210. To divide by a composite number, we may divide by its factors, as in division of simple numbers 24. Divide 396£ 2s. 3d. by 54. 25. Divide 725bush. Opk. 6qt. 1pt. by 81. 26. Divide 397 lb 113 73 19 4gr. by 63. 27. Divide 958m. 5fur. 5ch. 12 li. 5gin. by 48. 211. When the divisor is large and not composite, set down the work of dividing and reducing. There is no device for rendering the operation easier. 28. Divide 135bush. 3pk. 3qt. 1pt. by 47. bush. pk. qt. pt. 47) 135 3 3 1(2bush. 3pk. 4qt. 1pt., Ans. 94 4 1 bush. 26 pk. 8 211 qt. 188 23 qt. 2 Having found that 47 is contained twice in 135, multiply 17 by 2, and subtract the product, 94, from 135, which leaves a remainder of 41 bushels; reduce the 41 bushels to pecks, and add the 3 pecks, making 167 pecks; then divide the 167 pecks by 47, and so continue the process till the work is done. 4 7 pt. 47 210. Rule for dividing by a composite number? 211. Method of dividing when the divisor is large and not composite? Is there no easier mode? 29. If 587 yards of cloth cost 662£ 4s. 2d. 1qr., what is the price per yard? 30. Divide 1129gal. 1pt. 3gi. by 73. 31. A farmer raised 35334bush. 3pk. 3qt. 1pt. of corn on 643 acres of land; how much was the yield per acre? 32. Suppose a man should travel 10599m. Ofur. 14rd. 4yd. 2ft. 5in. in 313 days, what distance would he travel per day? 33. In 127 days a ship sails 11s. 9° 43′ 30′′; what is the distance per day? 212. To find the difference in the longitude of two places, when the difference of time is known. 34. When it is 12 o'clock at Washington, it is 23m. 543sec. past 12 at Boston; what is the difference in the longitude of the two places? First divide the 23m. by 4, because 4m. of time make a difference of 1° of longitude. This gives 5° and a remainder of 3m. The 3m. and 543sec. =2343sec. The 2343sec. divided by 4, because 4sec. of time make a difference of 1' of longitude, give 58′ and a remainder of 23sec. Finally, reduce the 23sec. to 60ths of a sec. and divide by 4, and the quotient is 39"; i. e. the difference in longitude between Boston and Washington, is 5° 58′ 39′′, Ans. Hence, RULE. Divide the difference in time, expressed in minutes, seconds, and 60ths of a second, by 4, and the quotient is the dif ference in longitude, expressed in degrees, minutes, and seconds. 35. Paris is 2° 20′ 15′′ east of Greenwich; how many degrees west of Greenwich is New York, the difference in time between Paris and New York being 5h. 5m. 213sec.? Ans. 74° 0′ 3′′. NOTE. The difference in longitude between Paris and New York is found to be 76° 20′ 18′′ and this diminished by 2° 20′ 15′′, the east longitude of Paris, gives 74° 0′ 3′′ for the west longitude of New York. 212 Rule for finding the difference in the longitude of two places, when the difference in time is known? 36. The difference in time between Philadelphia and Rome is 5 h. 50m. 30sec.; Philadelphia is 75° 9' west; what is the longitude of Rome ? Ans. 12° 28′ 40′′ east. 37. A message telegraphed from St. Petersburg, 29° 48′ east, at 12 o'clock, noon, was instantly received at Paris at 10h. 10m. 9sec., A. M., of the same day; what is the longitude of Paris? 38. At sun-rise in Astoria, Oregon, the sun is about 3h. 49m. 16sec. high at Eastport in Maine; what is the difference in longitude? 39. What is the difference in longitude between the Cape of Good Hope and Cape Horn, if a meteor seen at midnight at Good Hope is so high as to be seen at the same moment at Cape Horn, the time at Cape Horn being 17 minutes past 6 in the evening? Ans. 85° 45'. DUODECIMALS. 