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weled we required to be measured or laid down with he work, will be described among other instruments in a fitare chapter.

To measure the circumference of a given circle.

Following the method before employed in measuring #curved line (219) by means of a string or tape, the circumference of a circle which is accessible at every point may be measured. And this is the method most commonly employed, when practicable.

But it is so usual to consider a circle given, when its radius or diameter only is given, that it becomes necessary to measure circumferences of circles by determining what proportion the circumference bears to the radius; and as this proportion is proved to be the same for all circles, (98, Part 1.), that is, a constant quantity, the member which represents it is an important number in many mathematical calculations.

It was shewn in (98) that if C, c, represent the circumferences of any two different circles, and D, d, their diameters, Cc: D: d, or C: D: cd, that is, or the circumference of a circle bears an invari

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D d'

le ratio to its diameter, and therefore to its radius. This fixed number is commonly denoted by the Greek letter (read pi), the first letter of the Greek word periphery, or circumference.

с

So that if

D=", C=πxD;

the circumference of a circle is equal to its diameter linded by π.

Pat still the question remains, What is the numerical w? or, How many times is the diameter concircumference?

́ ́、 value, or number, is the same for every rious, that a single accurate measuresufficient to determine it. It might ht, that nothing can be more easy than tly constructed circle, and measure its ith a cord; and then measure the diaanner, and find the ratio of two measureill give the numerical value of π.

But, as in the case of the diagonal and side of a square before mentioned (228), so here also it is found, that the ratio of the circumference of a circle to its diameter cannot be exactly expressed in numbers. Each separately can be measured with perfect exactness; but both do not admit of being measured exactly by the same linear unit, however small that unit may be. This is another example of incommensurable lines.

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Practically, , or 3, is found to express for many purposes with sufficient exactness the value of π, the circular multiplier; is still nearer; and it will

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seldom be necessary to use a nearer approximate value than 3.1416.

Thus, for rough calculations,

circumference of circle=

22

x diameter;

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using whichsoever of the two is the most convenient. (See NOTE at the end of this section, p. 255).

N.B. Although the value of T cannot be expressed without some error; yet that error may be made as small as we please. For the value has been calculated to many places of decimals, and is found to be, up to 20 places, as follows:

3.14159265358979323846 &c.

If then for the value of π we use

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that is,

3.142..., it is plain that this differs from the true value by a quantity less than 01; that is, supposing the diameter to be 100 inches, the circumference, with this value of π, will be 3142... inches; but the true value is 314-15... inches; therefore the error in this circum

ference from using is less than one-tenth of an inch.

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(3) How many plots, each 11 yards square, can be obtained from 50 acres? Ans. 2000.

(4) The foot of a mast 120 ft. high is 25 ft. from the ship's side; find the length of a rope reaching from the side to a point in the mast at two-thirds of its height. Ans. 83-815...ft.

(5) A ten acre field is in the form of a square; find the cost of laying down a diagonal drain at 15d. per linear yard. Ans. £19. 8s. 10 d.

(6) The diagonal of a square board is 10 yards; what is the side of the square, and its area?

(1) Ans. 7071...yds. (2) Ans. 50 sq. yds.

(7) The sides of a rectangular plot are 108 ft. and 144 ft.; if the former dimension be shortened by 12 ft., how much must the latter be increased, so that the area may remain unaltered?

(8) In the preceding example, if the shortened by 12 ft., how much must the creased?

Ans. 18 ft.

longer side be other be inAns. 9 ft.

(9) Compute the lengths of the outer boundaries of each of the three rectangular plots in (7) and (8). Ans. 504; 516; 4997. (10) The sides of a triangle are 4, 5, and 6; alter the last dimension so that the triangle shall become rightangled. Ans. Add 403...to it.

(11) Out of a piece of metal, 15 inches square, as many circular portions as possible are cut, each 1 inch in diameter; how many will there be? Ans. 225.

(12) Find the areas of the trapeziums of which the dimensions are as follows:

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(13) The hypothenuse of a right-angled triangular field is 50 yds., and the other sides are in the ratio of 3 to 4; find its area, and cost at 80 guineas per acre.

(1) Ans. 600 sq. yds. (2) Ans. £10. 8s. 32d.

(14) Find the side of a square which cost £27. 1s. 6d. paving, at 8d. per square yard. Ans. 28 yds.

(15) How many square feet of flooring can be covered by a board whose length is 10 ft. 5 in. and the breadths of the two ends 24 ft. and 12 ft.?

Ans. 22 ft. 19 in. (16) Find the areas of the several lozenge-shaped parallelograms whose diagonals are as follows:

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(17) The diagonal of a square is 20 ft.; find the side approximately to 3 places of decimals. Ans. 14-142 ft.

(18) Two vessels sail from the same point, one due North at 9 knots per hour, and the other East at 11 knots; find how far they are apart in 12 hours.

Ans. 170:5512...miles.

(19) A ladder 40 feet high reaches to 3rds of the height of a building, when placed across a street 8 yds. wide; how much must the ladder be lengthened, so that it may reach the top of the building without changing its resting-place in the street? Ans. 13.665 ft.

(20) Prove that the rectangle, whose sides are 18 units and 8 units, has a longer diagonal than the square of the same area.

Note. The first two perpendiculars are upon the first diagonals.

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(1) Ans. 752.5. (2) Ans. 155-76. (3) Ans. 142·362. (26) The perimeter of a regular hexagon is 75 ft.; find the radius of the circumscribing circle.

Ans. 14 inches. (27) Find the areas corresponding to the following lengths and breadths of rectangular figures:

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(3) Ans. 61 sq. ft. 115 sq. in.

(4) Ans. 372 sq. ft. 231 sq. in.

(28) The following dimensions of rectangles are expressed in feet and decimal parts of a foot; or in inches and decimal parts of an inch. Find the areas.

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(2) Ans. 75 sq. ft. 32 sq. in.

(3) Ans. 095625 sq. in.

(4) Ans. 2 sq. ft. 23.2368 sq. in.

(29) Find the areas of the triangles, whereof one side and the perpendicular thereon from the vertex of the opposite angle are respectively as follows:

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