10. What will a load of wood 8 feet long, 34 feet wide, and 4 feet high, cost, at $1 per foot?

The foregoing remarks and illustrations may now be embraced in the following

RULES.

I. How do you find the contents of any solid or cube? A. Muliply the length, breadth and depth together.

II. When the length of wood is 8 feet, how can you find the number of cord feet it contains, without multiplying by 8 an! dividing by 16? A. Multiply the breadth and height together. and divide the product by 2; the quotient will be cord feet. III. How do you bring cord feet into cords? A. Divide by 8.

Note. If the wood is only 4 feet in length, proceed as last directed; then, as 8 feet in length is 2 times as much wood as only 4 feet in length, hence the result found, as above, will be the answer in cord feet; that is, divido by 2 twice, or once by 4.

Exercises for the Slate.

1. How many solid feet in a load of wood 8 feet long, 4 feet wide, and 3 feet high? 4×31427 cord feet, An: 2. How many feet in a load of wood 5 ft. 6 in. high, 3 ft. 9 in. wide, and of the usual length?

ft.

(Reduce the inches to the decimal of a foot.) A. 10312510 Perform this last example by reducing the inches of a foot to a common fraction. This method, in most cases, will be found preferable: thus, taking the last example :

5 ft. 6 in.5 ft.-; then, 3 ft. 9 in. 3 ft. x = 165 ÷ 2 = 185 = 105, Ans., as before.

=

3. In a block 8 ft. 6 in. in length, 3 ft. 3 in. wide, and 2 ft. 9 in. thick, how many solid feet? A. Decimally 75,96875 feet=75 feet. By common fractions; 7×13×¥= 24317532 feet, Ans., as before.

=

4. If a load of wood is 8 feet long and 3 feet wide, how high must it be to make 1 cord?

In this example, we know that the height multiplied by the width, and this product divided by 2, must make 8 cord feet, that is, 1 cord or load; hence, 8X2:

16÷3 5 feet, height, Ans.

3. If a load of wood is 5 feet high, and 8 feet long, how wido

must it be to make 2 cords?

2 cords 16 cord feet; then, 16 X 2 = 32 ÷ 5} 6 feet wide, Ans. 6. If a load of wood is 5 feet high and 8 feet long, how wide must it be to make 3 cords? (9) 4 cords? (12) 8 cords? (24) A. 45 feet.

7. How many solid feet of timber in a stick 8 feet long, 10 inches thick, and 6 inches wide? (34) 10 feet long, 12 inches thick, and 1 ft. 3 in. wide? (12) 20 ft. 6 in. long, 24 inches wide, and 1 ft. 9 in. thick? (71) A. 872 ft.

8. In a pile of wood 10 feet wide, 3 ft. 3 in. high, and 1 mile long, how many cord feet, and how many cords?

A. 10725 cord feet 1340g cord 9. How many tons of timber in 2 sticks, each 30 feet long. 20 inches wide, and 12 inches thick? A. 100 feet÷502 tons. 10. How many bricks 8 inches long, 4 inches wide, and 24 inches thick, will build a wall in front of a garden, which is to be 240 feet long, 6 feet high, and 1 foot 6 inches wide? A. 51840 bricks.

rived?

DUODECIMALS.

LXXXI. Q. From what is the word duodecimals deA. From the Latin word duodecim, signifying twelve. Q. In common decimals, we are accustomed to suppose any whole thing, as a foot, for instance, to be divided into ten equal parts; but how is a foot dividea in duodecimas? and what are the parts called? A. Into twelve equal parts, called inches or primes, and each of these parts into twelve other equal parts, called seconds; also each second into twelve equal parts, called thirds, and each third into twelve equal parts, called fourths, and so on to any extent whatever. Q. What, then, are duodecimals? A. Fractions of a foot. Q. What fraction of a foot is 1 inch? A. Q. What fraction of a foot is 1 second? A. Q. What fraction of a foot is 1 third? A. 1 QWhat fraction of a foot is 1 fourth?

ft.

=

2 of 12 = 144 ft.
of 1 of 1 = 1728 ft.

