£1983 Ans.

Ans. 525 livres. 525 as bef.

Note, that in England dollars are

4. To South Carolina and Geor- Bullion, that is, they are bought and

sold by weight, and their value varies as other articles of merchandize.

DUODECIMALS,

OR CROSS MULTIPLICATION,

IS a Rule, made use of by workmen and artificers in casting up the contents of their works.

Dimensions are generally taken in feet, inches and parts.

Inches and parts are sometimes called primes, seconds, thirds, &c. and are marked thus; inches or primes ('), seconds ("), thirds (), fourths (""), &c.

This method of multiplying is not confined to twelves; but may be greatly extended: For any number, whether its inferiour denominations decrease from the integer in the same ratio, or not, may be multiplied crosswise; and, for the better understanding of it, the learner must observe, that if he multiplies any denomination by an integer, the value of an unit in the product will be equal to the value of an unit in the multiplicand; but if he multiplies by any number of an inferiour denomination, the value of an unit in the product will be so much inferiour to the value of an unit in the multiplicand as an unit of the multiplier is less than an integer.

Thus, pounds, multiplied by pounds, are pounds; pounds, multiplied by shillings, are shillings, &c. shillings, multiplied by shillings are twentieths of a shilling; shillings, multiplied by pence, are twentieths of a penny; pence, multiplied by pence, are 240ths of a penny, &c.

RULE.*

1. Under the multiplicand write the corresponding denominations of the multiplier.

*The reason of this rule is evident by considering the denominations below the integer, as fractional parts of the integer, and multiplying as in Vulgar Fractions. Thus inches or primes are 12ths of a foot, seconds are 12ths of an inch, or 144ths of a foot, and so on. Then feet multiplied by inches would give inch

3 6

3 6

es, for 2 feet × 12-12-6 inches; inches by inches give seconds, for 12×12 13 1246 12

18

6

1 6

144 ̄12x12 ̄ ̄ ̄12x12=12×13+; 12X12 12144 1 inch and 6′′; inches in

6 3

13

12

+

6

to thirds gives fourths, for 12X14412X144 12X144 12x144 -1 and 6", and so on.

1

6

144+1728

A similar process will show the correctness of the Rule, when the denominations do not decrease uniformly by 12 or any one number, as in pounds, shillings and pence, where 1 shilling would be of a pound, and 1 penny, of a pound, and so on.

20

Note. It is evident that when the denominations decrease by any one num◄ her, as 12, the denomination of the product is the sum of the denominations of the factors. Thus primes into primes give seconds, 2 being the sum of 1+1, the denominations of the factors, seconds into thirds give fifths, 24-35; seconds into fourths give sixths, and so on.

2. Multiply each term in the multiplicand, beginning at the lowest, by the highest denomination in the multiplier, and write the result of each under its respective term, observing, in duodecimals, to carry an unit for every 12, from each lower denomination to its next superiour, and for other numbers accordingly,

3. In the same manner multiply all the multiplicand by the primes or second denomination in the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand.

4. Do the same with the seconds in the multiplier, setting the result of each term two places to the right hand of those in the multiplicand.

5. Proceed in like manner with all the rest of the denominations, and their sum will be the answer required.

3 7 6 Answer.

3. How many square feet in a board 17 feet 7 inches long, and 1 foot 5 inches wide?

Ans. 24ft. 10' 11" 4. How many cubick feet in a stick of timber 12 feet 10 inches long, 1 foot 7 inches wide, and 1 foot 9 inches thick?

Ans. 35ft. 6' 8' 6"

5. How many cubick feet of wood in a load 6 feet 7 inches long, 3 feet 5 inches high, and 3 feet 8 inches wide? Aus. 82ft. 5' 8".4""

6. There is a house with 4 tiers of windows, and 4 windows in a tier; the height of the first tier is 6ft. 8'; of the second, 5ft. 9'; of the third, 4ft. 6'; and of the fourth, 3ft. 10'; and the breadth of each is 3ft. 5'; how many square feet do they contain in the whole? Ans. 283ft.

The two following questions are Sexcessimals.

