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the head diameter, multiply 1 the sum by the length, divide

} as above for ale and wine gallons.

3. When the staves are very little curved, the cask is consider. ed to be of the third variety; then to the sum and half sum of the squares of the head and bung diameters, add to of the difference of their squares, multiply the sum by the length and divide as above for ale and wine gallons.

4. When the staves are straight between the bung and ends of the cask it is of the fourth variety; then from the sum and half sum of the squares of the head and bung diameters, subtract half the square of their difference ; then multiply the remainder by the length, and divide as above for ale and wine gallons.

A general rule for reducing casks to a cylinder. First, consider which of the varieties the proposed cask resembles, then from the bung diameter subtract the head diameter, and multiply by .7 for the spberoid, by .65 for the spindle, by .6 for the conoids, and by:55 for the cones ; add the product to the head diameter, the sum is a mean diameter, or the diameter of a cylinder of equal content and length with the cask.

EXAMPLE. 18. Suppose the bung diameter be 32 inches, the head die ameter 24, and the length of the cask 40 inches, the contents in ale and wine gallons, for each variety is required ? Ans. for the S 118.95 winegrations;} 97.44 .

by the first rule. spheriod 97.4 ale gallons. 119.15 wine gallons.

by the general rule.

. Ans. for the 114.89 winegallons

. } by the second rule. spindle 94.98 ale gallons. } 115.95 wine gallons. S

by the general rule. 90.71 ale gallons. ? Ans. for the / 110.83 wine gallons. S

by the third rule. conoids. 92.4 ale gallons 112.8 wine gallons.

by the general rule.
87.93 ale gallons,
Ans. For the) 107.34 wine gallons,

by the third rule.
89.85 ale gallons,
109.69 wine gallons,

by the general rule. By the Sliding Rule. Set the length of the cask in inches on the line C. to the gauge-points (for circular areas) on D. and against the mean diameter, for each form or variety on the line D.you bare the contents on C.

cones.

Diagonal Rod. X. The diagonal rod is generally used by those who are unac

quainted with the rules of gauging, but when it so happena that no rod is at hand, then it must be done by the following

RULE--With any straight rod, take the diagonal of the cask from the centre of the bung hole both ways, and make a mark on the rod, which may be measured with a carpenters rule, then multiply the cube of the diagonal in inches, by .002228 for ale, and by .00272 for wine gallons.

EXAMPLE 19. Suppose a cask to measure diagonally 36 inches : Rea. quired the contents in ale and wine gallons. Ans. {36x367362.01272= 126.90932 Wine Gallonsa,

Ale .
A TABLE
Of the Segments of a Circle, whose Area is Unity.
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1.001; 2 .0048 3 .0087 4 1.0134 5 1.0187

99 998
98 .9952
97 .991
96 1.9861
95 .981,

26 27 28 29 30

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74.7924 .2172 || 73 .7832 229311 72 1.7708 2407 .7593 .252 70 .7477

71

.024 7 .030€ 8.0373

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911.9755 93 969 92.962 91.9554 90 9486

31 32 33 34 35

2644 2759 2872 .299 .3119

69 7360 68 .7241 67 7122 66 .7002 65 1.6881

11 .0598 | 891.940% 36 .3241 64.6754 12 .068 88.932 37 .3364 63 !.6636 13 .0764 87

38 .3487 62 1.65131 14 ,0851 86.914 39 .3611 61.6389 15 1.0941 85 .9059||. 40.373.3||: 60 .6265

16 5.1030|| 84.8967 17.1127 83 1.8873 18.1224 82 .877' 19 .1323|| 81 1.867 20.1424 80.857t

41 3860|| 59.6140 42 1.3980 58

6014 41.1. 57 .5888

423 56 1.5762! 45 .436411 55.5636

21 .1526 22

.16311 23 .1737 24

.184 25 1.1955

79 .8474! 46
78 836 4?
77 .926 48
76 315 49
75 1.80451 50

1.4491 54.5509
.461 53 .5382
4745! 52 .5255
4873 51 5127
.500uli 50 .5000

XI. To find the Ullage of a Cask. Rule 1_Find such a mean diameter as you judge, will reduce the proposed cask to a cylinder, and then find its content.

2. From the bung diameter subtract the mean diameter, and take half the difference.

3. From the wet inches subtract the said half-difference ; reserve this difference, then use the proportion :

As the mean diameter is to 100

(the diameter of the tabular circle,) So is the reserv'd difference,

to a versed sine in the table. Then, if the tabular Segment be multiplied into the con. tent (as before) the product will be the quantity of liquor in the cask.

EXAMPLE 20. Let the cask be the same as in page 162, of the first form, where the bung-diameter is 32 inches, and the mean diameter-29.5, and the content 97.4 gallons : and suppose the wet inches 19, to find the quantity of liquor in the cask ? From 32

From 19.
Subtr. 29 6

Subtr. 1.2

Rom. 1.4

Rem. 17.8 reserved..

