places at the east of it, and before noon for all places at the west. Hence, if we find the difference of time between two places, and know the exact time at one of them, the corresponding time at the other will be found by adding this difference to the given time, if the place be East, or by subtracting it, if West.

224. The meridian of the Observatory of Greenwich, London, is the one from which longitude is reckoned; hence, the longitude of Greenwich is 0.

Longitude is estimated: West, 180°; and East, 180°.

1. Baltimore is in longitude 76° 37' west, and New York in longitude 74° 01' west. When it is 12 M. at Baltimore,

what is the time at New York?

2. The longitude of New York is 74° 1' west, and that of Philadelphia 75° 10' west: what is the time at Philadelphia when it is 12 M. at New York?

3. The longitude of Cincinnati, Ohio, is 84° 24' west: what is the time at Cincinnati, when it is 12 м. at New York?

4. The longitude of New Orleans is 89° 2′ west: what time is it at New Orleans, when it is 12 M. at New York ? 5. The longitude of St. Louis is 90° 15′ 10′′ west what is

223. What is the hour when the sun is on the meridian? When the sun is on the meridian of any place, how will the time be for all places East? How for all places West? If you have the difference of time, how do you find the time at either place?

-224. From what meridian is longitude reckoned? What is the longitude of this meridian? How is longitude reckoned from it?

the time at St. Louis, when it is 3 h. 25 m., P. M., at New York?

6. The longitude of Boston is 71° 4' west, and that of New Orleans 89° 2' west: what is the time at New Orleans, when it is 7 o'clock 12 m., A. M., at Boston?

7. The longitude of Chicago, Illinois, is 87° 30′ west: what is the time at New York, when it is 12 M. at Chicago?

225. Knowing the difference of time of two places, to find their difference of longitude.

1. Louisville, in Kentucky, is in longitude 85° 30′ west, and it is 9 o'clock, A. M., at the City of Mexico, when it is 9 hr. 54 min. 20 sec., A. M., at Louisville: what is the longitude of the City of Mexico?

2. Cincinnati is in longitude 84° 21' west, and it is 10 o'clock, A. M., at Cincinnati, when it is 21 min. 56 sec. past 10 at Buffalo: what is the longitude of Buffalo ?

3. By the chronometer, it is 5 hr. 6 min. 28 sec., P. M., at Greenwich, London, when it is 12 M. at Baltimore; Greenwich is in 0° longitude: what is that of Baltimore?

4. By the chronometer, it is 4 hr. 56 min. 42 sec., P. M., at Greenwich, when it is 12 M. at New York: what is the longitude of New York?

5. A captain, at sea, finds by his chronometer, that it is 2 hr. 15 min. 30 sec., P. M., at Greenwich, when it is 12 M. on board his vessel: in what longitude is the vessel?

DUODECIMALS.

226. If the unit, 1 foot, be divided into 12 equal parts, each part is called an inch, or prime, and marked, '. If an inch be divided into 12 equal parts, each part is called a second, and marked, ". If a second be divided, in like man ner, into 12 equal parts, each part is called a third, and marked, ""; and so on, for divisions still smaller.

The divisions of the foot, give

1' inch, or prime,

of a foot.

12

Hence: DUODECIMALS are denominate fractions, in which the primary unit is 1 foot, and 12 the scale of division.

NOTES.-1. Duodecimals are chiefly used in measuring surfaces and solids.

226. If 1 foot be divided into twelve equal parts, what is each part called? If the inch be so divided, what is each part called? What are Duodecimals? For what are Duodecimals chiefly used? What is the scale?

ADDITION AND SUBTRACTION.

227. The units of Duodecimals are reduced, added, and subtracted like those of other denominate numbers.

Examples.

1. In 185', how many feet?

2. In 250", how many feet and inches?

3. In 4367", how many feet?

4. What is the sum of 3 ft. 6' 3" 2''', and 2 ft. 1′ 10′′ 11"?

5. What is the sum of 8 ft. 9' 7", and 6 ft. 7' 3" 4"? 6. What is the difference between 9 ft. 3' 5" 6"", and 7 ft. 3' 6" 7"""?"

7. What is the difference between 40 ft. 6' 6", and 29 ft.

8. What is the difference between 12 ft. 7' 9" 6"", and 4 ft. 9' 7" 9""?

9. Reduce 6' 8" to the fraction of a foot.

10. Reduce 9' 10" 8" to the fraction of a foot.

11. Reduce 4' 5" 3" to the decimal of a foot.

12. Reduce 7" 6" to the decimal of a foot.

MULTIPLICATION OF DUODECIMALS.

228. MULTIPLICATION OF DUODECIMALS is an abbreviated method of finding the measure of surfaces or solids.

Two dimensions, multiplied together, produce square measure; and three dimensions, multiplied together, produce cubic or solid measure.

227. How do you add and subtract Duodecimals?

228. What is Multiplication of Duodecimals? What do two di mensions, multiplied together, produce? Three dimensions? Feet multiplied by feet, give what? Primes by primes? Primes by seconds? Seconds by seconds? Primes by feet? What index must a product have? Give the rule.

The multiplication of duodecimals is governed by the following principles :

1. Feet multiplied by feet, give square feet.

2. Primes x Primes =

3. Primes Feet =

11⁄2 ft. × 11⁄2 ft. =

ft., or seconds.

ft. x 1 ft. = 11⁄2 ft., or primes.

4. Primes Seconds ft. x 11728 ft., or thirds.

1

1

5. Seconds Seconds ft. ft. 20136, or fourths.

114 =

Since the parts of a foot are marked by accents, the foregoing principles give rise to the following law:

OPERATION.

The index of the unit of any product, is denoted by a number of accents equal to the sum of the indices of the factors. 1. Multiply 6 ft. 7' 8" by 2 ft. 9' ANALYSIS.-Multiply by 9'. Since 8" ft, and 9' ft., 8" x 9' ×12=78 ft., or thirds. Since 12 thirds make 1 second, 72"""=72÷÷12 =6"; therefore we put down 0"', and carry 6". 7', and 7' x 9'=×

6 ft. 7' 8"

2 ft.

9'

4 ft. 11' 9' 0"
3' 4"

13

9

12

14, or seconds; 63′′+6′′=69′′; 69" 12=5′ 9′′; we put down 9" and carry 5'. 6 ft. x 9' =

=

1, or primes; 54′+5'59'; 59′ ÷124 ft. 11'; which we set down, and then multiply by 2 feet. 8" x 2 feet = 16"; 16"÷12=1' 4"; we put down 4" under the seconds, and carry 1": 7' x 2 feet 12=1 ft. 3'; we put 3′ under the x 2 ft. = 12 sq. ft.; 12 ft. + 1 ft. the feet and add. The result is lowing

14'; 14' + 1′ = 15'; 15'÷ primes, and carry 1 ft.: 6 ft. 13 ft.; we set the feet under 18 sq. ft. 3' 1": Hence, the fol

Rule.

I. Write the multiplier under the multiplicand, so that units of the same order shall fall in the same column.

II. Begin with the lowest unit of the multiplier and the lowest of the multiplicand, and make the index of each product equal to the sum of the indices of the factors.

III. Reduce each product, in succession, to square feet, and 12ths of a square foot.