EXAMPLE. Ir a man weighing 150. rest on the end of a lever 12 feet long; what weight will he balance on the other end, supposing the prop 11⁄2 foot from the weight? 12 feet the Lever. 1,5 distance of the weight from the fulcrum. 10,5 distance from the Fulcrum to the man, Therefore, As 1,5 10,5 : 150: 1050 Ans. 2. Of the Wheel and Axle. As the diameter of the axle is to the diameter of the wheel: so is the pow er applied to the wheel, to the weight suspended by the axle, EXAMPLES. 1. A MECHANIC wishes to make a windlass in such a manner, as that 1lb. applied to the wheel, should be equal to 12 suspended from the axle ; now, supposing the axle 4 inches diameter, required the diameter of the wheel? lb. in. lb. As 1: 4 :: 12:148 4ns. or diameter of the wheel. 48 2. SUPPOSE the diameter of the axle inches and that of the wheel 60 inches; what power at the wheel will balance 10lb. at the axle ? 3. Of the Screw. Ans. llb. THE power is to the weight to be raised as the distance between two threads of the screw is to the circumference of a circle described by the power applied at the end of the lever. NOTE. 1. To find the circumference of the circle described by the end of the lever, multiply the double of the lever by 3,14159 the product will be the cir cumference. NOTE. 2. IT is usual to abate of the effect of the machine for friction. EXAMPLE. THERE is a screw whose threads are an inch asunder; the lever by which it is turned is 36 inches long, and the weight to be raised a ton, or 2240lb. What power or force must be applied to the end of the lever sufficient to tura the screw, that is, to raise the weight? The lever 36×2723,1419325194+the circumference. circumf. in. lb. 78. Then, as 216,194: 1: 22409909 Problems. 1. THE diameter of a circle being given to find the circumference, multiply the diameter by 3,14159; the product will be the circumference. 2. To find the area of a circle, the diameter being given; multiply the of the diameter by 785398; the product is the area. square 3. To measure the solidity of any irregular body whose dimensions cannot be taken; put the body into some regular vessel and fill it with water, then taking out the body, measure the fall of water in the vessel; if the vessel be square, multiply the side by itself, and that product by the fail of water which gives the solid content of the irregular body. SECTION IV. Miscellaneous Questions. In this Section there is nothing new to be proposed to the Scholar. Enough of Arithmetic has been taught him for all ordinary occurrences in life. It only remains to lead him into some reflections on the foregoing rules. For this purpose the following questions are subjoined. They are left without answers, that the Scholars's only resource of knowledge for working them should be in his own mind. Masters having wrought out these questions at a leisure hour, may transcribe them with their answers into a manuscript for their private use, to which on any occasion, without trouble or hindrance,they may readily advert to satisfy the enquiries of their pupils. 1. THE Northern Lights were first observed in London in 1560; how ma my years since ? 2. WHAT number, multiplied by 43 produces 88150? 3. Ir a cannon may be discharged twice with 6/b. of powder, how many times will 7C. 3qrs. 17b, discharge the same piece? 4. REDUCE 14 guineas and £75 138. 63d. to Federal Money. 5. What is the interest of Dolls. 79,49 one year, and five months? 6. A OWED B Dolls. 317,19, for which he gave his note, on interest, bearing date July 12th, 1797. On the back of the note are these several endorsements, viz. Oct. 17th, 1797, Received in cash, Dolls. 61, 10. March 20th, 1798, Received 17 Cwt. of Beef, at Dolls. 4, 33 per: Cwt WHAT was there due from A to B, of principal and interest, Sept. 18th 7. WHAT Cost 13 yards of flannel at 188 per yard? 8. WHAT must I give for 3 Cwt. 2 grs. 13 lb. of cheese, at 7 cents per lb 2 9. WHAT Will 35 yards of broadcloth cost at 23s6 per yard? 10. WHAT will be the cost of a line of veal, weighing 1636, at 23d per lb? 11. WHAT Will 874 lb. of tallow cost, at 94 d. per lb? 12. WHAT Will 196 yards of tape cost, at 3 farthings per yard ? 13. WHAT Will 56 bushels of oats cost, at 28. 31d. per bushel'? 14. AT £3 7s. 6d. per cwt. for sugar, what is that per lb. 15. How much in length of a board that is 10 inches wide will it require to make a square foot? 16. How many square feet in a board 1 foot, 3 inches wide, and 14 feet, 9 inches long? 17. How much wood in a load 9 feet long, 34 feet wide, and 2 feet 9 inches high? 18. Ar Dolls. 1, 33 per yard for cloth, what must I give for 72 yards? 23. Ar Dolls. 10,76 per cwt. for sugar, what is that per lb. ? 