The earth revolves round the sun once in 365 days, 5 hours, 48 minutes, 48 seconds: this period is therefore a Solar year. In order to keep pace with the solar year, in our reckoning, we make every fourth year to contain 366 days, and call it Leap year. Still greater accuracy requires, however, that the Leap day be dispensed with 3 times, in every 400 years. Whenever the number which denotes the year can be measured by 4, the year is Leap year, the centurial years excepted. The year is also divided into 12 months—See Almanac. THE CIRCLE. of 10 The divisions of the circle, C., are, the sign, S., the degree, (), the minute, (), the second, ("). This table is applied to the Zodiac; and by it are computed, planetary motions, latitude, longitude, &c. 60 seconds 1' 1” to of 1' 60 minutes 1° 1 30 degrees 1S. 10 of 1 S. 12 signs 10. is. 1 GEOGRAPHICAL MEASURE, The circumference of the globe-like every other circle—is divided in 360 equal parts, called degrees. Each degree is divided into 60 equal parts called miles, or minutes. Three miles are called a league. On the equator, 69} statute miles are equal to 60 geographical miles, or 1 degree, nearly: and, on the meridian, at a mean, 69 I statute miles are equal to a degree. REDUCTION OF COMPOUND NUMBERS. REDUCTION is the operation of changing any quantity from its number in one denomination, to its number in another denomination. RULE FOR REDUCTION. When a greater denomination is to be reduced to a smaller, multiply the greater denomination, by that number which is required of the smaller, to make a unit of the greater; adding to the product, so many of the smaller denomination as are erpressed in the given quantity. Perform a like operation on this product, and on each succeeding product, When a smaller denomination is to be reduced to a greater, divide the smaller denomination by that number which is required of the smaller, to make a unit of the next greater: the quotient will be of the greater denomination, and the remainder will be of the same denomination with the dividend. Perform a like operation on this quotient, and on each succeeding quotient. 1. Reduce £351 13s. Od. 1 qr. to its value in farthings. 2. How many pounds, &c. are there in 6169 pence? 3. In 59 lb. i3dwt. 5gr. Troy, how many grains ? 4. Change 20571005 drams to its value in tons, &c. 5. In 231 lb 33 03 03 5gr. how many grains ? 6. How many English ells are there in 352 nails? 7. Reduce 7 bushels and 6 quarts to pints. 8. How many bhds. are there in 9576 pints of wine ? 9. How many pints in 1 bl. 1 fir. 1 pt. of beer? 10. How many miles, &c. are there in 26431 rods? 11. In 3 square miles, how many square rods? 12. In 1259712 cubic inches, how many cubic yards ? 13. Reduce 1 solar year, 7d. and 10h. to seconds. ADDITION OF COMPOUND NUMBERS. The operation of adding compound numbers, differs from that of adding simple numbers, only, with respect to the irregular system of units, which determines the principles of carrying from one denomination to another. RULE. Write the numbers so that each denomination shall stand in a separate column. Add the numbers of the lowest denomination together, and divide their sum by that number which is required of this denomination to make a unit of the next higher: write the remainder under the column added, and carry the quotient to the next colunin. Thus proceed through all the denominations. 14. What is the sum of £9 8s. 4d., £250 8s. 5 d. 3gr., £9 7 s. 4 d., £20 16 s. 4 d., and 3s. 6d. 2 qr.? 15. Add together 10 oz. 14 dwt. 16 gr., 5lb. 9 oz. 6 d:v1 22 gr., 41b, 1 oz. 18 dwt. 9 gr., and 11 dwt., Troy 16. Add together 15 T. 19cwt. 3qr, 2 lb. 707., 25'T. 13cwt: 2qr. 20lb. 15 oz., and 3 gr. 26 lb. 17. How much is 18yd. 3qr. 3 na., 15yd. 2 qr. 3 na., 25 yd. Iqr. 2 na., and 57 yd. 3qr. 2 na. of cloth? 18. Add together 25 bu. 3pk. 7qt., 100 bu. 2 pk. 4 qt., 215 bu. 2 pk. 2qt. 1 pt., and 57 bu. 3 pk. of corn. 19. Add together 4p. 125 gal. 3qt., 75 gal. 2qt. J pt., 35 p. 92 gal., and 39 gal. 3qt. 1 pt. of wine. 20. How many acres are 13Ă: 3R. 38r., 87 A. 2 R. 33r., 28 A. 2R., 41 A. 2R. 28r., and 36 r.? 21. How much hewn timber is 9 T. 19 ft. 1725 ir 150 T. 39 ft. 1695 in., and 500 T. 31 ft. 915 in.? SUBTRACTION OF COMPOUND NUMBERS. RULE. Write the several denominations of the smaller quantity under the same denominations of the greater quantity: then, begin with the lowest denomination, and perform subtraction on each denomination separately. Whenever a number expressing a denomination in the upper line is smaller than the number under it, increase the upper number by as many as make a unit of the next higher denomination, and consider the number of the next higher denomination in the upper line, to be 1 less than it stands. 