COR. I. If the polygons are formed by continual. bisections of the circumference, then the perimeter of the polygon par is less than that of a square whose side is XY.

Now XY is double of OX, and .. double of NR ; but NR is <AN, for ▲ NAR is << OAR and .. < ▲ ARN; .. XY is < AB;

..the difference between the perimeters of the given. polygons,

and .. between either of them and the given circle is less than the perimeter of the square on a side of the inscribed polygon. (III. C)

COR. 2. Hence a polygon can be inscribed in a circle whose perimeter shall differ from that of the circumference of the circle by less than any assignable magnitude.

So also can a polygon be circumscribed about a circle, &c.

PROBLEM 16.

To find the area of a polygon circumscribed about a circle.

Let BFGHK be a polygon circumscribed about a circle. Join the centre Q with the angular points of the polygon, and also with the points of contact.

Then the area of ▲ BFQ is = rectangle (radius QM, and BF), (I. 32) so area of a FGQ is = rectangle (radius, and FG),

.. area of polygon = sum of the rectangles contained. by (radius, and BF), (radius, and FG) &c.

=rectangle (radius, and a straight line = the perimeter). (II. I)

COR. Now let the points of contact of the sides of the polygon be determined by continual bisections of the circumference, then ultimately the area of the polygon will differ from that of the circle by less than any assignable magnitude, as also will the perimeter of the polygon from the circumference of the O.

=

Hence the area of the circle is rectangle (radius, and a straight line = the circumference).

BOOK V.

RATIO AND PROPORTION.

INTRODUCTORY.

Two magnitudes of the same kind may be compared by considering how many times one contains the other.

If one does not contain the other a certain number of times exactly, it may happen that each contains some other magnitude a certain number of times, and then the two given magnitudes may be compared.

If each of two magnitudes does not contain some other magnitude a certain number of times, they are called incommensurable.

In order to institute a comparison, one of them may be divided into a number of equal parts as small as we please, and parts equal to these may be taken from the other till there remains less than one of these parts, and therefore all the parts together thus taken away differ from the whole magnitude by less than one of these parts. Hence the former magnitude may be compared with one which differs from the latter by less than any assignable quantity.

DEFINITION.

The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth when, if the first be divided into any number whatever of equal parts and the third be divided into the same number of equal parts, the second contains the same integral number of the former parts as the fourth does of the latter.

PROPOSITION I.

Straight lines have to one another the same ratio as rectangles of equal altitude described upon them.

Let AG, BH be the given straight lines,
and C the given altitude.

Then shall AG have to BH the same ratio as rectangle (AG, C) to rectangle (BH, C).

On AG, BH construct rectangles having altitudes each equal to C.

If now AG be divided into any number whatever of equal parts AM, MN, &c.,

and lines be drawn through M, N, &c. || to AK:

then KG will be divided into the same number of equal parts as AG.

If also from BH parts BP, PQ, &c., each equal to AM be cut off until there is no remainder or a remainder less than AM,

and lines be drawn through P, Q &c. || to LB: then LH will contain the same integral number of parts, each equal to KM as BH does parts each equal to AM; .. AG has the same ratio to BH as KG has to LH,

i.e. as rectangle (AG, C) has to rectangle (BH, C).

COR. In a similar manner it may be proved that straight lines have to one another the same ratio ass or as of equal altitude described upon them.

DEFINITION.

Magnitudes which have the same ratio are called portionals.

pro

The fact of four magnitudes being proportionals is expressed thus:

The first is to the second as the third is to the fourth, and written thus

the first

:

the second as the third: the fourth.

If two magnitudes are equal one will contain the same integral number of parts, each equal to any third magnitude, as the other. Hence

PROPOSITION II.

Proportions hold good if for the magnitudes involved others equal to them are substituted.

For the test expressed by the definition is satisfied.

PROPOSITION III.

If four magnitudes are proportional then any others which have the same ratio as the first and second are proportional to any others which have the same ratio as the third and fourth. For the test expressed by the definition is satisfied.

PROPOSITION IV.

If a magnitude have the same ratio to each of two others then these two must be equal.

For if A be the former and B, C'the two latter magnitudes, so that

A B as A: C,

then B must be equal to C.

Otherwise A might be divided into a number of equal parts, each less than the difference between B and C;

and B and C could not contain the same integral number of those parts; but they do (by the definition); .. B is equal to C.

C. G.

II