24. WHAT will be a man's wages for 6 months, at 43 cents per day, working 5-days per week? 25. WHAT must I give for pasturing my horse 19 weeks, at 33 cents per week? 26. How many revolutions does the Moon perform in 144 years, 2 days, 10 hours; one revolution being in 27 days, 7 h. 43 m? 27. WHAT will 7 pieces of cloth, containing 27 yards each, come to, at 15. 4 d. per yard? 28. A MAN spends 23 dollars 69 cents, 5 mills, in a year, what is that per day? 29. SUPPOSE the Legislature of this state should grant a tax of 7 cents 3 mills on a dollar, what will a man's tax be, who is 142 dollars 40 cents on the list? 30. A BANKRUPT, whose effects are 3948 dollars, can pay his creditors but 28 cents, 5 mills on a dollar : what does he owe? 31. SUPPOSE a cistern having a pipe that conveys 4 gallons, 2 qts. into it in an hour, has another that lets out 2 gallons, 1 qt. 1 pt. in an hour: if the cistern contains 84 gallons, in what time will it be filled? 32. IF 80 dollars worth of provisions will serve 20 men 25 days, what number of men will the same provision serve 10 days? 33. Ir 6 men spend 16 dollars, 7 cents, in 40 days; how long will 135 men be in spending 100 dollars? 34. A BRIDGE built across a river in 6 months, by 45 men, was washed away by the current; required the number of workmen sufficient to build another of twice as much worth in 4 months? 35. FOUR men, A, B, C, and D found a purse of money containing 12 dollars, they agree that A shall have one third, B one fourth, C one sixth, and D one eighth of it; what must each one have according to this agreement? 36. A CERTAIN usurer lent out 90l. for twelve months, and received principal and interest 957. 88. I demand at what rate per cent. he received interest? 37. IF a gentleman have an estate of 1000l. per ann. how much may he spend per day to lay up three score guineas at the year's end? 38. WHAT is the length of a road, which being 33 feet wide contains an acre? 39. REQUIRED a number from which if 7 be subtracted and the remainder be divided by 8, and the quotient be multiplied by 5, and 4 added to the product, the square root of the sum extracted, and three fourths of that root cubed, the cube divided by 9, the last quotient may be 24 ? 40. If a quarter of wheat affords 60 ten penny loaves, how many eight penny loaves may be obtained from it? 41. If the carriage of 7 cwt. 2qr. for 105 miles be 17. 58. how far may 5 cwt. I qr. be carried for the same money ? 42. IF 50 men consume 15 bushels of grain in 40 days, how much will 30 men consume in 60 days? 43. On the same supposition, how long will 50 bushels maintain 64 men? 44. A GENTLEMAN having 50s. to pay among his laborers for a days work, would give to every boy 6d. to every woman 8d. and to every man 16d. the number of boys, women and men, was the same; I demand the number of each? 45. A GENTLEMAN had 77. 17s. 6d. to pay among his laborers; to every boy he gave 6d. to every woman 8d. and to every man 16d. and there were for every boy three womer, and for every woman two men; I demand the number of each? 46. THREE Gardners, A, B and C, having bought a piece of Ground, find the profits of it amount to 1201. per annum. Now the sum of Money which they laid down was in such proportion, that as often as A paid 57. B paid 77. and as often as B paid 47. C paid 67. I demand how much each man must have per annum of the gain? 47. A YOUNG man received 2107. which was of his elder Brother's portion; Now three times the eldest brother's portion was half of the Father's Estate; I demand how much the estate was ? 48. Two men depart both from one place, the one goes North and the other South; the one goes 7 miles a day, the other 11 miles a day; how far are they distant the 12th day after their departure? 49. Two men depart both from one place and both go the same road; the one travels 12 miles every day, the other 17 miles every day; how far are they distant the 10th day after their departure ? 50. THE river Po is 1000 feet broad, and 10 feet deep, and it runs at the rate of 4 miles in an hour. In what time will it discharge a cubic mile of water (reckoning 5000 feet to the mile) into the sea? 51. If the country which supplies the river Po with water be 380 miles long and 120 broad, and the whole land upon the surface of the earth be 62,700,000 square miles, and if the quantity of water discharged by the rivers into the sea be every where proportional to the extent of land by which the rivers are supplied; how many times greater than the Po will the whole amount of the rivers be? 52. UPON the same supposition, what quantity of water altogether will be discharged by all the rivers into the sea in a year? 53. Ir the proportion of sea on the surface of the earth to that of land be as 101 to 5, and the mean depth of the sea be a quarter of a mile; how many years would it take if the ocean were empty to fill it by the rivers running at the present rate? 54. If a cubic foot of water weigh 1000 oz. avoirdupois, and the weight of mercury be 131⁄2 times greater than of water, and the heighth of the mercury in the barometer (the weight of which is equal to the weight of a column of air, on the same base, extending to the top of the atmosphere), be 30 inches what will be the weight of the air upon a square foot? a square mile? and what will be the whole weight of the atmosphere, supposing the size of the earth as in questions 51 and 53? 