By a continued application of these equations, we find the areas indicated in the following

Now, the areas of the last two polygons differ from each other by less than the millionth part of a unit, but the area of the circle differs from either by less than they differ from each other; hence, the value of the area of either will differ from that of the circle by less than a millionth part of a unit. Taking the fig ures as far as they agree, and denoting the number of units in the required area by л, we have, approximately,

π= 3.141592;

that is, the area of a circle whose radius is 1, is 3 141592.

Scholium. For practical computation, the value of π is taken equal to 3.1416.

The third method is taken from "Chauvenet's Elementary Geometry," and gives the approximate ratio of the circumference to the diameter, by two different methods, to 8192 sides each. The first is called the Method of Perimeters, and the second the Method of Isoperimeters. The above approximate ratio is followed by a chapter on the Doctrine of Limits taken from the same author.

PROPOSITION X.-PROBLEM.

25. Given the perimeters of a regular inscribed and a similar circumscribed polygon, to compute the perimeters of the regular inscribed and circumscribed polygons of double the number of sides.

Let AB be a side of the given inscribed polygon, CD a side of the similar circumscribed polygon, tangent to the arc AB at its middle point E. Join AE, and at A and B draw the tangents AF and BG; then AE is a side of the regular inscribed polygon of double the number of sides, and FG is a side of the circumscribed polygon of double the number of sides (4).

с

F

E

G

D

N

A

H

B

Denote the perimeters of the given inscribed and circumscribed polygons by pand Prespectively; and the perimeters of the required inscribed and circumscribed polygons of double the number of sides by p' and P' respectively. Since OC is the radius of the circle circumscribed about the polygon whose perimeter is P, we have (10),

and since OF bisects the angle COE, we have (III. 21),

Now FG is a side of the polygon whose perimeter is P', and is contained as many times in P' as CE is contained in P, hence (III. 9),

Again, the right triangles AEH and EFN are similar, since their acute angles EAH and FEN are equal, and give

Therefore, from the given perimeters p and P, we compute P' by the equation [1], and then with p and P' we compute p' by the equation [2].

26. Definition. Two polygons are isoperimetric when their perimeters are equal.

PROPOSITION XI.-PROBLEM.

27. Given the radius and apothem of a regular polygon, to compute the radius and apoihem of the isoperimetric polygon of double the number of sides.

Let AB be a side of the given regular polygon, O the center of this polygon, OA its radius, OD its apothem. Produce DO to meet the circumference of the circumscribed circle in ơ; join O'A, O'B; let fall OA' perpendicular to O'A, and through A' draw A'B' parallel to AB.

Since the new polygon is to have twice as many sides as the given polygon, the angle at its center must be one-half the angle AOB; therefore the angle AO'B, which is equal to one-half of AOB (II. 57), is equal to the angle at the center of the new polygon.

10

D'

B

Since the perimeter of the new polygon is to be equal to that of the given polygon, but is to be divided into twice as many sides, each of its sides must be equal to one-half of AB; therefore A'B', which is equal to one-half of AB (I. 121), is a side of the new polygon; O'A' is its radius, and O'D' its apothem. If, then, we denote the given radius OA by R, and the given apothem OD by

r, the required radius O'A' by R', and the apothem O'D' by ', we have

therefore, is an arithmetic mean between R and r, and R' is a geometric mean between R and r'.

MEASUREMENT OF THE CIRCLE.

The principle which we employed in the comparison of incommensurable ratios (II. 49) is fundamentally the same as that which we are about to apply to the measurement of the circle, but we shall now state it in a much more general form, better adapted for subsequent application.

28. Definitions. I. A variable quantity, or simply, a variable, is a quantity which has different successive values.

II. When the successive values of a variable, under the conditions imposed upon it, approach more and more nearly to the value of some fixed or constant quantity, so that the difference between the variable and the constant may become less than any assigned quantity, without becoming zero, the variable is said to approach indefinitely to the constant; and the constant is called the limit of the variable.

Or, more briefly, the limit of a variable is a constant quantity to which the variable, under the conditions imposed upon it, approaches indefinitely.

As an example, illustrating these definitions, let a point be required to move from A to B under the following conditions: it

shall first move over one-half of AB, that is to C;

then over one-half of CB, to C; then over one- A

half of CB, to C"; and so on indefinitely; then

the distance of the point from A is a variable, and this variable approaches indefinitely to the constant AB, as its limit, without ever reaching it.

As a second example, let A denote the angle of any regular polygon, and n the number of sides of the polygon; then, a right angle being taken as the unit, we have (8),

The value of A is a variable depending upon n; and since n may be taken so

4

great that shall be less than any assigned quantity however small, the value

n

of A approaches to two right angles as its limit, but evidently never reaches that limit.

29. PRINCIPLE OF LIMITS. Theorem. If two variable quantities are always equal to each other and each approaches to a limit, the two limits are necessarily equal. For, two variables always equal to each other present in fact but one value, and it is evidently impossible that one variable value shall at the same time approach indefinitely to two unequal limits.

30. Theorem. The limit of the product of two variables is the product of their limits. Thus, if x approaches indefinitely to the limit a, and y approaches indefinitely to the limit b, the product xy must approach indefinitely to the product ab; that is, the limit of the product xy is the product ab of the limits of x and y.

31. Theorem. If two variables are in a constant ratio and each approaches to a limit, these limits are in the same constant ratio.

Let x and y be two variables in the constant ratio m, that is, let x = my; and let their limits be a and b respectively. Since y approaches indefinitely to b, my approaches indefinitely to mb; therefore we have x and my, two variables, always equal to each other, whose limits are a and mb, respectively, whence, by (29), a = mb; that is, a and b are in the constant ratio m.

PROPOSITION XVI.—THEOREM.

42. The area of a circle is equal to half the product of its circumference by its radius.

Let the area of any regular polygon circumscribed about the circle be denoted by A, its perimeter by P, and its apothem which is equal to the radius of the circle by R; then (22),

A

D

B

Let the number of the sides of the polygon be continually doubled, then A approaches the area S of the circle as its limit, and P approaches the circumference C as its limit; but A and Pare in the constant ratio† R; therefore their limits are in the same ratio (31), and we have

43. Corallary I. The area of a circle is equal to the square of its radius multiplied by the constant number. For, substituting for C its value 2πR in [1], we have

S = R2.

44. Corollary II. The area of a sector is equal to half the product of its arc by the radius. For, denote the arc ab of the sector a O b by c, and the area of the sector by s; then, since c and s are like parts of Cand S, we have (III. 9),