12. A cellar 50 ft. long, 30 ft. wide, and 6 ft. deep was excavated by 5 men in 6 days; how many cubic yards did each man excavate daily? Ans. 11 cu. yd. 3 cu. ft. 13. If a town 5 miles square be divided equally into 150 farms, what will be the size of each farm?

Ans. 106 A. 2 R. 26 P. 20 sq. yd. 1 sq. ft. 72 sq. in. 14. How many times are 4 bu. 3 pk. 2 qt. contained in 336 bu. 3 pk. 4 qt.? Ans. 70.

15. A merchant tailor bought 4 pieces of cloth, each containing 60 yd. 2.25 qr.; after selling of the whole, he made up the remainder into suits containing 9 yd. 2 qr. each; how many suits did he make? Ans. 17.

LONGITUDE AND TIME.

224. Every circle is supposed to be divided into 360 equal parts, called degrees.

Since the sun appears to pass from east to west round the earth, or through 360°, once in every 24 hours, it will pass through of 360°, or 15° of the distance, in 1 hour; and 1° of distance in of 1 hour, or 4 minutes; and l' of distance in

225. To find the difference of time between two places, when their longitudes are given.

1. The longitude of Boston is 71° 3', and of Chicago 87° 30'; what is the difference of time between these two places?

Explain how distance is measured by time. longitude and time. Case I is what?

Repeat the table of

we multiply 16° 27′, the difference in longitude, by 4, and we obtain the difference of time in minutes and seconds, which, reduced to higher denominations, gives 1 h. 5 min. 48 sec., the difference in time. Hence the

RULE. Multiply the difference of longitude in degrees and minutes by 4, and the product will be the difference of time in minutes and seconds, which may be reduced to hours.

NOTE. If one place be in east, and the other in west longitude, the difference of longitude is found by adding them, and if the sum be greater than 180°, it must be subtracted from 360°.

EXAMPLES FOR PRACTICE.

2. New York is 74° 1' and Cincinnati 84° 24' west longitude; what is the difference of time? Ans. 41 min. 32 sec. 3. The Cape of Good Hope is 18° 28′ east, and the Sandwich Islands 155' west longitude; what is the difference of time? Ans. 11 h. 33 min. 52 sec. west, and St. Petersburg 30° difference of time?

4. Washington is 77° 1' 19' east longitude; what is their

Ans. 7 h. 8 min. 20 sec.

5. If Pekin is 118° east, and San Francisco 122' west longitude, what is their difference of time?

6. If a message be sent by telegraph without any loss of time, at 12 M. from London, 0° 0' longitude, to Washington, 77° 1' west, what is the time of its receipt at Washington?

NOTE. Since the sun appears to move from east to west, when it is exactly 12 o'clock at one place, it will be past 12 o'clock at all places east, and before 12 at all places west. Hence, knowing the difference of time between two places, and the exact time at one of them, the exact time at the other will be found by adding their difference to the given time, if it be east, and by subtracting if it be west.

Ans. 6 h. 51 min. 56 sec., A. M.

Give explanation. Rule.

7. A steamer arrives at Halifax, 63° 36' west, at 4 o'clock, P. M.; the fact is telegraphed to St. Louis, 90° 15′ west, without loss of time; what is the time of its receipt at St. Louis? Ans. 2 h. 13 min. 24 sec., P. M. 8. If, at a presidential election, the voting begin at sunrise and end at sunset, how much sooner will the polls open and close at Eastport, Me., 67° west, than at Astoria, Oregon, 124° west? Ans. 3 h. 48 min.

9. When it was 1 o'clock, A. M., on the first day of January, 1859, at Bangor, Me., 68° 47' west, what was the time at the city of Mexico, 99° 5' west?

Ans. Dec. 31, 1858, 58 min. 48 sec. past 10, P. M.

CASE II.

226. To find the difference of longitude between two places, when the difference of time is known.

1. If the difference of time between New York and Cincinnati be 41 min. 32 sec., what is the difference of longitude?

OPERATION.

min.

