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 one lay out?

Ans. $600.

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 per ann. A spends $175, & 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

f EXAMPLES. 1. What is the discount of 321,63

due 4 years hence at 6 per cent?

OPERATION.
Dolls.

6 interest of 100 dolls.

2. What is the present worth of $426 payable in 4 years and 12 days, discounting at the rate of 5 per cent ? Ans. $354,519.

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.

B b

13. GUAGING.

GUAGING is taking the dimensions of a cask in inches to find its contents 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; but 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 gallon

f

EXAMPLES.

1. How many ale or beer gallons will a cask hold, whose bung diameter 19 31 inches, head diameter 25 inches, and whose length is 36 inches?

NOTE 1. In taking the length of the casks, an allowance must be made for the thickness for both heads of one inch, 11 inches, or 2 inches, according to the size of the cask.

NOTE 2. The head diameter 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 larger casks, 5 or 6 tenths, and the sum will be very nearly the head diameter within.

14. MECHANICAL POWERS.

1. OF THE LEVER.

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.

EXAMPLE.

IF a man weighing 150lb. rest on the end of a lever 12 feet long, what weight will he balance on the other end, supposing the prop 14 feet from the weight?

12 feet the Lever.

1,5 distance of the weight from the fulcrum.

10,5 distance from the fulcrum to the man.

Feet. Feet. lb. lb.

Therefore,

As 1,5 10,5:: 150: 1050 Ans.

2. OF THE WHEEL AND AXLE. As the dam ། of the axle is to the pover applied to the wheel, to the weigh

EXAMPLES

meter of the heel, so is the pended by the axle.

1. A mechanic wishes to make a windlass in such a manner, as that 1lb. applied to the wheel should be equal to 12 suspended on the axle ; now supposing the axle 4 inches diameter, required the diameter of the wheel? lb. in. lb. in.

As 1 : 4 12 48 Ans. or diameter of the wheel.

2. Suppose the diameter of the axle 6 inches, and that of the wheel 60 inches, what power at the wheel will balance 10lb. at the axle ? Ans 1lb

3. OF THE SCREW

The power is to the weight to be raised as the distance between 2 threads of the screw is to the circumference of a circle described by the power applied at the end of the lever.

NOTE 1. To find the circumference of the circle described by the end of the lever, multiply the double of the lever by 3,14159, the product will be the circumference.

NOTE 2. It is usual to abate of the effect of the machine for friction.

183

EXAMPLES.

There is a screw whose threads are an inch asunder; the lever by which it is turned, is 36 inches long, and the weight to be raised a ton, or 2240lb. What power or force must be applied to the end of the lever, sufficient to turn the screw that is to raise the weight?

The lever 36X2=723,14159--226,194+the circumference.
circumf. in. lb. lb.

Then, as 226,194 : 1: 2240: 9,903

PROBLEMS.

1. The diameter of a circle being given to find the circumference, multiply the diameter by 3,14159; the product will be the circumference.

2. To find the area of a circle, the diameter being given, multiply the square of the diameter by ,785398; the product is the area.

3. To measure the solidity of any irregular body, whose dimensions cannot be taken, put the body into some regular vessel and fill it with water, then taking out the body, measure the fall of water in the vessel; if the vessel be square, multiply the side by itself, and the product by the fall of water, which gives the solid contents of the irregular body.