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Let w and y represent the two varying quantities. Let subscript numbers indicate particular values, the same subscript number being used for corresponding values.

We have to show that

Xy : 2,=Y, : 42 where

21

and X, are incommensurable values of x. If possible, let 21 : X2 = : 43

(1) where

Yz> Y2 and .:. Xz > 22. Suppose 2 to be divided into a number m of parts, each equal to a,

where a < X3 – 22.
Then, if na is the largest multiple of a less than 23,

na < xz but na > x2.
Call na, 24: then, since X4 <23, .. Y4 <43.
Again, since X and X4 (viz. ma and na) are commensurable,
..X : 24 =: 44

(2). Comparing (1) and (2), since

34 > X2 . . Y4 > Y3. But ye has been shown to be < oz.

Which is impossible. ..3: 2, is not unequal to : Y. i.e. it is equal to it. Q.E.D.

§ 2. THE ANGLE AND THE CIRCLE.

36. A circle may be described by the revolution in a plane of a straight line about one of its extremities which remains fixed.

The fixed extremity is called the centre; the moving extremity traces out the circumference.

Any part of the circumference is called an arc; and any part of the area, traversed by the revolving radius, is called a sector.

Now the length of the arc and the area of the sector clearly depend on two elements, viz. (1) The length of the revolving radius. (2) The amount of revolution described.

Hence we see how naturally a circle is connected with an angle as described in the first chapter.

The two fundamental propositions connecting angle, arc, and radius will now be proved. The first proposition is proved in a manner intended to show the connexion between Euclid's method of treating proportionals and the algebraical method.

37. In circles of equal radii, arcs which subtend unequal angles are to one another as the angles.

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Let the arcs AB, A'B of circles whose radii 04, O'A' are equal, subtend angles AOB, A'O'B' at the centres 0, O respectively. Then

arc AB : arc A'B' shall equal angle AOB : angle A'O'B'.

If not, let the fractions which measure these ratios be unequal: and suppose that p/q is a fraction lying between them, so that (say)

а

arc AB arc A'B

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P

angle AOB
and

р
9
angle A O'B'

9 Then q. arc AB <p.arc A'B'

and. q. angle AOB>p.angle A’O'B'. At O describe q angles (including AOB) each equal to AOB, and at O' describe p angles (including A'O'B') each equal to A'O'B'.

P Then, the whole angle AOQ=q.angle AOB; and the whole angle A'O'P' = p. angle A'O'B',

.. whole angle AOQ > whole angle A'O'P'. Again, by Euc. III. 26, the arcs AB, BC &c. which subtend equal angles are equal to one another.

;. the whole arc AQ=q .arc AB, and the whole arc A'P' =p.arc A'B',

:. whole arc AQ < whole arc A'P'. But again, by Euc. III. 26, since the radii 0A, O'A' are equal, and since angle AOQ > angle A'O'P',

.. arc AQ> arc A'P'. Which is absurd.

.. arc AB : arc A'B' = angle AOB : angle A'O'B'. Q.E.D. Or the proposition may be proved as follows:

Using the same figure, but considering the angles AOB, A'O'B' to be equal, it follows at once from Euc. III. 26, that

arc AQ : arc A'P' = angle AOQ : angle A'O'P' i.e. the ratio of any two commensurable values of the arc is equal to the ratio of the corresponding values of the angle,

:., by Art. 35, the same holds for all values.

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38. In circles of unequal radii, arcs which subtend equal angles are to one another as the radii.

Let the arcs AZ, az subtend equal angles at the centres O, o respectively. Then

arc AZ : arc az shall equal radius 0A : radius oa. Within angle AOZ, draw any number of radii OB, OC...OY :

:

and at o, make the angles aob, boc...xoy equal to AOB, BOC, ... XOY, respectively. Then

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.: whole angle AOZ = whole angle aoz, :. remaining angle YOZ= remaining angle yoz. Join AB, BC...YZ; ab, bc...yz. Then :: A0 : BO=ao : bo, and < AOB = 2 aob,

.. A AOB is similar to A aob. [Euc. VI. 6.

.:. AB : ab= AO : ao. Similarly BC : bc= BO : bo=A0: ao, and

CD:cd=CO : co =

:

AO: ao.

.. AB + BC + CD + : ab + bc + cd + ...... = AO : ao.

Now, as we increase the number of divisions of the arcs AZ, az, the rectilinear boundaries evidently approximate more and more closely to the arcs. Hence ultimately we may assume that

AB + BC + CD + = arc AZ; and

ab + bc + cd +
. . arc AZ : arc az = radius OA : radius oa*.

Q.E.D.

+

= arc az.

39. The above proposition is of the highest importance.

* A stricter proof of this proposition will be given later.

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Since arc and radius are both lengths, we may interchange the second and third terms of the proportion, and write

arc AZ : radius (A = arc az : radius oa,

when the angle AOZ = angle aoz. In other words, the ratio arc to radius is constant for the same angle described at the centre of a circle.

40. PROP. I. The ratio of the circumference to the diameter of a circle is the same for all circles.

For, in the last article, suppose the angles AOZ, aoz to be each four right-angles. Now the arc subtended by four right-angles, i.e. by a complete revolution, is the whole circumference. Hence,

circumference AZ: radius (A = circumference az : radius oa. :: circumference AZ: diameter 2.0A

circumference az : diameter 2.oa. i.e. the ratio of the circumference to the diameter in any one circle is equal to the ratio of the circumference to the diameter in any other circle.

Q.E.D. PROP. II. The angle which, at the centre of a circle, subtends

. an arc equal to the radius, is the same in magnitude for all circles.

For, in the last article, suppose the arc AZ to have been measured equal to the radius 04, and the angle aoz to be still equal to the angle AOZ. Then, since

,
arc AZ: radius 0 A = arc az : radius oa.
.. arc az is also equal to radius oa.

Thus, the angle, which, at the centre of any one circle, is subtended by an arc equal to the radius, is equal to the angle, which at the centre of any other circle, is subtended by an arc equal to the radius.

Q.E.D. 41. From the above two propositions, follow two definitions.

1. The ratio of the circumference to the diameter of any circle is called 7*.

:

* The Greek for circumference is tepuØépela, of which a is the initial letter.

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