86. Suppose a vessel to go up stream by the power of steam at the rate of 16 miles an hour, by sail at the rate of 4 miles, and to be set back by the current at the rate of 2 miles an hour; in what time will another, which is propelled forward at the rate of 25 miles an hour and is set back the same as the former, overtake her, if she starts 3 hours later?

DUODECIMALS*

214. Duodecimal Fractions, or Duodecimals, are fractions whose denominators are 12 or some integral power of 12. They may also be considered as a kind of compound numbers, the values of whose denominations vary by a uniform scale of 12.

Duodecimals are sometimes used in computing lengths, surfaces, and solids; but all examples in mensuration can be performed by the use of common or decimal fractions.

215. The denominations are feet, primes ('), seconds ("), thirds (""), fourths (""), fifths ("), &c. The foot is considered the unit; primes are 12ths of feet; seconds 12ths of primes or 144ths of feet; thirds 12ths of seconds, 144ths of primes, or 1728ths of feet, &c. Hence, in length, inches are represented by primes, in surface by seconds, and in solids by thirds.

The marks which indicate the degree of the denominations are called Indices.

Units X primes, seconds, &c. = primes, seconds, &c.
Primes (s.) X primes (s.) seconds (148)
Primes (s.) X seconds (148.) = thirds (17288.)
Primes (s.) thirds (17288.) = fourths (2368.)
Seconds (8.) X seconds (8) fourths (8.)

217. Duodecimals may be added, subtracted, multiplied, and divided like compound numbers, it being borne in mind that a unit of any denomination is 12 times one of the next lower denomination, and of one of the next higher.

1. What is the sum of the contents of 3 blocks of granite containing severally 92 ft. 11′ 7′′ 6"", 484 ft. 1′ 9", 472 ft. 6'?

Ans. 1019 ft. 7' 4" 6".

2. If from a board measuring 31 ft. 7', there be cut 19 ft. 11' 4" 93", what will remain ? Ans. 11 ft. 7' 7" 22". 3. Required the contents of 5 blocks of marble, each containing 4 ft. 3' 9".

Ans. 21ft. 6' 9". 4. There being 679 ft. 7' 6" of glass in 29 windows of equal size, how much glass does one window contain ?

Ans. 23 ft. 5' 20".

5. There are 3049 ft. 3' 0" 8" of glazing in my dwellinghouse and two equal green-houses. My dwelling-house contains 679 ft. 7' 6" 9". What is the quantity of glass in each green

house?

Ans. 1184 ft. 9' 8" 11" 6'"`. 218. It only remains to multiply and divide duodecimals by duodecimals. These operations can be easily performed, if we observe that the index of the product of any two terms equals the sum of the indices of the terms themselves, and the index of the quotient of one term divided by another, equals the difference of

the indices of the dividend and divisor. Thus, 2' x 4"=8"′′;

1234'.

219.

MULTIPLICATION.

ILL. EX. What is the area of a floor measuring 12 ft. 3' in length and 10 ft. 6' in breadth?

6' X 3'18"= 1'6". We write the 6" and reserve the 1' to add with the primes. 6′ × 12 = 72', which, with the 1' reserved 73′ = 6 ft. + 1', which we write in their proper places. 10 × 3′ = 30' 2 ft.+ 6'. We write the 6' and reserve the 2 ft. to add with the feet. 10 X 12 ft. 120 ft.; 120 Adding the partial products Hence the

= 122 ft.

128 ft. 7' 6" Ans. ft. + 2 ft.
we obtain for the answer 128 ft. 7' 6".

RULE FOR MULTIPLICATION OF DUODECIMALS. Beginning with the lowest denomination, multiply all the terms of the multiplicand by each term of the multiplier separately; divide each product by 12 (except when the product is feet); write the remainder, and reserve the quotient to add to the next product. Give to every term thus obtained an index equal to the sum of the indices of its two factors. The sum of the partial products will be the entire product.

EXAMPLES.

1. Multiply 7 ft. 4' by 5 ft. 2'.

2. Multiply 4 ft. 8′ 5′′ by 3 ft. 4'.

Ans. 37 ft. 10' 8". Ans. 15 ft. 8' 0" 8'".

3. How many feet of boards will be required to construct 50 boxes 2 ft. 3' long, 2 ft. 3' wide, 1 ft. 11' high, making no allowance for thickness of boards? Ans. 1368 ft. 9'.

