from the highest term to the lowest, remembering to place the remainder of each product, one place or degree to the right hand of the former, as its proper index will direct, and so continue, till you have multiplied all the terms of the multiplicand by each term of the multiplier, separately, then add the several products together, as in compound addition, and their sum will be the answer.

67 11 6 product by the feet in the multiplicand. 7 3 4 6"" product by the inches, or primes. 2 5 1 6"" product by the seconds.

Feet 75 5 3 7 6""" Ans.

2. What is the product of 8 feet, 4', 8", 6"", multiplied by 9', 4", 8"? Ans. 6 feet, 6', 9'', 6''', 5'''', 8"'""'.

3. What is the product of 12 feet, 6', multiplied by 9', 4", Ans. 9 feet, 8', 11", 1"'"', 6''''.

3""? 4. What is the product of 24 feet, 9', 8", multiplied by 1 foot, 64'? Ans. 37 feet, 8', 9", 2""', 1'"', 6''. 5. Multiply 19 feet, 11', into itself, and what is the product? Ans. 396 feet, 8', 1". 6. How many solid feet, in a block 3 ft. 8 in. wide, 3 ft. 5 in. thick, and 6 ft. 7 in. long?

7. How many square feet, in a board and 1 foot, 5 inches wide?

8. How many solid feet, in a cubic

Ans. 82 ft. 5', 8", 4"". 17 feet, 7 inches long,

Ans. 24 ft. 10', 11". block 6 feet, 9' square? Ans. 307 ft. 6', 6", 9"". 9. How many cubic feet, in a stick of timber 12 feet, 10 inches long; 1 foot, 7 inches vide; and 1 foot, 9 inches thick? Ans 35 ft. 6', 8", 6".

10. What must I give for 6 boards, each measuring 14 ft. in length, and 1 ft. 6 in. in breadth, at 24 cts. a ft.? Ans. $2,831. 11. What will a stick of timber be worth, measuring 36 feet, 9 inches long, 9 inches wide, and 7 inches thick, at 16 cents the cubic foot ? Ans. $2,75+cts.

12. Bought a stock of boards, 9 in number, each measuring 9 ft. 6 in. long, and 1 ft. 7 in. wide; what did they all cost at 24 cents a foot, and how many feet did they measure?

Ans. They measured 135 ft. 44 in. Cost $3,384 mills.

DUODECIMALS APPLIED TO MEASURING WOOD.

Q. How is wood estimated, in buying and selling?

A. By the cord, or by the cord-foot.

Q. What do you understand by a cord-foot of wood?

A. Sixteen solid or cubic feet; that is, 4 feet long, 4 feet wide, and one foot high, make a cord-foot.

Q. How many such cord-feet make one cord of wood?

A. Eight; that is, 8 feet long, 4 feet wide, and 4 feet high, make one cord.

Q. How do you measure a load of wood, in a cart or wagon? A. If the wood be of customary length, (8 feet,) multiply the average width by the height, and half the product will be the number of cord-feet.

Q. If the load be not 8 feet long, how do you measure it? A. Multiply the length by the average width, and that product by the height, then divide this last product by 16, and the quotient will be the number of cord-feet.

in

Q. How do you find the number of cords of wood, contained any number of cord-feet?

A. Divide the number of cord-feet by 8, and the quotient will be the number of cords.

EXAMPLES.

1. How many feet of wood, in a load 3 feet, 9 inches wide, 4 feet, 3 inches high, and of full length?

Ft.

Feet 7 11 7 Ans.

2. How many feet, in a load of wood, 7 feet long, 4 feet, 3 inches wide, and 5 feet, 9 inches high; and what must you pay for it, at $5,50 cents a cord?

Operation. 7 ft. X4 ft. 3'x5 ft. 9'÷16-11 ft. 5 in.+ Ans. 11 ft. 5 inches, and will cost $7,848 m. 2. How much wood in a load, and what must you give for asuring 3 ft. 4 in. in width, 2 ft. 6 in. high, and full paying at the rate of $6 the cord?

1

Ans. 4 ft. 2 in.

Must give $3,12).

4 How much wood in a load, 12 ft. long, 3 ft. 8 in. wide, and 4 ft. 10 in. high; and how much must I pay for it, at $6,50 cents a cord? Ans. 13 ft. 3 in. Must pay $10,799 m.

5. A poor woman bought a load of wood, measuring 3 ft. in length, 2 ft. 8 in. in width, and 23 feet in height; how much wood was there, and what must she pay for it, at the rate of $4,373 a cord? Ans. 1 ft. 4 in. Amount, 74 cents, 4 m.+

6. How much wood can be put into a wagon, whose body is 8 ft. 9 in. long, 4 ft. 3 in. wide, and the stakes 5 ft. 8 in. high; and what will it be worth, at $63 a cord?

Ans. 1 cord, 5 ft. 2 in. Is worth $10,49 cts.

