Double Position.

DOUBLE POSITION is that which discovers the true number, or number sought, by making use of two supposed numbers.

RULE.

1. TAKE any two numbers and proceed with them according to the conditions of the question.

2. PLACE each error against its respective position or supposed number; if the error be too great, mark it with+; if too small with

3. MULTIPLY them cross-wise, the first position by the last error, and the last position by the first error.

4. Ir they be alike, that is, both greater or both less than the given number, divide the difference of the products by the difference of the errors,and the quotient will be the answer; but if the errors be unlike, divide the sum of the products by the sum of the errors, and the quotient will be the answer.

EXAMPLES.

1. A MAN lying at the point of death, left to his three sons all his estate, viz. to F half wanting 50 dollars; to G one third; and to H the rest, which was 10 dollars less than the share of G. I demand the sum left and each Son's share.

Suppose the sum 300 dollars.

OPERATION.

Again, Suppose the sum 900 dollars.

Then, 900-2—50—400 F's part. 900 3300 G's part, G's part 300—10—290 H's part. Sum of all their parts 990

Errors. 10

90+

9000 27000

27000

Dolls.

100)36000 (360 Answer.

Error 90+

HAVING proceeded with the supposed numbers according to the conditions of the question, the sum of all their parts must be subtracted from the supposed number; thus, the 290 is subtracted from 300, the supposed number, &c.

The Divisor is the sum of the errors 90+and 10

2. There is a fish, whose head is 10 feet long; his tail is as long as his head and half the length of his body, and his body is as long as his head and tail; what is the whole length of the fish? Ans. 80 feet.

B b

1

3. A CERTAIN man having driven his Swine to market, viz. Hogs, Sows, and Pigs, received for them all 507. being paid for every hog 18s. for every sow 165. for every pig 2s : there were as many hogs as sows, and for every sow there were three pigs; I demand how many there were of each sort? Ans. 25 hogs, 25 sows, and 75 pigs.

4. A AND B laid out equal sums of money in trade; A gained a sum equal to of his stock, and B lost 225 dollars, then A's money was double that of B's; 'What did each lay out? Ans. 600 dollars.

5. A AND B have the same income; A saves of his; but B, by spending 30 dollars per annum more than A, at the end of 8 years finds himself 40 dollars in debt what is their income, and what does each spend per annum?

Ans, their income is 200 dolls per ann. A spends 175 dollars, & B 205 per ann.

§ 11. Discount.

DISCOUNT is an allowance made for the payment of any sum of money before it becomes due, and is the difference between that sum, due sometime hence, and its present worth.

THE present worth of any sum, or debt due some time hence, is such a sum, as, if put to interest, would in that time and at the rate per cent. for which the discount is to be made, amount to the sum or debt, then due."

RULE.

As the amount of 100 dollars, for the given time and rate is to 100 dollars, so is the given sum to its present worth, which subtracted from the given sum, leaves the discount.

EXAMPLES.

1. WHAT is the discount of Dolls 321,63 due 4 years hence, at 6 per Cent ?

OPERATION.

Dolls!

2. WHAT is the present worth of 426 dollars, payable in 4 years and 12 days, discounting at the rate of 5 per cent. Ans. Dolls. 354,515.

6 interest of 100 dolls. 1 year.
4 years.

24

100

124 amount.

Then, As 124: 100:: 321,63

321,63

124)32163,00(259,379

321,63 given sum.

259,379 present worth.

Ans. 62,251 discount.

§ 12. Equation of Payments.

EQUATION of payments is the finding of a time to pay at once, several debts due at different times so that neither party shall sustain loss.

RULE.

MULTIPLY each payment by the time at which it is due ; then divide the sum of the products by the sum of the payments, and the quotient will be the equated time.

§ 13. Guaging.

GUAGING is taking the dimensions of a cask in inches to find its content in gallons by the following

METHOD.

1. ADD two thirds of the difference between the head and bung diameters to the head diameter for the mean diameter; if the staves be but little curving from the head to the bung, add only six tenths of this difference.

2. SQUARE the mean diameter, which multiplied by the length of the cask and the product divided by 294, for wine, or by 359 for ale, the quotient will be the answer in gallons.

EXAMPLE.

1. How many ale or beer gallons will a cask hold, whose bung diameter is 31 inches, head diameter 25 inches, and whose length is 36 inches?

841 Square of mean diam. NOTE. 2. THE head di

36 Length.

5046 2523

ameter must be taken close to the chimes, and for small casks, add 3 tenths of an inch; for casks of 40 or 50 gallons, 4 tenths, and for

359)30276(84 galls. 1 qta. larger casks, 5 or 6 tenths,

and the sum will be very nearly the head diameter within.

To find what weight may be raised or balanced by any given power, Say, as the distance between the body to be raised or balanced, and the fulcrum or prop, is to the distance between the prop and the point where the power is applied; so is the power to the weight which it will balance or raise.