Half 1.2

29.6; 100 :: 17.8 :.60, the y. S. The Segment to 60 is .6265, which multiplied by 97.4, the content, the product is 61 gallons, the quantity of liquor in the cask.

By the Sliding Rule. 1st. Set the bung diameter 32 on the line of numbers, to 100 upon the line of Segments ; then against the wet inches 19 on the line of numbers is a fourth number 60 on the line of segments, which reserve.

2nd. Set 100 on the line A. to the whole content 97.4 on the line B, and against the reserved number 60 on A. is om gallons the answer on B.

The American Tutor's Guide, &c.

PART VI.

1656 years.

A COLLECTION OF QUESTIONS. 1. WRITE down two millions, five hundred and two thou. sand, two hundred and five.

2. Find how many years it was from the creation of Adam to the universal deluge in the days of Noah, called Noah's Flood ; by the 5th chapter, and 6th verse of the 7th chapter of Genesis.

Ans. 3. When the air presses with its full weight, in very fais weather, it may be demonstrated, that there presses upon a human body about 33905 pounds of that fluid matter; and in foul weather when the air is most light, but 30624 pounds. What difference of eight lies on such a body, in the two greatest alterations of the reather) ans. 3281 pounds.

4. Jacob by contract was to serve Laban for his two daughters 14 years; and when he had accomplished 11 years, 11 months, 11 weeks, 11 days, 11 hours, and 11 minutes : Pray how long had he to serve ?

Ans. ly. 11m. 3w. 2d. 12h. 49 min. 5. Moses was born Anno Mundi, 2433 ; Homer 832 after him ; Julius Cæsar lived 40 years before our Saviour, and Alexander 312 years before Cæsar; now as Christ was incarnate 4000 years after the creation, the sum of the intervals between Homer and the three great personages last men tioned is required? Ans. 1913 years, sum of the intervals.

6. From the Creation to the Flood was 1656 years ; tiience to the building of Solomon's Temple 1336 years ; thence to Mahomet, (who lived 622 years after Christ,) 1630 years : In what year of the world was Christ then born ?

Ins. Christ was born Anno Mundi, 4000. 7. Part 1500 ac.es of land, give B. 72 more than A. and C. 112 more than B.

414 A's)
Ans.
486; B's

Share.

598] C's S 8. The Spectator's club of fat people, tho' it consisted but of 15 persons, is said (No. 9) to weigh no less than 3 tons : How much on an equality was that per man?

Ana. 4 cws.

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9. A. B. and C. found a chest containing a certain number of pounds, and when they divided the booty, A. and B's share amounted to 471. B. and C's to 881. and A. and C's to 711. What sum was found, and each man's share?

Ans. found 1031. A's 15l. B's 321. G's 561. 10. If the human heart beat 70 times in a minute, and each pulsation transmit 4 oz. Averdupois, of blood, and the whole blood be a part of the weight of ibe body, in what time will the whole blood of a man, whose weight is 140 lb. circulate through the heart?

Ans: in 34 sec. 11. What number added to the 43d part of 4429, will make the sum 240 ?

Ans. 137. 12 What number deducted from the 26th part of 2262, will leave the 87th part of the same ?

Ans. 61. 13. What number divided by 419844, will quote 9494, and leave just } part of the divisor remaining ? Ans. 3986138884.

14. Divide S1000 betwixt A. B. and C. in such a manner, that A. may have 129 more than B. and B. 178 less than C.

Ans. $231 B's, 360 A's, 409 C's 15. Required to divide a prize taken at sea; value 203.701 the 6d. ainong the captain, 2 lieutenants, the surgeon and purser, 4 potty officers, and 90 private men, allowing each petty officer as much as 2 private men, the surgeon and purser each as much as 8 men, the lieutenants each as much as 16, and the captain as much as 50 men ?

Ans. A private man's share 1031. 18s. 7d. 16. The mean time of a lunation, that is, from new moon to new moon, is 29 days, 12 hours, 44 minutes, and 3 seconds: I demand how many lunations are contained in 19 Julian years,

of 365 days, 6 hours ?

Ans.235 lunations, 1 hour 28 m. 5 sec. 17. A. B. and C. play in concert at Hazard ; and at mak. ing up accounts, it appears, that A. and B. together, brought off 131. 10s. B. and C. together 121. 12s. and A. and C. together won 111. 16s. 6d. What did they severally get ?

Ans. 51. 98. 3d. A. 61. 78. 3d. B. and 71. 28. 9d. C. 18. Four persons advance in trade, as follows, viz: W. X. and Y. raised 3501. 10s.; W. X. and Z. 3441. 10s.; X. Y. and 2, contribute 4001. ; W. Y. and Z. 3731. 4s. In the conclusion they parted with their joint property for 450 guineas : What did they gain or loose by their adventure ?

Ans. Lost 181. 118. 4d. sterling. 19. How many trees may be planted on an English acre; at 6 feet distance ?

Ans. 1210. 20. A spring of water, which furnishes 16 gallons each minute, supplies a city of 5000 families. How much water has each family daily?

Ans. 4,04 gallons.

:

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