GAUGING OF CASKS.

677. Gauging is the process of finding the capacities of casks or other vessels.

Casks are generally considered to be of four varieties: 1. Having the staves nearly straight; 2. Having the staves very little curved; 3. Having the staves of a medium curve; 4. Having the staves considerably curved.

NOTE. Casks of the first variety approach very nearly the form of a cylinder; those of the third variety are of the shape of a molasses hogshead; those of the second variety have a curvature of stave between that of the first and third; and the fourth have a greater curvature than that of the third.

678. In gauging casks, it is necessary first to find the mean diameter. This is found by taking the end and middle diameters, and the length in inches; and then adding to the end diameter the product of the difference between the end and middle diameters by .55, .60, .65, or .70, as the cask may be of the first, second, third, or fourth variety.

679. To find the capacity of a cask in gallons.

Multiply the square of the mean diameter, in inches, by the length, in inches; and the product multiplied by .0034 will give the capacity in liquid or wine gallons.

NOTE 1.

If the capacity is required in ale or beer gallons, use for a multiplier .0028 instead of .0034. If imperial gallons are required, multiply the liquid or wine gallon, as found by the rule, by .833.

NOTE 2. The contents of any vessel being known in cubic inches, its capacity in liquid gallons may be found by dividing by 231; in ale or beer gallons, by dividing by 282; and in bushels, by dividing by 2150.42.

Ex. 1. Required the capacity in gallons of a cask of the fourth variety, whose middle diameter is 35 inches, head diameter 27 inches, and length 45 inches. Ans. 162.6.

2. What is the capacity in gallons of a cask of the third variety, whose middle diameter is 38 inches, head diameter 30 inches, and length 42 inches?

3. What are the contents in liquid measure of a tub 40 inches in diameter at the top, 30 inches at the bottom, and whose height is 50 inches? Ans. 209.66gal. 4. How many wine gallons will a cubical box contain, that is 10 feet long, 5 feet wide, and 4 feet high ?

5. How many ale gallons will a trough contain,

6 feet wide, and 2 feet high?

Ans. 1496 gal. that is 12 feet long, Ans. 88218gal.

6. How many bushels of grain will a box contain that is 15 feet long, 5 feet wide, and 7 feet high?

Ans. 421.8bu.

TONNAGE OF VESSELS.

680: The tonnage of a ship is the number of tons burden it will carry, with safety, under the ordinary circumstances of navigation.

The light-loaded water-line of a vessel is the line made by the water upon the outside of the hull as it floats without load; and the deeploaded water-line is that made in like manner when it is fully laden.

The number of cubic feet of the hull between these two waterlines, divided by 35, the number of cubic feet of sea-water which must be taken to weigh a ton, represents the weight of water displaced in sinking the vessel from the light to the deep-loaded waterline, and therefore its true tonnage.

681: The present government rule, or that adopted 1864, requires too many data properly to have a place in arithmetic. The former rule, although it does not always give the actual tonnage, yet as builders and others often make their estimates by it, is here given.

RULE.

FOR SINGLE-DECKED VESSELS. Take the length on deck from the forward side of the main stem to the after side of the stern post, and the breadth at the broadest part above the main wales; take the depth from the under side of the deck plank to the ceiling of the hold; and deduct from the length three fifths of the breadth; multiply the remainder by the breadth, and the product by the depth; and divide the last product by 95.

FOR DOUBLE-DECKED VESSELS. Proceed as with single-decked vessels, except for the depth take half the breadth.

NOTE.-The rule is differently construed. The length is usually taken in a line with the deck; the depth at the main hatch. But with regard to the breadth, there is a great want of uniformity among measurers; most take the breadth about 45 inches below the plank-sheer at the broadest part; some consider the upper wales, and others the lower, at the main wales, thus making a considerable difference in their results.

The rule for single-decked vessels operates very well, but the rule for double-decked vessels, which is also intended to include all vessels of more than one deck, often fails to give the true tonnage. A more accurate method would, for DOUBLE-DECKED VESSELS, take the breadth 5 feet below the upper deck, at the broadest part, and for THREE-DECKED VESSELS 7 feet below the upper deck; and in each case for depth of hold three fifths of the breadth.

