6. If a person travel 300 m. 2 fur. 25 p. in 9 days, at what rate will he travel per day? Ans. 33 m. 2 fur. 381 p. 7. A merchant has 10 pieces of cloth, of equal length, and together containing 575 yd. 2 qr. 3 na. What is the length of each piece? Ans. 57 yd. 2qr. 115 na. 8. A farmer having a tract of land containing 486 A. 2 R. 30 P., wishes to divide it into 12 fields of equal size. What quantity must there be in each field? Ans. 40 A. 2 R. 91 P. 9. A cellar measuring 1570 cu. yd. 18 cu. ft. was excavated by a man in 30 days. At what rate did he dig per day? Ans. 52 cu. yd. 93 cu. ft. 10. A manufacturer sold 13 pieces of cotton cloth, measuring in the aggregate 500 yd. 3 qr. Required the average length of each piece. Ans. 38 yd. 23 qr.

When the divisor and dividend are both Polynomials.

11. How many yards of silk at 7s. 6d. per yard may be purchased for 3£, 14 s. 10 d.?

The number of yards is the number of times that 7 s.6 d. is contained in 3£, 14s. 10 d.

Applying the 2d part of the Rule, we find,

7 s. 6 d. 90 d.; and £3, 14 s. 10 d.

898 d.;

Ans. 9.977' yd.

90 d. is contained in 898 d., 9.977' times. 12. How many hundred weight of iron, at 19 s. 8 d. per cwt. may be bought for 20£, 15 s.? Ans. 21.101' cut.

13. How many acres, of ground can be sown with 75 bu. 1 pk. of wheat, allowing 1 bu. 3 pk. to an acre?

Ans. 43 A.

14. In what time will a ship perform a voyage of 1000 L. 2 m., if she sail at the rate of 60 L. 1 m. per day?

Ans. 16.585' da. 15. What number of carpets, each to contain 34 yd. 3 qr., can be made out of 2 pieces of carpeting, each measuring 50 yd., and another piece measuring 49 yd. 2 qr.?

Ans. 4.302' carpets.

16. A silversmith makes 6 lb. 7 oz. 4 dwt. of silver into spoons weighing 3 oz. 6 dwt., each, and sells the spoons at 3 s. 6d. apiece. What does he get for his spoons ? Ans. 4£, 4 s.

17. A sugar planter makes 113 T. 9cwt. 2 qr. of sugar, which is to be put into hogsheads that will contain, on an average, 12 cut. 2 qr. How many such hogsheads will be requisite ? Ans. 181.56 hhd.

18. A traveler performed a journey of 975 m. 7 fur. The first 20 days he traveled 31 m. 2 fur. per day, and during the remainder of the journey 29 m. 5 fur. per day. How long was he on the journey? Ans. 31.843' da.

DUODECIMALS,

AND THEIR APPLICATION TO SQUARE AND CUBIC MEASURE.

§ 199. DUODECIMALS are a kind of polynomial quantities which result from supposing a linear, square, or cubic foot, to be divided into 12 equal parts, each of these parts again into 12 equal parts; and so on.

12ths of a foot

12ths of a prime

12ths of a second are called thirds; and so on.

Primes are denoted by an index of one accent; thus, 3', 3 primes. Seconds are denoted by an index of two accents; thus 4", 4 seconds.

Thirds are denoted by an index of three accents; thus, 7''', 7 thirds, &c.,

The polynomial 8 ft. 2′ 5′′'7''', for example, is 8 feet, 2 primes, 5 seconds, and 7 thirds.

How many fourths make one third? How many thirds make 1 second? How many seconds make one prime? How many primes make 1 foot?

Linear, Square, and Cubic Inches expressed

in Duodecimals.

§ 200. In linear measure, it is plain that inches are primes Thus 3 ft. 4 in. is 3 ft. 4', 3 ft. and 4 primes.

$201. In square measure, square inches are seconds.

For 1 square inch is 144 of a square foot, since 144 sq. in.= 1 sq. ft.; and 14 of a sq. ft, is 1" or 11⁄2 of 1' of a square foot. $ 202. In cubic measure, cubic inches are thirds.

For 1 cubic inch is T of a cubic foot, since 1728 cu. in.=1 cu. ft.; and 1723 of a cu. ft. is 1"'"' or ' of 12 of 11⁄2 of a cubic

foot.

In 9' In 5'? In 11'?
In 7′′′? In 8′ 3′′? Ir

In 7', linear measure, how many inches? In 5', square measure, how many sq. inches? 6', cubic measure, how many cu. inches? In 4'?

In 7′ 2′′?

Square and Cubic Measure—how found.

$203. Square measure, or measure of surface, is found by multiplying together length and breadth.

For example, 4 in. long and 3 in. wide, makes (4×3) 12 sq. inches.

§ 201. Cubic measure, or measure of solidity, is found by multiplying together length, breadth, and thickness.

For example, 4 in. long, 3 in. wide, and 2 in. thick, makes (4×3×2) 24 cubic inches.

Product of two Duodecimal Terms.

$ 205. The product of any two terms in duodecimals, has for its index the sum of the indices of the two terms;-a term in feet being understood to have no index.

For example, take 3 ft. in length, and 2′ in breadth.

3 ft. X2'=3 ft. X 1% ft. = 11⁄2 sq. ft. (§ 203),=6'′ sq. ft.

12

12

Again; taking 3′ in length, and 2′′ in breadth.

