37. Scholium. ARCHIMEDES (born 287 B.c.) was the first to assign an approximate value of . Вy a method similar to that above, he proved that its value is between 34 and 348, or, in decimals, between 3.1428 and 3.1408; he therefore assigned its value correctly within a unit of the third decimal place. The number 34, or 27, usually cited as Archimedes' value of (although it is but one of the two limits assigned by him), is often used as a sufficient approximation in rough computations.

METIUS (A.D. 1640) found the much more accurate value 55, which correctly represents even the sixth decimal place. It is easily remembered by observing that the denominator and numerator written consecutively, thus, 113|355, present the first three odd numbers each written twice.

More recently, the value has been found to a very great number of decimals, by the aid of series demonstrated by the Differential Calculus. CLAUSEN and DASE, of Germany (about a.d. 1846), computing independently of each other, carried out the value to two hundred decimal places, and their results agree to the last figure. The mutual verification thus obtained stamps their results as thus far the best established value to the two-hundredth place. (See SCHUMACHER'S Astronomische Nachrichten, No. 589.) Other computers have carried the value to over five hundred places, but it does not appear that their results have been verified. The value to fifteen decimal places is

π 3.141592653589793.

For the greater number of practical applications, the value π3.1416 is sufficiently accurate.

EXERCISES ON BOOK V.

THEOREMS.

1. An equilateral polygon inscribed in a circle is regular.

2. An equilateral polygon circumscribed about a circle is regular if the number of its sides is odd.

3. An equiangular polygon inscribed in a circle is regular if the number of its sides is odd.

4. An equiangular polygon circumscribed about a circle is regular.

5. The area of the regular inscribed triangle is one-half the area of the regular inscribed hexagon.

6. The area of the regular inscribed hexagon is three-fourths of that of the regular circumscribed hexagon.

7. The area of the regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.

8. A plane surface may be entirely covered (as in the construction of a pavement) by equal regular polygons of either three, four, or six sides.

9. A plane surface may be entirely covered by a combination of squares and regular octagons having the same side, or by dodecagons and equilateral triangles having the same side.

10. If squares be described on the sides of a regular hexagon, and their adjacent external vertices be joined, a regular dodecagon will be formed.

11. The diagonals of a regular pentagon form a regular pentagon. 12. The diagonals joining alternate vertices of a regular hexagon enclose a regular hexagon one-third as large as the original hexagon.

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13. The area of the regular inscribed octagon is equal to the product of the side of the inscribed square by the diameter. Suggestion. A quarter of the octagon is the sum of two triangles having as a common base the side of the inscribed square, and having the radius as the sum of their altitudes.

15. Prove the correctness of the following construction:

If AB and CD are two perpendicular diameters in a circle, and E the middle point of the radius OC, and if EF is taken equal to EA, then OF is equal to the side of the regular inscribed decagon, and AF is equal to the side of the regular inscribed pentagon. (v. III., 42.)

16. From any point within a regular polygon of n sides, perpendiculars are drawn to the several sides; prove that the sum of these perpendiculars is equal to n times the apothem.

Suggestion. Join the point with the vertices of the polygon, and obtain an expression for the area in terms of the perpendiculars: then see Proposition IV.

17. The side of the regular inscribed triangle is equal to the hypotenuse of a right triangle of which the sides of the inscribed square and of the regular inscribed hexagon are the sides. (v. IV., Proposition X.)

18. If a is the side of a regular decagon inscribed in a circle whose radius is R,

19. If a the side of a regular polygon inscribed in a circle whose radius is R, and a' = = the side of the

regular inscribed polygon of double the number of sides, then

a'

B

a

Hence

α AD

a2

and

; but AD

a'

R

D

20. If a = the side of a regular pentagon inscribed in a circle whose radius is R, then

21. If a = the side of a regular octagon inscribed in a circle whose radius is R, then

a = RV2 — √2.

22. If a = the side of a regular dodecagon inscribed in a circle whose radius is R, then

a = RV2√3.

23. The side of the regular inscribed pentagon is equal to the hypotenuse of a right triangle whose sides are the radius and the side of the regular inscribed decagon.

24. The area of a ring bounded by two concentric circumferences is equal to the area of a circle having for its diameter a chord of the outer circumference tangent to the inner circumfer

ence.

25. If on the legs of a right triangle, as diameters, semicircles are described external to the triangle, and from the whole figure a semicircle on the hypotenuse is subtracted, the remainder is equivalent to the given triangle.

26. If on the two segments into which a diameter of a given circle is divided by any point, as diameters, semi-circumferences are described lying on opposite sides of the given diameter, the sum of their lengths is equal to the length of a semicircumference of the given circle, and a line which they form divides the circle into two parts whose areas are to each other as the segments of the given diameter.

27. If a diameter of a given circle is divided into n equal parts, and through each point of division a curved line of the sort described in the last problem is drawn, these lines will divide the circle into n equivalent parts.

28. If a circle rolls around the circumference of a circle of twice its radius, the two circles being always tangent internally, the locus of a fixed point on the circumference of the rolling circle is a diameter of the fixed circle.

29. If a given square is subdivided into n2 equal squares, n being any given number, and in each of these smaller squares a circle is inscribed, the sum of their areas is equal to the area of the circle inscribed in the original square.

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