or, the polygon B' is equal to the quotient of twice the product of the given polygons divided by the sum of the inscribed polygons.

Thus, by means of the polygons A and B, it is easy to find the polygons A' and B', which have double the number of sides.

PROPOSITION XI.-THEOREM.

371. A circle being given, two similar polygons can always be formed, the one circumscribed about the circle, the other inscribed in it, which shall differ from each other by less than any assignable surface.

Let Q be the side of a square less than the given surface.

Bisect A C, a fourth part of the circumference, and then bisect the half of this fourth, and so proceed until an arc is found whose chord AB is less than Q. As this arc must be an ex

G

M

I

H

C

K

D

B

L

A

E

act part of the circumference, if we apply the chords A B, BC, &c., each equal to A B, the last will terminate at A, and there will be inscribed in the circle a regular polygon, ABCDE, &c. Next describe about the circle a similar polygon, G HIK L, &c. (Prop. VII.); and the difference of these two polygons will be less than the square of Q. Find the centre, O; from the points G and H draw the straight lines G O, HO, and they will pass through the points A and B (Prop. VII.). Draw also OM to the point of tangency, M; and it will bisect A B in N, and be perpendicular to it (Prop. VI. Cor. 1, Bk. III.). Produce AO to E, and draw B E.

Let P represent the circumscribed polygon, and p the inscribed polygon. Then, since these polygons are similar, they are as the squares of the homologous sides G II,

AB (Prop. XXXI. Bk. IV.); but the triangles GOH, AOB are similar (Prop. XXIV. Bk. IV.); hence they are to each other as the squares of the homologous sides OG and OA (Prop. XXIX. Bk. IV.); therefore

P:p::OG: OA or OM2.

Again, the triangles OG M, EA B, having their sides respectively parallel, are similar; therefore

P:p::OG: 0M2:: AE: BE';

and, by division,

P: P-p::AE: A E2-EB or A B2.

But P is less than the square described on the diameter AE; therefore Pp is less than the square described on AB, that is, less than the given square Q. Hence, the difference between the circumscribed and inscribed polygons may always be made less than any given surface.

372. Cor. Since the circle is obviously greater than any inscribed polygon, and less than any circumscribed one, it follows that a polygon may be inscribed or circumscribed, which will differ from the circle by less than any assignable magnitude.

.

PROPOSITION XII.-PROBLEM.

373. To find the approximate area of a circle whose radius is unity.

Let the radius of the circle be 1, and let the first inscribed and circumscribed polygons be squares; the side of the inscribed square will be 2 (Prop. IV. Cor.), and that of the circumscribed square will be equal to the diameter 2. Hence the surface of the inscribed square is 2, and that of the circumscribed square is 4. Let, therefore A 2, and B 4. Now it has been proved, in Proposition X., that the surface of the inscribed octagon, or, as it has been represented, A', is a mean proportional

=

=

between the two squares A and B, so that A' = √ 8 = 2.8284271; and it has also been proved, in the same proposition, that the circumscribed octagon, represented by B', ; so that B' 3.3137085. The

2 AX B
A+A

=

16

2+18

=

inscribed and the circumscribed octagons being thus determined, we can easily, by means of them, determine the polygons having twice the number of sides. We have only in this case to put A= 2.8284271, B = 3.3137085; and we shall find A' = AXB 3.0614674, and

=

In like manner may be determined the area of polygons of sixteen sides, and thence the area of polygons of thirtytwo sides, and so on till we arrive at an inscribed and a circumscribed polygon differing so little from each other, and consequently from the circle, that the difference shall be less than any assignable magnitude (Prop. XI. Cor.).

The subjoined table exhibits the area, or numerical expression for the surface, of these polygons, carried on till they agree as far as the seventh place of decimals.

It appears, therefore, that the inscribed and circumscribed polygons of 32768 sides differ so little from each other that the numerical value of each, as far as seven places of decimals, is absolutely the same; as the circle is between the two, it cannot, strictly speaking, differ from either so much as they do from each other; so that the number 3.1415926 expresses the area of a circle whose radius is 1, correctly, as far as seven places of decimals.

Some doubt may exist, perhaps, about the last decimal figure, owing to errors proceeding from the parts omitted; but the calculation has been carried on with an additional figure, that the final result here given might be absolutely correct even to the last decimal place.

374. Cor. Since the inscribed and circumscribed polygons are regular, and have the same number of sides, they are similar (Prop. I.); therefore, by increasing the number of the sides, the corresponding polygons formed will approach to an equality with the circle. Now if, by continual bisections, the polygons formed shall have their number of sides indefinitely great, each side will become indefinitely small, and the inscribed and circumscribed polygons will ultimately coincide with each other. But when they coincide with each other, they must each coincide with the circle, since no part of an inscribed polygon can be without the circle, nor can any part of a circumscribed one be within it; hence, the perimeters of the polygons must coincide with the circumference of the circle, and be equal to it.

375. Scholium. Every circle, therefore, may be regarded as a polygon of an infinite number of sides.

NOTE. — This new definition of the circle, if it does not appear at first view to be very strict, has at least the advantage of introducing more simplicity and precision into demonstrations. (Cours de Géométrie Élémentaire, par Vincent et Bourdon.)

376. The circumferences of circles are to each other as their radii, and their areas are to each other as the squares of their radii.

ence of the other circle, r its radius O B, A' its area; then will

and

C: CR:r,

A: A':: R2: 2.

Inscribe within the given circles two regular polygons of the same number of sides; and, whatever be the number of sides, the perimeters of the polygons will be to each other as the radii OA and OB (Prop. IX.). Now, conceive the arcs subtending the sides of the polygon to be continually bisected, forming other inscribed polygons, until polygons are formed of an indefinite number of sides, and therefore having perimeters coinciding with the circumference of the circumscribed circles (Prop. XII. Cor.); and we shall have

2

C: CR: r.

Again, the areas of the inscribed polygons are to each other as O A2 to O B2 (Prop. IX.). But when the number of sides of the polygons is indefinitely increased, the areas of the polygons become equal to the areas of the circles; hence we shall have

A: A': R2: r2.