213. DUODECIMALS are compound numbers in which the scale is uniformly 12. This measure is usually applied to feet and parts of a foot, and is used in determining distances, areas, and cubic contents. The denominations are feet (ft.), inches or primes ('), seconds The accents, " used to ("), thirds (""), fourths ("""), etc. designate the denominations are called indices. 214. The foot being the unit, the denominations have the relations indicated by the following Thus 12 of any lower denomination make 1 of the next higher; e. g. 12"""1"", 12" 1", 12" 1', 12' 1ft. 213. What are duodecimals? To what applied' For what used? The da pominations? How designated? 214. The unit, which denomination? ADDITION AND SUBTRACTION. 215. Addition and Subtraction of duodecimals are performed as the like operations of other compound numbers. Ex. 1. Add together 3ft. 6' 8" 4"" 7"""', 9ft. 7' 8" 2""' 5"""', and 4ft. 9' 8" 10" 8'""". of fourths, and add the 1"" to the thirds, and so proceed till all the columns are added, and so obtain 18ft. 0' 1" 5" 8"", Ans. 2. From 6ft. 8' 7" 9"" 3""" take 1ft. 6' 9" 2""' 8"'". OPERATION. Min., 6 8' 7" 9" Sub., 1 6 9 2 8 Rem., 5 1 10 6 7 3 Proof, 6 8 7 9 As 8""" cannot be taken from 3"""', add 12""" to the 3""", making 15", and then take 8""" from the sum, giving a remainder of 7"""; then take 3" from 9" or 2"" from 8"", giving 6"" by either process, and so proceed. 3. Add 10ft. 6′ 4′′, 12ft. 9′ 8′′, and 7ft. 10' 11". 4. Subtract 3ft. 8′ 4′′ 3"" from 9ft. 4' 6" 1"". MULTIPLICATION. 216. Multiplication of duodecimals is like multiplication of other compound numbers, except that, when both factors are in the form of compound numbers, it is required to find the denomination of the product. In this investigation, for the sake of convenience, we familiarly speak of multiplying feet by feet, feet by inches, inches by 215. Addition and Subtraction, how performed? 216. What in Multiplication is peculiar? What is the multiplier strictly? Why do we speak of multiplying feet by feet, feet by inches, etc.? inches, etc., though here, as everywhere (Art. 204), the multiplier is strictly an abstract number; e. g., suppose a board is 10 feet long and 1 foot wide, it evidently contains 10 square feet, and if it is 10 feet long and 2 feet wide, it as evidently contains 2 times 10 square feet 20 square feet (Art. 101), though it would be nonsense to affirm that it contains 2 feet times 10 feet; still, we are accustomed to say that the area of a board is equal to its length multiplied by its breadth. Again, if a board is 10 feet long and 1 inch wide, it contains as many square feet as it is feet in length; i. e. it contains of 10 square feet: 13-q. ft. 10'; and if the board is 10ft. long and 2in. wide, it contains of 10sq. ft. of a sq. ft. = 18sq. ft. 8. This illustration can be carried to any extent. = 28 = = 1 ft. and 217. Since 1'=7'2ft., 1′′=1}4ft., 1′′′′= 17'58ft., etc., whether the measure is linear, square, or cubic, it follows that 1', in linear measure, is a line, of a foot in length; in square measure, l' is an area, 1 foot long and 1 inch wide, and 1" is an area 1 inch square; in cubic measure 1' is a solid, 1 foot long, 1 foot wide, and 1 inch deep, 1" is a solid, 1 foot long, 1 inch wide, and 1 inch deep, and 1"" is a cubic inch; etc. 218. Let us now determine the denomination of the product obtained by multiplying any two denominations together. 17. What is 1' in linear measure? 1' in square measure? 1" in square measure 1' in cubic measure? 1" 1"? 1"""? |