number for the second operation; the bottom number of the fuppofition forms the first number, and the bottom number of the demand the third number; then the answer, or fourth number of this fecond operation, is the true answer to the queftion; as in the following example, which is the foregoing one at large.

If 12 months gain 77. what will 9 months?

The answer would have been the fame if the number of months had been the 'uppermost numbers inftead of the pounds principal; in which cafe the first question would be, if 12 months give 57. intereft, what will 9 months give? and the answer is 37. 15s.; then the fecond queftion would be, if fool. gain 37. 155. what will 1407. gain? and the answer is as before, 51. 55.

1

In this example, the work is ftated as in the former; the two first numbers in the fuppofition are multiplied together, and the two numbers in the demand are also multiplied together, then thefe two products are made, the one the first, and the other the third number, and the second number in the first question is also the second number. This rule is the moft fure and practical method of proving the double rule of three direct, when wrought by two fingle

ones.

The foregoing example worked both ways will be suffi cient to inftruct the learner. I fhall, therefore, give a few queftions, with their answers, omitting the operation.

Qu. 2. Suppofe 468 men confume 175 quarters of wheat in 168 days, I demand how many quarters will serve 5612 men 58 days?-Anf. 724 quarters, and 4 of a quarter, or a little more than half a quarter.

024

627

Qu. 3. Suppofe So acres of grafs be mowed by 8 men in 14 days, I deinand how many acres 28 men will mow in 12 days?—Anf. 240 acres.

Qu. 4. Suppose the wages of 12 men for 6 days amount to 7. 45. what are the wages of 25 men for 40 days?-Anf. 100%.

Qu. 5. If 150l. principal put out to interef for 9 months

be increased, principal and intereft, to 1564. 15s, I demand how much is that per cent. per annum ?—Anf. gl.

SECT. X.

OF THE DOUBLE RULE of thREE INVERSE.

THE double rule of three inverfe is when there are five given numbers to find a fixth, in an inverted proportion.

Rule

Rule. Place the numbers as directed in the laft fection. Multiply the lower number of the firft place by the upper one of the third, and make the product the first number; next multiply the upper term of the first place by the lower one of the third, for the third number: then if the inverse proportion be found in the three upper numbers, the answer is given by one operation in the rule of three direct; but if the inverse proportion be found in the lower numbers, the work is performed by the inverse rule (for of every sum in this rule one question is direct and the other inverfe)."

Example 1. If 1004. gain 57. iutereft in 12 months, what principal will gain 51. 55. in 9 months?

The pounds intereft being reduced to fhillings, and multiphed by the number of months, the question will stand, and operation be performed as follows:

If the number of months had been made the upper terms, the upper proportion would then have been direct, and would have been required to have been worked by the direct method. It would in that case stand thus:

*The rule laid down in this fection will be found quite general, and fufficient for working all questions in the double rule of three both direct and inverfe; and is fo obvious, as to require no demonstration. Nevertheless, in confequence of receiving advice from fome teachers (not the most competent) of the mathematics, that I had not given the improved method of working this rule, I shall state this much-approved rule, verbatim, from a well-known treatise, and fhew its fallacy.

Qu. 2. If 481. ferve for the maintenance of 12 men S days, how long will 2881. ferve for 4 men?-Anf. 144 days. Qu. 3. If, when a bufhel of wheat cofts 6s. 8d. a penny loaf weighs 6 ounces, how much will a loaf weigh that costs 10d. when the wheat is 10s. the bufhel?-Anf. 42 ounces.

"Rule 1. Let the principal caufe of lofs or gain, intereft or decrease, action or paffion, be put in the first place.

66 2. Let that which betokeneth time, diftance, or place, and the like, be in the second place; and the remaining one in the third.

"3. Place the other two terms under their like in the supposition.

re

?

4. If the blank falls under the third term, multiply the firft and fecond terms for a divifor, and the other three for a dividend; but,

"5. If the blank falls under the first or fecond term, multiply the third and fourth terms for a divifor, and the other three for the dividend; and the quotient will be the answer.

66 Proof. By two fingle rules of three.

"Example 1.

If 14 horfes eat 56 bufhels of oats in 16 days, how many bushels will be fufficient for 20 horfes for 24 days?.

That this rule is not founded on mathematical principles is evident from infpection; for by a different ftatement of the queftion (though exactly agreeable to the rules) a different anfwer will arife. Thus, how eafily might the learner, required to work this queftion by one ftate. ment, order the numbers as follows:

hor.

bu. da.

14: 56: 16

20: -:24

56×24 × 14.
=584
16X20

According to this ftatement the answer will be 58 bushels; whereas, the true anfwer is 120 bufhels. The queftion is, nevertheless, ftated

in this laft cafe agreeable to the rules.

Qu.

Qu. 4. If 96 pioneers in 24 days caft a trench 96 yards long, how many pioneers will caft a trench 336 yards long in 8 days?-Anf. 1008 pioneers.

Qu. 5. If 10 men mów 40 acres in 8 days, how many days will it require 3 men to mow 150 acres?-Anf. 100 days.

This rule may be proved by two fingle rules.

The following rules in this chapter, except vulgar fractions and the extractions of roots, are wrought either by fome of the rules of proportion, delivered in the four preçeding fections, or by the rule of practice, hereafter to be spoken of.

SECT. XI.

OF FELLOWSHIP.

FELLOWSHIP is that rule whereby merchants and others, trading in company, and employing a joint capital stock, are enabled to afcertain each partner's particular loss or gain, according to his fhare in the fame joint stock.

This rule alfo ferves to divide a bankrupt's eftate among his creditors; to pay legacies, when there is a deficiency of the teftator's effects: and, in fine, to divide the lofs or profit of any joint concern among the lofers or proprietors.

The rule of fellowship is either fingle or double, that is, without regard to time, or with time.

Fellowship without time, or fingle fellowship, is when different perfons employ their refpective ftocks for the fame time.

Rule.