PROBLEM II.

Division by the lines A and B.

RULE.

1. As the divisor on B is to unity on A, so is the dividend on B to the quotient on A.

Or,

2. As the divisor on A is to unity on B, so is the dividend on A to the quotient on B.

Note 1. When the divisor on B is set to unity on A, if the dividend cannot be properly expressed on B, according to the true numeration of the Rule, you must divide it by some power of 10, and multiply the quotient by the same number by which you divide the dividend. (See the first Note in the last Problem.)

2. If the divisor and dividend be both whole numbers, and the divisor less than the dividend, the quotient will be a whole, or a mixed number; but if the divisor be greater than the dividend, the quotient will be a decimal. Also, if the divisor be a whole number, and the dividend a decimal, the quotient will be a decimal; if the divisor be a decimal, and the dividend a whole number, the quotient will be either a whole, or a mixed number; and if both the divisor and dividend be decimals, the quotient will be a decimal when the divisor is greater than the dividend; but when it is equal to, or less than the dividend, the quotient will be either a whole, or a mixed number.

3. When the divisor is greater than the dividend, you must conceive a competent number of cyphers to be placed on the right of the dividend, as decimals; and you must always remember that there will be as many decimal places in the quotient as the dividend contains more than the divisor.

4. It is sometimes difficult to know how many whole numbers, and how many decimals there should be in the quotient; but this may easily be ascertained by inspection: Thus, if it be required to divide 37 by 3; it is evident that there will be two whole numbers in the quotient, and the rest of the figures will be decimals.

Again, let it be required to divide 32 by 46. Here it is evident that there must be a cypher placed on the right of the dividend, before it can contain the divisor; hence, all the quotient will be decimals.

Lastly, let it be required to divide 22 by 365. Here it appears that two cyphers must be affixed to the dividend, before we can divide it by the divisor; hence all the quotient will be decimals, and the first figure on the left will be a cypher.

EXAMPLES,

1. What is the quotient of 8 divided by 2?

Set 2 on B to 1 on A, then against 8 on B, is 4 on A ;
Or, by Proportion,

As 2 on B: 1 on A:: 8 on B: 4 on A; Or, As 2 on A:

1 on B :: 8 on A: 4 on B.

2. Divide 42 by 7.

Ans. 6.

3. Required the quotient of 96 divided by 8.

Ans. 12.

4. Divide 1536 by 24.

Ans. 64.

5. What is the quotient of 33920, divided by 128?

Ans. 265.

6. Divide 45.05 by 5.3.

Ans. 8.5.

7. What is the quotient of 16.45 divided by 4.7?

Ans. 3.5.

8. Divide 14.88 by 2.4.

Ans. 6.2.

9. Required the quotient of 1.62 divided by .9.

Ans. 1.8.

10. What is the quotient of 2122.16 divided by 32.8?

11. Divide 35 by 125.

Ans. 64.7.
Ans. 28.

12. What is the quotient of 15 divided by 632?

Ans. .0237.

PROBLEM III.

To find a fourth proportional to three numbers; or to perform the Rule of Three by the lines A and B.

RULE.

As the first term on A, is to the second on B; so is the third term on A, to the fourth on B.

Note. The method of finding a third proportional is exactly the same; the second number being twice repeated: Thus, if a third proportional be required to 80 and 60; it will be as 80 on A : 60 on B: 60 on A: 45 on B, the third proportional sought.

EXAMPLES.

1. If 12 yards of cloth cost 56s. what will 18 yards cost at the same rate?

As 12 on A: 56 on B:: 18 on A ; 84 on B. 2. If 44 gallons of ale cost 5ēs. what will 8 gallons cost? Ans. 10s.

3. If 24 candles weigh a pound, what is the weight of 1200 candles? Ans. 150 lb. 4. What is the fourth proportional to the three numbers 12, 24, 36?

Ans. 72.

5. If 31 bushels of malt cost 28s. 6d. what will 8{ bushels cost? Ans. 76.73s. 6. If a gallon of rum cost 19s. 9d. what will 451 gallons cost? Ans. 898.6s.

PROBLEM IV.

Inverse Proportion by the lines A and B.

RULE.

As the third term on A, is to the first on B, so is the second on A, to the fourth on B.

EXAMPLES.

1. If 20 workmen can build a brew-house in 180 days; how many workmen can build it in 90 days?

da.

men.

da.

men.

As 90 on A: 20 on B:: 180 on A: 40 on B. 2. How many gallons of rum at 16s. per gallon, are equal in value to 8 gallons of brandy at 20s. per gallon? Ans. 10 gallons,

3. If 8 men can do a piece of work in 18 days, in how many days can 48 men do it? Ans. 3 days. 4. If 2 Excise Officers can gauge and fix a certain number of utensils in 12 days, in how Officers gauge and fix them?

PROBLEM V.

many days can 6 Ans. 4 days.

To reduce a vulgar fraction to a decimal, by the lines

A and B.

RULE.

As the denominator on A, is to unity on B, so is the

numerator on A, to the required decimal on B.

EXAMPLES.

1. Reduce to a decimal fraction. Denom. Unity. Num.

Decimal.

As 4 on A 1 on B:: 1 on A: .25 on B.

Having a divisor given, to find, by the lines A and B, a factor that shall perform the same by Multiplication as the divisor would do by Division.

.RULE.

As the given divisor is to unity, so is unity to the factor sought.

EXAMPLES.

1. If the divisor be 125, what will be the factor to that number?

As 125 on A: 1 on B: 1 on A: .008 on B, the factor sought.

2. If the divisor be 282, what will be the factor?

3. The divisor is 231, what is the factor?

Ans. .003546.

Ans. .004329.

4. What is the factor to the divisor 2150.42?

Ans. .000465.

PROBLEM VII.

Having a factor given to find a divisor by the lines A and B.

RULE.

As the factor is to unity, so is unity to the divisor.

EXAMPLES.

1. Find a divisor to the factor .008.

As .008 on A; 1 on B :: 1 on A: 125 on B, the divisor sought.

2. The factor is .003546, what is the divisor?

3. If a factor be .004329, what is the divisor?

Ans. 282.

Ans. 231.

4. Find a divisor to the factor .000465. Ans. 2150.42.

PROBLEM VIII.

To square any number by the lines C and D.

RULE.

Set 1 on D to 1 on C, then against the given number on D, is its square on C.

EXAMPLES.

1. What is the square of 3?

Set 1 on D to 1 on C; then against 3 on D, is 9 on C.

2. Required the square of 12. 3. What is the square of 17.5?

4. Find the square of 125.8.

PROBLEM IX.

Ans. 144. Ans. 306.25.

Ans. 15825.64.

To extract the Square Root of any number by the lines C and D.

RULE.

Set 1 on C to 1 on D, then against the given number on C, is its root on D.

EXAMPLES.

1. What is the square root of 9?

Set 1 on C to 1 on D; then against 9 on C, is 3 on D.

E