15. Reduce

of a year to integers.

16. Reduce of a day to integers. of an hour to

integers

17. From

of a day subtract

of an hour.

of a ton subtract of a cwt.

18. To of a day add 7 of an hour.

19. From

20. To of a ton add of a cwt.

LONGITUDE AND TIME.

260. Longitude and Time are employed in determining,

a. The difference of longitude between two places, when their difference of time is known.

b. The difference of time between two places, when their difference of longitude is known.

261. The solution of these two problems in longitude and time depends on two facts:

a. The fact that the earth revolves on its axis once in twenty-four hours.

b. The fact that 360° make one circumference.

From these two facts it follows:

1st.

That 1 hour of time corresponds to 15° of longitude.

That 1 minute of time corresponds to 15' of longitude.

That 1 second of time corresponds to 15" of longitude.

2d.

That 1° of longitude corresponds to 4 minutes of time.

That l' of longitude corresponds to 4 seconds of time.

That 1" of longitude corresponds to seconds of time.

This constant relation which subsists between the difference of longitude and the difference of time between any two points on the Earth's surface, enables us to determine

Their difference of time from their difference of longitude.

Their difference of longitude from their difference of time.

262. To reduce Longitude to Time.

1. In 5° of longitude how many minutes of time? Suggestion. — In 1° there are how many minutes of time? Then, 5° equal how many minutes of time?

2. In 5' of longitude how many seconds of time? Suggestion. —l' of longitude equals how many seconds of time?

Then, 5' of longitude equal how many seconds of time?

3. In 5° 5' of longitude there are how many seconds of time?

4. In 5° 5' 5" of longitude there are how many seconds of time?

263. To Reduce Time to Longitude.

1. Reduce 20 minutes of time to longitude.

Suggestion.

1 minute of time equals how many minutes of longitude?

Then, 20 minutes of time equal how many minutes of longitude?

2. Reduce 20 seconds of time to longitude.

Suggestion.-1 second of time equals how many seconds of longitude?

Then, 20 seconds of time equal how many seconds of longitude?

MISCELLANEOUS EXERCISES.

1. The city of Washington is in west long. 77° 2′ 48", and Boston is in west long. 87° 4' 9". Find their difference in time.

2. At 10 A. M. in Boston, is it earlier or later in Washington.

3. Columbus is 83° 3′ west long.; and when it is 37 min. 33 sec. past 1 P. M., it is 11 o'clock A. M. at San Francisco. Find the longitude of the latter place.

4. Is it always earlier, or later, at all points east of us? Why? At all points west of us? Why?

5. A celestial phenomenon which was observed at the same time at Boston, Greenwich, and St. Petersburg, occurred at 45 min. 46 sec. past 7 o'clock P. M. at Boston, and at 30 min. past midnight following, at Greenwich. Taking the longitude of Greenwich as 0°, and that of St. Petersburg as 30° 19' east, at what time did it occur at St Petersburg? What is the longitude of Boston?

DUODECIMALS.

264. Duodecimals are a species of fractions sometimes used in linear, superficial, or cubic measure.

a. The denominator of a common fraction may be any number.

b. The denominator of a decimal is 10, or some power of 10.

c. The denominator of a duodecimal is 12 or some power of 12.

265. In decimal fractions, the denominator, though unwritten, is determined by the decimal point.

In duodecimals, the denominator, though unwritten, is determined by certain characters, called indices. Twelfths of a foot or inches are read primes, and marked thus: '.

Twelfths of an inch are read seconds, and marked thus: ".

Twelfths of a second are read thirds, and marked thus: "".

Twelfths of a third are read fourths, and marked thus: """.

GENERAL LAWS.

1. Twelve units of any denomination make one unit of the next higher denomination.

2. One unit of any denomination makes twelve units of the next lower denomination.

Hence, The denominator of primes is 12.

The denominator of seconds is 144.

The denominator of thirds is 1728.

266. All the operations in duodecimals are performed in the same manner as the corresponding operations in compound numbers, generally, the scale in this case being 12.

a. The product of a duodecimal by any abstract number expresses length.

b. The product of two duodecimals expresses superficial contents.

c. The product of three duodecimals expresses cubic

contents.

EXAMPLES.

1. Find the superficial contents of the floor of a room 12 ft. 8' in length, and 11 ft. 9' in width.

FIRST OPERATION.

12 ft. 8' Multiplicand.
11 ft. 9' Multiplier.

9 ft. 6' 139 ft. 4'

1st. Partial product. 2d. Partial product.

Explanation.-The product of 8' by 9' is 72". Since 12" make 1', 72" make 6'. Multiplying 12 ft. by 9' the product is 108'. Since 12' make 1 ft., 108' make 9 ft. Multiplying 8' by 11 ft. the product is 88'. Since 12' make 1 ft., 88' make 7 ft. 4'. product of 12 ft. by 11 ft. is 132 ft.; 132 ft. plus 7 ft. gives 139 ft. Finally, 9 ft. 6', and 139 ft. 4′ make 148 ft. 10'.

The

148 ft. 10' Total product.

The product of ft. by 11 ft. is........ 74 ft. 2d.
The product of 12 ft. by ft. is........ 9 ft. 3d.
The prod. of ft. by 12 ft. is, or..

Σ

Partial prod.

Partial prod.

ft. 4th. Partial prod.