502. To compute the ratio of the circumference of a circl to its diameter approximately.

TO FIND the ratio of its circumference to its diameter approximate ly, or the value of π.

Since the ratio is the same for all circles (§ 491), it is sufficient to compute it for any one.

We select a circle of which the diameter is unity.

The radius of this circle will be and the side of a regu lar inscribed hexagon will be ; and of a circumscribed square I.

Using the formula x= V

2

($500), we form the

following table giving the length of the scribed polygons of 6, 12, 24, etc., sides. perimeter is obtained by multiplying the by the number of sides.

sides of regular inThe length of the length of one side

following table giving the length of the sides and perimeters of regular circumscribed polygons of 4, 8, 16, etc., sides.

But the length of the circumference must be intermediate between the lengths of the circumscribed and inscribed poly

gons. Hence it must be intermediate between 3.141558 and 3.141632. Hence 3.1416 is the nearest approximation to four decimal places.

Since the diameter of the circle is 1, the ratio of the cir

503. Exercise.-By means of the value of

just found

and the formulas for the circumference and area of a circle, find the circumference and area of a circle whose radius is 23.16 inches.

355
113

*The earliest known attempt to obtain the area of the circle or to " square the circle" is recorded in a MS. in the British Museum recently deciphered. It was written by an Egyptian priest, Ahmes, at least as early as 1700 B.C., and possibly several centuries earlier. The method was to deduct from the diameter of the circle one-ninth of itself and square the remainder. This is equivalent to using a value of equal to 3.16. Archimedes (about 250 B.C.), the greatest mathematician of ancient times, proved, by methods essentially the same as those employed in the text, that the true value of π lies between 34 and 340, i. e., between 3.1429 and 3.1408. Ptolemy (about 150 B.C.) used the value 3.1417. In the 16th century Metrus, of Holland, using polygons up to 1536 sides, obtained the easily-remembered approximation (write 113355 and divide last three by first three), which is correct to six places of decimals. Romanus, also of Holland, using polygons of 1,073,741,324 sides, soon after computed sixteen places. With the better methods of higher mathematics various mathematicians have extended the computations gradually, until Mr. Shanks, in 1873, published a result to 707 places, the first 411 of which have been verified by Dr. Rutherford. The following are the first figures of his result. #=3.141,592,653.589,793,238,462,643, 383, 279, 502,884, 197, 169, 399, 375. 105,8. How accurate a value this is may be inferred from Prof. Newcomb's remark that ten decimals would be sufficient to calculate the circumference of the earth to a fraction of an inch if we had an exact knowledge of the diameter.

The Greeks sought in vain for a perfectly accurate result or geometrical construction for obtaining a square equivalent to the circle, as did many medieval mathematicians. "Circle squarers" still exist among the ignorant, although Lambert (about A.D. 1750) proved π incommensurable, i. e., inexpressible as a finite fraction, and Lindemann, in 1882, proved it is also transcendental, i.e., inexpressible as a radical or root of any algebraic equation with integral coefficients.

PROBLEMS OF DEMONSTRATION

504. The angle at the centre of a regular polygon is the supplement of any angle of the polygon.

505. If the sides of a regular circumscribed polygon are tangent to the circle at the vertices of the similar inscribed polygon, then each vertex of the circumscribed figure lies in the prolongation of the apothem of the inscribed.

506. If the sides of a regular circumscribed polygon are tangent to the circle at the middle points of the arcs subtended by the sides of a similar inscribed polygon, then the sides of the circumscribed figure are parallel to those of the inscribed, and the vertices lie in the prolongation of the radii.

507. If from any point within a regular polygon of n sides perpendiculars are drawn to the several sides, the sum of these perpendiculars is equal to ʼn times the apothem.

Hint.-Apply § 495.

508. The area of a circumscribed square is double that of an inscribed square.

509. The side of an inscribed equilateral triangle is equal to one-half the side of a circumscribed equilateral triangle, and the area of the first is one-fourth that of the second.

510. The apothem of an inscribed equilateral triangle is equal to half the radius.

511. The apothem of a regular inscribed hexagon is equal to half the side of the inscribed equilateral triangle.

512. The radius of a regular inscribed polygon is a mean proportional between its apothem and the radius of the similar regular circumscribed polygon.

513. The area of the ring included between two concentric circles is equal to that of a circle whose radius is one half a chord of the outer circle drawn tangent to the inner.

514. In two circles of different radii, angles at the centre subtended by arcs of equal length are to each other inversely as their radii.

515. Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.

M

Hint.-Prove the triangles ABC and BCM similar (§ 275). Then prove AM=AB=BC (§ 77), and substitute in the proportion derived from the first step.

PROBLEMS OF CONSTRUCTION

516. Having given a circle, to construct the circumscribed hexagon, octagon, and decagon.

517. Upon a given straight line as a side to construct a regular hexagon.

518. Having given a circle and its centre, to find two opposite points in the circumference by means of compasses only.

519. To divide a right angle into five equal parts. 520. To inscribe a square in a given quadrant.

521. Having given two circles, to construct a third circle equivalent to their difference.

522. To divide a circle into any number of equivalent parts by circumferences concentric with it.