A. 1 of 1 of 12 of 12= 20736 ft. Q. Now, since 12ths multiplied by 12ths make 144ths, and make 12, also, 144ths multiplied by 12ths make 1728ths, and 12 make TT, it is

plain that we may write the fractions without their denominators, by making some mark to distinguish them. What marks are generally used for this purpose? A. 12ths, inches, or primes, are distinguished by an accent, thus; 8′ signifies, 8 inches, or 8 primes; 7" 17, or 7 seconds; 6'1725, or 6 thirds, &c.

=

Q. We have seen that 12ths multiplied by 12ths produce 144ths; what, then, is the product of 5' (inches or primes) multiplied by 7' (inches)? A. 35", that

Q. What is the product of 5" (seconds) multiplied by 7' (inches)? A. 35", that is, 35 thirds.

Q. What is the product of 5" (seconds) multiplied by 7 (seconds)?

A. 35, that is, 35 fourths.

Q. How may the value of the product always be determined? A. By placing as many marks or accents at the right of the product as there are marks at the right of both multiplier and multiplicand counted together.

Q. What, then, would 7 (fifths) multiplied by 8 (sixths) produce? A. 56, that is, 56 elevenths.

Q. What would 7" (seconds) multiplied by 5'' (thirds) produce? A. 35, that is, 35 fifths.

Q. What would 8" multiplied by 3" produce? A. 24, (fourths.) Q. From the preceding, what appears to be the value of feet multiplied by primes or inches, or what do feet multiplied by primes give? A. Primes. Q. What do primes multiplied by primes give? A. Seconds. Q. What do primes multiplied by seconds give? Q. What do seconds multiplied by seconds give?

A. Thirds.

A. Fourths. Q. What do seconds multiplied by thirds give? A. Fifths.

Q. What do thirds multiplied by thirds give? A. Sixths.

Note. This might be extended in the same manner to any indefinite length. The following table contains a few of these denominations.

Repeat the

12" (seconds) . 1' (inch or prime.)

12' (inches or primes) 1 foot.

A. In

Q. How may duodecimals be added and subtracted? the same manner as compound numbers; 12 of a less denomination always making 1 of a greater, as in the foregoing table.

MULTIPLICATION OF DUODECIMALS.

Q. What are duodecimals used for? A. For measuring any thing respecting which length and breadth, also depth, are considered. 1. How many square feet in a board 10 ft. 8 in. long, and 1 ft. 5 in. broad?

We have seen how such an example may be performed by common decimals we will now perform it by duodecimals.

OPERATION.

Length, 10 ft. 8'
Breadth, 1

4/

=

=

8 inches or primes of a foot and 5′ (primes) = 11⁄2 of a foot· then, X4 of a foot, that is, 40" (seconds) =3' (inches) and 4" (seconds): we now write down 4" at the right of the inches, reserving the 3' to be ca ried to the inches. In multiplying 10 feet by or 50' (inches), and the 3' we reserved 12 12

5'

4"

the 5', we say,

10X5:

=50

makes 53',4 feet and 5', which we place under feet and

inches in their proper places. Then, multiplying 10 ft. 8' by 1 ft. makes 10 ft 8', which we write under the 4 ft. 5'. W now proceed to add these two products together, which, by carrying 12, after the manner of compound rules, make lo ft. 1' (inch) 4 (seconds), the Answer.

It will be found most convenient in practice to begin by multiplying the multiplicand first by the feet, or highest denomination of the multiplier, then by the inches, &c., thus:

OPERATION.

10 t. 8/

1

5/

10

8'

4

15

5/ 4"

1X88', and 1 ft. X 10 ft. = 10 ft Then, 58403', (to carry,) and 4", (to write down); 10X550+3' (to carry)=53′ =4 ft. and 5', which we write down underneath the 10 and 8'. Then, the sum of these two products, added together as before, is 15 ft. 14" Ans., the same result as the other.

Note. Had we been required to multiply 15 ft. 1/4" by feet and inches again, we should have proceeded in the same manner, carrying (thirds) one place farther towards the right, and (fourths) another place still, and

30 on.

From these examples we derive the following

RULE.

How do you multiply in Duodecimals?

