14. What is the difference between .177. and .7s.? Ans. 2s. 8d. 1.6 q.

15. What is the difference between .41 of a day, and .16 of an hour? Ans. 9h. 40 min. 48 sec.

CASE III.

To reduce any given quantity to a decimal of a higher denomination.

RULE.

I. When the given quantity consists of several denominations, proceed as follows:

1. Write the given numbers under each other, for dividends; proceeding orderly from the least denomination to the greatest.

2. Opposite to each dividend, on the left hand, place such a number, for a divisor, as will, (according to the rule for Reduction Ascending,) bring the dividend to the next superior denomination.

3. Divide each dividend, beginning at the top of the column, by its divisor, and write the quotient of each division as decimal parts on the right hand of the dividend next below it, and the last quotient will be the decimal sought.

Note 1.-In dividing in this manner, you must find as many decimal figures in each quotient as are required to be in the decimal sought, when the true quotients will contain so many figures.

II. When the given quantity consists of one denomination only, it may be reduced to a decimal as directed above; or it may be divided, at once, by such a number as will reduce it to the decimal required.

Note 2.-Cases 2d and 3d prove each other.

EXAMPLES.

1. Reduce 15s. 6d. to the decimal of a pound. Operation.

12 6 d.

Here, I first set down the 15s. below the 6d. and place the proper divisors against these numbers. Inext divide the 6d. by 12, to reduce them to the decimal of a shilling, and the quotient is .5 s., which I annex to the

15s. I then divide the 15.5 s. by 20, to reduce them to the decimal of a pound, and the quotient, .7757., is the answer.

The student must remember to prefix ciphers to the quotients, when necessary, to make up the proper number of decimal places, according to the rule for the division of decimals.

2. Reduce 11 oz. 8.832 dr. to the decimal of a pound, Avoirdupois. Ans. .722 lb.

3. Reduce 7d. 2q. to the decimal of a shilling.

Ans. .625s,

4. Reduce 10 rods to the decimal of a mile. By the 2d part of the Rule thus, 10-320 (the number

of rods in a mile)=.03125 m. Ans.

5. Reduce lin. 1.5b. c. to the decimal of a foot.

Ans. .125 ft.

6. Reduce 2 roods, 1 sq. rod, to the decimal of an acre.

Ans. .50625 A.

7. Reduce 27 sq. inches to the decimal of a sq. foot.

Ans. .1875 sq. ft.

8. Reduce 16 cub. feet to the decimal of a cord.

Ans. .125 Cd.

9. Reduce 1 pt. 1.824 gil. to the decimal of a gallon.

Ans. 182 gal.

10. Reduce 5h. 2min. 24 sec. to the decimal of a day.

Ans. .21 da.

11. Reduce 54 seconds to the decimal of a degree. Ans. .015 deg.

QUESTIONS ON THE FOREGOING.

1. What is a decimal fraction? 2. In what manner are decimal fractions written? 3. What is a mixed number? 4. How are decimals enumerated? 5. What does the first figure to the right hand of the decimal point denote? the second? the third? &c. 6. What effect have ciphers when placed at the right or left hand of the significant figures of a decimal? 7. What is the rule for the addition of decimals? 8. How is the subtraction of decimals performed? 9. How are decimals multiplied together; and what is the rule for placing the decimal point in the product? 10. What is the shortest method of multiplying a decimal, or mixed

number, by 10, or 100, &c.? 11. How is the division of decimals performed; and what is the rule for placing the decimal point in the quotient? 12. What is the shortest method of dividing a decimal, or mixed number, by 10, or 100, &c.? 13. What are the denominations of Federal Money? 14. In what ratio do these denominations increase? 15. If dollars be considered as integers, what will each of the lower denominations be? and what will eagles be? 16. Which of these denominations is usually considered the integer; and how are sums of federal money usually written? 17. Is it customary, in reckoning in federal money, to mention the names of eagles and dimes? 18. How do we write down, in the decimal form, a part of a dollar which consists of any number of cents less than 10, or any number of mills less than 10? 19. How are Addition, Subtraction, Multiplication, and Division of Federal Money performed? 20. What is the method of reducing a vulgar fraction to a decimal? 21. What is the rule for reducing a decimal fraction of any of the higher denominations of money, &c. to its value in the lower denominations? 22. What is the method of reducing a compound quantity to an equivalent decimal of a higher denomination? 23. How is a simple quantity reduced to a decimal of a higher denomination?

