2. Compute the radius of the base and the altitude of a cylinder, given the convex surface and the volume.

3. Find the convex surface of a frustum of a cone, the radii of whose bases are 5.4 feet and 3.6 feet, and whose altitude is 2.4 feet.

4. Find the volume of a frustum of a cone, the radii of whose bases are 5.4 feet, and 3.6 feet, and whose side is 3 feet.

5. Compute the volume of a cone, the radius of whose base is 8 feet, and whose side is 10 feet.

6. Compute the convex surface of a cone, whose altitude is 16 feet, and the radius of whose base is 12 feet.

7. Compute the volume and whole surface of a cylinder, the radius of whose base is 16 feet 9 inches, and whose altitude is 20 feet.

8. The radius of the base of a cone is 9 feet, and its altitude is 12 feet. If this cone is rolled out completely on a plane (see Prop. VIII., Schol. 1), find the radius and angle of the circular sector which is the developed convex surface of the cone.

9. The side of a cone being 1.8 yards, find the parts into which it is divided by a plane parallel to the base of the cone which divides the convex surface, First, Into two equivalent parts, Secondly, Into two parts proportional to the numbers 3 and 5.

10. Find the surface of the sphere whose radius is 20 inches.

11. The surface of a sphere being 1000 square feet, find its vol

ume.

12. The meter is one ten-millionth part of a quadrant of the earth's great circle (earth being regarded as a sphere). Find the surface of the earth in square kilometers. 1,600,000,000

Ans.,

π

13. The diameter of the earth (regarded as a sphere) is 7912.5 miles, find its surface in square miles.

14. Find the volume of a sphere whose radius is 6 feet.

15. The length of one second on the arc of a great circle of a sphere being one foot, find the volume of the sphere.

16. The radii of two spheres are respectively 6 feet and 8 feet. Find a sphere whose surface shall be equivalent to the sum of the surfaces of these two spheres.

17. Find the volume of a spherical shell, the internal radius being 6 inches and the external radius 9 inches.

18. Find the radius of a sphere whose volume is 12 cubic feet.

19. The diameters of the earth, the moon, and the sun are to each other as the numbers 1, 3, and 108. Compare the surfaces and

volumes of these bodies.

20. Find the surface of a zone whose altitude is 4 feet, the radius of the sphere being 10 feet.

21. Find the surface of a zone of one base, the radius of the sphere being 20 feet, and the radius of the base of the zone being 12 feet.

22. The angles of a triangle on the earth's surface (regarded as a sphere) are 87°, 72°, and 21° 0' 1". 5, find the area of the triangle in square miles, the diameter of the earth being 7912 miles.

23. Find the volume of a spherical segment of one base, whose altitude is of a foot, the radius of the sphere being one foot.

24. Find the volume of a spherical sector generated by the revolution of a circular sector whose arc is 30° about one of its radii, the radius of the sphere being 2.4 feet.

25. Given the volume generated by the revolution of an equilateral triangle about one of its sides to be 27 cubic feet, find the side of the triangle.

26. Compute the volume generated by the revolution of an equilateral triangle whose side is 2 inches, about a perpendicular to its base produced, which is at a distance of 2 inches from its nearest Ans., 33.648068 cubic inches.

vertex.

27. The side AB of a parallelogram, ABCD, is 10 inches, the side BC 4 inches, and the angle ABC = of a right angle. Find the volume generated by this parallelogram, First, When it revolves about AB. Secondly, When it revolves about BC.

28. If we join the middle points of two of the sides of a triangle and then revolve it about the third side, what will be the ratio of the volumes generated by the two parts of the triangle?

29. Given the radius of a sphere, find the sides respectively of its inscribed regular tetraedron, cube, and octaedron.

HINTS TO SOLUTIONS OF EXERCISES.

BOOK I.

1. MO is the difference of AO and MA, and also MO is the difference of MB and OB; hence, result. For second part, MO is the sum of MB and OB, and MO is the difference of MA and OA, hence, the conclusion.

