( 29 ).

chains 38 acres 2 roods and 33-152 poles the area of the field.

ACKNOWLEDGEMENTS, &c.

The following is an accurate list of the names and places of residence of the ingenious gentlemen who favoured us with true solutions to the questions proposed in No. 1. The figures at the ends of the lines refer to the questions answered by each as numbered in the work.

The prize Medal has been determined to Robert Adrain, York-Town, Pennsylvania.

N. B. We are sorry that the nature of this work, has com pelled us to acknowledge many excellent solutions, which otherwise would have been published at full length. It will always be our particular study to display the talents of our ingen ious correspondents, according to their respective merits: those therefore who have been unavoidably neglected in their first attempt, may depend on being remembered in their next, as far, as is consistent with the general plan and credit of our work.

As no person who is an editor of this work can receive prize medal, the merits of the different editors, are not to be Considered in awarding the prizes.

ARTICLE I. continued from page 15.

§ 21. In the Rule of Proportion, the conditions of the question, and the nature of proportional numbers, must shew that the answer required, is à fourth proportional to three numbers which are given in the question.

Stating the question, is the art of placing the three given numbers in such an order, that the first may be to the second, as the third is to the answer; this is done by the following

GENERAL RULE.

Write that number for the third term, which is of the same kind with the answer or number sought. Consider from the nature of the question whether this third term is greater or less than the answer; and if greater write the greater of the other two given numbers for the first term and the less for the second; but if less write the less for the first term and the greater for the second, and the question is stated.*

To find the answer or fourth term,

GENERAL RULE.

Case 1. If the first term be a multiple or a part of the second, the third will accordingly be a like multiple, or a like part of the fourth; and the answer will be found by division or by multiplication.

Case 2. If when a part of the first term is added to or subtracted from the first, the sum or the remainder be equal to the second, a multiple of the second, or a part of the second; then accordingly a like part of the third term being added to or sub

The truth of this rule is demonstrated in Ast. 1, § 7, of the Mathematical Correspondent.

tracted from the third, the sum or the remainder will be equal to the fourth term, a like multiple of the fourth, or a like part of the fourth. Hence the answer will be known or readily found by case 1. of this rule.

Case 3. In other cases reduce the first and second terms to the same denomination and consider them as pure numbers; then the product of the second and third terms divided by the first, gives the answer in the denomination of the third term.

REMARK,

In the 2d and 3d cases of this rule, the operations will frequently be rendered much shorter, by using like multiples or like parts of the first and second, or of the first and third terms instead of the terms themselves.*

PRACTICAL QUESTIONS.

1. Bought 63 yards of cloth for 1431. 6s. 6d. how much must be paid for 9 yards of the same cloth ?

1. s. d.

9

In this question it is evident that 1 yard is a part of 9 yards, and that the price of one yard is a like part of the price of 9 yards; also that 63 yards are a multiple of 1 yard, and that the price of 63 yards is a like multiple of the price of 1 yard. Hence (by art. 1. § 6 case 3 Math. Corr.) a fourth proportional to the three given numbers is here required: and consequently this and all similar questions belong to

*The first case of this rule depends on Art. 1. § 6 case 1 and 2 M. Corr. The second case is deduced from Art. 1. sections 8, 9, 10, 11, 12 and 13 Math. Corr.

The third case is founded on Art. 1. § 19 Math. Corr.

The remark is drawn from Art. 1. sections 15, 16, 17 and 18 M. Corr,

the Rule of Proportion. Those who are desirous of acquiring a rational knowledge of this rule, ought, before they attempt the solution of a question, thus to demonstrate to the satisfaction of their own minds, that such a question actually belongs to the rule.

In stating this question, it is easily perceived that the answer required is money; and therefore the given sum of money is here written for the third term. Again, it is evident, that this third term being the price of 63 yards, is greater than the answer or price of 9 yards; consequently 63 the greater of the other two given numbers is written for the first term, and the less 9 for the second and hence the question is stated according to our general rule.

In finding the answer, since the first term 63 is a multiple of the second 9, of consequence the third term is a like multiple of the fourth; therefore the third term multiplied by 7 gives the answer as above. divided

2. If 13 bushels of corn cost 12 doll. 75 cents, how much money will purchase 65 bushels at the same rate?

Here a part of the first term added to the first is equal to the second, and therefore a like part of the third term added to the third gives the answer.

4. When I can purchase 24 yards of linen for 97. 13s. 4d, how much ought I to pay for 21 yards of the same?

The question being stated according to the rule, it is easy to see that a part of the first term subtracted from the first, leaves a remainder equal to the second; and consequently a like part of the third term subtracted from the third leaves the answer.

5. I have 45cwts. of sugar, the value of which is £64; what quantity of this sugar.must I give to discharge a debt of £12?

Since a part of the first term, added to the first, is a multiple of the second; a like part of the third term, added to the third, is a like multiple of the

answer.

6. Bought 48 tons of logwood for 3960 dollars, how much money would have purchased 7 tons of the