PART III. cyphers to unity, and dividing by the number of inches in the wine or ale gallon. The multipliers for round veffels are got, by dividing 7853982 by the number of inches in the feveral measures in which it is required that the contents fhould be expreffed. For example, .7853982, divided by 282 the number of inches in an ale-gallon, gives .0027851, the multiplier for ale-gallons; therefore, instead of multiplying by .7854, in finding the content, multiply by .0027851, and the content will be found in ale-gallons. In the fame manner, .0034 is found to be the multiplier for wine-gallons: and fince these multipliers are to one another, nearly as 9 to 11, it follows, that 9 ale-gallons are nearly equal to 11 wine-gallons. And as there are 2150 folid inches in a malt-bufhel, the multiplier in this cafe will be .0003653: and if the Scots pint be 103.4 inches, the multiplier for it will be .0076; and for the Scots wheat-firlot of 21 pints, or 2197 inches, the multiplier is .000358; or for the barley-firlot, of 31 pints, it is .000245; fo that 107 wheatfirlots are nearly equal to 110 malt-bufhels; and there are 92 bolls in 47 quarters of the fame measure, or 196 bolls nearly in 100 quarters. By the help of these multipliers, the contents of veffels may be found by the following rules. 1. To find the content of a cylindric veffel; multiply the fquare of the diameter of the bafe, by the height, both being expreffed in inches; then, to find the content in wine-gallons, multiply the product by .oc34; or to have it in ale-gallons, multiply by .0027851; or to have it in malt-bufhels, multiply it by .0003653; or again, to have the content in Scots pints, multiply the faid product by .0076; or if it be required in Scots wheat-firlots, multiply by .000358; or in barley-firlots, by .000245. If the bafe of the veffel be not a circ1 tiply the product of the greatest and } height, and then by the proper multiplier. t an ellipfe, muldiameters, by the Let the diameter of the base be 48 inches, and the height 54 inches; the fquare of 48, viz. 2304, multiplied by 54, gives 124416, the common product, which, multiplied by .0034, gives 423.0144 wine-gallons for the content; or if it be multiplied by .0027851, the content will be 346.511 ale-gallons. 2. When the veffel may be confidered as the fruftum of a cone; from the fquare of the fum of the greatest and leaft diameters, fubtract their product, and multiply of the remainder by the length, and then by the multiplier for the measure. Thus, if the length of fuch a cafk be 40 inches, and the bungdiameter 32 inches, and the head-diameter 26 inches; their fum is 58, from the fquare of which, 3364, fubtract their product 832, and the remainder is 2532; and of it 844, multiplied PART III. by the length 40, gives 33760, which, multiplied by .0034, gives 114.7154, the content in wine-gallons. 3. When the veffel is of the kind mentioned in Prob. 3. which is called a fruftum of a parabolic conoid; take the diameter in the middle, between the greatest and leaft widths, and fquare it; or add the squares of the greatest and leaft diameters, and take half of their fum; then multiply by the length, and the multiplier. 4. If the veffel be the fruftum of a sphere; from the fquare of the greatest diameter, fubtract of the square of the diftance between the greatest and leaft diameters, and multiply the remainder by the length, and by the multiplier. 5. If the veffel be the fruftum of a spheroid; to twice the fquare of the greatest diameter, add the fquare of the least, and multiply of the fum by the length, and the multiplier: but if the bafes be unequal, find a fourth proportional to the fquare of the axis, the fquare of conjugate axis of the ellipfe, and 13 of the fquare of the length, and add this fourth proportional to the fquares of the greatest and leaft diameters, and multiply the fum by the length, and by the multiplier. 6. When the veffel is the fruftum of a hyperbolic conoid; find a fourth proportional to the fquare of the axis, the square of the conjugate axis, and 1 of the fquare of the length, and fubtract it from the fum of the fquares of the greatest and least diameters; then multiply of the remainder by the length, and by the multiplier. 7. When the veffel is the fruftum of a parabolic fpindle; to twice the fquare of the greatest diameter, add the fquare of the leaft, and from the fum fubtract of the fquare of the diffe rence of the fame diameters, and multiply of the remainder by the length, and by the multiplier for the meafure required. The most common forms of cafks contain lefs than what is given by the 4th, 5th, or 7th rule; but more than what is given by any of the others. The following method feems to agree best with common cafks, and is nearly a mean betwixt the 3d and 7th rules. 8. From 12 times the head-diameter, fubtract 7 times the bung-diameter, and multiply the remainder by twice the bungdiameter, and fubtract the product from the fquare of 5 times the fum of these diameters; then multiply the remainder by the length, and by .00034, and the product, divided by 9 for wine, or by 11 for ale, gives the content in gallons. The dimenfions being as in the 2d rule, the content by this rule would be 118 wine-gallons, or 97 ale-gallons. 9. But the best way is to take a diameter in the middle, be PART III. tween the head and bung, and to add together the fquares of the head and bung diameters, and of twice the middle diameter for of the fum reduces the cask to a cylinder; that is, when multiplied by the length, and the proper multiplier, it gives the content, whatever be the form of the cafk. 10. Large veffels are measured by distinguishing them into fruftums, and taking the diameter at the middle of each fruftum, with which the content of the fruftum is found, as if it were a cylinder; and the contents of all the fruftums added together give the content of the veffel: after which, they form a table of the contents at each inch deep, by fubtracting fucceffively the areas of the circles belonging to thofe mean diameters, each as often as there are inches in the altitude of the fruftum to which it belongs; and from this table it readily appears how much liquor is contained in the veffel, by only knowing the depth of the liquor. 11. To mealure round timber; find the mean circumference, or girth, and multiply its fquare by .0795774715, which is the area, when the circumference is 1, the product is the content. Let the mean girth of a tree be 10.3 feet, and the length 24 feet, then the fquare of 10.3, is 106.09, which, multiplied by 40, and by .0795775, gives 202.617 feet for the content. But the common way is to take of the girth, and to multiply its fquare by the length for the content; according to this method, the tree, in this example, would be only 159.135 feet, which is 43.482 feet too little. Square timber is measured by multiplying the mean breadth and thickness together, and the product by the length. 12. To find a fhip's tonage; mu tiply the length of the keel by the breadth of the fhip in the middle, and then multiply the product by the depth of the hold, and the product, divided by 94, gives the tonage; but neither this, nor any other rule that can be given, will ferve for all forts of fhips. OF THE SLIDING RULE, AND THE DIAGONAL ROD. HE officers of the revenue generally find the contents of TVE lines on thefe inftruments are divided into parts, proportional to |