4) 41

sec.

32

ANALYSIS. Since 4 minutes of time make a difference of 1° of longitude, and 4 seconds of time, a difference of 1' of longitude, there will be as many degrees of longitude as there are minutes as many minutes of longitude as there are seconds of time. Hence,

10° 23', Ans.

of time, and

RULE. Reduce the difference of time to minutes and seconds, and then divide by 4; the quotient will be the difference in longitude, in degrees and minutes.

2. What is the difference of longitude between the Cape of Good Hope and the Sandwich Islands, if the difference of time be 11 h. 33 min. 52 sec.? Ans. 173° 28'.

3. What is the difference of longitude between Washington and St. Petersburg, if their difference of time be 7 h. 8 min. 20 sec.? Ans. 107° 20'.

Case II is what? Give explanation. Rule.

4. When it is half past 4, P. M., at St. Petersburg, 30° 19' east, it is 32 min. 36 sec. past 8, A. M., at New Orleans, west; what is the difference of longitude? Ans. 119° 21' A sea cap5. The longitude of New York is 74° 1' west. tain leaving that port for Canton, with New York time, finds that his chronometer constantly loses time. What is his longitude when it has lost 4 hours? 8 h. 40 min.? 13 h. 25 min.? Ans. 14° 1' west; 55° 59' east; 127° 14' east.

6. When the days are of equal length, and it is noon on the 1st meridian, on what meridian is it then sunrise? sunset? midnight? Ans. 90° west; 90° east; 180° east or west.

227.

DUODECIMALS.

Duodecimals are the divisions and subdivisions of a unit, resulting from continually dividing by 12, as 1, 1, T T72, &c. In practice, duodecimals are applied to the measureinent of extension, the foot being taken as the unit.

If the foot be divided into 12 equal parts, the parts are called inches, or primes; the inches divided by 12 give seconds; the seconds divided by 12 give thirds; the thirds divided by 12 give fourths; and so on.

From these divisions of a foot it follows that

What are duodecimals? To what applied? Explain the divisions

of the foot. Repeat the table.

NOTE. Duodecimals are really common fractions, and can always be treated as such; but usually their denominators are not expressed, and they are treated as compound numbers.

ADDITION AND SUBTRACTION OF DUODECIMALS.

228. We add and subtract duodecimals the same as other compound numbers.

EXAMPLES.

1. Add 13 ft. 4' 8", 10 ft. 6′ 7′′, 145 ft. 9′ 11′′.

Ans. 169 ft. 9' 2".

2. Add 179 ft. 11' 4", 245 ft. 1' 4", 3 ft. 9′ 9′′.

Ans. 428 ft. 10' 5".

3. From 25 ft. 6' 3" take 14 ft. 9' 8". Ans. 10 ft. 8' 7". 4. From a board 15 ft. 7' 6" in length, 3 ft. 8' 11" were sawed off; what was the length of the piece left?

Ans. 11 ft. 10' 7".

MULTIPLICATION OF DUODECIMALS.

229. Length multiplied by breadth gives surface, and surface multiplied by thickness gives solid contents (198). 1. How many square feet in a board 11 feet 8 inches long and 2 feet 7 inches wide?

OPERATION.

11 ft.

8'

2

7'

6 ft.

23

4'

30 ft. 1'

8"

9' 8"

ANALYSIS. We first multiply by the 7'. 7 twelfths times 8 twelfths equals 56 one hundred forty-fourths, which equals 4 twelfths and 8 one hundred forty-fourths. We write the 8 144ths-marked with two indices to the right, and add the 4 12ths to the next product. 7' times 11 equals 77', which added to 4' equals 81', equal to 6 feet and 9'. We write the 9' under the inches, or 12ths, and the 6 under the feet, or units. 2 times 8' equals 16', or 1 foot and 4'. We write the 4' under the 9', and add the 1 foot to the next product. 2 times 11 feet are 22 feet, and 1 foot added make 23 feet, which we write under the 6 feet. Add

How are duodecimals added and subtracted? Give analysis of example 1.