7. If 2 places differ in longitude 2° 12'; what is their difference. of time?

Mult. 2° 12' 00" 00′′

by

3′ 59′′ 20" the time in which the sun passes through 1°

8' 46" 32" Answer.

3. Two places differ in longitude 31° 27' 30"; What is the dif ference, in time, of the sun's coming to the meridian of those places, the sun passing through 15° in an hour?

31° 37' 30"

4' 00" In 4' of a solar day, or day of 24 hours, the sun passes 1o

2° 6' 30" 00" Answer.

9. Bought a load of wood, which was 3 feet wide, 2 feet 8 inches high, and 8 feet long; what part of a cord of wood did it contain ? Ans. Half a cord. 10. A load of wood was 4 feet 6 inches wide, 3 feet 10 inches high, and 7 feet 8 inches long; how many feet more than a cord did it contain? Ans. 4 feet.

11. A stick of timber is 1 foot 8 inches in depth, and 2 feet 3 inches in width, and 42 feet 8 inches long; how many solid feet of timber does it contain? Ans. 160.

8d.x £2=16d. = 0 1 £3X58. 15s.

£3x £2= £6

=6 0 0

Gs.X £212s. = 0 12

0

4

=

= 0 15

0

69. X5s.

=

38.

- 0 1

6

13. A, B and C bought a drove of sheep in company; A paid £14 5s. B, £13 10s. and C, £11 5s. They agreed to dispose of them at the market; that each man should take 18s. as pay for his time, &c. and that the remainder should be divided in proportion to their several stocks: At the close of the sale, they found themselves possessed of £46 5s. what was each man's gain, exclusive of the pay for his time, &c.

£14 5£13 10+ £11 5= £39, and £46 5— £39- £7 5, and £75-18s.X3= £4 11s. whole gain, and £4 11÷÷39=2s. 4d. gain

THE SINGLE RULE OF THREE,

IS so called, because three numbers are given to find a fourth, which shall have the same ratio to one of the given numbers, as there is between the other two. It is usually distinguished into Direct and Inverse. The reason of this distinction, and the particular rules, will be given hereafter. It will be more easy however, for the student to proceed according to the following General Rule for stating and working questions in the Rule of Three.

GENERAL RULE.*

1. Place that number, which is of the same name or quality as the answer sought, for the second term.

2. Consider whether the answer should be greater or less than the second term. If it must be greater, place the greater of the two remaining numbers in the question on the right for the third

This Rule, on account of its great and extensive usefulness, is sometimes called the Golden Rule of Proportion: For, on a proper application of it and the preceding rules, the whole business of Arithmetick, as well as every mathematical enquiry depends. The rule itself is founded on this obvious principle, that the magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause: Thus, the quantity of goods bought, is in proportion to the money laid out; the space gone over by an uniform motion, is in proportion to the time, &c.

As the idea, annexed to the term, proportion, is easily conceived, the truth of the rule, as applied to ordinary inquiries, may be made evident by attending to principles, already explained.

It has been shewn, in Multiplication of Money, that the price of one, multiplied by the quantity, is the price of the whole; and in Division, that the price of the whole, divided by the quantity, is the price of one: Now, in all cases of valuing goods, &c. where one is the first term of the proportion, it is plain that the answer found by this rule, will be the same as that, found by Multiplication of Money; and, where one is the last term of the proportion, it will be the same as that found by Division of Money.

In like manner, if the first term be any number whatever, it is plain, that the product of the second and third terms will be greater than the true answer, required, by as much as the price in the second term exceeds the price of one, or as the first term exceeds a unit; consequently, this product, divided by the first term, will give the true answer required.

Note 1. When it can be done, multiply and divide as in Compound Multiplication, and Compound Division.

2. If the first term, and either the second or third can be divided by any number without a remainder, let them be divided and the quotient used instead of them.

The following methods of operation, when they can be used, perform the work in a much shorter manuer, than the general rule.

1. Divide the second term by the first: Multiply the quotient into the third, and the product will be the answer.

2. Divide the third term by the first; multiply the quotient into the second, and the product will be the answer.

3. Divide the first term by the second, and the third by that quotient, and the last quotient will be the answer.

4. Divide the first term by the third, and the second by that quotient, and the last quotient will be the answer,

R