36. A granite column is 50 feet high, and each end 8 feet in cir cumference. Required its solidity.

A. 254.65 nearly. trigonal ends, and side A. 32.9sq. ft. nearly.

37. A stick of timber is 19 feet long, with each 2 feet wide. Required its solidity. 38. The largest Egyptian pyramid has, according to Herodotus, an altitude of 800 feet, and a square base whose perimeter is 3200 feet. Required its content. A. 170,666,6663 ft.

39. The same author says the construction of the above pyramid occupied 100,000 men nearly 20 years. What then would it have Post at the rate of $1 per day for each man, (he boarding himself,) allowing 26 working days to the month? A. $936,000,000.

40. What is the solidity of a globe 12in. in diameter? A. 904,+ 41. The earth is about 25,000 miles in circumference Required its solidity.

A. 198,943,750sq. m. : 263,857,570,390 cubic miles, nearly. 42. The diameter of the moon is about 2,180 miles. What is its solidity? A. 5,424,617,475+ sq. miles. 43. A frustrum of a pyramid is 12 feet high, the base 9 feet square, and the other end 6 feet square. Required its solidity. (See XCIX. 69.) A. 684 s. ft. 44. If a round stick of timber be 30 feet long, and its extreme peripheries 4ft. 6in., and 3ft. 9 in., what is its solidity? The ratio of 4ft. 6in. to 3ft. 9in. is, or which is more convenient. (xcix. 76.) A. 40.0732+ s. ft.

45. One end of a rectangular frustrum is 60 feet by 40, the other end 40 feet by 30, and 120 feet in length. Required its solidity. A. 212,000 s. ft. nearly.

46. A stick of hewn timber is 45 feet long, and its ends are 24in. by 20, and 30in. by 24. Required its solidity. A. 186 s. ft. nearly. 47. If the base of a wedge be 30 by 8, and the length 60, what is its solidity? A. 7,200. 48. Allowing the earth's diameter to be 8,000 miles, what is the side of the largest cube that can be inscribed in it?

A. 4,618.8 miles nearly.

49. To find the solidity of any irregular body whose dimensions cannot be ascertained.-Immerse the solid in a regular vessel of water, and carefully note the difference between the height of the water before the inmersion, and afterwards; for the requisite dimensions, with which proceed according to previous rules.

50. A solid immersed in a vessel 18 inches square, raised the water 9in. Required the content of the given solid. A. 1.6875 s. ft. 51. A boy boasting of his knowledge of arithmetic, was asked by his father "If he had got so far that he could measure a brush heap?" Oh, certainly, says he, caly chop it up fine and throw it into the cider vat, and it is done. But, rejoined the father, suppose the vat is 6ft. square, and the cider is raised by the brush 2 inches, let us see, after all, if you can calculate the contents of the brush. A. 6 s. ft.

SIMILAR FIGURES.

52. When two figures vary in size, but are alike in shape or form they are called Similar Figures.

53. Similar Figures have the angles of the one equal to the cor. responding angles of the other, each to each. The sides opposite to equal angles are called homologous sides.

RULES OF PROPORTION.

54. The homologous sides of all similar figures are proportional. 55. All similar figures, whether they be triangles, quadrangles, or polygons, are in proportion to each other as the squares of their homologous sides.

56. The circumferences of circles are in proportion to each other as the radii or diameters of the circles. The same is true of the arcs and chords of similar segments.

57. Circles, or their areas are to each other as the squares of their radii, diameters or circumferences.

58. To find the area of a regular polygon, or any regular figure: Multiply the square of one of its sides by the area of a similar figure of which the side is a unit, as in the following:

59. Cubes, globes, and all similar solids are to each other as the

cubes of their similar dimensions. FIG. 20.

60. In figure 20 the perpendicular is B 40 rods, and the base 30 rods; what is the base of figure 21, the perpendicular being 28 rods; what is the hypothenuse of each figure, and what the sum of the areas of both? [See 54, 55.] A. BC =50rd.: a c=21rd. :bc=25rd. Areas 5A. 2R. 14sq. rd.

30

FIG. 21

Q. What are similar figures? 52. Wherein are they equal? 53. Which are the homologous sides? 53. Which sides are proportional? 54. What propor tion have similar and rectilineal figures, or their areas, to each other? 55. What proportion have the circumference of different circles? 56.

61. Wanting to know the height of the cathedral at York, I measured the length of its shadow, and found it to be 200 feet. At the same time a staff 5 feet long cast a shadow of 4 feet required the height of that elegant and magnificent structure :[See 55.] A. 250 feet.

B

62. Being desirous of finding the height of a steeple, I placed a looking glass at the distance of 100 feet from its base on the horizontal plane, and walking backwards 5 feet, I saw the top of the steeple appear in the centre of the glass; required the steeple's height, my eye being 5 feet 6 inches from the ground? [See 54.] A. 110ft.

B

63. When the sides of a figure are each 25 rods, what would be its area in square rods, if it were

64. There is a circle whose diameter is 6 inches, required the diameter of one two times as large ?-of one three times as large?-of one ten times as large? Ratios 2, 3, and 16; therefore, 6'x2=72; and ✓72 8.485in. the diameter of one two times as large.