2. The same reasoning applies in this case. 3. By reductio ad absurdum from Prop. VI.

4. By reductio ad absurdum from Prop. VI. 5. By Props. V and VI.

6. By superposition (or by drawing diagonals and making comparison of equal triangles).

7. By superposition or by comparison of triangles, Prop. VII.

8. Construction given. Compare equal triangles and use Prop. IX.

9. Get three inequalities, as in 8, and add.

10. Construction given. Use Prop. IX. Sum the inequalities thus obtained and cancel equals on the two sides of resulting inequality.

II. Same method as in 10.

12. Use Prop. X. for first and Prop. IX. for last, getting in each case three inequalities, and adding them respectively.

13. By comparison of triangles, Prop. VII.

14. By comparison of triangles, Prop. VII.

15. By comparison of triangles, Props. XII. and VII.

16. By comparison of equal triangles.

17. By similar method.

18. Construction given. Assume any other point and join it to

two given points and extremity of produced perpendicular. Then use Prop. IX.

19. Draw AA' and BB' and CC', meeting xy in points M, N, and O. Now conceive the part of figure above xy revolved around xy, the points A, B, and C will fall on A', B', C', and thus proposition is proved. This revolution is merely a convenient mode of superposition.

20. Corollary of 18.

21. By comparison of triangles, Props. XXIV. and VII.

22. Let fall perpendiculars from A and C on the parallel BD, and compare the two triangles thus formed. Then use Prop. XIV.

23. Use Props. XIII. and XXIV.

24. Compare the triangles formed by the construction with the original triangle.

25. An easy corollary of 24. The converse by the reductio ad absurdum.

26. Follows directly from 25.

30. The bisectrix of the exterior angle at B is the locus of all points within this angle, equidistant from BC and AB, produced. The bisectrix of the exterior angle, C, is the locus of points within this angle, equidistant from BC and AC, produced. These bisectrices must meet (Prop. XXV., Scholium, etc.).

32. Use Prop. XXIX.

33. Draw from O a line, OM, parallel to side AB, and then use 25 (converse), and compare the triangles AOM and COM.

34. Use Prop. XIII. in each one of the isosceles triangles, AOC and AOB, and then converse of Prop. XXIX., Cor. 5.

35. Triangle BCD, formed by given construction, is an equilateral triangle, and perpendicular CA bisects angle A, etc.

36. At C make angle ACD = BCA, and complete triangle ACD. Then BCD is equilateral, etc.

37. Let the bisectrices of the exterior angles B and C meet in O; of exterior angles A and C meet in N; of exterior angles A and B in M. Then use Prop. XXIX. to prove semiexterior angle A in triangle ABM equal to angle O in triangle OCB. Using, also, triangles NCA and OMN.

38. Follows directly from 37, or equality of angle O and semiexterior angle A.

39. Angle DAO = a right angle minus angle AOD, and AOD 2C C+C=C+ a right angle diminished by B, whence result.

=

40. Apply Prop. IV. at each vertex and then Prop. XXX. for the sum of interior angles.

41. Apply Corollary of Prop. XXX. and Prop. XXIX.

42. Same method.

43. Angle of equiangular hexagon is of a right angle. Hence, the triangles formed by sides of squares and lines joining the corners are equilateral, etc.

44. Let number of sides be x; then sum of angles (Prop. XXX., Cor. 4): = 2x 26, whence x.

4=

46. Apply Prop. XXX., Cor. 4, as above.

47 and 48. Compare the angles of these polygons and of others. with four right angles.

50. The sides are parallel, by XXIV.

51. Let quadrilateral be ABCD. and B meet in M; those of angles C

Let also bisectrices of angles A and D meet in N; first, angle angle N = two right angles

− (C + D), etc. Hence M + N = two right angles. Second, if ABCD is a parallelogram, angle M two rights minus the half of two rights, etc. (Prop. XXV.) Third. A rectangle with its diagonals at right angles to one another, hence a square. Prop. XXXIV., Scholium 3.

52. See (21). Then draw diagonals and divide the surfaces to be compared into triangles, which compare by VII.

53. See Exercises 25 and 24. This parallelogram is a rhombus when the given quadrilateral is a rectangle; and a rectangle when the quadrilateral is a rhombus ; a square when the quadrilateral is a square.

55. The figure is a parallelogram (Def.). Compare the areas by means of equal triangles.