. The area of a circumscribed polygon is equal to half the product of its perimeter by the radius of the circle. 64. A tangent is a mean proportional between a secant drawn from the same point and the part of the secant without circle. Join AD, DC. (54; 21.) (II. 57.) 65. The angle formed by two secants, two tangents, or a secant and a tangent cutting each other without the circle, is measured by half the difference of the intercepted arcs. Join CF. (I. 39.) (21.) 66. By moving the point of intersection, show that (21) can be deduced from (65). Show also that (46) can be deduced from (65). 67. Two secants drawn from the same point are to each other inversely as the parts of the secants without the circle. Join CF, DG. (21.) (II. 57.) A B D G B E D 68. Two tangents drawn to a circumference from the same point without this circumference are equal. Join BE. Figure in (66.) (54.) B 69. A perpendicular from a circumference to the diameter is a mean proportional between the segments of the diameter. C Join AB, BC. (23.) (II. 26.) 70. If from one end of a chord a diame ter is drawn, and from the other end a per pendicular to this diameter, the chord is a mean proportional between the diameter and the adjacent segment of the diameter. Join A B. (23.) (II. 25.) 71. The sum of the opposite sides of a circumscribed quadrilateral is equal to the sum of the other two sides. (68.) BOOK IV. GEOMETRY OF SPACE. PLANES AND THEIR ANGLES. DEFINITIONS. 1. A straight line is perpendicular to a plane when it is perpendicular to every straight line of the plane which it meets. Conversely, the plane, in this case, is perpendicular to the line. The foot of the perpendicular is the point in which it meets the plane. 2. A line and a plane are parallel when they cannot meet though produced indefinitely. 3. Two planes are parallel when they cannot meet though produced indefinitely. THEOREM I. 4. A plane is determined, 1st. By a straight line and a point without that line; 2d. By three points not in the same straight line; 3d. By two intersecting straight lines. 1st. Let the plane MN, pass- M ing through the line AB, turn' upon this line as an axis until it contains the point C; the position of the plane is evidently determined; for if it is turned in A C -B N either direction it will no longer contain the point C. |