448. To find the difference of longitude between two places, when the difference of time is known. 1. When it is 9 o'clock at Washington, it is 7 min. 4 sec. past 8 o'clock at St. Louis. Find the diff. of longitude. as many degrees many minutes of Since 4 min. of time make a difference of 1° of long., and 4 sec. of time a difference of 1' of long., there will be of long. as there are minutes of time, and as long. as there are seconds of time. 2. The difference in the time of Washington and of St. Petersburgh is 7 hr. 9 min. 19 sec. What is the difference in their longitudes? 3. When it is 12 o'clock M. at Rochester, N. Y., it is 9 hr. 1 min. 37 sec. A. M. at San Francisco. The long. of Rochester being 77° 51′ W., what is the long. of the latter? RULE.-Multiply the difference of time expressed in hours, minutes, and seconds by 15; the product will be the difference of longitude in degrees, minutes, and seconds. Or, Reduce the difference of time to minutes and seconds, then divide by 4; the quotient will be the difference of longitude in degrees and minutes. 4. Noon comes 1 hr. 5 min. 42 sec. sooner at Quebec than at Chicago, whose longitude is 87° 37′ 45′′. What is the longitude of Quebec? 5. When the days and nights are of equal length, and it is noon on the first meridian, on what meridian is it then sunrise? Sunset? Midnight? 449. The following table of the Longitude of places is compiled from the Records of the U. S. Coast Survey. Albany..... Ann Arbor. Astoria, Or. Boston..... 73° 44′ 50′′ W. New York....... 74° 3' W. 83° 43' W. New Orleans... 90° 2' 30" W. Berlin.. Richmond, Va... 77° 25′ 45′′ W. San Francisco....122° 26' 45" W. | St. Paul, Minn... 95° 4′ 55′′ W. St. Louis, Mo.... 90° 15′ 15′′ W. Univ. of Virginia. 78° 31′30′′ W. West Point, N. Y. 73° 57' W. 99° 5' W. Washington, DC. 77° 0' 15" W. 450. To find the difference of time between two places, when their longitudes are given. 1. Find the diff. in the time of Cinn. and of St. Paul. and 15", a difference of 1 sec. of time (446), there are as many hours, minutes, and seconds of time as there are degrees, minutes, and seconds of longitude. Since 1° of long, makes a diff. of 4 min. of time, and 1' makes a diff. of 4 sec. of time (446), there is a diff. of 4 times as many minutes and seconds of time as there are deg., min., and sec. of long. 2. Find the difference in the time of Ann Arbor, Mich., and of Cambridge, Mass. ? * 3. When it is half-past 3 o'clock P.M. at West Point, N. Y., what time is it at Bombay ? RULE.-Divide the difference of longitude expressed in degrees, minutes, and seconds, by 15; the quotient will be the difference of time in hours, minutes, and seconds. Or, Multiply the difference of longitude by 4, and the product will be the difference of time in minutes and seconds, which may be reduced to hours. Find the difference in time of 4. Washington, and Rome. 8. Richm'd, and St. Louis. 5. Chicago, and Paris. 9. New York, and Mexico. 6. N. Orleans, and N. York. 10. Ann Arbor, and Berlin. 7. Albany, and Jefferson C'y. 11. Mexico, and San Fran. 12. When it is 6 A. M. at Boston, what time is it at Cincinnati? At Chicago? At St. Louis? 13. When it is 6 P.M. at the University of Va., what time is it at Berlin? At St. Paul? At Astoria, Or.? 14. How much later does the sun rise in New York than in Rome? Than in Paris? 15. In sailing from San Francisco to Bombay, will a chronometer gain or lose time, and how much? * Take from the Table the required Longitude of the different places. DUODECIMALS. 451. Duodecimals are fractions of a foot formed by successively dividing by 12; as, 1, T4, Tres, etc. 452. The Unit of measure is 1 foot, which may be a linear, a square, or a cubic foot. The scale is uniformly 12. 453. In the duodecimal divisions of a foot, the different orders of units are related as follows: The marks', ", "", "", are called Indices. Duodecimals are used by artificers in measuring surfaces and solids. ADDITION AND SUBTRACTION. 454. Duodecimals are added and subtracted in the same manner as compound numbers. 1. Add 14 ft. 7' 8", 16 ft. 3′ 5′′, and 21 ft. 9′ 11′′. 2. Add 140 ft. 10' 7" 9"", 71 ft. 8", and 107 ft. 4' 11" 3"". 3. From 54 ft. 9' 5" subtract 30 ft. 10' 8". Duodecimals are not much used. The subject is fully treated and applied in "Robinson's Higher Arithmetic." MULTIPLICATION. 455. In the multiplication of duodecimals, the product of two dimensions is area or surface, and the product of three dimensions is solidity or volume. (344, 349.) WRITTEN EXERCISES. 456. 1. Multiply 9 ft. 8' by 4 ft. 7. OPERATION. 9 ft. 8' 4 ft. y' 5 ft. 7' 38 ft. 8' 8" ANALYSIS.-Begin at the right. 8' x 7'56" = 4' 8". = == Write the 8" one place to the right, reserving the 4' to add to the next product. Then 9 ft. x 7' 63'; 63' + 4′ = 67′ = 5 ft. 7', which write in the places of feet and primes. Next multiply by 4 feet; 8′ × 4 ft. = 32′ : 2 ft. 8'. Write the 8' in the place of primes, reserving the 2 ft. to add to the next product. 36 ft.; 36 ft.+ 2 ft. 38 ft., which write in the place of feet. Adding the partial products, the sum equals 44 ft. 3' 8", the product required. 44 ft. 3' 8" Then 9 ft. x 4 ft. 2. How many square feet in 4 boards, each 12 ft. 9' long, and 1 ft. 4' wide? RULE.-I. Write the terms of the multiplier under the corresponding terms of the multiplicand. II. Multiply each term of the multiplicand by each term of the multiplier, beginning with the lowest order of units in each. Reduce each product to higher denominations when possible, and write in their proper places. The sum of the partial products will be the product required. 3. Multiply 10 ft. 6' 4" by 5 ft. 3′ 8′′. 4. Find the area of a floor 14 ft. 8' wide and 16 ft. 5' long. 5. What are the solid contents of a block of marble 6 ft. 10' long, 4 ft. 3' wide, and 1 ft. 9' thick? |