A. Total, 37.85in. +. 65. There is a circle with a diameter of 12 inches,-what is the diameter of one only half as large ?- as large ?- as large ?-} as large as large ?- as large? A. Total, 44.938in. +

66. If 113.097 be the area of a given circle, what will be the area of one 4 times as large, and the area of one whose diameter is 4 times as large? (Retain 3 decimal figures.) A. 452.388: 1809.552. 67. If a ball 3 inches in diameter weigh 4lb., what will a ball of the same metal weigh, whose diameter is 6in? 33: 63: 4lb. A. 32lb.

·

Q. What, the area of circles? 57. What, cubes and ali similar figures 59. How is the area of a regular polygon found? 58.

68. There are two little globes, one of them is 1 inch in diameter and the other two inches; how many of the smaller globes will make one of the larger? A. 8 globes.

69. If the diameter of the planet Jupiter is 12 times as great as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? A. 1728 globes. 70. If the sun be 1,000,000 times as large as the earth, and the earth 8,000 miles in diameter, what is the diameter of the sun?

A. 800,000 miles

CXIII.

GAUGING.

1. GAUGING is the process of ascertaining the capacity of

any regular vessel, in bushels, gallons, &c.

2. The ale gallon contains 282 cubic inches.

3. The wine gallon contains 231 cubic inches.

4. The bushel contains 2,150.4 cubic inches.

5. A cubic foot of pure water weighs 1,000 ounces=62 pounds avoirdupois.

6. To find what weight of water may be put into a given vessel.— Multiply the cubic feet by 1000 for the ounces, or by 621 for the pounds, avoirdupois.

7. What weight of water can be put into a cistern 71⁄2 feet square ? A. 26,367lb. 3oz. 8. What weight of water will fill a circular fish pond that is 15 rods in circumference, and has an uniform depth of 4 feet?

A. 609T. 6cwt. 2qr. 4lb. 10oz. 12dr.

9. To find the number of gallons or bushels that a given vessel may contain.-Calculate the content in inches, which divide by 282 for the ale gallons; by 231 for the wine gallons, and by 2,150.4 for the bushels.

10. How many barrels of ale will a vat 8 feet square hold? A. 87bl. 55gal.

11. What will the oats cost at 62 cents a bushel, that will fill 25 bins, 12 of which are cylindrical, being 18 feet in circumference, and 7 feet deep; and the rest 7 feet square? A. $3327.18+

12. A cellar 20 feet long, 15 feet wide, and 8 feet deep, became, during a heavy rain, filled with water. What would be the expense, when labor is $14 per day of 8 hours, of removing the water, allowing that one man can empty three buckets, in 2 minutes, each bucket to hold 23 gallons, (wine measure)? A. $11688+.

13. To find the number of gallons in a cask, or to gauge it:CXIII. Q. What is Gauging? 1. What is the number of cubic inches in an ale or wine gallon? 2, 3. In a bushel? 4. What is the rule for finding what weight of water may be put into a given vessel? 6. What weight of water may be put into a vessel 1 foot square? Into one 2 feet square? Into one 3 feet deep and 2 feet square at each end? What is the rule for ascertaining the capacity of vessels? 9. 24

RULE.

14 Take the dimensions in inches, viz., the diameter of the bung and head, and the length of the cask, and find the difference between the bung and the head diameter.

15. If the staves of the cask be much curved between the bung and the head, multiply the difference (found above) by .7; if not quite so much curved, by .65; if they bulge yet less, by .6; and if they are almost straight, by .55; add the product to the head diameter; the sum will be the mean diameter, by which the cask is reduced to a cylinder. 16. Square the mean diameter thus found, then multiply it by the length; divide the product by 359 for ale or beer gallons, and by 294 for wine.

17. There is a certain cask, whose bung diameter is 35 inches; head diameter 27 inches; and length 45 inches. Required its capacity in ale and wine gallons.

18. Thus, 35-27-8x.7-5.6+27=32.6×32.6=1062.76×45 =47824.20÷359 and 294— A. 133.21gal. ale, and 162.66gal. wine. 19. What is the content of a cask in wine and ale gallons, whose bung diameter is 36 inches, head diameter 30 inches, and length 48 inches? A. 153.65+ale gal.; 187.62+wine gal.

20. To find the capacity of a vessel, which is in the form of a lower frustrum of a cone, that is round, and larger at one end than at the other.

RULE.

21. To the product of the diameters add of the square of their difference; which result multiply by the height, and divide as above directed.

22. What is the capacity both in wine and ale gallons, of a tub 40 inches in diameter at the top, 32 inches at the bottom, and its per pendicular height 48 inches?

A. 174 ale gal. nearly+; 212.46+ wine gal.

TONNAGE OF VESSELS.

CXIV. 1. SHIP CARPENTERS' RULE.

For single-decked vessels, multiply the breadth of the main beam, the depth of the hold, and the length together, and divide the product by 95; the quotient will be tons. For double-decked vessels, take one half of the breadth of the main beam for the depth of the hold, with which proceed as before.

2. A single-decked vessel is 80 feet long, 25 feet broad, and 12 feet deep. Required its tonnage. A. 2521 tons.

3. Required the tonnage of a double-decked vessel, whose length is 80 feet, and breadth 26 feet. A. 284. 4. GOVERNMENT RULE. "If the vessel be double-decked, take the length thereof from the fore part of the main stern, to the after part

Q. What is the rule for ascertaining the capacity of casks? 14, 15, 16. CXIV. Q. What is the rule employed by ship carpenters in estimating .ba .onnage of vessels? What is the government rule? 4.