PROPOSITION XI.

THEOREM. A circle is equivalent to the rectangle contained by lines equal to the radius and half the circumference.

Let us represent the rectangle of the radius and semicircumference of the circle ABCD by P: we are to show that this rectangle is equal in surface to the

circle.

A

B

D

C

If the rectangle P be not equivalent to the circle, it must be either greater or less. Suppose it to be greater, and let us represent the excess by Q. Then, by the corollary to last proposition, a polygon may be circumscribed about the circle, which shall differ therefrom by a magnitude less than Q, and must consequently be less than the rectangle P. But the area of every circumscribed polygon is equivalent to the rectangle of the radius and half its perimeter, (B. V, Prop. VIII,) and the perimeter exceeds the circumference of the circle: consequently the rectangle of the radius of the circle and semi-perimeter of the polygon must be greater than P, the rectangle of the same radius and semicircumference of the circle; but it was shown above to be less, which is absurd. Hence the hypothesis that P is greater than the circle, is false.

But suppose the rectangle P is less than the circle,

and let us represent the defect by the same letter Q. Then, by the same corollary, a regular polygon may be inscribed in the circle, which shall differ from it by a magnitude less than Q, and must consequently be greater than the rectangle P. But every inscribed regular polygon is equivalent to the rectangle of the perpendicular drawn from the centre to one of the sides, into half its perimeter; and this perpendicular is less than the radius. of the circle, and the perimeter is less than the circumference consequently the rectangle of this perpendicular and semi-perimeter of the polygon must be less than P, the rectangle of the radius and semicircumference of the circle; but it was shown above to be greater, which is absurd. Hence the second hypothesis also is false.

As therefore the circle can be neither greater nor less than the rectangle P, it must necessarily be equivalent

to it.

PROPOSITION XII.

THEOREM. Circles are to each other as the squares of their radii.

Let the circles ABCD, abcd, be compared: we shall have the proportion

AO2: ao2 :: circle ABCD circle abcd.

For if this proportion has not place, let there be

AO ao circle ABCD: P,

P being some magnitude either greater or less than the circle abcd. Suppose it to be less, and let us represent the defect by Q. Then, (B. V, Prop. xi,) a polygon may

be inscribed in the circle abcd, which shall differ from it by a magnitude less than Q, and will therefore exceed the magnitude P. Let abcde, &c., be such a polygon; and describe a similar polygon ABCDE, &c., in the other circle. Then, since regular polygons of the same number of sides are similar, we have, (B. IV, Prop. XVI,)

:

AO ao pol. ABCDE, &c. pol. abcde, &c. Hence, by equality of ratios, we have

circ. ABCD: P:: pol. ABCDE, &c. : pol. abcde, &c. Now in this proportion the first antecedent is greater than the second, consequently the first consequent is greater than the second; that is, P is greater than the polygon abcde, &c. ; but it has been shown to be less, which is absurd. Therefore P cannot be less than the circle abcd.

But suppose that P is greater than the circle abcd. Then, still representing the difference by Q, a polygon may be circumscribed about the circle abcd, which shall differ from it by a magnitude less than Q, or be less than P. Suppose such a polygon to be described, and that a

similar one is formed about the circle ABCD; then these polygons being to each other as the squares of the radii of their respective circles, it will evidently result, by comparing, as in the preceding case, this proportion with that advanced in the hypothesis, that the circle. ABCD is to P as the polygon about this circle to the polygon about the other; in which proportion the first antecedent is less than the second, and consequently the first consequent is less than the second, that is, P is less than the polygon circumscribed about the circle abcd; but it was shown above to be greater, which is absurd. Hence P can neither be less nor greater than the circle abcd; consequently it must be equal to it, and therefore

AO: ao2:: circle ABCD circle abcd.

:

Cor. 1. Since every circle is equivalent to the rectangle of its radius and half its circumference, the above proportion may be expressed thus :

AO ao AO.

whence, AO ao

circ. ABCD: ao. circ. abcd;

circ. ABCD: circ. abcd.

Consequently the circumferences of circles are to each other as their radii, and therefore their surfaces are as the squares of the circumferences.

Cor. 2. It follows, also, that similar arcs are to each other as the radii of the circles to which they belong; for they subtend equal angles at the centres, (B. III. Def. 12,) and each angle is to four right-angles as the arc which subtends it is to the whole circumference; consequently the one arc is to the whole circumference, of

which it forms a part, as the other arc to the circumference of which it is part; and as the circumferences are as the radii, we have alternately the one arc to the other as the radius of the former to that of the latter.

Cor. 3. Therefore also similar sectors are to each other as the squares of their radii; for each sector is to the circle as the arc to the circumference: consequently the one sector is to its circle as the other sector to its circle; and as the circles are as the squares of the radii, we have alternately the one sector to the other as the square of the radius of the former to the square of that of the latter.

Cor. 4. It readily follows that similar segments are also as the squares of the radii; for they result from similar sectors, by taking away from each the triangle formed by the chord and radii, which triangles, being similar, are also to each other as the squares of the radii; therefore the sectors and triangles being proportional, it follows that the segments also are as the sectors, or as the squares of the radii, or indeed as the squares of their chords.

PROPOSITION XIII.

PROBLEM. The surface of a regular inscribed polygon and that of a similar circumscribed polygon being given, to find the surfaces of regular inscribed and circumscribed polygons of double the number of sides.