Miscellaneous Matter in Mensuration. 1. Purchased a box of window-glass containing 256 panes of glass, 8 by 10 inches; required the number of square feet, and the cost at 5c. per foot. Ans. 8x10=80 in. : 256 X 80=20480= 144=142.22 square feet, D7.11.1. 2. Bought a box of window-glass containing 320 square feet of glass 10 x 12 inches; required the number of panes. Ans. 10X12=120 : 320 x 144=46080--120=384 panes. 3. In a pile of wood 7.5 feet long, 61 feet high, and cut 4.75 feet, how many cords, and how much will it come to at D5 per cord ? Ans. 7.5 X 6.25 X 4.75=22265625-128=1.73950 cords x D5=8.69.75. 4. From a mahogany plank 26 inches broad, a yard and a half is to be sawn off; what distance from the end must the line be struck ? Ans. 6.23 feet. 5. A joist is 8 inches deep, 3.5 broad; what will be the dimensions of a scantling just as large again as the joist, that is 44 inches broad ? Ans. 12.52 inches deep. 6. How many 3-inch cubes can be cut out of a 12-inch cube? Ans. 64. h 7. A may-pole, whose top was broken off by a blast of wind, struck the ground at 15 feet distance from the bottom of the pole ; what was the height of the whole may-pole, supposing the length of the broken piece to be 39 feet? Ans. 75 feet. 8. The diameter of a circle is 25 rods ; required the length of a stone wall that will enclose it. Ans. 78.54 rods. 9. How many men may stand on one acre of land, allowing 12 square feet to each man? Ans. 3630. 10. How many solid yards of earth will be thrown out to make a cellar 48 feet long, 27 wide, and 6 deep? Ans. 288. 11. There is an island 50 miles in circumference, and three men start together to travel the same way about it; A. goes 7 miles per day, B. 8, and C. 9; when will they come together again? Thus, 50 X 7-+50X8+50 X 97+8+9= 50 days, time of meeting, and A. will travel 350 miles, B. 400, C. 450. Ans. 12. A general disposing his army into a square battalion, found he had 231 men remaining, but increasing each side with one soldier, he wanted 44 to fill the square ; of how many men did his army consist ? Thus: 231 +44=275 and 275-1;2=137, then 137 x 137+231=19000. Ans. Proof, 138x138=19044 : 719044=138, 13. Suppose a lighthouse built on the top of a rock; the distance between the place of observation and that part of the rock level with the eye is 620 yards; the distance from the top of the rock to the place of observation is 846 yards; and from the top of the lighthouse, 900 yards; the height of the lighthouse is required. 111 Ans. V900 x 900—620 x 620–846 X 846—620 x 620= 76.77 yards. 14. Suppose one of those meteors called fire-balls, to move parallel to the earth's surface, and 50 miles from it, at the rate of 20 miles per second : in what time would it move round the earth? Thus: the earth's diameter is 7964 miles, then 7964+50 X2 =8064=the diameter of the circle described by the ball, then 8064 x 3.1416=25333.8624 miles its circumference, and 25333.8624 -20=1266.69312 sec'ds=21 6 41 35 13 55 12 Ans. 15. The mean distances of the planets from the sun, in English miles, are as follows: Mercury 36686617.5, Venus 68552135.83, Earth 94772980 (or 95000000), Mars 144404783.33, Jupiter 492912533.33, Saturn 903957657.5; now, as a cannon-ball at its first discharge flies a mile in 8 seconds, and sound 1142 feet in a second, in what time, at the above rate, would a ball pass from the earth to the sun, and sound move from the sun to Saturn ? Thus : 94772980 X 8''=758183840=24 years, 15 days, 6 hours, 27 minutes, 20 seconds, for the passage of the ball. And 903957657.5 X 5280=4772896431600 feet, and 4772896431600 =1142=132 years, 192 days, 21 hours, 42 minutes, 21441 seconds, for the passing of sound from the sun to Saturn. Ans. 16. Light passes from the sun to the earth in 8.2 minutes ; in what tiine would it pass from the sun to the Georgium Sidus (Herschel), it being 1803930416.66 English miles? Ans. As 94772980 : 8.2 :: 1803930416.66 : 2 h. 36 m. 4" 50". 17. If a cubic foot of iron were hammered or drawn into a square bar, an inch about, that is, į of an inch square, required its length, supposing there is no waste of metal. Thus, 12 x 12 x12-25 X.25 X4=6912 inches=576 ft. Ans. 18. How many square feet of boards in a load consisting of 150, 124 feet in length, and 11 inches in width ? Thus : 12.5 X11=137=12=11.46 feet in one board ; 11.46 X 150=1719 square feet (nearly), Ans. 19. The solid content of a square stone is found to be 136 feet, its length is 91 feet; what is the area of one end? and if the breadth be 3 feet 11 inches, what is the depth ? 136.5 x 1728 2069.0526 Thus: = area 2069.0526 in. and 9.5 x 12 47 =44.022 inches. Ans. 20. How many perches of stone are there in a wall 48.25 feet in length, 1.5 in width, and 12 feet in height? Ans. 35.09+ perches. 21. Required the quantity of wood in a tier 64.5 feet in length, 4 feet in width, and 4 feet in height. Ans. 8.0625=815 cords. 