Ex. 1. A. & G. T. Sampson, of East Boston, have contracted to build a clipper ship 191 feet long, 365 feet wide, 22 feet deep; what is the tonnage of the ship? Ans. 11842 tons. Meridian, whose length is Ans. 128447 tons. long, 39 wide, and 27 Ans. 13971 tons.

feet

2. What is the tonnage of the ship 184, width 3811, and depth 28 feet? 3. The ship Mattakeeset is 195 deep; what is the tonnage of the same? 4. Required the tonnage of a single-decked vessel, whose length is 78 feet, width 21 feet, and depth 9 feet. 5. What is the tonnage of a double-decked vessel, whose length is 159 feet, and width 30 feet? Ans. 6671 tons.

Ans. 1305 tons.

6. What is the tonnage of Noah's ark, admitting its length to have been 479 feet, its breadth 80 feet, and its depth 48 feet? Ans. 1451717 tons.

MISCELLANEOUS EXAMPLES.

1. What number is that to which, if ‡ of § be added, the sum will be 1 ? Ans.

2. A certain gentleman, at the time of his marriage, agreed to give his wife of his estate, if, at the time of his death, he left only a daughter, and if he left only a son, she should have of his property; but as it happened, he left a son and a daughter, in consequence of which the widow received in equity $ 2400 less than she would have received if there had been only a daughter. What would have been his wife's dowry if he had left only a son? Ans. $ 2100.

3. A gentleman being asked what o'clock it was, said that it was between 5 and 6; but, to be more particular, he said that the minutehand had passed as far beyond the 6 as the hour-hand wanted of having reached the 6; that is, that the hour and minute-hands made equal acute angles with a line passing from the 12 through the 6. Required the time of day. Ans. 32m. 18s. past 5.

4. Divide 97deg. 55m. 7fur. 35rd. 4ft. 6in. by 6.

5. A, B, and C are to share $100,000 in the proportion of 1, 1, and 3, respectively; but C's part being lost by his death, it is required to divide the whole sum properly between the other two.

Ans. A's part is $ 57,142%, and B's $ 42,8574.

6. A father devised of his estate to one of his sons, and of the residue to the other, and the remainder to his wife. The difference of his sons' legacies was found to be 257£. 3s. 4d. What money did he leave for his widow? Ans. 635£. Os. 1032d.

7. In the walls of Balbec, in Turkey, the ancient Heliopolis, there are three stones laid end to end, now in sight, that measure 61 yards in length, one of which is 63 feet long, 12 feet thick, and 12 feet broad; what is its weight, supposing its specific gravity to be 3 times that of water? Ans. 850 tons.

8. A burden of 200lb., suspended on a pole 4ft. in length, the point of suspension being 6in. from the middle, is carried by two men, the ends of the pole resting on their shoulders; how much of this load is borne by each man? Ans. 125lb. and 75lb.

9. The court-house in Boston has eight pillars of granite, each 25ft. 4in. in length, 4ft. 5in. in diameter at one end, and 3ft. 5in. in diameter at the other end. How many cubic feet do they contain, and what is their weight, allowing a cubic foot to weigh 3000 ounces ? Ans. 2455.03 cub. ft.; 230.15+tons.

10. A father, dying, left his son a legacy, of which he spent in 8 months; of the remainder lasted him 12 months longer, after which he had only $410 left. What amount did his father bequeath him? Ans. $956.663

11. A merchant sold goods to a certain amount, on a commission of 4 per cent., and, having remitted the net proceeds to the owner, received per cent. for prompt payment, which amounted to $ 15.60. What was the amount of his commission?

Ans. $260.

12. A, of Boston, remits to B, of New York, a bill of exchange on London, the avails of which he wishes to be invested in goods on his account. B, having disposed of the bill at 7 per cent. advance, received $9675; and having reserved for himself per cent. on the sale of the bill, and 2 per cent. for commission, he invests the remainder. What is the amount invested, and for how much was the bill drawn? Ans. Investment, $9461.58; the bill was £ 2025. 13. Bunker Hill Monument is 30ft. square at its base, and 15ft. square at its top; its height is 220 feet. From the bottom to the top, through its centre, is a conical avenue 15ft. in diameter at the bottom, and 11ft. at the top. How many cubic feet are there in the monument? Ans. 86,068.518+ ft.

14. A hired a house for one year for $300; at the end of 4 months he takes in M as a partner, and at the end of 8 months he takes in P. At the end of the year, what rent must each pay ?

Ans. A pays $1831; M pays $ 831; P pays $333.

