12. This follows immediately from Prop. XXIII.

13. See Exercise 7, and Prop. XXIV.

14. The volume generated consists of a cylinder whose side is a, together with the two solids generated by the revolution of two triangles (one above and one below the rectangle which generates the cylinder) about an axis passing through their vertices.

15. For the side of the circumscribed equilateral cone and the diameter of its base, see Book IV., Prop. VI., Scholium. Prop. XXII.

See also

16. For the altitude of the inscribed cylinder and the diameter of its base, see Book IV., Prop. V. For the side and diameter of the base of the inscribed equilateral cone, each, see Book IV. Prop. VI., Scholium. Then apply General Scholium, 1, 2, and 4.

17. Let R = radius of sphere. Then the side of equilateral cone and diameter of base (See Book IV., Prop. VI., Scholium), etc. Then apply General Scholium, 1, 2, and 4.

18. Find altitudes and radii of bases by reference to the Props. V. and VI. of Book IV., and then apply General Scholium, I, 2, and 4.

19. See Prop. XIX., Cor.

20. Let a and b be the two adjacent sides, and h and h' the corresponding altitudes. Then the volume generated about a is πh2 × a, and that about b is πh'2 × b, but ah

Loci.

=

bh', etc.

1. The convex surface of a cylinder of revolution having this line for an axis.

2. The convex surfaces of two cylinders of revolution having the same axis with the given cylinder.

3. See Book V., Prop. IX.

4. This locus is the intersection of two loci.

5.

The circumference of a circle, or the circumferences of two circles the intersection of two loci.

6. The curve of intersection of a plane and a cylinder, or the curves of intersection of two planes and a cylinder.

7. A circumference the intersection of two spheres (See Book II., Prop. XXIII., Cor., and Prop. XX., Cor. 2).

8. A plane.

9. Circumference of a circle. See Book VII., Exercises, Locus I.

PROBLEMS.

1. See General Scholium, 1. The radius of the base is equal to the given altitude.

2. Use General Scholium, 1.

3. The side is the hypothenuse of a right angled triangle, whose altitude is the given altitude of the frustum, and whose base is the difference of the radii of the bases of the frustum. This being found, apply General Scholium, 3.

4. Find the altitude from the triangle, the construction of which is given in Problem 3, and then apply General Scholium, 3.

5. Find the altitude from the data, and apply General Scholium, 2. 6. Find the side from the data, and apply General Scholium, 2. 7. General Scholium, 1.

8. Find the side of the cone from the data. the sector.

This is the radius of

Then apply Prop. XIX., Book IV., to find the angle.

9. Let SA = 1.8 yards be the side of the given cone, and Sx the unknown side of the cone cut off. Then the two convex surfaces are to each other as SA2: Sx2, whence by the data Sx can be found in both cases.

IO. II. 12. 13. 14. 15. Apply General Scholium, 4.

16. See Prop. XV., Cor. 3.

17. See these Exercises, Theorem 10.

18. Apply General Scholium, 4.

19. See Prop. XV., Cor. 3, and Prop. XXI., Cor. 1.

20. Apply General Scholium, 5.

21. Find the altitude of the zone from the data, and then apply General Scholium, 5.

22. Apply General Scholium, 7.

23. Find the radius of the base from the data, and then apply General Scholium, 6.

24. Find the altitude of the zone of the sector from the data, and then apply General Scholium, 5.

25. Let x be the required side.

Then Prop. XVII. gives the expression for the given volume, whence x can be found.

26. Let ABC be the triangle, MN the axis, C the nearest vertex, BCN perpendicular to MN, so that CN= 2 inches. Let AO be the altitude of the triangle. Then ON = 3 inches, and AO = √3, and AM, parallel and equal to ON, = 3 inches.

Then

vol. ABC vol. ABNM - vol. ACNM.

Apply General Scholium, 3.

27. See Exercises, Theorem 20.

First. nah, but h=b. Secondly. nb'h', but h'a. (See Book I., Exercise 36.)

28. Let ABC be the triangle, and DE the line parallel to BC bisecting the sides AB, AC at D and E and the altitude AM at the point N.

We are to compare the volumes described by the revolution of ADE and DECB about BC, or what is more simple, to compare vol. DECB with the vol. ABC. From D and E draw DO and EP per

pendicular to BC, then

vol. DECB= vol. DEPO + vol. DOB + vol. EPC,

which is easily computed and compared with vol. ABC.

29. Let the radius of the sphere = R, then the side of the regular in

scribed tetraedron =

4 R
3

2 R

the side of the inscribed cube =

3,

Hence

and the side of the inscribed regular octaedron is R√2.

the volumes can be easily computed. (See Exercises, Appendix to Book VI., Problems.)