4. Which will contain more, and how much more, a box 3 ft. 9' by 1 ft. 6' by 2 ft. 3', or a box 2 ft. 6' each way?

Ans. The latter by 2 ft. 11' 7" 6".

5. What will be the cost of polishing a piece of marble on one side and all the edges at $.334 per square foot, the marble being 3 ft. 7' by 1 ft. 9' and 1' thick? Ans. $2.387. 6. How many cubic feet of masonry in a wall 16 rods long, 8 ft. 9' high, and 2 ft. 2' thick?

Ans. 5005 ft.

7. What is the cost of laying two floors, each 16 ft. 8' by 12 ft. 6', at 18 cts. per sq. yd.? Ans. $8.33

8. Find the price, at $24 per thousand ft., of 3 boards measuring as follows: 17 ft. 11' by 1 ft. 2', 19 ft. 4' by 1 ft. 11', and 22 ft. 8' by 1 ft. 9'. Ans. $2.343.

9. How many feet, board measure, in 6 planks 2 in. thick, each 25 ft. 9' long, 6' wide? (See Art. 213, Ex. 43, note.)

220. DIVISION OF DUODECIMALS.

Ans. 154 ft.

ILL. EX. A plat of ground contains 65 ft. 0' 7"; its width is

RULE FOR DIVISION OF DUODECIMALS. Divide the highest term in the dividend by the highest term in the divisor; the quotient will be the first term in the answer. Multiply the entire divisor by that term, and subtract the product from the dividend. Divide as before, and thus proceed till all the terms of the dividend are divided. Should there be a remainder, it may be reduced to numbers of lower denominations and divided, or annexed to the quotient in a fractional form, having for its denominator the divisor expressed in units.

EXAMPLES.

1. Divide 54 ft. 7' 4" 6" by 4 ft. 1'.

Ans. 13 ft. 4' 6".

Ans. 3 ft. 4'.

2. What is the width of a table, 4 feet 3' long, which contains 14 ft. 2/? 3. How many feet of joist, 4 inches wide and 3 inches thick, allowing nothing for waste by sawing, can be made from a piece of timber 44 ft. 5' long, 1 ft. 3′ wide, and 1 ft. 4' thick?

Ans. 888 ft. 4!

4 How many blocks of stone containing 1 ft. 11′ 5′′ 6'" can be sawed from a block containing 11 ft. 8′ 9′′? Ans. 6 blocks. 5. What is the thickness of a block of granite, one of whose surfaces contains 75 ft. 10' 8", and whose solid contents are 107 ft. 6' 1" 4""? Ans. 1 ft. 5'.

221. GENERAL REVIEW, No. 4.

1. Reduce 7 £ 3 s. 6 d. to farthings,

2. Reduce 4876 gr. to lb., etc., Troy.

3. 4 tb, 93, 7 3, 29, 8 gr. + 3 lb, 63, 23, 29, 8 gr. =? 4. 3 T. 1 cwt. 2 qr. 1 lb.

5. 1 m.

-

1 T. 2 cwt. 3 qr. 7 lb. 8 dr. =?

6 f. 16 r. 3 yd. 1 ft. 8 in. =?

6. Multiply 2 m. 30 ch. 12 1. by 8.

7. Multiply 5 y, 212 d. 10 h. 15 m. by 20. (3651 days to the year.)

8. Divide 4 A. 3 R. 24 r. by 9.

9. In & c. 1. how many feet?

10. What part of 1 A. is 3 R. 13 r. 51 ft.?

11. Reduce & cu. yds. to feet and inches.

12. Reduce 8' 53" to the fraction of a degree.

13. What cost 12 bu. 2 pks. of plums at $.06 a pint? 14. What cost 2 qts. 13 pts. oil at $1.12 per gallon?

15. Required the number of square feet in a garden 4 rds long and 1 rd. 15 ft. wide.

16. How many cu. ft. of space in a cellar measuring on the inside of the wall 5 yd. 1 ft. in length, 4 yds. in width, and 10 ft in depth?

17. What is the difference of time in two places whose longi tudes differ 7° 8' 4"?

18. When the difference of time is 3 h. 4 m. 6 s., what is the difference of longitude between two places?

19. How many days from Jan. 5, 1864, to March 3, 1865? For changes, see Key.