the head diameter, multiply the sum by the length, divide as above for ale and wine gallons. 3. When the staves are very little curved, the cask is considered to be of the third variety; then to the sum and half sum of the squares of the head and bung diameters, add 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 spheroid, 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 diameter 24, and the length of the cask 40 inches, the contents în ale and wine gallons, for each variety is required? 97.44 ale gallons. 118.95 wine gals Ans. for the spindle Ans. for the conoids. S 94.11 ale gallons. 114.89 wine gallons. 94.98 ale gallons. 115.95 wine gallons. S 90.71 ale gallons. 110.83 wine gallons. 92.4 ale gallons 112.8 wine gallons. by the first rule. by the general rule. by the second rule. by the general rule.. by the third rule. by the general rule. Ans. For the 107.34 ale Ballons, by the third rule. cones. 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 have the contents on C. Diagonal Rod. X. The diagonal rod is generally used by those who are unacquainted with the rules of gauging, but when it so happens 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 : Required the contents in ale and wine gallons. Ans. S36X36X36X.002228=103.949568 Ale Gallons. A TABLE Of the Segments of a Circle, whose Area is Unity. V.Seg 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 con tent. 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 content (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-29.6, and the content 97.4 gallons: and suppose the wet inches 19, to find the quantity of liquor in the cask? From 32 Rem. 1.4 Half 1.2 From 19. Subtr. 1.2 Rem. 17.8 reserved.. 29.6 100 17.8 .60, the V. 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 61 gallons the answer on B. A The American Tutor's Guide, &c. PART VI. 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. 1656 years. 3. When the air presses with its full weight, in very fair 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 weight lies on such a body, in the two greatest alterations of the weather? 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. 1y. 11m. 3w. 2d. 12h. 49min.. 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. 1813 years, sum of the intervals. 6. From the Creation to the Flood was 1656 years; thence 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? Ans. Christ was born Anno Mundi, 4000. 7. Part 1500 acres of land, give B. 72 more than A. and C. 112 more than B. 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 cw. 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 887. and A. and C's to 71. What sum was found, and each man's share? Ans. found 1037. A's 15. B's 321. C'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 part of the weight of the body, in what time will the whole blood of a man, whose weight is 140 lb. circulate through the heart? Anst 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? 61. 13. What number divided by 419844, will quote 9494, and leave just part of the divisor remaining? Ans. 3986138884. 14. Divide $1000 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 203.70! 106d among the captain, 2 lieutenants, the surgeon and purser, 4 petty 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 1037. 18s. 71d. 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 making up accounts, it appears, that A. and B. together, brought off 137. 10s. B. and C. together 12. 12s. and A. and C. together won 117. 16s. 6d. What did they severally get? Ans. 51. 9s. 3d. A. 61. 7s. 3d. B. and 71. 2s. 9d. C. 18. Four persons advance in trade, as follows, viz: W. X. and Y. raised 3501. 10s.; W. X. and Z, 344/. 10s.; X. Y. and Z, contribute 400/.; W. Y. and Z. 3781. 4s. In the conclusion they parted with their joint property for 450 guineas: What did they gain or loose by their adventure?" Ans. Lost 187. 11s. 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. 304 48 gallons. |