If the diameter be given, find the circumference by Art. 12. If the circumference be given, find the diameter by Art. 13. EXAMP. A circle whose diameter is 12, and circumference is 37.7, given, to find the area?

ART. 15. The Diameter being given to find the Area of a Circle without finding the Circumference.

RULE. Multiply the square of the diameter by 7854, and the product will be the area of the circle, whose diameter was given. EXAMP. The diameter of a circle being 12, to find the area?

Set 1 on A to the diameter on B, then find 7854 (which expresses the area of a circle whose diameter is 1) on A, against which on B is a 4th number, then find this 4th number on A, against which on B is the area.

By Gunter.

The extent from 1 to the length of the diameter reaches from •7854 to a 4th number, and from that 4th number to the area.

ART. 16. The Circumference of a Circle being given, to find the Area without finding the Diameter.

RULE.-Multiply the square of the circumference by 07958, and the product will be the area of the circle.

EXAMP. The circumference of a circle being 37.7, to find the area.

ART. 17. The Dimensions of any of the parts of a Circle being given, to find the side of a Square equal to the Circle.

RULE. If the area of the circle be given, extract the square root of the area, which will be the side of a square equal to the circle :

If the diameter or circumference be given, find the area by Art. 15 or 16, and then extract the square root, as before. And this is a general rule to find the side of a square equal to any superficial figure, regular or irregular: for the square root of the area of any figure whatever, is the side of a square equal to the given figure. But with regard to circles, if the diameter be given; multiply it by 886, and the product will be the side of an equal square: or, as 13.545 is to 12, or 1354 to 1200: so is the diameter of a circle to the side of a square equal to the given circle. And, if the circumference be given, multiply it by 282 for the side of an equal square. Or, divide it by 3.545, and the quotient will be the side of an equal square.

ART. 18. The Area of a Circle being given, to find the Diameter. RULE. Multiply the given area by 1.2732, and the product will be the square of the diameter; then, extracting the square root of the product, you will have the diameter.

EXAMP. The area of a circle being 113.09, to find the diameter.

ART. 19. The Area of a Circle being given, to find the Circumference.

RULE.-Multiply the given area by 12.566, and extract the square root of the product, which root will be the circumference required. EXAMP. The area of a circle being 113.03 to find the circumfe

rence.

ART. 20. The Side of a Square being given, to find the Diameter of a Circle equal to the Square, whose Side is given.

RULE.-Multiply the given side by 1.128, and the product will be the diameter of a circle, whose area is equal to the area of the given square. Or, if the side of the square be divided by 886, the quotient will be the diameter. Or, as 12 to 13.54, so is the side of any square to the diameter of an equal circle.

EXAMP. The side of a square being 10-635, to find the diameter of a circle equal to that square?

10.635×1·128 = 12 nearly. Or, 10·635÷88612= diameter. Or, as 12: 13.54:: 10-635: 12 nearly.

ART. 21. The Side of a Square being given, to find the Circumference of a Circle equal to the given Square.

RULE.-Multiply the given side by 3.545 and the product will be the circumference required. Or, divide it by 282, and the quotient will be the circumference.

EXAMP. The side of a square being 10631, to find the circumference of a circle equal to that square.

10.631×3-545-37.686-circum. Or, 282) 10.631(37-698 circum. ART. 22. To find the Area of a Semicircle, the Diameter being given. RULE. Find the area of the circle by Art. 15, and take the half of it.

In the same manner may the area of a quadrant, or a quarter of a circle, be found, by taking a fourth part of the area of the whole

circle.

But with regard to measuring a sector, or a segment of a circle, it will be necessary first to show how to find the length of the arch line of a sector, and the diameter of the circle to a given segment. ART. 23. A Segment of a Circle being given, to find the length of the Arch

Line.

RULE. Divide the segment into two equal parts; then measure the chord of the half arch, from the double of which subtract the chord of the whole segment; and one third of that difference, being added to the double of the chord of the half arch, will give the length of the arch line, EXAMP.

ART. 24. The Chord and versed Sine of a Segment being given, to find the Diameter of a Circle.

RULE.-Multiply half the chord by itself, and divide the product by the versed sine; then add the quotient to the versed sine, and the sum will be the diamèter of the circle.

ART. 25. To measure a Seator.

Definition. A sector is a part of a circle, contained between an arch line, and two radii or semidiameters of the circle.

RULE. Find the length of half the arch by Art. 23: Then multiply this by the radius or semidiameter, and the product will be the

ART. 26. To find the Area of a Segment of a Circle.

Definition. A segment of a circle is any part of a circle cut off by a right line drawn across the circle, which does not pass through the centre, and is always greater or less than a semicircle.