PROPORTION,

Is the relation which one quantity has to another. Numbers are compared to each other in two different ways: One comparison considers the difference of the two numbers, and is named Arithmetical Relation, and the difference is sometimes called the Arithmetical Ratio: the other considers their quotient, which is called Geometrical Relation, and the quotient, the Geometrical Ratio.* So, of these two numbers, 8 and 2, the difference, or arithmetical ratio, is 8-2, or 6, but the geometrical ratio is 8+2, or 4.

*It may be proper to inform the learner that, in this Work, where mention is made of the ratio of two quantities, the geometrical ratio is intended, excepting where it is otherwise expressed.

There must be two numbers to form a comparison: the number which is compared, being placed first, is called the antecedent; and that to which it is compared, the consequent. So, of the two numbers above, 8 is the antecedent, and 2 the consequent.

Note.-Arithmetical and Geometrical Proportions will be treated of at large in a subsequent part of this Treatise; but I shall here explain so much of the latter as is necessary to give the learner a correct idea of the nature of the following Rule of Proportion, called the Rule of Three.

A geometrical ratio is usually denoted by writing a colon between the two terms, (that is between the antecedent and the consequent,) of the ratio. Thus, 3: 5, denotes the ratio of 3 to 5.

Four quantities are said to be in geometrical proportion, when the ratio of the first to the second is equal to the ratio of the third to the fourth. Thus, the four numbers 8, 4, 14, 7, are proportional, because the ratio 8: 4 is equal to the ratio 14:7; that is, 8÷4 is equal to 14÷7, equal to 2. The equality or identity of two such ratios is usually denoted by writing a double colon between the ratios: Thus, 2:36:9, denotes that the ratio of 2 to 3 is the same with, or equal to, the ratio of 6 to 9: read thus, as 2 is to 3, so is 6 to 9. Such a series, consisting of four terms in geometrical proportion, is called an Analogy; the first and last terms being called the extremes, and the other terms, the means.

In any analogy, the product of the extremes is equal to the product of the means; that is, the product of the first and fourth terms is equal to the product of the second and third terms.* Thus, in the analogy 2: 4 :: 3 : 6, the pro

* This proposition, or assertion, may be demonstrated as follows: Let a, b, and r, represent any three quantities whatever; then, a: ar :: b: br, will denote any analogy, or any four quantities in geometrical proportion; r being the ratio, or quotient, of the first and second, and of the third and fourth terms. Now, axbrar b=abr; that is, the product of the extremes is equal to the product of the means. Q. E. D.

Corollary-Hence, if the product of the two mean terms of any analogy be divided by either of the extremes, the quotient will be the other extreme; and, if the product of the extremes be divided by either of the mean terms, the quotient will be the other mean; for it is evident bat if the product of any two numbers be divided by one of the num

duct 2×6 is equal to the product 4×3, equal to 12.-Hence, if the product of the two mean terms of any analogy be divided by either of the extremes, the quotient will be the other extreme; which is the foundation of the following Rule of Simple Proportion, commonly called the Rule of Three.

The Rule of Proportion is divided into Simple and Compound.

Simple Proportion is a single analogy, or the equality of the ratio of two quantities to that of two other quantities; as 2: 6 :: 8:24.

Compound Proportion is the equality of the ratio of two quantities to another ratio, the antecedent and the consequent of which are respectively the products of the antecedents and consequents of two or more ratios; as 2:3

:: 6:7, in which the ratio 6: 7 is equal to a ratio 9:75 whose antecedent is 2X9, or 18, and its consequent 3×7, or 21.

SIMPLE PROPORTION, OR

THE SINGLE RULE OF THREE.

The Rule of Simple Proportion, or Rule of Three, teaches how to find the fourth term of any analogy from three given terms. It is called the Rule of Three, because three terms or numbers are given to find a fourth; and, because of its great and extensive usefulness, it is sometimes called the Golden Rule.

In every question belonging to the Rule of Three, two of the three given terms, or numbers, are contained in a supposition, and the other in a demand; and hence the former are called the terms of supposition, and the latter the term of demand. One of the terms of supposition is always of the same kind or quality with the demanding term, and the bers, the quotient will be the other number. Consequently, if any three of the terms of an analogy be given, or known, the other term may be found; which admits of four cases; viz. 1st, when the two mean terms and the first term are given, to find the fourth: 2d, when the two mean terms and the fourth term are given, to find the first: 3d, when the extremes and the second term are given, to find the third: 4th, when the extremes and the third term are given, to find the second.