5. To find the circumference of a circle, multiply the diameter by 3-1416. Or as 7 is to 22, so is the diameter to the circumference; or as 113 is to 355 so is the diameter to the circumference.

Not?. The ratio of the diameter of a circle to its circumference has never yet been exactly det rmined in numbers; but any of the above methods will give a result sufficiently correct for all practical purposes.

6. To find the area of a circle, multiply the square of the diameter by 7854; or multiply half the circumference by half the diameter; or multiply the circumference and diameter together, and divide the product by 4.

7. To find the area of an illipsis, multiply the product of the two axis by 7854.

8. To find the surface of a sphere, multiply the square of its diameter by 3·1416.

859. The calculation of the solid contents of figures.

9. To find the contents of a rectangular solid, multiply its length, breadth, and thickness into each other.

10. To find the solid contents of a cylinder, multiply the area of its base by its height.

11. To find the solid content of a pyramid or cone, multiply the area of its base by of its height.

12. To find the solid content of the frustum of a pyramid or cone add the area of the two ends, and the square root of their product, together, and multiply the sum by the perpendicular.

13. To find the solid content of a sphere, multiply the cube of its diameter by 5236.

14. The solid content of round timber is found by multiply. ing the square of of the mean circumference, or as it is termed the quarter girt by the length.

15. The solid contents of squared timber are found by multiplying the mean breadth by the mean thickness, and this product by the length.

860. EXAMPLES,

1. What are the contents of a rectangular field measuring 1640 links by 720 links ?

2. What is the area of a field in the form of a regular rhomboid, one side of which is 2185 links, and the perpendi cular 1236 links?

3. What is the side of a square, the area of which is 132496 ?

4. The base of the largest Egyptian pyramid is square, and the side is said to measure 693 feet. How much ground does it cover?

5. How many yards of carpet, yard wide, will cover a rectangular room floor, which measures 28 feet 6 inches, by 18 feet 9 inches?

6. What is the area of a triangular field, the base of which measures 3568 links, the perpendicular 1689, and the distance between the right hand end of the base and the place of the perpendicular 1490 links?

7. What is the area of a triangular field, the sides of which severally measure 2486, 2340, and 2052 links?

8. In the right angled triangle A, B, C, are given, the base, AB 38, and the perpendicular, B C = 30, to fine the hypothenuse, A. C.

9. The distance from the ridge to the eaves of a house being 16 feet, and the perpendicular height of the gable 9 feet, what is the breadth of the building?

10. A ladder 50 feet in length, being placed in a street, reached a window 30 feet from the ground, and by turning it over, without removing the bottom, it reached another window, on the other side of the street, 35 feet from the ground, what was the breadth of the street?

11. The parallel sides of a piece of ground measure 1021, and 824 links, and their perpendicular distance is 836 links, what is the area?

12. Required the area of a pentangular field, the first side of which measures 926, the second 536, the third 835, the fourth 628, and the fifth 587 links? The diagonal from the first angle to the third measures 1194, and that from the third to the fifth 1223 links.

13. Required the circumference of a circle, the diameter of of which is 14.

14. Find the area of a circle, the diameter of which is 106, and the circumference 333 feet.

15. How many square yards are there in the floor of a circular building, the diameter of which is 73 feet 3 inches?

16. The diameters of an elliptical piece of ground are 330 and 220 feet, what is the area?

17. A ceiling measures 43 feet 10 inches in length, and 25 feet 6 inches in breadth, how many square yards does it

contain?

18. A plate of glass measures 2 feet 8 inches, by 1 foot 6 inches, what will 12 such plates cost at 14s. 6d. per square foot?

19. A block of marble measures 10 feet long, 4 feet 4

inches broad, and 2 feet 1 inch thick, how many cubic feet does it contain?

20. What would be the cost of sinking a well 45 feet deep, and 3 feet 9 inches diameter, at 2s. 6d. the cubic yard?

21. The altitude of a square pyramid is 15 feet 6 inches, and the side of its base 5 feet 9 inches, what is its solidity?

22. What is the solid content of a piece of timber, the length of which is 24 feet 3 inches, and the mean breadth and thickness, each 22 inches?

23. What is the solid content of an oak tree, its length being 22 feet, and its mean girt 6 feet 2 inches?

24. What is the expense of digging out a foundation, the length of which is 45 feet 6 inches, the breadth 26 feet 9 inches, and the depth 10 feet, at 2s. 6d. the cubic yard?

25. What is the expense of lining a rectangular cistern with lead at 34d. per pound, the lead weighing 6lbs. per square foot, the length of the cistern being six feet 4 inches, its breadth 4 feet 2 inches, and its depth 4 feet 6 inches; and how many imperial gallons will such a cistern contain, there being in a gallon 277-274 cubic inches?

861. QUESTIONS FOR EXAMINATION

UNDER CHAPTER 32.

1. How is the area of a rectangular surface found? 2. How is the area of a parallelogram found?

3. How is the area of a triangle found?

4. How is the area of a polygon, that is, of a figure having more sides than four, found?

5. How is the area of a circle found?

6. How is the area of an ellipsis found?

7. How is the surface of a sphere found?

8. How are the contents of a rectangular solid found?

9. How are the solid contents of a cylinder found?

10. How are the solid contents of a pyramid or cone found? 11. How are the solid contents of the frustum of a pyramid or cone found?

12. How are the solid contents of a sphere found?

13. How are the solid contents of round timber found? 14. How are the solid contents of squared timber found?

335

CHAPTER XXXIII.

QUADRATIC EQUATIONS.

862. If, when all the unknown quantities except one are exterminated from an equation, both that unknown quantity and its square should be found in it, such an Equation is called a Quadratic Equation, and may be resolved by the following Rule.

RULE.

863. Having brought the terms involving the unknown quantity to one side of the equation by themselves, divide both sides of it by the coefficient of the square of the unknown quantity, if it have one; then add to both sides of it the square of half the coefficient of the unknown quantity, which will complete the square of the side containing the unknown quantity; after which extract the square root of both sides, and the equation will be reduced to a simple one, which may then be resolved by the rule given in paragraphs 496 to 505.

Ex. Given 2x2 8x90 to find x.

By dividing by the coeffi

4x

cient of x2x2 By completing the square r2 By extracting the square

root

By transposing

a

45

X

As the square root of x2 - 2 a x + a2 is either x → a or x, the root of the known side of the equation may have both and before it. Sometimes both these signs give the proper solution; at other times only one of them.

43 1.633468 44 1.643453 45 1-653213

93 1.968483

143 2-155336

95 1.977724

145 2.161368

193 2.285557 194 2.287802 195 2.290035

46

1.662758 96 1.982271 146 2.164353 196 2.292256 47 1.672098 97 1.986772 147 2-167317 197 2.294466 48 1-681241 98 1.991226 148 2-170262 198 2.296665 49 1.690196 99 1.995635 149 2.173186 199 2.298353 50 1.698970 100 2.000000 150 2.176091 200 2.301030

FINIS.

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