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29. Thus T stands always for the exact value of a certain incommensurable number, whose approximate value is 3:14159265, which number is the ratio of the circumference of any circle to its diameter.

It cannot be too carefully impressed on the student's memory that a stands for this number 3:14159265...&c., and for nothing else; just as 180 stands for the number one hundred and eighty, and for nothing else.

*30. We proceed to prove that the ratio of the circumference of a circle to its diameter is the same for all circles.

The proof depends on the following important principle;

The length of the circumference of a circle is that to which the length of the perimeter of a regular inscribed polygon approaches, as the number of its sides is continually increased,

*31. We take for granted that the straight line is the shortest line that can join two points.

* 32. Let ABCDEF be any regular polygon inscribed in a circle.

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Then the side AB is shorter than any other line joining AB, so that the side AB is less than the arc AB; therefore the perimeter of the polygon, viz.

AB + BC + CD + DE + EF+ FA is less than the sum of the arcs, that is, is less than the circumference of the circle.

.

Now let each of the arcs AB, BC, etc. be bisected in a, B, y, d, e, n, and let the lines Aa, aB, BB, BC, etc. be joined ;

then the figure AaBBCy etc. is a regular polygon of twice as many sides as the first polygon.

And since the sides Aa + aB are together greater than the side AB, it follows that the perimeter of the second polygon, viz.

Aa +aB+ BB+ BC + Cy+yD+ etc. is greater than the perimeter of the first.

But the perimeter of the second polygon is less than the circumference of the circle,

because each side is less than the corresponding arc.

Hence the perimeter of the second polygon is nearer the circumference, but is less than the circumference.

By bisecting the arcs of the second polygon we should get a third polygon, whose perimeter is nearer the circumference of the circle than the second.

It is clear that by continuing this process, we can get a polygon whose perimeter is as near as we please to the circumference of the circle. Hence,

The length of the circumference of a circle is that to which the length of the perimeter of a regular inscribed polygon approaches, as the number of its sides is continually increased.

* 33. PROP. The ratio of the circumference of a circle to its diameter is the same for all circles.

Let ABCDEF, abcdef be any two circles.

Let a regular polygon of any, the same number of sides be inscribed in each of them.

Join A, B, C, etc., a, b, c, etc. the angular points of the polygons to the centres 0, o respectively.

a

B

E

D

Then AOB any one of the isosceles triangles in the first figure is equiangular with*, and therefore similar to, aob any one of the isosceles triangles in the second figure.

.. AB : 0A = ab : oa, [Euc. VI. 4.] and

BC : 0A = bc : oa, and so on.

AB + BC + CD + etc. :0A = ab + bc + cd + etc. : 00; or, the perimeter of the first polygon is to the radius of the

* For at O and at o, four right angles are each divided into the same number of equal angles, so that the vertical angles AOB, aob of the isosceles triangles AOB, aob are equal.

first circle as the perimeter of the second polygon is to the radius of its circle.

This is true whatever be the number of the sides of the two polygons.

And therefore it is true however great be the number of sides of the two polygons.

But the circumferences of the circles are what the perimeters of the polygons become, when the number of the sides is indefinitely increased. Therefore the circumference of the first circle is to its radius OA as the circumference of the second circle is to its radius oa.

Thus the ratio of the circumference of any circle to its radius is equal to the ratio of the circumference of any

other circle to its radius.

circumference So that the ratio

is the same numerical

diameter quantity for all circles.

Q. E. D.

34. We said above (Art. 29) that this number is 3•14159 etc., and that it is denoted by T. Hence the circumference of any circle of radius r

= (3.14159265 etc.) x its diameter = 2mr.

35. We may notice that * = 3.142857.

So that and differ by less than a thousandth part of their value.

Also i15 = 3•1415929 etc. So that 315 may be used for 7 with sufficient

for any practical purpose. Hence , 3.14159 and 115 are each used for a according to the degree of accuracy required.

accuracy

36. Of these 3.14159 is the most frequently used. The student should notice however that in dividing by a it will be more convenient to use

113

than 3.14159.

3 5 5

37. The following results are instructive.

The ratio of the perimeter of a regular polygon inscribed in a circle to the diameter of the circle, when the polygon has four sides, is 2/2

=2.8284... six sides, is 3

=3 eight sides, is 4/(2-J2)

=3:0614... ten sides, is 1 (1/5 1)

=3:0901... twelve sides, is 3/2 (13 - 1)

=31058... twenty sides, is 5{J(3+5)-(5-J5)} =3.1287...

15 sixty sides, is

-1(10+2/5) (13 – 1)}=3•1401... These numbers approach 3:14159 as the number of sides is increased, while the surd expression becomes more complicated.

The first of these results the student will be able to verify; the second is proved in Euclid iv. 15. The rest will be proved later on.

2/2 {(15 – 1)(3+1)

Example 1. The driving wheel of a locomotive engine is 5 ft. 6 in. high. What is its circumference ? Here we have a circle whose diameter is 51 feet; .: its circumference=1 X 5.5 feet,

=(3:14159...) x 5.5 feet,

=17.278.., feet. The circumference is 17 ft. 3 in. approximately.

Example 2. If a piece of wire 1 foot long be bent into the form of a circle, what will be the diameter of the circle ?

Here the circumference=1 foot, that is

a x diameter=I foot,

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