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 1 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. 5018660 8465 79057637 76 92709227 8881 8827 151 8680 8493 7937 7681 77| 9296 9258 8944 8874 52 8700 85207970/7725 78 9324 9290 8967 8922 538720 8548 8002 7768|79| 9352 9320 9011 8970 154 8740 8576 8036 7813|| 80 9380 9352 9055 9018 55 8760 8605 80707858 81 9409 9383 9100 9066 5687818633 8104 7902 82 9438| 9415| 9144 9114 57 8802 86628140 7947 83 9467) 9446 9189 9163 158 8824 8690 8174 7992 84) 9496 9478 9234 9211 59 88468720 8210 8037 85 9526 95101 9280 9260 1608869 8748 8246 8082 86 9556 9542 9326 9308 61 8892 8777 8282 8128 87 9586 9574 9372 9357 1628915 8806 8320 8173 88 9616 9606 9419 9406 638938 8835 8357 8220 9647| 9638) 94 55 164 8962 8865 8395 8265 90 96781 9671 9513 9504 65 8986 8894.84338311|| 91 97101 9703 9560 9553 6619010 8924 84728357 92 97401 9736 9608 9602 6719034 8954 8511 8404|| 93 9772 9768 9656 9652 168 9060 8983 8551 8450|| 94 9804 9801 9704 9701 169 9084 9013 8590 8497 95 9836 9834 9753 9751 709110 9044 86318544|| 96 9868 9867 9802 9800 71/9136 9074 86728590 97 9901 9900 9851 9850 729162 9104 8713 8637 98) 9933| 9933 99001 9900 739188 9135 8754 8685| 99 9966 9966 9950 9950 74.9215 91661879618732! 100 10000/10000 10000 10000 45192429196 8838 8780 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 .9242which being each (.29.5744 ) for the corres.9196 multiplied by 32, ) 29.4272 ponding mean .8838 produce respect- 28.2816 diameters re.8780 ) ively 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 which being 29.60 ) for the respective mean di5.10 added to 24, 29.10 | ameters; all of which are 4.56 28.56 there result too great, except the second, 4.12 28.12 ) wbich 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, 29.57 97.4 119.5 on C, as near as can 29.43 96.5 118.0 | be judged; which 28.28 89.1 108.8 | agree nearly with the 28.10 88.0 107.3 contents computed in the preceding chapter. against SCHOLIUM. 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 tbat 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. Ez. 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. Take the length of the lower deck, from the rabbet of the stem to that of the stern-post, and from it subtract 32 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 3 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. The motion described by bodies freely descending tz 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 . 1 2 3 4 &c. The velocities will be also as 1 2 3 4 &c. The spaces as their squares. 1 4 9 16 &c. and the spaces for each time, as 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: 2. e. 1" 2" 3" &c. Velocities in feet will be . 327 613 961 &c. Spaces in the whole time will be 1672 643 1447 &c. Spaces for each second will be . 161: 48} 80 &c. . . |