15. A merchant receives on commission three kinds of flour; from A he receives 20 barrels, from B 25 barrels, and from C 40 barrels. He finds that A's flour is 10 per cent better than B's, and that B's is 20 per cent. better than C's. He sells the whole at $6 per barrel. What in justice should each man receive?

Ans. A receives $139141; B, $ 158172; C, 211111.

16. Bought 100 barrels of flour, at $5 per barrel, and immediately sold it on a credit of six months. The note which I received for pay I got discounted at the Suffolk Bank, and, on examining my money, I found that I had gained 20 per cent. on my purchase. What did I receive per barrel for the flour? Ans. $6.181698

1939

17. Required the greatest possible number of hills of corn that can be planted on a square acre, the hills to occupy only a mathematical point, and no two hills to be nearer than three and a half feet. Ans. 4165. 18. Lent a friend $ 700, which he kept 20 months. Some years after I borrowed of him $ 300; how long should I keep it to balance the favor? Ans. 463 months.

19. John Lee gave of his estate to his wife, of the remainder to his oldest son, and of the residue to his oldest daughter, and of what then remained, which was $1500, was to be equally distributed among his other children, who received $150 each; required the number of his children, and the value of his estate.

20. A and B set out to travel round a certain island, which is 80 miles in circumference. A travels 5 miles a day, and B 7 miles a day. How far must B travel to overtake A? Ans. 280 miles.

21. If 24.4 cubic inches of lead weigh 16 pounds, required the number of feet of lead pipe that can be made from 80 pounds of lead, the caliber of the pipe to be 1 inch, and the thickness of it of an inch. Ans. 10.35 feet.

22. How long a tube can be made from a cylinder of lead 8 inches long and 2 inches in diameter, and through the centre of which is a hole of an inch in diameter; the tube or pipe to be of an inch in caliber, and of an inch in thickness? Ans. 16.29 in.

23. Four men, A, B, C, and D, bought a stack of hay, containing

8 tons, for $100. A is to have 12 per cent. more of the hay than B, B is to have 10 per cent. more than C, and C is to have 8 per cent. more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 feet high, and 12 feet square at its base, it being an exact pyramid; and it is agreed that A shall take his share first from the top of the stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sum shall each pay? Ans. A takes 13.22+ft., and pays $28.9312287; B takes 3.14+ft., and pays $ 25.8311917; C takes 2.06+ft., and pays $ 23.4818132; D takes 1.58+ft, and pays $ 21.7417088.

24. A, B, and C bought a grindstone, for which they paid $ 10.60. B paid 20 per cent. more than A, and 10 per cent. less than C. The diameter of the stone was 65 inches, and the diameter of the place for the shaft 3 inches. What sum did each pay, and how much must each grind off from the semidiameter to obtain his proper share of the stone?

Ans. A paid $3, B $ 3.60, and C $ 4. 7 inches, and C 18 inches.

A grinds off 5 inches; B

25. A servant draws off a gallon on each day, for 20 days, from a cask containing 10 gallons of wine, each time supplying the deficiency by the addition of a gallon of water; and then, to escape detection, he again draws off 20 gallons, supplying the deficiency each time by a gallon of wine. How much water still remains in the cask?

Ans. 1.0679577 gallons, or more than a gallon and half a pint. 26. The dimensions of a bushel measure are 18 inches wide, and 8 inches deep; what should be the dimensions of a similar measure that would contain 8 bushels? Ans. 37in. wide, 16in. deep.

27. What is the weight of a hollow spherical iron shell 5 inches in diameter, the thickness of the metal being 1 inch, and a cubic inch of iron weighing 128 of a pound? Ans. 13.2387lb.

28. At a certain time between 2 and 3 o'clock, the minute-hand was between 3 and 4. Within an hour after, the hour-hand and minute-hand had exactly changed places with each other. What was the precise time when the hands were in the first position? Ans. 2h. 15m. 5622s. 29. Required the contents of the largest cube that can be inscribed in a sphere 20 inches in diameter. Ans. 1539.58 cu. in.

92

30. If in a pair of scales a body weigh 90 pounds in one scale, and only 40 pounds in the other, required the true weight, and the proportion of the lengths of the two arms of the balance-beam on each side of the point of suspension.

Ans. Weight 60lb., and the proportions 3 to 2. 31. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner wheel made but one; the wheels were each 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axlctree, what was the circumference of the track described by the outer wheel? Ans. 62.83 feet.

32. The ball on the top of St. Paul's Church is 6 feet in diameter. What did the gilding of it cost, at 34d. per square inch?