22. Subtract 1 lb. 10 oz. 16 dwt. from 3lb., Troy. 23. From 6 T. 3cwt. take 7 cwt. 2 qr. 15 lb., Avoir. 24. From 2 11 73 take 73 63 25gr., Apoth. wt. 25. Subtract 3qr. 3na. from 5yd. 2qr. 1 na. of cloth 26. Subtract 8 bu. 1 pk. 6qt. 1 pt. from 50 bu. of corn. 27. From 3 hhd. 25 gal. take 41 gal. 2 qt. of wine. 28. From 6 bl. 1 kil. take 1 fir. 6 gal. 3qt. of beer. 29. Subtract 3yd. 10 in. from 5yd. 2 ft. 2 in., Long mea. 30. Subtract 57 A. 2R. 31 r. from 1 m., Square mea. 31. Subtract 2 Y. 90 d. 4 h. 55m. from 4 Y., Time. "MULTIPLICATION OF COMPOUND NUMBERS ROLE. Begin with the lowest denomination, and mul tiply each denomination separately; divide each product by the number which is required of its own denomination to make a unit of the next higher; write the remainder under the denomination multiplied, and carry the quotient to the product of the next higher denomination. 32. Multiply £215 19 s. 6 d. by 72 or its factors. 33 Multiply 2 lb. 5 oz. 7 dwt. 10gr., Troy, by 56. 34. What is 16 times 18 cwt. 3qr. 15 lb., 14 oz.? 35. What is 81 times 36 bu. 3 pk. 6 qt. 1 pt., Dry mea. 36. Multiply 4p. 105 gal. 3qt. of wine by 60. 37. Multiply 2m. 7 fur. 35r., Long mea., by 63. 38. Multiply 4m. 320 A. 1 R. 9 r., Šquare mea., by 15. 39. Multiply 2 Y. 250d. 14 h. 30 m., Time, by 96. DIVISION OF COMPOUND NUMBERS. RULE. Divide each denomination separately, beginning with the highest. Whenever a remainder occurs, reduce it to the next lower denomination, add it to the number expressed in the lower denomination, and divide their sum. 40. Divide £251 15 s. 7d. 2qr. into 46 equal parts. 41. Divide 15 lb. 3oz. 7 dwt. 5 gr., Troy, by 13. 42. Divide 12T. 27 lb. 15 oz., Avoirdupois, by 5. 43. Divide 136 E. e. 3qr. 3 na. of cloth by 31. 44. Divide 1621 bu. 2 pk. of corn into 50 equal parts 45. Divide 1 pipe of wine equally among 9 owners. 46. Divide a Leap year into 100 equal parts. FEDERAL MONEY. The denominations of Federal Money are, the eagle, the dollar, the dime, the cent, and the mill. 10 mills make 1 cent, 10 cents 1 dime, 10 dimes 1 dollar, and 10 dollars 1 eagle. Dollars, $, and Cents, cts. are the only denominations commonly mentioned in business- eagles being counted as tens of dollars, dimes being counted as tens of cents, and mills not being denoted. 100 cents $1 || 1 cent .. The cents in any number of dollars are expressed by the same figures which express the dollars, with two ciphers annexed; $15=1500 cents. The dollars in any number of cents are distinguished by cutting off two figures from the right for cents; 325 cts. = $3.25. Operations on numbers expressing Federal money, are performed as on simple numbers; care must however be taken, in addition and subtraction, to place dollars under dollars, and cents under cents; these denominations being separated by a point. 47. What is the sum of $34.21, $ 7064.04, 36 cts., $ 10004.85, $ 96, $ 900.10, $ 14, $1.99, and $76529? 48. Subtract $4926 from $12262.37. 49. Subtract $ 297.18 from $100000. 50. Suppose $295.48 to be a multiplicand, and 25 the multiplier; what is the product ? In multiplication, only one of the factors can be Federal money, and the product will be of the same denomination as this factor. If, therefore, there be cents in either factor, two figures must be pointed off for cents, from the right of the product. 51. What is the product of 96 cts. multiplied by 43 ? 52. What is the value of 1304 pounds of coffee at 9 cents per pound? 53. How many times $7 are there in $29.46 ? In division, when both the dividend and divisor are Federal money, they must both be of the same denomiGation. If therefore, one of the numbers contain cents, and the other dollars only, the latter number must have two ciphers annexed to it. 54. How niany barrels of flour, at $4.36 per barrel, can be purchased for $4370? 55. Divide $4279.50 into 746 equal parts. 56. If 407 pounds of Hyson tea cost $395, what is the cost of 1 pound ? 57. How many times are 95 cts. contained in $ 56 ? 58. A merchant sold 1248 yards of cloth, at such price as to gain 1 cent on every nail. How much did he gain ? 59. What is the gain on a hogshead of molasses, sold at an advance of 3 cents per gallon? 60. A jeweller sold a silver pitcher 3 lb. 8 oz. 16 dwt., at 7 cents a pennyweight: What did it ansount to? |