a circle, the opening or chord of the recess being 15 ft. and its greatest width 4 ft.; find the area of the whole room.

(7) An arched gateway is 30 ft. wide, and measures 20 ft. from the ground to the spring of the arch; and the distance also from the ground to the crown of the arch is 30 ft. Find how many square feet of timber will be required for blocking it up.

(8) Find how many sq. ft. of brickwork are used in blocking up one of the arches in a railway viaduct :

The span of the arch is 60 ft., the height above the piers is 20 ft., and the distance from the ground to the spring of the arch is 20 ft.

XVII. THE ELLIPSE.

Definitions.-The ellipse is one of the conic sections obtained from cutting a cone through both sides of it by a plane which is not parallel to the base.

But in simple language it may be defined to be a flattened circle, or a circle with a por

same, and is equal to AB the major axis. o, the middle point between F and F', is the centre of the ellipse: and any straight line drawn through o is a diameter.

The greatest of these diameters is AB, which is called the transverse diameter or major axis; whilst the least of them CD is called the conjugate diameter or minor axis.

AB and CD are also called the principal axes.

RULES.-(1) To find the circumference or perimeter of an ellipse which is not very eccentric.

Multiply half the sum of the major and minor axes by 22.

(2) To find the area of an ellipse which is not very

eccentric.

Multiply the product of the major and minor axes by H.

Note 1.—To find either of the principal axes, when one of them and the circumference of the ellipse are given. Subtract the given axis from 7 of the circumference.

Note 2.-To find either of the principal axes, when one of them and the area of the ellipse are given.

of the area by the given axis.

Divide

Note 3. The circumference and the area of an ellipse, which is not very eccentric, are very nearly equal to the circumference and the area respectively of a circle whose diameter is a mean proportional between the major and minor axes; that is, whose diameter is

AB+ CD

2

Or, in other words, the ellipse, when not very eccentric, may be considered a mean proportional between its circumscribed and inscribed circles; that is, between circles described about the major and minor axes. For instance,

if a circle were described having AB for its diameter, and another circle were described having CD for its diameter, then the area of the ellipse is equal to half the sum of the areas of these two circles. The same remarks will also apply to the circumference of an ellipse.

Note 4.-By the rules given above, the circumference and the area of an ellipse, which is not very eccentric, will be found with sufficient accuracy for all practical purposes. If, however, greater accuracy is desirable, then in Rule 1 substitute 3.1416 as multiplier, instead of 22; and in Rule 2, 7854 for 11.

When the ellipse is very eccentric, then the results obtained by the above rules will be far from correct.

Note 5.-The area of an elliptical ring is the difference between the area of the outer and inner ellipse.

Example 1.-Find the perimeter of an ellipse whose major axis is 28 ft. 6 in., and whose minor axis is 20 ft. 6 in.

By Rule 1:

Circumference (major axis+minor axis) × 22
=(28 ft. 6 in. +20 ft. 6 in.) × 2
=×49 ft. x 22=77 ft.

Example 2.-Find the area of an ellipse, whose major axis is 196 ft., and minor axis is 120 ft.

By Rule 2: Area 196 x 120 x 11-18480 sq. ft.

Example 3.-The perimeter of an ellipse is 28 yds. 1 ft. 3 in., and its minor axis is 22 ft. 9 in. ; find the major axis. Now, 28 yds. 1 ft. 3 in.=1023 in.; and 22 ft. 9 in.=273 in. By Note 1:

The major axis of the circumference-minor axis =(1 of 1023)-273-651-273=378 in. =31 ft. 6 in.

Example 4.-The area of an ellipse is 28 sq. ft. 60 sq. in., and its major axis is 7 ft. 9 in.; find the minor axis.

Now, 28 sq. ft. 60 sq. in.=4092 sq. in.; and 7 ft. 9 in. =93 in.

By Note 2: the minor axis=1 of the area÷given axis =11 of 4092÷93=5208÷93=56 in.=4 ft. 8 in.

Formulæ

I. Circum. =

=

major axis+minor axis

2

II. Area (major axis x minor axis) × 14.

=

III. Req. axis of the circum.—given axis.
IV. Req. axis=11 of the area÷given axis.

EXAMPLES.

Find the circumference of the ellipse whose major and

minor axes are, respectively

(1) 48 in., and 36 in.

(3) 87 yds. 1 ft., and 34 yds.

(2) 13 ft., and 8 ft.

(4) 3 ch. 80 lks., and 3 ch. 20 Iks.

Find the minor axis of an ellipse, when its circumference

and its major axis are, respectively

(5) Circumference 286 ft., and major axis 121 ft.

(6) Circumference 385 ft., and major axis 145 ft. (7) Circumference 47 ft. 8 in., and major axis 16 ft. 8 in. (8) Circumference mile, and major axis 165 yds.

Find the area of the ellipse, when its major and minor axes are, respectively—

(9) 110 ft., and 98 ft. (10) 40 ft. 10 in., and 29 ft. 2 in. (11) 42 yds., and 32 yds. 2 ft.

(12) 102 yds. 2 ft., and 93 yds. 1 ft.

(13) 2 ch. 38 lks., and 2 ch.

Find the minor axis of the ellipse, when its area and its major axis are, respectively—

(14) Area 550 sq. ft., and major axis 35 ft.

(15) Area 15 sq. ft. 40 sq. in., and major axis 5 ft. 10 in. (16) Area 7 ac. O rood 20 poles 11 sq. yds., and major axis 224 yds.

(17) Area 2 roods 8 poles, and major axis 2 ch. 80 lks. (18) In a rectangular plot of land, which is 100 yds. long and 70 yds. broad, is dug a fish-pond, in the shape of an ellipse, whose major axis is 98 yds. and minor axis is 60 yds.; the remaining part is to be gravelled, at 3d. per sq. yd. Find the expense.

(19) The rental of a field, in the shape of an ellipse, whose major axis is 140 yds., at £6 1s. per acre, is £15 88.; find the minor axis.

(20) The building of a wall round an elliptical plot of land, whose major axis is 106 yds., at 2s. 9d. per yard, costs £36 68. find the length of its minor axis.

:

(21) A lawn, in the shape of an ellipse, whose major and minor axes are, respectively, 98 ft. and 58 ft., is surrounded by a walk 1 yd. wide; find the cost of gravelling it, at 6d. per sq. yd.

SOLIDS.

DEFINITIONS.

1. The volume, solidity, or solid content of a solid is the space (cubic yds., ft., or in.) that it occupies.

2. The surface of a solid is its outside area.

Thus the

3. Parallel planes are flat or even surfaces, which are everywhere equally distant from each other. floor and ceiling of a room are parallel planes.

4. Straight lines which are perpendicular to the same plane are parallel to each other; and two parallel straight lines comprised between two parallel planes are equal.

5. The different solids have been so fully explained in the following chapters, that it has not been thought necessary to give any definition of them here. The Student must try to understand the explanations given of each solid before attempting the Examples.

1 cubic ft. of water weighs 1000 oz. avoir., very nearly. 1 cubic ft. contains 6 gallons, very nearly.

1 gallon contains 277 cubic in., or, more nearly, 277-274

cubic in., or 277 cubic in.

A cubic in. of gold weighs