the number of men, or other creatures, it would ferve a certain time, to find the number of men, or other creatures, it would ferve any other time; or, from the quantity of provifion and the number of confumers, to find the time it would ferve any other number of confumers (of this nature is the fixth example).

SECT. IX.

OF THE DOUBLE RULE OF THREE DIRECT

In this rule there are five given numbers to find a fixth, which fhall bear the fame proportion to the product of the fourth and fifth numbers, as the third number bears to the product of the first and fecond.

Questions in this rule are refolved either by two operations in the fingle rule of three direct, or the rule of three compofed of five given numbers.

Each question in this rule confifts of two parts, the Suppofition and the demand.

Rule 1. By two operations. Place that number which is of the fame nature with the fixth number, or anfwer, in the fecond place in the firft operation; and the two other numbers in the fuppofition in the firft place, the one over the other; and the two numbers in the demand in the third place, one over the other, in like manner as the two in the first place; obferving that the bottom numbers in the first and third places be of like nature, as will also the top ones.

* Some modern writers compound the two double rules of three into one; I have, however, given them diftinctly, being more conforant to the true theory of science. In the next Section is, neverthelefs, fhewn an infallible method of working both by one rule.

The

The work is then performed by two operations in the fingle rule of three direct; the answer to the first operation forming the second number in the fecond operation.

2. By one operation. Multiply the two numbers which ftand one over the other in the supposition, together, for the first number; the two numbers in the demand for the third number; and the second number in the first question will be alfo the fecond number in the work. Then the answer is found by one operation in the fingle rule of three direct, as in the following example:

Example 1. If 100l. principal gain 57. interest in 1 yeàr, what will 1407. gain in 9 months? The Question ftated.

If 100l. gain 57.

What will 140%.

gain in 9 months?

in 12 months, In this example, the numbers 100, 5, and 12, belong to the fuppofition, and 140 and 9 is the demand; for the meaning of the work is, fuppofe 100l. gain 51. interest in 12 months, (then follows the demand) I demand to know how much 140% will gain in 9 months at the fame rate of intereft?

Thus the queftions are ftated according to the foregoing directions: the 51. being the intereft of the money (and of the fame nature with the fixth number or anfwer) must be the second number; and the two other numbers in the fuppofition 100 and 12 are placed one above the other, as are the two numbers 140 and 9 in the demand. It matters not which of these two numbers is uppermoft, provided that the numbers in each, which are of the fame nature, occupy the corresponding places respectively: thus, in the fuppofition, the pounds principal is the uppermost number, fo it is in the demand, and the number of months is undermoft in both.

The question being thus ftated, the work is wrought by two operations of the fingle rule of three direct. The three uppermost numbers are the numbers for the firft operation, and the fourth number, or answer to thefe forms, the fecond

number

number for the fecond 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.

Firft Operation.

If 100. gain 51. what will 1407.

1,00)7,00(7

Second Operation.

If 12 months gain 77, what will 9 months?

The answer would have been the fame if the number of months had been the uppermoft numbers instead of the. pounds principal; in which cafe the firft queftion would be, if 12 months give 57. intereft, what will 9 months give? and, the answer is 3. 155.; then the fecond queftion would be, if 100%. gain 37. 155. what will 14c. gain? and the answer is as before, 51. 5s.

By one Operation.

If rool. gain 51. what will 1407. gain? months 12

1200

9 months 1260

In this example, the work is stated as in the former; the two firft numbers in the fuppofition are multiplied together, and the two numbers in the demand are also multiplied together, then these 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 fufficient to inftruct the learner. I fhall, therefore, give a few queftions, with their anfwers, omitting the operation.

Qu. 2. Suppofe 468 men confume 175 quarters of wheat in 168 days, I demand how many quarters will ferve 5612 men 58 days?-Anf. 724 quarters, and 64 of a quarter, or a little more than half a quarter.

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

Qu. 4. Suppofe the wages of 12 men for 6 days amount to 77. 45. what are the wages of 25 nien for 40 days?—Anf. 100%.

Qu. 5. If 150l. principal put out to intereft for 9 months be increased, principal and intereft, to 1561. 15s. I demand how much is that per cent. per annum ?—Anf. 9l.

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 fectior. Multiply the lower number of the first 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 inverfe proportion be found in the lower numbers, the work is performed by the inverfe rule (for of every fum in this rule one question is direct and the other inverfe).

Example 1. If 100/. gain 57. intereft in 12 months, what principal will gain 57. 5. in 9 months?

The pounds intereft being reduced to fhillings, and mul tiplied 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 dire& method. It would in that cafe stand thus:

The rule laid down in this fection will be found quite general, and fufficient for working all queftions in the double rule of three both direct and inverfe; and is fo obvious, as to require no demonftration. 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 shew its fallacy.