To find the diameter, when the circumference is 100.

take a smaller are; as, for instance, suppose that of 30°, or tr of the circumference.

part

Then, since the tangent of 30°, to radius 1, is √}, the geri

&c. whole circumference, and so for any oth

Those who would wish to see the methods of Machin, Euler, &c. may consult Dr. Hutton's Mensuration, and a paper of his in the Philosophical Transactions, upon this subject.

3. If the diameter of the earth be 7958 miles, as it is very nearly, what is the circumference, supposing it to be exactly round? Ans. 2508528 miles.

4. To find the diameter of the globe of the earth, supposing its circumference to be 25000 miles.

PROBLEM X.

Ans. 7957 nearly.

To find the length of any are of a circle.

RULE I.

As 180 is to the number of degrees in the arc,
So is 31416 times the radius to its length.

Or,

Or, As 3 is to the number of degrees in the arc,
So is 05236 times the radius to its length.

EXAMPLES.

1. To find the length of an arc ADB of 30 degrees, the radius being 9 feet.

[See Figure under Problem VIII. page 50.]

From 8 times the chord of half the arc subtract the chord of the whole arc, and of the remainder will be the length of the arc nearly.

And therefore 8 times the chord DC=8++ 74,

755

&c.

164

2. The chord AB of the whole arc being 465874, and the chord AD of the half arc 234947; required the length of the arc.

But the length of the arc DC, whose sine is s, is known to be

&c. and therefore the arc DE=25+

6r

+

355 40+3'

40r+

&c. which differs from 2+

672 + 755
48,4

+

3r

&c. only by a small quantity, and shews the rule to be very near the truth.

COR.

rule will be

Q. E. D.

When the chord of the whole arc is given the

3

A great number of approximating rules might be given for finding the arc of a circle, but the two given in the text, and the three following ones, will be found sufficient.

RULE 1.

01745, &c. X rad. X number of degrees in any arc to the length of that arc.

3. Required the length of an arc of 12 degrees 10 minutcs, the radius being 10 feet. Ans. 2 1234

i

8

4. To find the length of an arc, whose chord is 6, and the chord of its half is 3. Ans. 73

5. Required the length of the arc, whose chord is 8, and height PD 3. Ans. 103.

6. Required the length of the arc, whose chord is 6, the radius being 9. Ans. 611706.

PROBLEM XI.

To find the area of a circle.

The area of a circle may be found from the diameter and circumference together, or from either of them alone, by the following Rules.

RULE

Multiply half the circumference by half the diameter. Or, Take of the product of the whole circumference and diameter.

RULE 2.†

Multiply the square of the diameter by 7854.

RULE

* DEMONSTRATION. A circle may be considered as a regu lar polygon of an 'infinite number of sides, the circumference being equal to the perimeter, and the radius to the perpendicular. But the area of a regular polygon is equal to half the perimeter multiplied by the perpendicular, and consequently the area of a circle is equal to half the circumference multiplied by the radius, or half the diameter. QE. D.

+ DEMONSTRATION.

All circles are to each other as the

squares of their diameters. (Euc. XII. 2.)

But