PROPOSITION V

353. The area of a regular polygon is equal to half the product of its perimeter and its apothem.

Proof. If all the vertices of the polygon are joined to the centre, the triangles so formed are all equal, and the area of each equals half the product of a side and the apothem. (Art. 306.) Therefore the area of the whole polygon equals half the product of its perimeter and its apothem.

PROPOSITION VI

354. If the number of sides of a regular inscribed polygon be doubled the perimeter will be increased, but if the number of sides of a regular circumscribed polygon be doubled the perimeter will be diminished.

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Let ABCDE be any regular polygon inscribed in a circle, and FGHJK a regular polygon circumscribed about the same circle. For convenience, we may let the two polygons have the same number of sides and be so arranged that the vertices of the inscribed coincide with the points of contact of the circumscribed.

It is required to prove that if the number of sides of the inscribed polygon be doubled, say by joining the mid-points of the arcs of the circle to the vertices, the perimeter will be increased; but if the number of sides of the circumscribed polygon be doubled, say by drawing tangents at the mid-points of the arcs of the circle, the perimeter will be diminished.

Proof. Let L be the mid-point of the arc AE, and PQ be tangent to the circle at L.

First, in ▲ALE, the sum of AL and LE is greater than AE. (Art. 70.)

Therefore the perimeter of the regular polygon of which AL and LE are two adjacent sides is greater than the perimeter of the polygon of which AE is a side.

Next, in ▲ FPQ, the side PQ is less than the sum of PF and FQ.

Therefore the sum of AP, PQ, and QE is less than the sum of AF and FE.

Hence the perimeter of the regular polygon of which PQ is one side is less than the perimeter of the given circumscribed polygon.

PROPOSITION VII

355. The area of a regular inscribed polygon is increased, and the area of a regular circumscribed polygon is diminished, when the number of sides is doubled.

Proof. The area of a regular polygon equals half the product of its perimeter and its apothem.

(Prop. V.)

When the number of sides of a regular inscribed polygon is doubled, the perimeter is increased (Prop. VI), as is also the apothem [why ?]. Hence the area is increased.

When the number of sides of a regular circumscribed polygon is doubled, the perimeter is diminished (Prop. VI), while the apothem remains unaltered. Hence the area is diminished.

SECTION II

MEASUREMENT OF THE CIRCLE

356. In the preceding chapters, whenever we have spoken of a length, we have had in mind a straight line distance; it has been in every case the measure of a line-segment.

What is meant by the length of a curved line is not so evident, and hence it is necessary to give this expression a meaning by definition.

357. If any regular polygon is inscribed in a given circle, and the number of its sides is repeatedly doubled (or is indefinitely increased in any regular way), the polygon can be made as nearly as you please to coincide with the circle.

In other words, the circle is the limit which the regular inscribed polygon approaches, as the number of its sides is indefinitely increased.

The apothem of the inscribed polygon approaches the radius of the circle as its limit.

358. DEFINITION. The length of a circle is defined to be the limit of the perimeter of an inscribed regular polygon, as the number of sides of the polygon is indefinitely increased.

The fact that the perimeter of a variable inscribed regular polygon has a limit admits of a formal proof, the essential points of which are: (1) the series of perimeters is constantly increasing (Prop. VI); (2) the perimeter never exceeds a fixed finite quantity, for example, the perimeter of a particular circumscribed polygon.

The length of a circle is called its circumference.

When the number of sides of an inscribed regular polygon is indefinitely increased, the 'circle' is the limit of the 'polygon'; the circumference of the circle' is the limit of the 'perimeter of the polygon.'

The length of an arc of a circle is defined in the same way to be the limit of the sum of chords in the arc when the number of such chords is indefinitely increased in some regular way.

The area of a circle is the surface en-
It is equal to the limit of the area of

359. DEFINITION. closed by the circle. a regular inscribed polygon as the number of its sides is indefinitely increased.

360. The circumference of a circle could just as well be defined as the limit of the perimeter of a regular circumscribed polygon when the number of its sides is indefinitely increased, since the limit of the perim

eter of the circumscribed polygon is the same as the limit of the perimeter of the inscribed polygon.

This statement again admits of a formal proof, which however involves a greater knowledge of algebra than the pupil is supposed to have at this stage. The assertions that the circle is the limit of the regular inscribed polygon (Art. 357), and also of the regular circumscribed polygon (Art. 360) may be taken as postulates.

The radius of the circumscribed polygon approaches the radius of the circle as its limit, and the area of the polygon the area of the circle as its limit.

361. Since the perimeter of the inscribed polygon continually increases and the perimeter of the circumscribed polygon continually decreases as the number of their sides is indefinitely increased (Prop. VI):

1. The circumference of a circle is greater than the perimeter of any regular polygon inscribed in it.

2. The circumference of a circle is less than the perimeter of any regular polygon circumscribed about it.

PROPOSITION VIII

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

Let O and O' be the centres of any two circles whose circumferences are denoted by C and C', their diameters by d and d', and their radii by r and r'.

Proof. Inscribe in the two given circles regular polygons ABCDE and A'B'C'D'E' of the same number of sides. The perimeters of these polygons are in the same ratio as their radii, i.e. as the radii of the circles in which they are inscribed. (Prop. IV, Cor. II.)

or

Let P and P' be the perimeters of the two polygons.
Then

P: P'=r: r',

Suppose now the number of sides in each polygon is doubled, and let the perimeters of the polygons so formed be denoted by P1 and P.

If this process is repeated indefinitely, the perimeter P will approach the circumference C as its limit, and the perimeter P' will approach the circumference C' as its limit. (Art. 357.)