DUODECIMALS.

452. Duodecimals are equal parts of a linear, square or cubic foot, formed by successively dividing by 12. Hence the following:

be marked so as to indicate whether the number represents linear, sur

face or cubic measure.

Thus, if the feet are marked ft., the lower denominations denote length; if marked sq. ft., surface; if marked cu. ft., volume.

2. Each of the following definitions should be carefully studied by drawing a diagram representing the unit defined. The diagram can be made on the blackboard on an enlarged scale.

453. A Linear Prime is one-twelfth of a foot; a Linear Second, one-twelfth of a linear prime; and a Linear Third, one-twelfth of a linear second.

454. A Surface Prime is one-twelfth of a square foot, and is 12 inches long and 1 inch wide, and is equal to 12 square inches.

455. A Surface Second is one-twelfth of a surface prime, and is 1 foot long and 1 linear second wide, which is equal to 1 square inch. Hence square inches are regarded as surface seconds.

456. A Surface Third is one-twelfth of a surface second, and is 1 foot long and 1 linear third wide, which is equal to 12 square seconds. Hence a square second is regarded as surface fourths.

45%. A Cubic Prime is one-twelfth of a cubic foot, and is 1 foot square by 1 inch thick, and is equal to a board foot.

458. A Cubic Second is one-twelfth of a cubic prime, and is 1 foot long by 1 inch square, and is equal to 12 cubic inches or a board inch.

459. A Cubic Third is one-twelfth of a cubic second, and is 1 foot long, 1 inch wide, and 1 linear second thick, and is equal to a cubic inch.

460. Illustrate the following by diagrams on the blackboard:

1. 5 feet multiplied by 7 in. equals 35 surface primes.
2. 8 ft. multiplied by 4" equals 32 surface seconds.
3. 7 feet multiplied by 6'"' equals 42 surface thirds.

4. 3 in. multiplied by 5 in. equals 15 surface seconds.

5. 4' multiplied by 3" equals 12 surface thirds.

6. From these examples deduce a rule for multiplying feet, inches, seconds, etc., by feet, inches, seconds, etc.

Multiply and explain the following:
7. 17 ft. 5' 8" by 8 ft. 9' 7".
8. 32 ft. 9' 4" by 6 ft. 5' 11".
9. 15 ft. 6' 10" by 9 ft. 4' 8".

10. 25 ft. 9' 3" by 14 ft. 7' 2".
11. 18 ft. 7' 9'' by 12 ft. 8′ 5′′.
12. 34 ft. 8' by 26 ft. 4' 9".

13. 19 ft. 8' 7 by 2 ft. 5' 9" by 3 ft. 2' 4".

14. 48 ft. 9' by 1 ft. 7' 9" by 2 ft. 8' 5".

Duodecimals are added and subtracted in the same manner as other compound numbers. Division being of little practical utility, is omitted. The pupil may, if desired, deduce a rule for division as was done for multiplication.

LONGITUDE AND TIME.

461. Since the earth turns on its axis once in 24 hours, of 360°, or 15° of longitude must pass under the sun in 1 hour, of 15°, or 15' must pass under it in 1 minute of time, of 15', or 15", must pass under it in 1 second of time. Hence the following

and

and

TABLE OF EQUIVALENTS.

A difference of 15° in Long. produces a diff. of 1 hr. in time.

Hence the following rule to find the difference of time between two places, when their difference of longitude is given:

462. RULE.-Divide the difference of longitude of the two places by 15, and mark the quotient hours, minutes, and seconds, instead of degrees, minutes and seconds.

To find the difference of longitude when the difference of time is given.

463. RULE.—Multiply the difference of time between the two places by 15, and mark the product degrees, minutes, and seconds, instead of hours, minutes, and seconds.

Since the earth revolves from west to east, time is earlier to places west and later to places east of any given meridian.

464. Find the difference in time between the following: 1. Albany West Long. 73° 44' 50" and Boston W. Long. 71° 3′ 30′′.

When the given places are on the same side of the first meridian, the difference of longitude is found by subtracting the lesser from the greater longitude.

2. Bombay East Long. 72° 54′ and Berlin East Long. 13° 23′ 45′′.

3. New York W. Long. 74° 3' and Chicago W. Long. 87° 37' 4".

4. San Francisco W. Long. 122° and St. Louis W. Long. 90° 15' 15".

5. Calcutta E. Long. 88° 19′ 2′′ and Philadelphia W. Long. 75° 9' 54".

Observe, that when the given places are on opposite sides of the first meridian, the difference in longitude is found by adding the longitudes.

6. Constantinople E. Long. 28° 59′ and Boston W. Long. 71° 3' 30".

7. The difference in the time of St. Petersburg and Washington is 7 hr. 9 min. 19 sec. What is the difference in the longitude of the two places?

8. When it is 12 o'clock м. at New York, what time is it at a place 50° 24' west?

9. In sailing from New Orleans to Albany, the chronometer lost 1 hr. 5 min. 10 sec. The longitude of Albany is 73° 44' 50". What is the longitude of New Orleans?

10. An eclipse is observed by two persons at different points, the one seeing it at 8 hr. 30 min. P. M., the other at 11 hr. 45 min. P. M. What is the difference in their longitude?

REVIEW AND TEST QUESTIONS..

465. 1. Define Related Unit, Denominate Number, Denominate Fraction, Denomination, and Compound Number. 2. Repeat Troy Weight and Avoirdupois Weight.

3. Reduce 9 bu. 3 pk. 5 qt. to quarts, and give a reason for each step in the process.

4. In 9 rd. 5 yd. 2 ft. how many inches, and why?

5. Repeat Square Measure and Surveyors' Linear Measure. 6. Reduce 23456 sq. in. to a compound number, and give a reason for each step in the process.

7. Define a cube, a rectangular volume, and a cord foot. 8. Show by a diagram that the contents of a rectangle is found by multiplying together its two dimensions.

9. Define a Board Foot, a Board Inch; and show by diagrams that there are 12 board feet in 1 cubic foot and 12 board inches in 1 board foot.

10. Reduce of an inch to a decimal of a foot, and give a reason for each step in the process.

11. How can a pound Troy and a pound Avoirdupois be compared ?

12. Reduce .84 of an oz. Troy to a decimal of an ounce Avoirdupois, and give reason for each step in the process.

13. Explain how a compound number is reduced to a fraction or decimal of a higher denomination. Illustrate the abbreviated method, and give a reason for each step in the process.

BUSINESS ARITHMETIC

SHORT METHODS.

466. Practical devices for reaching results rapidly are of first importance in all business calculations. Hence the following summary of short methods should be thoroughly mastered and applied in all future work. The exercises under each problem are designed simply to illustrate the application of the contraction.

When the directions given to perform the work are not clearly understood, the references to former explanations should be carefully examined.

467. PROB. I.-To multiply by 10, 100, 1000, etc.

Move the decimal point in the multiplicand as many places to the right as there are ciphers in the multiplier, annexing ciphers when necessary (91).

468. PROB. II. To multiply where there are ciphers at the right of the multiplier.

Move the decimal point in the multiplicand as many places to the right as there are ciphers at the right of the multiplier, annexing ciphers when necessary, and multiply the result by the significant figures in the multiplier (93).