22. In a pile of stone 25 feet in length, 121 feet in width, and 5! feet in height; required the cubical feet, the number of perches, and the value at 75c. per perch? Ans. 1718.75 cubical feet=69.444 perches, value D52.08.3+ 23. How much wood in a pile 141 feet in length, 61 feet high, and 31 feet in width; and value at D4) per cord ? Ans. 2.47802 cords ; D11.15.1090 value. 24. What is the breadth of a piece of cloth which is 36 yards long, and which contains 63 square yards? Ans. 63-36=1 yards. 25. How many yards of carpeting 11 yards wide will cover a floor 30 feet long and 221 feet broad? Thus ; 30 x 22.5=675 ft. ;-9=75 yds. -11=60 yds. Ans. 26. What is the side of a square piece of land containing 289 square rods? 27. If the diameter of the earth be 7930 miles, what is the circumference ? Ans. 24912.8. 28. How many miles does the earth move in revolving round the sun, supposing the orbit to be a circle whose diameter is 190 millions of miles ? Ans. 596902654 29. What is the circumference of a circle whose diameter is 769843 rods ? 30. If the circumference of the sun be 2800000 miles, what is the diameter ? Ans. 891267. 31. What is the diameter of a tree which is 5 feet round ? 32. If the circumference of the moon be 6850 miles, what is her diameter ? Ans. 2180 miles. 33. If the whole extent of the orbit of Saturn be 5650 millons of miles, how far is he from the sun ? 34. What is the area of a circle whose diameter is 623 feet : Ans. 304836 square feet. To measure stone in a well. In taking the diameter of the wall, measure from centre to contre, or to the clear; add the thickness of the wall; and when the diameter is taken, or obtained, find the circumference, which is the length of the wall. RULE. As 7 : is to 22 :: so is the diameter (taken): to the circumfer ence required; having found the length of the wall, and measured its thickness and depth, it is then prepared for calculation as explained above. 35. A well walled with stone 1 foot 2 inches thick, its diameter in the clear 2 feet 4 inches, and depth 50 feet; required the length of the wall, and the quantity of stone. As 7: 22 :: 3.5 : 11 feet circumference, or length; then 1 ft. 2 inches=1.1666+6 the thickness, 11 length. 50 depth. 1.1666+6 11 Is what the well measures in the clear (inside the walls); add the thickness of the wall, and you have the diameter of the wall or circle. Ans. 25.925+ perches. 36. Required the number of perches of stone in a well, whose diameter is 4 feet 9 inches in the clear, the depth of the well 20 feet, and the thickness of the wall 15 inches. Ans. 19.047 perches, oro1-9+ perches. 37. How many acres are there in a circular island whose diameter is 124 rods? Ans. 75 acres, 76 rods. 38. If the liameter of a circle be 113 and the circumference 355, what is the area ? Ans. 10029. 39. What is the diameter of a circle whose area is 380.1336 feet? Thus: 380.1336 =.7854=484 and 484=22. Ans. 40. What is the area of a square inscribed in a circle whoso area is 159 ? Thus : .7854 : 1 :: 159 : 101.22. Ans 41. What is the side of a square whose area is equal to that of a circle 452 feet in diameter ? Ans. ✓ (452)2 x.7854=400.574 42. What is the diameter of a circle which is equal to a squaro whose side is 36 feet ? Ans. ✓(36)2=0.7854=40.6217. 43. If the height of a square prism be 21 feet, and each side of the base 103, what is the solidity ? Ans. the area of the base 103 x 103=106] square feet; and the solid content = 1063 x2=240 cubic feet. 44. How many gallons ale are there in a cistern, which is 11 feet, 9 inches deep, and whose base is 4 feet, 2 inches square ? The cistern contains 352500 cubic inches-- 282=1250 gallons. Ans. 45. Suppose à cellar dug 16 feet in length, 16 in width, and 6 feet in depth ; required the quantity of stone to enclose a wall 1.5 feet thick, and 6 feet in height, in side of said cellar. Ans, 21 perches. (247 ft.=1 perch.) 46. A room is 11 feet high, 21 feet long, and 16 feet wide; how many cubic feet of space in it? Ans. 3696 cubic feet. 47. How many square feet in a stack of 15 boards, 12 feet, 8 inches in length, and 13 inches wide ? Ans. 2058=205 ft. 10 in. 48. Suppose a ship sails from latitude 43°, north, between north and east, till her departure from the meridian be 45 leagues, and the sum of the distance and difference of latitude be 135 leagues ; I demand her distance sailed ? and latitude come to ? Ans. 135x135 – 45 X 45 = 60 leagues, and 60x3=180 135 X 2 miles=3 degrees, the difference of latitude 135-60=75 leagues, the distance. Now, the vessel is sailing from the equator, and consequently the latitude is increasing; therefore, to the latitude sailed from 43o add the difference of latitude 3°, and the sum is the latitude come to, that is, 46 degrees. 49. Four men, E, F, G, and H, purchased a grindstone of 60 inches in diameter; how much of its diameter must each grind off, to have an equal share of the stone, if one first grind his share, and then another, till the stone is ground away, making no allowance for the eye. RULE. Divide the square of the diameter by the number of men, subtract the quotient from the square, and extract the square rooi of the remainder, which is the length of the diameter after |