A common and more scientific name for the longer axis of an ellipse, is the transverse, or major, and for the shorter, the conjugate or minor.

Ex. 1 What is the area of an ellipse whose longer axis AB is 70 feet, and whose shorter DC is 50 feet?

Ex. 2. What is the area of an ellipse whose axes are 16

and 12?

Ans. 150.79.

PROBLEM XV.

To find the Circumference of an Ellipse.

ART. 31. Rule.-Square the two axes, and multiply the square root of half their sum by 3.14159; the product will be the circumference, nearly.

Ex. 1. What is the circumference of an ellipse whose transverse and conjugate axes are 16 and 18 feet?

-2 -2

OPERATION.

16+18-580-sum of the squares of the axes.
And, 290-half sum.

Then,

290X3.14159-911.064-circumference.

Ex. 2. Required the circumference of an ellipse whose axes are 24 and 20 feet.

Ans. 69.39.

PROBLEM XVI.

To find the area of an Elliptic Segment cut off by a line perpendicular to either axis.

ART. 32. Rule.-Find the area of a corresponding cir

cular segment having the same height and the same vertical axis or diameter. Then say, as the vertical axis is to the other axis, parallel to the segment's base, so is the area of the circular segment before found, to the area of the elliptic segment sought.

Ex. 1. What is the area of an elliptic segment, cut off parallel to the shorter axis, whose height is 10 and whose axes are 25 and 35? Ans. 162.03.

Ex. 2. What is the area of an elliptic segment, cut off parallel to the longer axis; whose height is 5 and the axes 25 and 35? Ans. 97.84.

PROBLEM XVII.

To find the area of a Parabola.

ART. 33. Rule.-Multiply the base by the height, and two-thirds of the product will be the arca.

Ex. 1. What is the area of a parabola whose base AB is 26 inches, and height e

Ex. 2. What is the area of a parabola whose base is 12

feet and height 18 feet?

Ans. 144 ft.

Ex. 3. What is the area of a parabola whose base is 4 feet and height 10 feet? Ans. 31.50 ft.

PROBLEM XVIII.

To find the area of a Frustrum of a Parabola, cut off by a line drawn parallel to the base.

ART. 34. Rule.-Multiply the difference of the cubes of the two ends of the frustrum by twice its altitude, and divide the product by three times the difference of their

squares.

Ex. 1. What is the area of a frustrum of a parabola whose height Db is 12 feet, and its upper end ef 12 feet, and base AB 20 feet? (See last problem.)

Ex. 2. What is the area of a frustrum of a parabola

whose height is 8 feet, and its ends 16 and 10 feet?

Ans. 105.8.

PROBLEM XIX.

To find the area of a Hyperbola.

ART. 35. Rule.-1. To five-sevenths of the abscissa add the transverse diameter; multiply the sum by the abscissa, and extract the square root of the product.

II. Then, multiply the transverse diameter by the abscissa, and extract the square root of that product.

III. Then, to 21 times the first root, add 4 times the second root; multiply the sum by double the product of the conjugate and abscissa, and divide by 75 times the transverse; this will give the area, nearly.

Ex. 1. If the base of a hyperbola be 24 feet, the height 10, and the transverse axis 30; what is the area?

Ans. 151.69.

PROMISCUOUS EXAMPLES.

1. What is the area of a circle whose diameter is 26/?

Ans. 546.35.

2. What is the circumference of a circle whose diameter is 145 ?

Ans. 45.94.

3. Required the area and circumference of a circle whose diameter is 561. Ans. 2474.0-area. 176.3-circum.

4. If the area of a circle be 2839.2, what is its diameAns. 60.125.

ter?

5. What is the length of an arc of 30 degrees in a circle whose radius is 58 feet? Ans. 30.36.

6. What is the length of the side of a square, inscribed in a circle whose circumference is 300 feet? Ans.

7. If the diameters of two circles are 16 and 10, what will be the area included between the circumferences?

Ans. 122.52.

8. What will be the expense of papering the sides of a room, at 10 cents a square yard, if the room be 21 feet long, 18 feet broad, and 12 feet high, and if there be deducted 3 windows, each 5 feet by 3, two doors, 8 feet by 41, and one fire place, 6 feet by 41? Ans. $8.80.

9. What is the area of a circular segment whose height is 9 and base 24?

ans.

10. If a circular piece of land be enclosed by a fence, in which 10 rails make a rod in length, and if the field contains as many square rods as there are rails in the fence, what is the value of the land at 120 dollars per acre? Ans. $942.48.

11. What is the area of a square inscribed in a circle whose diameter is 36 feet?

Ans.