GAUGING. 162 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. 76.4 of 28.6=17.16 17.16 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 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, anii ote 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. ifin. 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. 497=.0034 EXAMPLES. 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 ? 26.8 x 26.8=718.24 square of the mean diameter X 40 length 30 bung 20 head X.68 1149184 6.8 861888 +20. head diameter Ans. 97.68064 gallons. 26.8 mean diameter. 2. What is the content of a cask of whiskey whose bung diameter is 35 inches, head diameter 25 inches, and length 40 inches, it being considerably curved, as seen by the diamters? Ans. 137.52864 gals. 3. What is the content in gallons, in a cask of large curvature, whose bung diameter is 36 inches, head diameter 24 inches, and length 40 inches? Ans. 142.76736 gals. 4. Required the content in gallons, of a cask of moderate curvature (.62) whose bung diameter is 30 inches, head 24 inches, and length 38 inches. Ans. 99.277 +gals. 5. Required the content in gallons, of a cask of a small curvature (.6), whose bung diameter is 31 inches, head 26 inches, and length 36 inches. Ans. 102.9384. A short method to find the content of a cask of a common curvature without the help of instruments. RULE. With any straight rod take the diagonal of the cask from the centre of the hung-hole correctly, and make a mark on the rod, which may be measured with a carpenter's rule; then multiply the cube of the diagonal in inches, by .00272 for the content in gallons. Or, Divide the cube of the diagonal, by 368 for the answer required. EXAMPLES. 1. Suppose a cask to measure diagonally 36 inches : required the content in gallons. 36 x 36 x 36 x .00272=126,9+gals. Or, by rule 2d. 36 x 36 x 36-368=126.782 +gals. 2. Required the content of a cask of apple brandy, whose diagonal measure at the bung is 30 inches. Ans. by rule 1st, 73.44gals. by rule 2d, 73.369+gals. ANNUITIES. An annuity is a sum of money payable either yearly, half yearly, or quarterly, for a certain number of years, or for ever. When the debtor keeps the annuity in his own hands beyond the time of payment, it is said to be in arrears. The sum of all the annuities for the time they have been forborne, together with the interest due upon each, is called the amcunt. If an annuity is to be bought or paid all at once, at the beginning of the first year, the price which ought to be given for it is called the present worth. To find the amount of an annuity at simple interest. RULE. 1. Find the sum of the natural series of numbers, 1, 2, 3, &c. to the number of years less one. 2. Multiply this sum by one year's interest of the annuity at the given rate, and the product will be the whole interest due upon the annuity. 3. To this product add the product of the annuity and time, and the sum will be the amount sought. ANNUITIES AT COMPOUND INTEREST. CASE 1. When the annuity, the time it continues, and the rate per cent. are given, to find the amount. RULE. 1. Make 1 the first term (of a geometrical progression,) and the amount of $1, at the given rate per cent. the ratio. 2. Carry the series to as many terms as the number of years, and find its sum. ANNUITIES AT COMPOUND INTEREST. 166 3. Multiply the sum thus found by the given annuity, and che product will be the amount sought. Or, multiply the amount of $1 for one year, at the given ate per cent., into itself as many times as there are years given ; from the product subtract one, then divide the remainder by the interest of $1 for one year, at its given rate per cent., and multiply the quotient by the annuity for the mount required. If the payments be half yearly or quarterly, multiply the mount found as above, by the proper number in the folowing TABLE. The construction of this Table is as follows: For half yearly payments take one from the ratio, and from the square root of the ratio ; half the quotient of the first remainder divided by the latter, will be the tabular number. For quarterly payments, use the 4th root as above, and take one quarter of the quotient. Or, without the help of the table, find the amount of the whole number of years as before, then find the interest of that amount for the given parts of a year; add this interest to the former amount, and it will give the whole amount cequired. W ye the EXAMPLES. (the the 1. What will an annuity of 300 dollars amount to in four years, at 5 per cent., compound interest? Ans. $1293 03 cents 7.5 mills. by of C |