when 30 men dig a trench 350 feet long, 6 wide and 3 deep, in 259-2 days? Ans. 7 hours.

The following Rule for the Double Rule of Three, involves the consideration of Direct and Inverse Proportion. Though the General Rule will enable the student to solve all questions with ease, this Rule is retained for the satisfaction of those who might desire to use it.

RULE.

Always place the three conditional terms in this order: That number, which is the principal cause of gain, loss or action, possesses the first place; that, which denotes the space of time, distance of place, rate, medium or mean of action, the second; and that, which is the gain, loss or action, the third: This being done, place the other two terms which move the question, under those of the same name, and if the blank place, or term sought, fall under the third place, then the question is in direct proportion: therefore,

RULE 1.*

Multiply the three last terms together, for a dividend, and the two first for a divisor:-But, if the blank fall under the first or second place; then, the proportion is inverse; therefore,

RULE II.

Multiply the first, second and last terms together for a dividend, and the other two for a divisor, and the quotient will be the an

swer.

EXAMPLES.

1. If $100 gain $6 in a year; what will $400 gain in 9 months? D. P Mo. D. Int.

100 12

6 Terms in the supposition, or conditional terms. 400: 9 Terms which move the question.

Here, the blank falling under the third place, the question is in direct proportion, and the answer must be found by the first Rule; therefore,

400× 9×6=21600 For the dividend, and,

1. When the blank falls under the third term by this mode of statement, it is obvious on inspecting the statement that the proportion is direct, and the same terms are taken to form the dividend and diviser as in the preceding rule, or, by two statements in the Single Rule of Three direct.

2. But in Example 2nd, and when the blank falls under the first or second term, the proportion is inverse. In this Example, more principal and interest require less time, and, every statement according to the rule will make more require less. The operation by the rule is the same as from two statements by The Single Rule of Three Inverse. These statements on this Example would be thus

2. If $100 will gain $6 in a year; in what time will $400 gain $18? D. Mo. D.

100 12 6 Terms in the supposition.

400 : :: 18 Terms which move the question.

Here, the blank falling under the 2d place, the question is in reciprocal or inverse Proportion, and the answer must be sought by the second rule; therefore,

100×12×18=21600 For the dividend.

Here, the blank falling under the first place, the proportion i inverse, and the answer found by the second rule, as in the last example.

5. If 8 men spend £32 in 13 weeks; what will 24 men spend in 52 weeks?

Ans. £384. 6. If the freight of 9hhds. of sugar, each weighing 12cwt. 20 leagues, cost $50; what must be paid for the freight of 50 tierces ditte, each weighing 24cwt. 100 leagues? Ans. $289 35c. 123m.

7. There was a certain edifice completed in a year by 20 workmen; but the same being demolished, it is necessary that just such an one should be built in 5 months. I demand the number of men to be employed about it? Ans. 48 men.

3. If 6 men build a wall 20 feet long, 6 feet high and 4 feet thick, in 16 days, in what time will 24 men build one 200 feet long, 3 feet high, and 6 feet thick? Ans. 80 days.

COMPARISON OF WEIGHTS AND MEASURES.

EXAMPLES.

1. If 78 pence Massachusetts be worth 1 French crown, how many Massachusetts pence are worth 320 French crowns?

F. cr. d. F. cr.

As 1 78: 320

78

2560

2240

24960 Ans.

2. If 24 yards at Boston make 16 ells at Paris, how many ells at Paris will make 128 yards at Boston?

16ells: 128yds.: 851ells, Ans.
at Amsterdam, how many pounds

3. If 60 at Boston make 561 at Boston will be equal to 350 at Amsterdam?

Ans. 375 Boston.

4. If 95 Flemish make 100 American, how many American pounds are equal to 550 Flemish ?

Ans. 5781 American.

CONJOINED PROPORTION,

IS when the coins, weights or measures of several countries are compared in the same question; or, in other words, it is joining many proportions together, and by the relation, which several antecedents have to their consequents, the proportion between the first antecedent and the last consequent is discovered, as well as the proportion between the others in their several respects.

