§ 13. Guaging. GUAGING is taking the dimensions of a cask in inches to find its contents in gallons by the following METHOD. 1. Add two thirds of the difference between the head and bung diameters to the head diameter for the mean diameter; but if the ftaves be but little curving from the head to the bung, add only fix tenths of this difference. 2. Square the mean diameter, which multiplied by the length of the cask and the product divided by 294, for wine, or by 359 for ale, the quotient will be the answer in gallons. EXAMPLE. 1. How many ale or beer gallons will a cask hold, whofe bung diameter is 31 inches, head diameter 25 inches, and whofe length is 36 inches ? NOTE. 1. In taking the length of the cask, an allowance must be made for the thickness for both heads of one inch, 11⁄2 inch, or 2 inches according to the fize of the cask. NOTE. 2. The head diameter must be taken clofe to the chimes, and for small casks, add 3 tenths of an inch; for casks of 40 or 50 gallons, 4 tenths, and for larger casks, 5 or 6 tenths, and the fum will be very nearly the head diameter within. § 14. Mechanical Powers. 1. OF THE LEVER. TO find what weight may be raised or balanced by any given power, Say, as the distance between the body to be raised or balanced, and the fulcrum or prop, is to the distance between the prop and the point where the power is applied; fo is the power to the weight which it will balance or raife. EXAMPLE. If a man weighing 150%. reft on the end of a lever 12 feet long, what weight will he balance on the other end, fuppofing the prop 1 foot from the weight? As the diameter of the axle is to the diameter of the wheel, fo is the pow er applied to the wheel, to the weight fufpended by the axle. EXAMPLES. 1. A mechanic wishes to make a windlas in fuch a manner, as that 1 lb. applied to the wheel, fhould be equal to 12 fufpended on the axle ; now, fuppofing the axle 4 inches diameter, required the diameter of the wheel? lb. in. lb. in. As 14 12:48 Ans. or diameter of the wheel. 4 2. Suppofe the diameter of the axle 6 inches and that of the wheel 60 inches, what power at the wheel will balance 10 lb. at the axle ? Ans. 1lb. 3. OF THE SCREW. The power is to the weight to be raised as the distance between two threads of the fcrew is to the circumference of a circle defcribed by the power applied at the end of the lever. NOTE 1. To find the circumference of the circle defcribed by the end of the lever, multiply the double of the lever by 3,14159, the product will be the circumference. NOTE 2. It is ufual to abate of the effect of the machine for friction. 3 EXAMPLES. There is a fcrew, whose threads are an inch afunder; 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 fufficient to turn the fcrew, that is, to raife the weight? The lever 36x2 =72 +3,14159=226,194+the circumference. 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 fquare of the diameter by ,785398; the product is the area. 3. To measure the folidity of any irregular body whose dimensions cannot be taken, put the body into fome regular veffel and fill it with water, then taking out the body, measure the fall of water in the veffel; if the veffel be fquare, multiply the fide by itself, and the product by the fall of water, which gives the folid contents of the irregular body. IN this Section there is nothing new to be propofed to the scholar. Enough of Arithmetic has been taught him for all ordinary occurrences in life. It only remains to lead him into fome reflections on the foregoing rules. For this purpose the following queftions are fubjoined. They are left without anfwers, that the fcholar's only resource of knowledge for working them fhould be in his own mind. Mafters having wrought out thefe queftions at a leisure hour, may transcribe them with their answers into a manuscript for their private use, to which on any occafion, without any trouble or hindrance, they may readily advert to fatisfy the inquiries of their pupils. 1. The Northern Lights were first observed in London in 1560; how many years fince? 2. What number multiplied by 43 produces 88150 ? 3. If a cannon may be discharged twice with 6lb., of powder, how many times will 7C. 3qrs. 17lls. discharge the fame piece ? 4. Reduce 14 guineas and £75 13s. 6d. to Federal Money. 5. What is the interest of $79,49 one year and five months? 6. A owed B $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. October 17, 1797, Received in cafh, 61,10. March 20th, 1798, Received 17 cwt. of Beef, at $4,33 per cwt. January 1st, 1800, Received in cafh, 84 dollars. What was there due from A to B, of principal and intereft, Sept. 18th, 1801 ? 7. What coft 13 yards of flannel at Is. 8 per yard? 8. What must I give for 3 cwt. 2 qrs. 13lb. of cheese at 7 cents per lb. ? 9. What will 35 yards of broadcloth coft at 23s. 6d. per yard? 10. What will be the coft of a line of veal, weighing 161b. at 21d. per bl.? 11. What will 874lb. of tallow coft, at 94d. per lb. ? 12. What will 196 yards of tape coft, at 3 farthings per yard? 13. What will 56 bufhels of oats coft, at 2s. 34d. per bufhel? 14. At 63 7s. 6d. per cwt. for fugar, 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 fquare 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, 3; feet wide, and 2 feet 9 inchés high? 18. At $1,33 per yard for cloth, what must I give for 72 yards? 23. At $10,76 per cwt. for fugar, 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 pafturing my horfe 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 15s, 4d. per yard? 28. A man fpends 23 dollars 69 cents, 5 mills, in a year, what is that per day? 29. Suppofe the Legiflature of this State fhould 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 lift? 30. A Bankrupt, whofe effects are 3948 dollars, can pay his creditors but 28 cents 5 mills on the dollar; what does he owe? 31. Suppofe a ciftern 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 ciftern contains 84 gallons, in what time will it be filled? 32. If 80 dollars worth of provifions will fer ve 20 men 25 days, what number of men will the fame provifions ferve 10 days? 33. If 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 workinen fufficient 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 fhall have one third, B one fourth, C one fixth, and D one eighth of it; what muft each one have according to this agreement ? 36. A certain ufurer lent 901. for twelve months, and received principal and interest 951. 8s. I demand at what rate per cent. he received interest ? 37. If a gentleman have an eftate of 1000l. per ann. how much may he fpend per day to lay up three fcore 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 fubtracted and the remainder be divided by 8, and the quotient be multiplied by 5, and 4 added to the product, the fquare root of the fum extracted, and three fourths of that root cubed, the cube divided by 9, the laft 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. 5s. how far may 5 cwt. 1 qr. be carried for the fame money ? 42. If 50 men confume 15 bushels of grain in 40 days, how much will 30 men confume in 60 days? 43. On the fame fuppofition, how long will 50 bushels maintain 64 men? 44. A gentleman having 50s. to pay among his laborers for a day's 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 fame, 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 women, and for every woman two men ; I demand the number of each ? |