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PROP. XII. To find an approximate expression for the ratio of the circumference of a circle to its diameter.

We shall take the diameter of the circle as given, and determine the perimeters of an inscribed and similarly circumscribed polygon. We shall then determine the perimeters of inscribed and circumscribed polygons of double the number of sides. Taking the last found perimeters as given, we will then determine the perimeters of polygons of double the number of sides by the same method, and so on. As the number of sides increases, it may be shewn that the perimeters approach nearer and nearer to the circumference; hence their successively determined values will be nearer and nearer approximations to the value of the circumference.

Taking the diameter of the circle as 1, we will begin by inscribing and circumscribing a square, and finding their perimeters.

2=

Here we have p = 2 √2 = 2·8284271,

and

P = 4.

Having found P and p, we find by Proposition X.

P' and p';

thus

P'

2p+ P

P+p

-=3.3137085,

and

'p' = √√p× P' = 3·0614675.

Then taking P' and p' as given quantities, we put

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and find, by the same formulæ, for the polygon of 16

sides,

P' = 3.1825979,

p=3.1214452.

Continuing this process, the results will be found as in the following table:

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From the last two numbers of this table, we learn that the circumference of a circle whose diameter is

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represents the circumference to within a unit of the seventh decimal place.

It is usual to represent this number by the symbol

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... circumference: = × diameter.

If we convert the expression for into a series of continued fractions,* we will find for it

3 22 333 355 103993
I'7' 106' 113'

It is usual, in practice, to

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take for , 22, or 37, as

this is sufficiently near for all practical purposes.

Let C denote the circumference,

D the diameter,

r the radius.

We then have the following formulæ :

* See author's treatise on the Theory of Arithmetic.

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Cor.-Hence the ratio of two circumferences is the same as the ratio of their radii.

Let C and be the circumference and radius of one circle, and C' and the circumference and radius of

another.

Then

and

C = 2πr,
C' = 2xr';

C r
Cr

THE MEASURE OF ANGLES.

PROP. XIII.-The numerical measure of an angle at the centre of a circle is the same as the numerical measure of its intercepted arc, if the adopted unit of angle is the angle at the centre which intercepts the adopted unit of arc. Let AOB be an angle at the centre O, and AB its intercepted arc. Let AOC be the angle which is adopted as the unit of angle, and let its intercepted arc AC be the arc which is adopted as the unit of arc. (Euclid, VI. 33),

Then

angle AOB

A

arc AB

=

angle AOC arc AC

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B

But the first of the above ratios is the measure of the angle AOB, referred to the unit angle AOC; and the second ratio is the measure of the arc AB, referred to the unit arc AC. Therefore, with the adopted units, the numerical measure of the angle AOB is the same as that of the arc AB.

Remark 1.-This proposition, being of frequent application, is usually more briefly though inaccurately

expressed, by saying that an angle at the centre is measured by the arc on which it stands. In this statement of the proposition, the condition that the adopted units of angle and arc correspond to each other is understood, and the expression 'is measured by' is used for has the same numerical measure as.'

Cor. 1.-An angle, ABC, at the circumference of a circle is measured by half the arc AC on which it stands, since it is equal to half the angle AOC at the centre (Euclid, III. 20).

Cor. 2. An angle, ABC, standing within the circle, but not at the centre, is measured by half the sum of the arcs AC and A'C'. For joining A'C, we have

but and

angle ABC = angle AA'C + angle C'CA';

angle AA'C is measured by half-arc AC,
angle C'CA' is measured by half-arc A'C';
arc AC + arc A'C'

.. angle ABC is measured by

B

C

2

;

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but

and

angle ABC angle AA'C-angle C'CA';
angle AA'C is measured by half-arc AC,
angle C'CA' is measured by half-arc A'C' ;
arc AC-arc A'C'

.. angle ABC is measured by

2

;

or, in other words, the angle ABC is equal to that angle at the centre which stands on an arc equal to half the difference of the intercepted arcs AC and A'C'.

Remark 2.-The measurement of angles, then, is reduced to the measurement of arcs, and the arc which is taken for the unit is an arc in the circle which is equal to the radius. The unit angle will be an angle at the centre subtended by the unit arc; and we have seen that the ratio of the circumference of a circle to its radius is an invariable number, so that, in a circle of any radius whatever, the angle at the centre, which is subtended by an arc equal to the radius, is always of the same magnitude. That is, in the circles ABC, DEF,

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of different radii, AO and DQ, the angles AOB, DQE, subtended at their centres by arcs equal to their radii, are equal, whatever be the magnitudes of the circles.

When an angle is measured by referring it to the unit angle, or, which is the same thing, when its arc is measured by referring it to the unit arc, the measure we obtain is said to be the circular measure of the angle. There is another method adopted for measuring angles, and this we now explain.

The right angle is, by its nature, the simplest unit

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