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An Ellipse or Oval is a curve line, which returns into itself like a circle, but has two diameters of unequal A length, the longest of which is called the transverse, and the shortest the

C 8

D

conjugate axis; thus, ABCD (fig. 8) is an ellipse.

PROBLEM I.

B

To find the circumference of a Circle when the diameter is given.

ART. 16. Rule I.-Multiply the diameter by 3.14159, and the product will be the circumference. Or,

Rule II. As 7: 22 diameter to the circumference; that is, Multiply the diameter by 22 and divide the product by 7. Or,

Rule III.-Multiply the diameter by 355, and divide the product by 113.

NOTE. The learner may have the curiosity to enquire why we use the number 3,14159. or 3,1416, as is sometimes used, instead of any other number. by which to multiply the diameter of a circle to find its area. It is the result of the efforts of the most distinguished mathematicians to SQUARE THE CIRCLE. Archimedes made the earliest approxi mation to the ratio of the circumference of a circle to its diameter. He demonstrated that the ratio of the perimeter of a regular inscribed polygon of 96 sides, to the diameter of the circle. is greater than

: 1; and that the ratio of the perimeter of a circumscribed polygon of 192 sides, to the diameter, is less than 32: 1; that is, than 22: 7.

Metius next followed, and gave the ratio of 355: 113, which is a little more accurate than any other expressed in small numbers.

Now, to show how they obtained these numbers, and how near they are to the truth, let us suppose the subjoined Dodecagon to contain 1556 sides instead of

12; we shall then have a regular polygon of 1556 Duodecagon

sides inscribed in the circle, and if another similar polygon be circumscribed about the circle, it may be demonstrated that the perimeter of the inscribed

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polygon is 6,2831788, and that of the circumscribed polygon 6,2831928. Now, the circumference of the circle being greater than the perimeter of the inscribed polygon, and less than that of the circumscribed, it must consequently be greater than 6,2831788, and less than 6,2831928, and must therefore be nearly half their sum, which is 6.2831858. Hence, the circumference being 6,2831858, when the diameter is 2, it will be the half of that, or 3,1415928, when it is 1; to which the ratio, in the Rule, viz: 1 to 3,14159 very nearly.

Ex. 1. What is the circumference of a circle whose diameter EF is 24 feet?

E

In this operation we simply multiply the diameter, 24, by the number 3.14159, and the product gives the circumference.

OPERATION.

24x3.14159-75.39816,

which is the circumference.

Ex. 2. What is the circumference of the earth, the diameter being 7930 miles? Ans. 24912.8 miles.

Ex. 3. Required the circumference of a circle whose diameter is 73. Ans. 231.6924.

Ex. 4. What is the circumference of a circle whose diameter is 40 feet?

Ans. 125.65.

PROBLEM II.

To find the Diameter of a Circle when the circumference is given.

ART. 17. Rule I.-Divide the circumference by 3.14159, and the quotient will be the diameter. Or,

Rule II.—Multiply the circumference by 7 and divide the product by 22. Or,

Rule III.-Multiply the circumference by 113 and divide the product by 355.

Ex. 1. If the circumference of a circle be 14 feet 5 inches, what is its diameter ?

We simply divide the circumference by 3.14159, and the quotient, 4.21, is the diameter.

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OPERATION.

14 ft. 5 in.=173 in.

And 173÷3.14159-50.56 in. which-4.21 ft. the diameter.

Ex. 2. If the circumference of the Sun be 2800000 miles, Ans. 891267 miles.

what is his diameter ?

Ex. 3. What is the diameter of a cylinder whose circumference is 146.084 ?

Ans. 46.5.

Ex. 4. What is the diameter of the Moon if her circumference be 6850 miles?

Ans. 2180.

Ex. 5. Required the diameter of a tree whose circumference is 5 feet. Ans. 21 inches.

ART. 18. The same result may be obtained more conveniently, by exchanging the divisor, 3.14159, for a multiplier, which will give the same answer.

Now, in the proportion 3.14159: 1 :: Circ. : Diam. the fourth term, may be directly found by dividing the second by the first, and multiplying the quotient into the

third. Thus, 1÷3.14159-0.31831. Therefore, if the circumference of any circle be multiplied by the decimal .31831, the product will be the diameter.

In many cases there will be a decided saving of labor by exchanging the divisor for a multiplier, as will be seen in the following example.

Ex. 1. What is the diameter of a circle whose circumference is 50?

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Ex. 2. The circumference of a circle is 69.115 yards:

what is the diameter ?

Ans. 22 yds.

Ex. 3. If the whole extent of the orbit of Saturn be 5650 million miles, how far is he from the Sun ?

Ans. 899225750 miles.

PROBLEM III.

To find the area of a Circle when the diameter and circumference are both known.

ART. 19. Rule 1.-Multiply the square of the diameter by .7854. Or,

Rule II. Multiply the diameter into the circumference, and divide the product by 4; in either case the product will be the area.

The area of a circle is equal to the product of the diameter into the circumference.

Now, we have just seen, (Art. 18,) that if the diameter be 1, the circumference will be 3.14159, and one fourth of

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this is 0.7854, nearly. Consequently, the area of any circle is found by multiplying the square of the diameter by .7854, which is the area of a circle whose diameter is 1.

Ex. 1. Required the area of a circle whose diameter is 623 feet.

We square the diameter, which gives us 388129, and this number we multiply into the decimal.7854, which gives the area.

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OPERATION.

623-388129

.7854

1552516

1940645

3105032

2716903

304836.5166 Ans.

Ex. 2. What is the area of a circular pond whose diameter is 40 rods?

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40-1600, and 1600x.7854-1256.64, Ans.

Ex. 3. What is the area of a circle whose diameter is 33,25 inches? Ans. 868,30 sq. in.

Ex. 4. How many square yards are there in a circle Ans. 2.18. whose diameter is 5 feet?

Ex. 5. What is the area of a circle whose diameter is 4.5?

Ans. 15.904.

PROBLEM IV.

To find the area of a Circle when the Circumference only is given.

ART. 20. Rule.-Multiply the square of the circumference by the decimal .07958, and the product will be the area very nearly.

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