3'x2" ft. X1åa fl. = 1728 $q. ft. (§ 203), =6"" sq. ft.

In these examples, the products 6′ and 6'' have their indices (' and ''') equal, respectively, to the sums of the indices of the two terms multiplied together.

Reduction, Addition, &c., of Duodecimals.

§ 293. Reduction, Addition, &c., are performed in Duodecimals in the same manner as in other polynomials.

We have here, however, a distinct case-that of multiplying one duodecimal polynomial by another-from the manner of performing it, sometimes called Cross Multiplication.

RULE XLI.

$ 207. For Duodecimal or Cross Multiplication.

1. Multiply each term of the multiplicand, from right to left, by each term of the multiplier, noting the denomination of each product term, by means of indices (§ 205), and setting similar terms one under another.

2. When any product term (excepting ft.) is 12 or more, divide it by 12; set down the remainder, if any, and add the quotient to the product of the next term.

3. Add up the similar product terms, as in polynomial addition, for the entire product.

EXAMPLE.

To find the number of square feet in a plank 16 ft. 8 in. long, and 2 ft. 5 in. wide.

Denoting inches or primes, by an index', and multiplying, we have 8'5' 40", the product 40 having an index to the sum of the indices of the two terms 8′ and 5′, (§ 205.)

equal

Dividing the 40" by 12, since 12" make 1', we get 3' and 4". We set the 4" towards the right; and then multiplying the 16 ft., we find 16ft.×5′=80'; adding the 3', we have 83′; dividing by 12, we get 6ft. and 11', which we set in the places of ft. and primes, respectively.

Next, 8'×2 ft. =16'=1 ft. 4'; setting the 4' under 11', and adding the 1 ft. to 16 ft. X2 ft., we find 33 ft.

The two rows of products are added together, for the entire product. We thus find 40 sq. ft. 3' 4". And since, in square measure, seconds are square inches, (§ 201), by reducing the 3' to seconds,

we find 3'x12+4"-40" or 40 sq. in.

Without employing duodecimals,—

16 ft. 8 in.=163⁄41⁄2 ft., and 2 ft. 5 in. =2{1⁄2 st.;

then 16X2=4015 sq. ft. =40 sq. ft. and 40 sq. inches.

18

Or, 16 ft. 8 in. 16.666' ft.,.and 2 ft. 5 in. =2.416' ft.; then 16.666'2.416'=40.265056' sq. ft. =40 sq. ft. 38.168' sq. in.; the number of square inches falling a little below the true number 40, by reason of the imperfect decimals .666' and .416'.

If the 16 ft. 8 in., and 2 ft. 5 in. were reduced to inches, and then multiplied together, we should find the product in square inches, which, divided by 144, would be reduced to square feet.

The measure of a surface, as expressed in sq. in., sq. ft., &c., is called its area. Thus the area of the plank in the preceding example, is 40 sq. ft. 3′ 4′′, or 40 sq. ft. 40 sq. in.

EXERCISES.

1. How many square feet are there in a pavement 30 ft. 10 in. long, and 7 ft. 5 in. wide? Ans. 228 sq. ft. 8′ 2′′. 2. How many square feet of plank will make a close fence 80 ft. 8 in. long, and 6 ft. 4 in. high? Ans. 510 sq. ft. 10′ 8′′. 3. How many square feet, and also how many square yards are in a ceiling 18 ft. 5 in. long, and 12 ft. 10 in. wide?

Ans. 236 sq. ft. 50 sq. in.=26 sq. yd. 2 ft. 50 in. 4. How many square yards are contained in a floor which measures 25 fl. in length, and 16 ft. 7 in. in breadth? Ans. 46 sq. yd. 84 sq. in. 5. How many square yards of plastering would be required for one side of a wall which is 50 ft. 6 in. in length, and 20 ft. 4 in. in height? Ans. 114sq. yd. 120 sq. in.

6. Required the number of cubic feet in a piece of timber 9 ft. 10 in. long, 3 ft. 4 in. wide, and 2 ft. 6 in. thick.

Multiplying the length and breadth together, we get the product 32 sq. ft. 9' 4".

Multiplying this product by the thickness, we get the product 81 cu. ft. 11' 4" for the solidity.

Since, in cubic measure, thirds are cubic inches, by reducing the 11' 4" to thirds, 11'× 12+4′′=136′′, and 136′′ ×12= 1632" 1632 cu. in.

Without employing duodecimals, the given dimensions might be taken in feet and fractions of a foot. Or the dimensions might all be reduced to inches, and the final product divided by 1728, since 1728 cu. in. make 1 cu. ft.

7. How many cubic feet are there in a hewn log 22 ft. 8 in. long, 1 ft. 10 in. wide, and 1 ft. 2 in. thick?

Ans. 48 cu. ft. 5′ 9′′′ 4′′”. 8. How many cubic feet are there in a piece of scantling 15 ft. long, 1 ft. 2 in. wide, and 8 inches thick?

Ans. 11 cu. ft. 8'. 9. How many cubic feet were dug from a cellar which measures 42 ft. 10 in. long, 12 ft. 6 in. wide, and 8 feet deep? How many cubic yards?

ft. 4'.

Ans. 4283 cu. ft. 4'=158 cu. yd. 17 cu.. 10. It is required to find how many cubic yards of earth were excavated from a ditch which measures 100 ft. in length, 4 ft. 8 in. in breadth, and 3 ft. in depth.

Ans. 51 cu. yd. 23 cu. ft.