it is to being horizontal, the broader or more circular is the ellipse.

This will be a convenient place for referring to two other conic sections, as they are called. The curve made by a section of a cone, parallell to one of its sides, is a parabola; the curve formed by a section which makes a greater angle with the base than the side of the cone makes, is a hyperbola.

It would lead to too great a digression to go into the subject more fully, and therefore we shall only add two definitions of the name ellipse, which you will understand better when you have gone farther into the subject.

1. An ellipse is a figure bounded by a regular curve which corresponds to an oblique projection of a circle, or an oblique section of a cone through its opposite sides.

2. An ellipse is a curve, such that the sum of the distances of any point in it, from two given points, is equal to a given line.

The longer diameter AB of an ellipse is called the major axis, or the transverse diameter; the shorter CD is called the minor axis, or the conjugate diameter.

The area of an ellipse is found by a method similar to that for finding the area of a circle.

To find the area of an ellipse, multiply the product of the two axes by 7854; or multiply the product of the semi-axis by 3.14159.

It has been already stated that the area of a circle is equal to the square containing it multiplied by 7854, or to the square of the radius multiplied by 3.14159. This cor

responds to the rule given above, which is only another way of saying that the area of an ellipse is equal to the area of the rectangle containing it multiplied by 7854, or to the rectangle contained by its longer and shorter radii multiplied by 3.14159.

In other words, an ellipse is equal to a circle whose diameter is a mean proportional between the two axes.

EXERCISES (J).

1. The axes of an ellipse are 6 ft. and 10 ft.; find its area. 2. The axes of an ellipse are 6 ft. and 7 ft.; find its area.

3. The conjugate diameter of an ellipse is 25 yds., and the transverse diameter 34 yds.; find the area.

4. A plantation in the shape of an ellipse has diameters which measure respectively 110 yds., and 73 yds. 1 ft.; how many stakes will be wanted to fence it if they are placed 10 inches apart?

PART III.

MENSURATION OF SOLIDS.

LESSON XIV. THE CUBE AND PARALLELOPIPED.

WE have already said that a line has only one dimension, viz., length, and that a superficies has two dimensions, viz., length and breadth; hence in calculating area we always proceed by multiplying two dimensions together.

We now pass on to speak of solid figures, which have three dimensions, viz., length, breadth, and thickness, and you will notice that in every case we shall calculate their volume or solidity by multiplying three dimensions together.

Every superficies is bounded by lines; every solid is bounded by superficies.

A cube, or solid square, is a solid figure contained by six equal squares (A).

A parallelopiped, or solid rectangle, is a solid figure having six rectangular sides, every opposite two of which are equal and parallel (B).

called rectangular parallelopipeds; the regular solid C, con

structed on a rhombus or rhomboid, is also a parallelopiped, and may be treated by the same rules as the rectangle, only remembering that the thickness of the solid must then

be measured by the perpendicular distance (DE) of the sides.

A definition of the term as used in this wider sense is: A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel.

To find the volume of a cube or parallelopiped, multiply together its length, breadth, and thickness.

The truth of this may be proved experimentally by actually constructing the figures by means of solid cubical blocks, like the toy bricks of children. But the rule is also evident from the figures given above, which represent a cube and a parallelopiped divided out in this way, one cube being removed from the corner of each to show how they are arranged.

Each side of the cube measures 4 ft., so that by the rule (the length, breadth, and thickness being the same), the volume is 4 x 4 x 4 = 64 cubical ft. But from the figure it is evident that in the square base there are 16 square ft., formed by the square sides of the 16 cubical blocks at the bottom, and upon each of these square feet is a pile of four blocks, so that altogether there are 16 x 4 blocks, each of one

cubic foot; and the solidity of the whole figure is therefore 64 cubic ft.

In the same way the volume of the parallelopiped may be shown to be 4 × 6 × 3 = 72 cubic ft., for at the base there are 24 square surfaces (4 × 6), upon each of which is a pile of 3, making altogether 3 x 24 72; and the volume is 72 cubic ft. From what has already been said, it is evident that the entire surface of a cube is equal to six times the area of one of its square sides. The surface of a parallelopiped is the sum of the areas of the six quadrangles which contain it.

The side of a cube may be found by extracting the cube root of its volume.

Similarly, if the volume of a parallelopiped be divided by the product of any two of its dimensions, the quotient is the other dimension.

EXERCISES (K).

Find the solid content of the cubes, the side of each being as follows:

1. 2 ft. 6 in.

2. 7 yds. 1 ft.

3. 7 ft. 6 in.

4. 6 ft. 8 in.

5. 6.5 ft.

6. 12 ft. 9 in.

7. 2 ft. 10 in.

8. 2 ft. 9 in.

9. 3 ft. 4 in.

10. 15 ft. 4 in.

11. 10 ft.

12. 7 ft. 4 in.

Find the surface of the cubes, the side of each of which is as follows:

13. 5 ft. 10 in.

14. 10 ft. 3 in.

15. 6 ft. 3 in.

16. 5 ft. 9 in.

Find the solid content of the parallelopiped, whose length, breadth and thickness respectively are as follows:

17. 70 yds.; 14 yds.; 14 yds.

18. 24 65 yds.; 1.56 yds.; 2.82 yds.

19. 12 ft. 9 in.; 4 ft. 3 in.; 12 ft. 6 in.

20. 30 yds.; 28 ft.; 2 yds.

21. 15 ft.; 1 ft. 9 in.; 1 ft. 9 in. 22. 10 yds.; 5.75 yds.; 3.5 yds.

23. 42 ft. 8 in.; 18 ft. 9 in.; 21 ft. 4 in. 24. 1 yd.; 2 yds.; 5 ft. 9 in.

25. The surface of a cube is 216 sq. yds.; find the length of its side.

26. The volume of a cubical block of iron is 1 cubic ft.; how long is its edge.

27. A store closet is 18 ft. long, 8 ft. broad, and 12 ft. high; what would be the cost of painting it at 94d. per square yard?

28. A pit is 25 ft. long, 15 ft. broad, and 7. ft deep; what was the cost of digging it at 74d. per cubic yard?

29. A rectangular log of wood measures at each end 1 ft. 9 in. by 1 ft. 3 in.; what length must be cut off to contain 7 cubic ft.?

30. A tank is 34 ft. 4 in. long, 4 ft. 3 in. broad, and 6 ft. 5 in. deep; how many gallons of water will it hold?

31. A cistern is 12 ft. long and 8 ft. wide; if when it is full of water 1000 gallons be drawn off, how much will the surface of the water sink?

32. A tank is 20 yds. long and 14 yds. broad, and will hold 1680 cubic yds. of water; find its depth.

LESSON XV.-THE PRISM.

A prism is a solid whose sides are parallelograms, and its ends two similar and parallel plane figures.

If the sides of a prism are at right angles to its ends (A and B), it is a rectangular prism, and all its sides are rectangles; if the sides are not at right angles to the ends it is an oblique prism (C), and all its sides, except perhaps two, are in the form of a rhombus or rhomboid.

Prisms are generally named from the form of their ends; thus, one which has triangular ends (A) is a triangular prism; one which has square ends a square prism (i.e., either a cube or parallelopiped); one which has pentagonal ends (B) a pentagonal prism; or hexagonal ends (C) a hexagonal prism.