55. A BEGAN trade June 1, with 40 dollars, and took in B, as a partner, Sept. 8, following, with 120 dollars; on Dec. 24, A put in 190 dollars more, and continued the whole in trade till May 5, following, when their whole gain was found to be 82 dollars; what is each partner's share? 56. If I give 80 bushels of potatoes, at 21 cents per bushel, and 240lb of flax, at 15 cents per lb. for 64 bushels of salt, what is the salt per bushel? 57. WHAT is the present worth of 482 dollars, payable 4 years hence, discounting at the rate of 6 per cent ? 58. I have owing to me as follows, viz. Dolls. 18,73 in 8 months; Dolls. 46 in 5 months; and Dolls. 104, 84 in three months; what is the mean time for the payment of the whole? 59. If I sell 500 deals at 15d. a piece, and lose L. 9 per cent. what do I lose the whole quantity? 60. IF I buy 1000 Ells Flemish of linen for £90 what may I sell it per El in London to gain £ 10, in the whole ? Сс Pleasant and diverting Questions. 1. THERE was a well 30 feet deep; a Frog at the bottom could jump up 3 feet every day, but he would fall back two feet every night. How many days did it take the Frog to jump out? 29 Days answer. by Pontion 2. Two men were driving sheep to market, says one to the other, give me one of yours and I shall have as many as you; the other says, give me one of yours and I shall have as many again as you. How many had each ? 10.2 1428 3. As I was going to St. Ives, I met seven wives, Ans one had 5 Every wife had seven sacks, Every sack had seven cats, Every cat had seven kits, Kits, cats, sacks, and wives, How many were going to St. Ives? 4. The account of a certain School is as follows, viz. geometry, learn grammar, learn arithmetic, to read; I demand the number of each ? of the boys learn learn to write, and 9 learn 5. A MAN driving his geese to market, was met by another, who said, Goodmorrow, master, with your hundred geese; says he I have not an hundred, but if I had half as many as I now have, and two geese and an half beside the number I now have already, I should have an hundred: How many had he? 65 6. THREE travellers met at a caravansary, or inn, in Persia; and two of them brought their provision along with them, according to the custom of the country; but the third not having provided any, proposed to the others that they should eat together, and he would pay the value of of his proportion. This being agreed to, A produces 5 loaves, and B 3 loaves, which the travellers eat together, and C paid 8 pieces of money as the value of his share, with which the others were satisfied, but quarrelled about the dividing of it. Upon this the affair was referred to the judge, who decided the dispute by an impartial sentence. Required his decision? an Ahady pieces by had th 7. SUPPOSE the 9 Digits to be placed in a quadrangular form: I demand in what order they must stand, that any three figures in a tight line, may make just 15, 8. A COUNTRYMAN having a Fox, a Goose, and a peck of Corn, in his jour ney came to a river, where it so happened that he could carry but one over at a time. Now as no two were to be left together that might destroy each other; so that he was at his wits end how to dispose of them: for, says he, the corn can't eat the goose, nor the goose eat the fox, yet the fox can eat the goose, and the goose eat the corn. The Question is how he must carry them over, that they might not devour each other. 9. THREE jealous husbands with their wives, being ready to pass by night over a river, do find at water side a boat which can carry but two persons at once, and for want of a waterman they are necessitated to row themselves over the river at several times: The Question is, how those 6 persons shall pass by 2 and 2, so than none of the three wives may be found in the company of, one or two men, unless her husband be present? ` 15. Two merry companions are to have equal shares of 8 gallons of wine, which are in a vessel containing exactly 8 gallons; now to divide it equally between them, they have only two other empty vessels, of which one contains 5 gallons, and the other 3; the Question is, how they shall divide the said wine between them by the help of these three vessels, so that they may have 2 gallons a-picce ? |