CIR

163

CIR

only. n

a c j r " c | , r- Wallis was the first who expressed the area ofeh 0vv ' n terms of the diameter, by an infinite series, andthat, if the square of the diameter was 1 , the area

^ 3 X 5 x 5 X 7 X 7

Other* 4 . x 4 X 6 X 6 X 8 8 24 48

higtyjo series were also found by Lord Brounker, Sir Isaacexn re ■’ an< ^ Or. Gregory. The most convenient forms of8 tat em ' n ® t3le circumference, are shown in the followingbei n g u • w ‘ lei 'e c represents the circumference, the diameter

&c. or ■

25X —24

49

, &c.

4 * +

V 3 5

N/12 X

=8

=8

23

4 V2 x

(-

<rn

( 2 X

5

1

jh3*

1111— 4- — —» — —

7 9 11 13

+

+

i

i

1.3.5

1

l7~1

5.3*

1

3^7

1.3

1

L3*

1

1

ai*

1

31.3.5

-(>

4.6.8.2*1 1

4.6.91 1

2 XX

5.7.9. 7,9.111.3.5

&c.

1

I?

&c.)

&c.

)

4.6.8.111 1

5 4.2’ X 7

—>&c. ^

11 )

1 1.3

)

1.3

' 4.6.2 J

_ X — — _ X— —

2 3 2.4 5 2.4.6

X - -

1

*9

1.3.5

2.4.6.8

xl

)

‘wo 8tra . ”‘j le most useful properties of a circle is that whenthe 8e '® lt ^’ nes > or chords, cut each other, the rectangle8e grnenta f 60 *" 3 ‘^ e one line * s equal to the rectangle of the

V . et| ientl v 0t tbe °ther line. This property may be very con-c j r cle ) a PP‘> e d to finding the centre of the segment of a®{*>e of tij e 6 len S‘h of the radius, the chord and the versed. e 8 qo are a ^° being given. The rule is as follows : Divide8 * n e to th °* t3le . half chord by the versed sine, add the versedClr cle. e quotient, and half the sum is the radius of the

Th

^ re *ice; * 8 ^ ae ^ er a circle being given, to find the circum-f UtQ feren 0 „^’ "^ a ^ is to 22 , so is the diameter. to the cir-f° r ‘be most 1631 ^' ^bis rule will be sufficiently accuratepaired m Practical purposes; but if greater accuracy bej/°duet’^ii^b'Pty the given diameter by 3.1416, and the6 c ' r cumf be t b® circumference, or very near. When" " eren '*“ — the diameter, divide the

the quotient will be the dia-^ -.v.,. — , so is the circumference to the

<j find t

8 e 8 reea h e ;^ e J en gth of an arc, the radius and number of3 i^'cird^ ? lven ’ say, As 180, the number of degrees inch 416 times to ‘b e number of degrees in the arc, so isSuk 4 0 ‘' the n ra< ^ us to the length of the arc. When thehalf ^t the and ‘he chord of half the arc are given;—the i^* e a rc a j r< !. 0 f tbe arc from eight times the chord ofkn 0 en gth ’ nea | ui^ide the remainder by 3, which will giveVe ra V 1 ’"-^irstl c ^ben the chord and versed-sine areof ' s me mni?’ 4 ‘b e diameter, from which subtract theth e ? Ver sed-s‘ by .82, and divide the remainder by ■§■Ca °rd, w i'- lae > add 1 to this quotient, and multiply bywdl gj ve the length of the arc.

The area of a circle, whose diameter is unity, has beenfound to be .7854 nearly; and since the area of a circle is asthe square of its diameter, the areas of all circles will there-fore be ascertained by the following rule: Multiply the squareof the diameter by .7854, and the product will be the area otthe circle.

But it may sometimes appear, that the circumference onlycan be ascertained, the diameter being inaccessible: in thiscase, also, the areas of circles are as the squares of their cir-cumferences. It has been found, that when the circumferenceof a circle is unity, the area of the circle is .07958. There-fore the rule may be thus: Multiply the square of the cir-cumference by .07958, and the product will be the area of thecircle nearly.

Though either the diameter or the circumference is suffi-cient to find the area of the circle, yet, if the dimension ofeach can be easily ascertained, the operation of finding the areais much shorter. This may be done by multiplying the radiusinto the semi-circumference; the product will be the area.

By this rule, the standard area, when the diameter or thecircumference is unity, as in the two preceding rules, will beeasily ascertained, by halving the ratio of the diameter 1 tothe circumference 3.1416: for,

2)3.1416

1.5708 half the circumference.

.5 half the diameter, or the radius.

.78540 the area of the circle by the last

rule, when the diameter is unity.

Again, when the circumference is unity, the diameter willbe found to be .31830 nearly.

Then, .31830 2 = .15915 the semi - circumference,

and 1. - 4 - 2 = .5 the radius.

Therefore .15915 X .5 = .079575 or .07958 nearly.

To find the area of a sector, when the number of degreesin the arc are known; say, As 360, the number of degrees inthe whole circle, is to the number of degrees in the arc of thesector, so is the area of the circle to the area of the sector.When the radius, and the whole or half the length of the arcare given; multiply the diameter by the arc of the sector, anddivide the product by 4 ; or, multiply the radius by half thelength of the arc; in either case, the result will be the areaof the sector.

To find the area of a segment. Having found the area ofthe sector, subtract from it the area of the triangle, when thesegment is less than a semi-circle, and add, when greater, or;Divide the cube of the versed-sine by twice the chord, andadd the quotient to two-thirds of the product of the chordand versed-sine.

As we must necessarily frequently recur to the subjectof this article hereafter, we think it convenient to refer thereader for further information thereon to Geometry, Per-spective , &c.

Circular Roofs, are those roofs whose horizontal sectionsare circular.

Circular Winding Stairs, are such as have a cylindricalcase, or walled enclosure, with the planes of the risers of thesteps tending to the axis of the cylinder.

Circular Work, a term applied to all work with cylin-drical surfaces.

Circular - Circular, or Ctlindro - Ctlindric Work,whatever work is formed by the intersection of two cylinders,whose axes are not in the same direction. The line formedby the intersection of the surfaces, is called hy mathema-ticians, a line of double curvature .