and the altitude 0 G of the segment O E G of one base, be 5 feet, how many solid feet does this segment contain ? 22. If the radius H G (fig. 125) of the sphere be 8F125 feet, and the altitude O P of the segment 0 EF P of two bases, be 2 feet, the greater base being at the distance of H O or 3 feet from the centre, what is the solidity of the segment O EFP? 23. If the radius of a spherd be 10 feet, and the altitude of a segment of two bases 4 feet; the greater base being 2 feet from the centre, and both in the same hemisphere; what is the solidity of the segment? 24. If the radius of a sphere be 12 feet, and the altitude of a segment of two bases 6 feet; the centre being between the bases, and one base being 4 feet from the centre; what is the solidity of the segment ?2638 Note. In finding the solidity of the greater segment of one base the cone must here be added to the sector. Comparison of Similar Surfaces and Solids. . 192. In articles 116, 117, 164, 167, 168, the following propositions were demonstrated. 1. Two similar polygons are to each other as the squares of their homologous sides 2. Two circles are to each other as the squares of their radii or diameters*3.—The surfaces of two spheres are to each other as the squares of their radii 4. -The folidities of two spheres are to each other as the cubes of their radii 5. -Two similar polyedrons are to each other as the cubes of their homologous sides op - Two similar cones or cylinders are to each other as 193. The following questions plication of the rules in the preceding article. 1. The side of one triangle being 11 inches, and the corresponding side of a similar triangle being 3 inches, what is their ratio in numbers? 2. The dimensions of a field being found in rods, and the plan being projected upon the scale of 10 rods to an inch, what is the ratig of the plan to the field, expressed in numbers? + the "cubes of the radio en todos may be solved by the ap 3. The engraving of a painting 10 feet square is made upon a surface 10 inches square. What is the ratio of reduction ? 1:144 4. The homologous sides of two similar figures are as 8 to 5, and the area of the first is 120 square feet. What is the area of the second ? H678 5. The radii of two circles are as 8 to 10. What is the ratio of the circles ? 64.100 6. Two circles are to each other as 12 to 20. What is the ratio of their radii? 9.6: 4.4 7. The radius & the earth is 3960 miles, and that of Mars 2000 miles. What is the ratio of their surfaces ? What is the ratio of their solidities? 7. The diameter of Jupiter is 89000 miles, what is the ratio of the surfaces of Jupiter and the earth ? What is the ratio of their volumes ? 8. The sides of two similar polyedrons are to each as 3 to 9. What is the ratio of their solidities ? 2): 729 9. A model of the temple of Minerva is made upon the scale of 6 feet to an inch. What ratio does the magnitude of the model bear to that of the original ? 173 47881 10. The radii of two similar cylinders are to each other as 3 to 9. What is the ratio of their solidities ? 27379 11. The solidities of two similar cylinders are as 27 to 64. What is the ratio of their radii? 9:4 12. The radii of two similar cones are to each other as 4 to 7. What is the ratio of their solidities ? . 13. It is required to make a model or copy of a given cone, upon the scale of 16 feet to an inch. What ratio will the copy bear to the original ? 726 98Hi! Junini. Sunils Junius Junus Lunu, Anu ENIS imu ginis of 1. What is a point? Can you make or perceive a geo metrical point ? 2. What is a line? What are its extremities ? 3. What is a straight line? Can you prove that it is the shortest distance between two points ? 4. What is a linear unit? 5. Can the ratio or value of lines be expressed in num bers ? How? 6. What is a broken line ? 7. What is a curved line? What is it composed of? 8. What is the circumference of a circle ? Radius ? Di ameter? Arc? Sector ? Segment? Chord ? 9. What is a degree? Minute ? Second ? 10. What is an angle? How is it read? How measured ? 11. What is a right angle ? Acute ? Obtuse ? 12. What is the sum of all the angular space about a point ? 13. What is the supplement of an angle ? Complement? 14. What are vertical angles ? Are they equal Why? 15. When is a line perpendicular to another? When ob lique ? 16. If a perpendicular be erected on the middle of a line what follows ? What follows if that line be a chord ? 17. Can you find the centre of a given arc ? Can you find a circumference that will pass through any three points not in a straight line? 18. What measures the shortest distance from a point to a straight line? 19. Can there be more than one perpendicular at a given point ? 20. When are two lines parallel ? Can they ever meet? 21. When two parallels meet a third line how are the an gles named ? Why are they so called? and what is their property? 22. What are two interior angles on the same side, and what is proved of them? 23. What is proved of parallels comprehended between parallels ? 24. What is proved of two angles which have their sides parallel and directed the same way? 25. What is proved of two parallel tangents or secants ? 26. What is proved of every angle which has its vertex in the circumference ? 27. What are inscribed angles and what is proved of them? 28. What is a triangle ? Can it always be inscribed ? Why? 29. To what are the three angles always equal ? Why? 30. Can a triangle have more than one right angle? Why? 31. What is a right triangle? What is the side opposite to the right angle? 32. Do two angles of a triangle determine the third angle ? 33. What is an exterior angle, and what is proved of it? 34. What is an isosceles triangle and what is proved of it ? 35. What is an equilateral triangle and what is proved of it ? 36. What is proved of the greater side of every triangle ? 37. What are the four cases in which two triangles are equal ? 38. There are six things in a triangle; how many are ne cessary to determine the triangle ? 39. Do three angles determine a triangle? Why? 40. What is a ratio ? How is it written ? 41. What is a proportion? How is it written? How is it read ? 42. What are the extremes ? Means? Antecedents ? Con sequents ? 43. Are the products of the means and extremes equal ? 44. What if two proportions have a common ratio ? 45. May the means or the extremes change places ? 46. May either ratio be multiplied or divided by the same number? 47. May one proportion be multiplied by another or by itself? 48. What is the ratio of the sum of the two first terms to the sum of the two last, and of the difference of the two first to that of the two last? 49. In a continued proportion, what is the ratio of the sum of the antecedents to that of the consequents ? 50. What is proved of a line drawn through the sides of a triangle, parallel to the base ? 51. What are the problems that are solved upon this prin ciple ? 52. What are similar triangles ? In what three cases are triangles similar 53. What important proposition is demonstrated of similar triangles ? 54. What is proved of a perpendicular let fall from the circumference to the diameter ? 55. What is proved of a tangent and secant drawn from the same point to a circle ? 56. What is meant by dividing a line in extreme and mean ratio? 57. What is a figure of four sides called ? 58. What is a parallelogram? Trapezoid ? Trapezium? 59. What is a right parallelogram? Square? Oblong or Rectangle ? 60. What is an oblique parallelogram? Rhombus ? Rhom boid? 61. What is proved of the diagonal of a parallelogram? 62. What is a polygon ? Regular? Irregular? Similar? 63. To what is the sum of the interior angles of a polygon equal ? Why? 64. What is proved of two polygons composed of the same number of similar triangles ? 65. What is proved of two regular polygons of the same number of sides ? 66. Can every regular polygon be inscribed ? 67. How is a square inscribed in a given circle ? 68. How a regular hexagon ? An equilateral triangle? 69. How a regular polygon of 10 sides? Of 15 sides? Of 5 sides? 70. What sort of a polygon is the circle demonstrated to be ? 71. What is proved of the perimeters of regular polygons of the same number of sides ? 72. What is the ratio of the circumferences of circles ? 73. What is a surface? What are its boundaries ? 74. How may we conceive it generated ? 75. How many kinds of surfaces are there? |