EXAMPLES.

1. Divide $1000 so that B may have twice as much as A wanting $80, and C three times as much wanting $150the share of each is required.

Suppose $100 A

120 B

100 × 2-80

100x 3-150-150 C

1000-370-630 error

Suppose $200 A 200×2-80 =320 B 200x3150=450 C

1000-970- 30 error

600 diff. of errors

200 × 630=126000 product

100 × 30= 3000 do.

6,00)123,000 diff. of products

205 A's share

205 × 2-80 330 B's share

205 × 3-150=465 C's share

Proof 1000 whole sum

2. 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.

3. A farmer having driven his cattle to market, received for them all $320, being paid at the rate of $24 per ox, $16 per cow, and $6 per calf-there were as many oxen as cows, and four times as many calves as cows-how many were there of each sort? Ans. 5 oxen, 5.cows, and 20 calves. 4. What number is that, which being increased by its, its 4, and 5 more, will be double?

Ans. 20.

5. A gentleman going into a garden, met a number of ladies, and said to them, "Good morning to you ten ladies.” "Sir, you mistake," answered one of them, "we are not ten, but if we were thrice as many as we are, we should be as many above ten, as we are now under"-how many ladies were there?

Ans. 5.

6. 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 as long as the head and tail-what is the whole length of the fish? Ans. 80 feet.

7. Three men, A, B, and C, playing at cards, staked $324; but they happened to disagree;-each seized as many of the dollars as he could; A got a number unknown, B as many as A and 15 over, C got a fifth part of their sums added together-how many dollars did each get?

RULE. 120,164,8,0

For singlecked vessels, multiply the length, breadth at the main beam, and depth of the hold together, and divide the product by 95; the quotient will be the tonnage required.

For double-decked vessels, take half the breadth of the main beam for the depth of the hold, and work for singledecked vessels.

NOTE.-95 may be considered an arbitrary number. In England 94 is used.

EXAMPLES.

1. I demand the tonnage of a single-decked vessel, measuring as follows, viz. length 60 feet, breadth 20 feet, and the depth of the hold 8 feet.

Ans. 101 tons, or 101.0526 tons. NOTE. This is the usual method of tonnaging a single-decked vessel, having her deck bolted to the wale. But if it be required that the deck be bolted at any height above the wale, the custom is to pay the carpenter for one half of the additional height, (to which the deck may be thus raised) that is, one half of the different e being added to the former depth, gives the depth to be used in calculating the tonnage.

2. Required the tonnage of a double-decked vessel whose length is 80 feet, breadth 26 feet.

Ans. 2848 tons, or 284.631+ 3. What is the tonnage of a double-decked vessel, whose length is 65 feet, and breadth 21 feet 6 inches; and what will it amount to at 16 dollars per ton?

Ans. 158.138 tons; and $2530.21.

The weight of a ship's burthen is the difference between the bodies of water displaced by her, when in ballast, and when laden.

To ascertain a ship's legal tonnage (by government.)

If the ship be double-decked, take the length thereof from the fore part of the main stem to the after part of the stern-post, above the upper deck; the breadth thereof at the broadest part above the main wales, half of which breadth shall be accounted the depth of the vessel; then deduct from the length three-fifths of the breadth, multiply the remainder by the breadth, and the product by the depth, divide by 95, the quotient is deemed the tonnage of such vessel.

If the ship be single-decked, take the length and breadth as above directed, deduct from the length, three-fifths of the breadth, take the depth from the under side of the deck plank to the ceiling in the hold, and multiply and divide as aforesaid for the tonnage.

(Laws of United States, chapter 128, sec. 64.) Though this act does not indicate by what specific measures this length, breadth, and depth are to be taken, yet it has always been understood here to be in feet and tenths of a foot. [Vide Adams' Report on Weights and Measures, p. 65.]

EXAMPLES.

1. Required the legal tonnage of a double-decked vessel, the length being 110.5 feet, and the breadth 30.6 feet?

of 30.6 18.36, this subtracted from the length, thus
110.5-18.36=92.14

2)30.6

2. What is the legal tonnage of a single-decked ship, which measures 76.4 feet in length, 28.6 feet in breadth, and 12.3 feet in depth? Ans. 219.362+tons.

59.24 X 28.6 x 12.3=20839.4472-95=219.362+

3. I demand the government tonnage of a double-decked vessel, whose length is 87.5 feet, and breadth 29.2 feet. Ans. 314.04+ tons. 4. Required the government tonnage of a single-decked vessel, whose length is 66 feet, breadth 20 feet, and depth 9 feet. Ans. 102,31+tons.

NOTE 1.-For ships of war, divide the (continual) product of the length, breadth, and depth, in feet, by 100, and the quotient will be the tonnage thereof.

Ex. Required the tonnage of a ship of war, length 97 feet, breadth 31, and depth 15.5. Ans. 466.085 tons.

NOTE 2.--To find the length of the mast of a ship, multiply the length of the keel by 2, and divide the product by 3; to the quotient add the breadth of the beam, and the total will be the length of the main-mast.

Ex. Required the length of the main-mast for a ship of 108 feet keel, whose breadth of beam is 40 feet.

Ans. 112 feet.

GAUGING.

GAUGING is the art of measuring all kinds of casks or vessels used for liquor, and of determining the quantity they will contain.

The instruments used in gauging are, the gauging rod, callipers, the sliding rule, and Gunter's scale.

RULE.

Take the dimensions of the cask in inches, viz. the diameter at the bung and head, and the length of the cask; subtract the head diameter from the bung diameter, and note the difference.

If the staves of the cask be much curved or bulging between the bung and head, multiply the difference between the bung and the head diameter by .7; if not quite so much curved, .68; if they curve yet less, .66, .64, .62, or .60; and if almost or quite straight, by .56, and add the product to the head diameter; the sum will be a mean diameter, by which the cask is reduced to a cylinder.

Square the mean diameter thus found, then multiply it by the length; and that product by .0034 which product will be the content in gallons.

Or, multiply the square of the mean diameter by .7854, and the product multiplied by the length; the last product. will be the content in cubic inches, which being divided by 231, the quotient will be the content in gallons.

Observations.-The length and head diameters are usually taken by callipers, allowing for the thickness of both heads, lin. 1in. or 2 inches, according to the size of the cask.

The head diameter must be taken close to its outside, and for small casks, add three-tenths of an inch (.3 inch); for casks of 30, 40, or 50 gallons, four-tenths; and for larger casks five or six-tenths, and the sum will be very near the head diameter within.

The bung diameter is usually taken by the gauging rod, and in taking it, observe, by moving the rod backward and forward, whether the staves opposite the bung are all of the same thickness; if they are not, make allowance accordingly.

1. How many gallons (wine or beer reckoned similarly in this country) in a cask whose bung diameter is 36 inches, head diameter 27 inches, and length 45 inches? Ans. 166.617.

2. What is the content in gallons, of a cask, considerably curved, whose bung diameter is 30 inches, head diameter 20 inches, and length 40 inches?