7. Bought a load of wood, measuring 7 ft. 8 in. in length, 3 ft. 9 in. in width, and 5 ft. 7 in in height; how many feet of wood in the load, and what must I pay for it, at the rate of $7 a cord. Ans. 1 cord, 2 feet. Must pay for it $8,75+. 8. If I buy a load of wood, measuring 7 ft. 8 in. in length, 4 ft. 6 in. wide, and 3 ft. 10 in. high, how many feet does it contain, and what must I pay for it, at the rate of $6,75 cents the cord? Ans. 84 feet. Must pay $6,96.

9. Two persons bought a load of wood together; A.'s end of the load measured 3 ft. 9 in. wide, 4 ft. 6 in. high, and 4 ft. 3 in. long; B.'s end measured 3 ft. 7 in. wide, 4 ft. 11 in. high, and 4 ft. 3 in. long; how much wood had each, and what must each pay, at the rate of $5,50 a cord?

Ans.

A. had 4 feet, 5 inches+. Must pay $3,07+.
B. had 4 feet, 8 inches+. Must pay $3,20.+

ARITHMETICAL PROGRESSION.

Q. What is an Arithmetical Progression?

A. It is any rank of numbers, more than two, increasing by a common excess, or decreasing by a common difference.

Q. What is such a rank of numbers, as 2:4:6:8:10: 12, &c. called?

A. It is called an ascending arithmetical series.

Q. What is such a rank of numbers, as 12:10: 8: 6:4:2 &c. called?

A. It is called a descending arithmetical series.

Q. What is to be understood, when you speak of the terms of an arithmetical progression ?

A. It is the numbers, or rank of numbers, that form the series.

Q. How many parts are always included in an arithmetical series?

A. Five parts, viz.: 1. The first term. 2. The last term. 3. The number of terms. 4. The common difference. 5. The sum of all the terms.

Q. What is meant by the extremes of a progression ?

A. The first term and the last term are called the two extremes. The other terms are called the means.

Q. How many parts of an arithmetical series, must always be given, to find any other part required?

A. Three. By having any three parts given, the other two may be easily found.

CASE FIRST.

Q. What is the first case in arithmetical progression?

A. It is when one of the extremes, the common difference, and the number of terms of an arithmetical series, are given, to find the other extreme.

Q. What is the RULE in this case?

A. Subtract one from the number of terms, and multiply this remainder by the common difference; then (if the greatest or last term be required) add the first, or least term to the product, and the sum will be the last term, or answer; but, if the first or least term be required, subtract the product from the greatest or last term, and the remainder will be the first term.

EXAMPLES.

1. If the least term of an arithmetical series be 3, the common difference 2, and the number of terms 9, what will be the greatest or last term?

Operation. 9-1-8 number of terms less by 1.

X2 common difference.

16

+3= the first or least term.

Ans. 19= the greatest or last term.

2. If the greatest term of a series be 70, the common difference 3, and the number of terms 21, what is the least term? Operation. 21-1-20= number of terms, less by 1. x3= common difference.

Therefore

70-60-10 the first term, or answer.

3. If a man pay one dollar the first week, 3 dollars the secondok, and continue to increase his payment 2 dollars, evfor the term of one year; what will be the amount

52d payment?

Ans. $103.

4. A debt was discharged in a year, by paying a certain sum the first week, and increasing each weekly payment by $2, the last payment being $103; what was the first? Ans. $1.

5. A. owes to B. a certain sum of money, which he agrees to pay in arithmetical progression; the first payment to be $3, the number of payments 24, and the common difference 2; what will be the amount of the last payment? Ans. $49. 6. If a man travel 15 days, and increase each day's journey 4 miles, and the last day travel 75 miles, how far must he have travelled the first day? Ans. 19 miles.

CASE SECOND.

Q. What is the second case in arithmetical progression ? A. It is when the two extremes and common difference are given, to find the number of terms?

Q. What is the RULE in this case?

A. Divide the difference of the extremes, by the common difference, add 1 to the quotient, and the sum will be the number of terms.

EXAMPLES.

1. If the two extremes are 3 and 27, and the common dif ference 2, what is the number of terms?

Operation. 27-3÷2+1=13 the number of terms.

2. A man travelled 15 miles the first day, and increased his journey 3 miles a day, until he travelled 75 miles in a day; how many days did he travel? Ans. 21 days.

4. A man discharged a debt by weekly payments, and paying one dollar the first week, increasing his payment 5 dollars each week, until his last payment was 81 dollars; how many payments did he make ?

CASE THIRD.

Ans. 17.

Q. What is the third case in arithmetical progression? A. It is when the extremes and number of terms are given, to find the common difference.

Q. What is the RULE in this case?

A. Divide the difference of the extremes, by the number of terms, less 1, and the quotient will be the common difference.

EXAMPLES.

1. In an arithmetical series, the extremes are 3, and the number of terms, 13; what is the common differen Operation. 27-3-13-1-2, the common differen