EXAMPLE I. Supposing the bung diameter 32, and content 92 ale gallons; to find the ullage for 8 wet inches. 32)8(.25, whose tab. seg. is .153546 92 307092 1381914 14.126232 is 3.531558 17.657790 Ans. EXAMPLE II. Taking the length of the cask 40, bung diameter 32, head diameter 24; and supposing the wet inches to be 8. What is the ullage? Ans. 18 ale gallons. Of Gauging Casks by their Mean Diameters. PROBLEM I. To find the Mean Diameter of a Cask of any of the four varieties, having given the bung and head diameters. DIVIDE the head diameter by the bung diameter, and find the quotient in the first column of the following table, marked Qu. Then if the bung diameter be multiplied by the number on the same line with it, and in the column answering to the proper variety, the product will be the true mean diameter, or the diameter of a cylinder of the same content with the cask proposed, cutting off four figures for decimals. Qu Var. 2 Var. 3 Var. 4 Var. Qu 1 Var. 2 Var. 3 Var. 4 Var. 50 8660 8465 7905 7637 51 8680 8493 7937 7681 52 8700 8520 7970 7725 53 8720 8548 8002 7768 54 8740 8576 8036 7813 55 8760 8605 8070 7858 56 8781 8633 8104 7902 57 8802 8662 8140 7947| 58 8824 8690 8174 7992 59 8846 8720 8210 8037 60 8869 8748 8246 8082 61 8892 8777 8282 8128 62 8915 8806 8320 8173 53 8938 8835 8357 8220 64 8962 8865 8395 8265 65 8986 8894 8433 8311 91 66 9010 8924 8472 8357|| 92 9740 9736 9608 9602 67 9034 8954 8511 8404|| 93 9772 9768 9656 9652 68 9060 8983 8551 8450 94 9804 9801 9704 9701 69 9084 9013 8590 8497 95 9836 9834 9753 9751 70 9110 9044 8631 8544 96 9868 9867 9802 9800 71 9136 9074 8672 8590|| 97 9901 9900 9851 9850 72 9162 9104 8713 8637 98 9933 9933 9900 9900 73 9188 9135 8754 8685 99 9966 9966 9950 9950 74 9215 9166 8796 8732 100 10000 10000 10000 10000 75 9242 9196 8838|8780| 76 9270 9227 8881 8827 77 9296 9258 8944 8874 78 9324 9290 8967 8922 79 9352 9320 9011 8970 80 9380 9352 9055 9018 81 9409 9383 9100 9066 82 9438 9415 9144 9114 83 9467 9446 9189 9163 84 9496 9478 9234 9211 85 9526 9510 9280 9260 86 9556 9542 9326 9308 87 9586 9574 9372 9357 88 9616 9606 9419 9406 89 9647 9638 9466 9455 90 9678 9671 9513 9504 9710 9703 9560 9553 EXAMPLE. Supposing the diameters to be 32 and 24, it is required to find the mean diameter for each variety. Dividing 24 by 32, we obtain .75; which being found in the column of quotients, opposite thereto stand the numbers .9242) which being each (29.5744) for the corres.9196multiplied by 32, .8838 produce respect- 28.2816 .8780 jively 29.4272 ponding mean diameters re 28.0960 quired. BY THE SLIDING RULE. Find the difference between the bung and head diameters on the fourth face of the rule, or inside of the third slider; and opposite thereto is, for each variety, a number to be added to the head diameter, for the mean diameter required. So, in the above example, against 8, the difference of the diameters, are found the numbers 5.60 5.10 4.56 4.12 which being added to 24, 29.60 for the respective mean di29.10ameters; all of which are 28.56 too great, except the second, 28.12 which is too little. So that this method does not give the true mean diameter. PROBLEM II. To find the content of a cask by the mean diameter on the Sliding Rule. Set the length on C, to the guage point, 18.95 for ale, or 17.15 for wine, on D; then against the mean diameter on D, is the content on C. EXAMPLE. If the bung diameter be 32, the head 24, and the length 40 inches, Having found the mean diameters, as in the last problem, and set 40 on C, to 18.95 or 17.15 on D, Having delivered the necessary rules for measuring casks, &c., I do not suppose that any thing more of the subject of gauging is wanted to be given in this book. For, as to cisterns, couches, &c. tuns, coolers, &c. coppers, stills, &c. which are first supposed to be in the form of some of the solids in the former parts of this work, and then measured accordingly, no person can be at a loss concerning them, who knows any thing of such solids in general; and to treat of them here would induce me to a long and tedious repetition only for the sake of pointing out the proper multipliers or divisors; which is, I think, a reason very inadequate to so cumbersome an increase of the book. I shall only just observe, that when tuns, &c. of oval bases are to be gauged; as those bases really measure more than true ellipses of the same length and breadth, they ought to be measured by the equi-distant ordinate method. And that when casks are met with which have different head diameters, they may be deemed incomplete casks, and their contents considered and measured as the ullage of a cask. TO FIND THE TONNAGE OF A SHIP. The length is taken in a straight line along the rabbet of the keel, from the back of the main stern-post to a perpendicular from the fore part of the main stem, under the bowsprit, from which subtract of the breadth, the remainder is the length. The breadth is taken at the broadest part of the ship, from the outside to the outside. RULE. Multiply the square of the breadth by the length, and divide the product by 188, the quotient will be the tonnage. Ex. 1. Required the tonnage of a ship, of which the length is 75 feet, and the breadth 26 feet. Ans. 26 x 26 x 75÷÷188-270 tons, nearly. Ex. 2. Length 96, and breadth 33 feet? Ans. 556 tons. Note. This rule is very erroneous, and no other general rule can be given which is perfectly accurate; the best way is to find the quantity of water displaced by the ship when she is loaded; but as this must be done by means of ordinates, the operation is laborious. It is easier to load her with ballast, weighing the load as it is put on board. The following rule is a near approximation for ships of burden. 55 Take the length of the lower deck, from the rabbet of the stem to that of the stern-post, and from it subtract of it, for the length. Take the extreme breadth from outside to outside, and add it to the length of the lower deck, of the sum is the depth. Set up this depth from the limber strake, where the extreme breadth was taken, and at this height take a breadth from outside to outside, take another breadth at of this height, and a third at of the height, add these three to the extreme breadth, and of the sum is the mean breadth. Multiply the length, breadth, and depth, and divide three times the product by 110 for the tonnage. FALLING BODIES. 1 2 3 4 &c. The motion described by bodies freely descending ty their own gravity is, viz.-The velocities are as the times, and the spaces as the squares of the times. Therefore, if the times be as the numbers . The velocities will be also as The spaces as their squares. and the spaces for each time, as 1 2 3 4 &c. 1 4 9 16 &c. 1 3 5 7 &c. namely, as the series of the odd numbers, which are the differences of the squares, denoting the whole spaces; so that if the first series of numbers be seconds of time: i. e. 1" 2" 3" &c. 32 64 96 &c. 16 641 144 &c. ΤΣ 1 16 481 805 &c. 12 |