This rule may generally be so abridged by cancelling equal quantities on both sides, and abbreviating commensurables, that the whole operation may be performed with very little trouble, and it may be proved by as many statings in the Single Rule of Three, as the nature of the question may require.

CASE I.

When it is required to find how many of the first sort of coin, weight, or measure, mentioned in the question, are equal to a given quantity of the last.

RULE.

Place the numbers alternately, that is, the antecedents at the left hand, and the consequents at the right, and let the last number stand on the left hand; then multiply the left hand column continually for a dividend, and the right hand for a divisor, and the quotient will be the answer.

EXAMPLES.

1. Suppose 100 yards of America=100 yards of England, and 100 yards of England 50 canes of Thoulouse, and 100 canes of Thoulouse 160 ells of Geneva, and 100 ells of Geneva 200 ells of Hamburgh: How many yards of America are equal to 379 ells of Hamburgh?

Therefore,

379X5

8=2363yds. of America 379 ells of Hamburgh.

ILLUSTRATION.

The two 100s of both sides cancel each other. Let the last cyphers of the next three antecedents and consequents be cancelled, which is dividing by 10. Then divide the second antecedent and consequent by 5, and the quotients will be 2 on the side of the antecedents, and 1 on the side of the consequents; then 2 will measure the third antecedent and consequent, and the quotients will be 5 and 8. 10 will measure the 4th antecedent and consequent, and the quotients will be 1 and 2. Now, there being 2 left on each side, they cancel each other, and as there is no farther room for abridging by reason of the odd number 379, the operation is finished, and the answer found, as before.

2. If 20b at Boston make 23b at Antwerp, and 155 at Antwerp make 180 at Leghorn: How many at Boston are equal to 144 at Leghorn? Ans. 10713.

at Amsterdam

3. If 12b at Boston make 10ib at Amsterdam, 10 12b at Paris: How many pounds at Boston are equal to 80ib at Paris? Ans. 801b.

4. If 140 braces at Venice be equal to 150 braces at Leghorn, and 7 braces at Leghorn be equal to 4 American yards: How many Venetian braces are equal to 32 American yards?

Ans. 52.

5. If 40 at Newburyport make 36 at Amsterdam, and 901 at Amsterdam make 116 at Dantzick: How many pounds at Newburyport are equal to 260 at Dantzick? Ans. 224

CASE II.

When it is required to find how many of the last sort of coin. weight or measure, mentioned in the question, are equal to a given quantity of the first.

RULE.

Place the numbers alternately, beginning at the left hand, and let the last number stand on the right hand; then multiply the first row for a divisor, and the second for a dividend.

EXAMPLES.

1. Suppose 100 yards of America=100 yards of England, and 100 yards of England 50 canes of Thoulouse, and 100 canes of Thoulouse 160 ells of Geneva, and 100 ells of Geneva 200 ells of Hamburgh: How many ells of Hamburgh are equal to 2367 yards of America?

This needs no further illustration. The learner will readily see, that this case being the reverse of the former, they are proofs to each other.

2. If 2015 at Boston make 23b at Antwerp, and 155 at Antwerp make 180 at Leghorn: How many at Leghorn are equal to 141 at Boston ? Ans. 114. 3. If 12 at Boston make 10b at Amsterdam, and 100b at Ainsterdam 120 at Paris: How many at Paris are equal to 80lb at Boston? Ans. 80.

4. If 140 braces at Venice be equal to 150 braces at Leghorn, and 7 braces at Leghorn be equal to 4 American yards: How many American yards are equal to 52 Venetian braces?

Ans. 32 yards.

5. If 401b at Newburyport make 36 at Amsterdam, and 90 at Amsterdam make 116 at Dantzick: How many pounds at Dantzick are equal to 241 at Newburyport ? Ans 23312.

ARBITRATION OF EXCHANGES.

By this term is understood how to choose, or determine the best way of remitting money from abroad with advantage; which is performed by conjoined proportion: Thus,