4. Begin with the highest denomination of the multiplier and the lowest denomination of the multiplicand, placing the first figure in each product one place further towards the right than the former, recollecting to carry by 12, as in com pound rules.

More Exercises for the Slate.

2. How many feet in a board 2 ft. 6' wide, and 12 ft. 3' long? ins. 30 ft. 7′ 6′′.

3. In a load of wood 8 ft. 4' long, 2 ft. 6' high, and 3 ft. 3 wide, how many solid feet? A. 67 ft. 8′ 6′′.

Note. Artificers compute their work by different measures. Glazing and mason's flat work are computed by the square foot; painting, paving, plastering, &c. by the square yard; flooring, roofing, tiling, &c. by the square o 100 feet; brick work by the rod of 16 feet, whose square is 272; the contents of bales, cases, &c. by the ton of 40 cubic feet; and the tonnage of ships by the ton of 95 feet

4. What will be the expense of plastering the walls of a room 8 ft. 6' high, and each side 16 ft. 3' long, at $,50 per square yard? A. $30,694.

5. How many cubit feet in a block 4 ft. 3' wide, 4 ft. 6′ long, and 3 ft. thick? A. 57 ft. 41 6".

6. How much will a marble slab cost, that is 7 ft. 4' long, and 1 ft. 3' wide, at $1 per foot? A. $9,163.

7. How many square feet in a board 17 ft. 7 long, 1 ft. 5' ide? A. 24 ft. 10' 41".

8. How many cubic feet of wood in a load 6 ft. 7' long, 3 ft. 5' high, and 3 ft. 8' wide? A. 82 ft. 5' 8" 4".

9. A man built a house consisting of 3 stories; in the upper story there were 10 windows, each containing 12 panes of glass, each pane 14' long, 12′ wide; the first and second stories contained 14 windows, each 15 panes, and each pane 16 long, 12 wide: how many square feet of glass were there in the whole house? A. 700 sq. ft.

10. What will the paving of a court yard, which is 70 ft. long, and 56 ft. 4' wide, come to, at $,20 per square?

A. $788,663. 11. How many solid feet are there in a stick of timber 70 ft. long, 15' thick, and 18' wide? A. 131 ft. 3.

Questions on the foregoing.

1. How many pence are there in 1 s. 6 d.? How many cents? 2. What will 4 yards of cloth cost, in cents, at 1 s. 6d. per yard? At 3 s. per yard? At 4 s. 6 d.? At 6 s.? At 9 s.? At 10 s. 6 d. ?

3. If a man consume 1 lb. 9 oz. of bread in a week, how much would he consume in 1 month?

4. At 4 cents for 1 oz., what would 1 lb. cost?

5. At 4 cents for 2 oz., what would 1 lb. cost?

6. At 4 cents for 8 oz., what would 2 lbs. cost?

7. If a man spend $2 per day, how many days would he be n spending $4 ? $63? $124 $20 ?

8. How many marbles, at 4 cents apiece, must be given for 24 apples, at 2 cents apiece?

9. How many yards of cloth, at $4 per yard, must be given for 6 bbls. of cider, at $2 per bbl.? For 8 bbls.? For 12 bbls.? For 18 bbls.?

10. What part of 1 month is 1 day? 2 days? 4 days? 5 days? 6 days? 7 days? 10 days? 20 days? 29 days?

11. What is the interest of $1 for 12 mo.? 10 mo.? 9 mo.? 6 mo.? 3 mo.? 1 mo.? 15 days?

12. What is the interest of $6 for 1 yr. 2 mo.? 2 yrs.? 1 yr. 1 mo.? 9 mo.? 2mo.? 1 mo.? 15 days? 10 days? 6 days?" 5 days? 1 day?

13. What is the amount of $1 for 6 mo.? 3 mo.? 2 mo.? 1 mo.? 15 days?

14. Suppose I owe a man $115, payable in 1 yr. 6 mo., withcut interest, and I wish to pay him now, how much ought I to pay him?

15. What is the discount of $115, for 2 yrs. 6 mo. ?

16. William has } of an orange, and Thomas ; what part of an orange do both own?