PROBLEM 42.

To find the area of an ellipse.

RULE.-Multiply the two axes together, and their product by the decimal 7854, and the result will be the required area. 1. Required the area of an ellipse whose transure axes A B = 70 feet, and the conjugate axes D E = 50 feet.

3500 × 7854 2748.9 area.

=

2. What is the area of an ellipse whose

axes are 35 and 25?

A

Ans. 687-225. 3. What is the area of an ellipse whose axes are 50 and 45? Ans. 1767.15.

REVIEW.

1. How will you find the area of a square? 2. How will you find the length of the side of a square? 3. When the diagonal is given, how will you find the area of a square? 4. When the area of a square is given, how will you find the diagonal? 5. When the diagonal of a square is given, how will you find the side? 6. How will you cut off a given area from a square parallel to either side? 7. When the length and breadth of a rectangle are given, how will you find the area? 8. When the area and either side of a rectangle are given, how will you find the other side? 9. When the area and the proportion of the two sides of a rectangle are given, how will you find the sides? 10. When the sides of a rectangle are given, how will you cut off a given area parallel to either side? 11. How do you find the area of a rhombus? How will you find the area of a rhomboid? How will you find the area of a parallelogram? 12. When the area of a rhombus or rhomboid, and the length of the side are given, how will you find the perpendicular height? 13. When the base and perpendicular are given, how will you find the area? 14. The three sides of a triangle being given, how will you find the area? 15. When two sides of a right-angled triangle are given, how will you find the third side? Rule second,? Rule third? 16. When the sum of the hypotenuse and perpendicular, and the base of a right-angled triangle are given, how will you find the hypotenuse and perpendicular? 17. How will you find the area of an equilateral triangle? 18. When the side of an equilateral triangle is given, how will you find the perpendicular? When the area of an equilateral and the perpendicular are given, how will you find the side? When the area of an equilateral triangle is given, how will you find the side? 19. How will you find the area of an isosceles triangle when the length of the side is given? When the area and base of an isosceles triangle are given, how will you find the length of the equal sides? 20. How will you find the area of a scalene triangle, the base and perpendicular being given? 21. When the area and base of any triangle are given, how will you find the perpendicular height? 22. When the base and perpendicular of any plane triangle are given, how will you find the

side of the inscribed square? 23. How do you find the area of a trapezium? 24. How will you find the area of a trapezoid? 25. How will you find the area of a regular polygon? How will you find the area of a regular polygon when one of its equal sides only is given? How will you find the areas of polygons by the table? 26. When the area of a regular polygon is given, how will you find the side? 27. How will you find the area of an irregular figure bounded on one side by a straight line? 28. How will you find the circumference of a circle when the diameter is given; or the diameter when the circumference is given? 29. How will you find the area of a circle? 30. When the area of a circle is given, how will you find the diameter or circumference? 31. How will you find the area of a circular ring, or the area between two concentric circles? 32. When the diameter or circumference of a circle is given, how will you find the side of an equivalent square? 33. When the diameter or circumference of a circle is given, how will you find the side of an inscribed square? 34. How will you find the diameter of a circle, equal in area to any given superficies? 35. When the diameter of a circle is given, how will you find another containing a proportionate quantity? 36. How will you find the length of a circular arc when the number of degrees and radius are known? 37. How will you find the length of the arc of a circle when the chord and radius are given? 38. When the chord and versed sine are given, how will you find the diameter of a circle? 39. When the versed sine of an arc and the diameter of the circle are given, how will you find the chord? 40. When the chord and versed sine are given, how will you find the area of a sector? 41. How will you find the area of a segment of a circle? 42. How will you find the area of an ellipse?

NOTE. It is very important that the pupil should pass a strict examination in the reviews, by having a certain portion assigned him for a daily lesson, and on no account should this duty be neglected by the instructor.

PART SECOND.

Mensuration of Solids.

DEFINITIONS.

THE measure of any solid body is the whole capacity or content of that body, when considered under the triple dimensions of length, breadth, and thickness.

A cube whose side is one inch, one foot, or one yard, &c., is called the measuring unit; and the content or solidity of any figure is computed by the number of those cubes contained in that figure.

1. A cube is a solid contained by six equal square sides; as A B C D E F.

2. A parallelopipedon is a solid contained by six rectangular plane sides or faces, every opposite two of which are equal and parallel, as A B C D E F.

Fig. 2.

A

B

3. A prism is a solid whose ends are two equal, A parallel, and similar plane figures, and its sides parallelograms; as ABCDEF.

It is called a triangular prism when its ends are triangles; a square prism when its ends are square, B &c.

Fig. 3.

4. A cylinder is a solid described by the revolu tion of a rectangle about one of its sides, as an axis, which remains fixed; as A B C D.

5. A cone is a solid described by the revolution of a right-angled triangle about one of its legs, which remains fixed; as A B C.

Fig. 5.
A

6. A pyramid is a solid whose sides are all triangles meeting in a point at the vertex, and the base any plane figure; as ABCDE.

When the base is a triangle, it is called a triangular pyramid, &c.

E

7. A sphere is a solid described by the revolution of a semicircle about its diameter, which remains fixed; as A B C D.

The centre of a sphere is a point within the figure, equally distant from every part B of its convex surface.

A diameter of the sphere is a straight

line passing through its centre.

The axis of a sphere is any line about

Fig. 7.

A

which it revolves; and the points at which the axis meets the surface are called the poles.

8. A circular spindle is a solid generated by the revolution of a segment of a circle about its chord, which remains fixed; as A B D C.

Fig. 8.
A

D

9. A speroid or ellipsoid is a solid generated by the revolution of a semiellipsis about one of its axes, which remains fixed; as A B D C.

A spheroid is called prolate when the revolution is made about the transverse axes, and oblate when it is made about the conjugate axes.

10. The segment of a pyramid, sphere, or any other solid, is a part cut off from the top of it by a plane parallel to the base of the figure.

11. A frustum or trunk is a part that remains at the bottom after the segment is cut off.

12. The zone of a sphere is that part which is intercepted between the parallel planes; and when those planes are equally distant from the centre, it is called the middle zone of the sphere.

13. The height of a solid is a perpendicular drawn from its vertex to the base, or to the plane on which it is supposed to stand.

14. A wedge is a solid, having a rectangular base, and two of its opposite sides meeting in an edge.

15. A prismoid is a solid, having on its ends two rectangles parallel to each other; and its upright sides are four trapezoids.

The mensuration of solids is divided into two parts:

1. The mensuration of the surfaces of solids.

2. The mensuration of their solidities.

PROBLEM 1.

To find the area of the surface of a cubc.

RULE.-Multiply the square of the length of one side by the number of sides, and the product will be the area of the surface.