42 min. 21 sec. 8 = 42 213

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. 5. Chicago, and Paris.

8. Richm'd, and St. Louis.

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, 12, 144, 1728, 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:

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. 7'

5 ft.

ANALYSIS.—Begin at the right. 8' x 7'56" 48". 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', ' 8" which write in the places of feet and primes. 38 ft. 8' Next multiply by 4 feet; 8′ x 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?

9. DUODECIMALS.

1. DEFINITION.

2. UNIT OF MEASURE.

3. TABLE.

4. ADDITION AND SUBTRACTION.

1. Product of two dimensions.

5. MULTIPLICATION. 2. Product of three dimensions.

3. Rule, I, II.

8. LONGITUDE

AND TIME.

2. First Meridian.

2. COMPARISON OF LONGITUDE AND TIME.

3. RULES.

1. To find diff. of long. when diff

of time is given.

2. To find diff. of time when diff. of long. is given.

1. Longitude.

1. DEFINITIONS.

7. DIVISION.

66

5. SUBTRACTION

66

2. UPON WHAT PRINCIPLES BASED.

4. ADDITION OF COMP. NUMBERS.

1. How PER

FORMED.

458. Measurements involve a practical application of the Weights and Measures to various operations required in the mechanic arts, and to the common business of life.

RECTANGULAR SURFACES.*

459. A Rectangle is a plane figure bounded by four sides, having all its angles right angles.

It has two dimensions-length and breadth.

Rectangle.

When all its sides are equal, it is called a Square.

460. The Area of a rectangle is the surface included within the lines which bound it, and is expressed by the number of times it contains a given unit of measure.

461. The Unit of Measure for surfaces is a square each side of which is a unit of some known length.

Thus, the unit of square inches is 1 square inch; of square feet, 1 square foot; of square yards, 1 square yard, etc.

* Measurements of plane figures requiring a knowledge of Involution and Evolution are treated at the close of this book under the head of " Mensuration."