zo being divided into 1000 equal parts, the number 2 is placed PART III. at 301 of these parts, and the number 3 at 477 of them ; and so on; because these are the logarithms of 2 and 3, when the logarithm of 10 is 1000. The distance between 1 and 10, on the line marked D, is three times as great as that between 1 and 10 on the line E, and twice as great as that on any of the other lines A, B, C, N, or MD; and therefore the numbers on E are the cubes of those opposite to them on D, and the numbers on the other lines are the squares of those on D. Any proportion may be wrought on the sliding rule, by setting the first term on the rule opposite to the third on the slider; and then, opposite to the second term on the rule, will be found the fourth term, or answer on the slider : Or it may be wrought on the rod, by extending the compaffes from the first to the third term, and that extent, laid the same way from the second term, will reach to the fourth term. The lines, marked SS, and SL, are used for ullaging standing and lying calks ; set the length of a standing calk, or the bung-diameter of a lying caik on N, opposite to 100 on the other lines, and opposite to the wet inches on N, take off the number on SS, if it be a standing calk, or on SL, if it be a lying calk; then having placed the content of the cak on A, opposite to 100 on B, look on B, for the number taken off, and opposite to it, on A, is the quantity of ale- gallons in the calk. In like manner, on the rod, extend the compaffes on the line of numbers, froin the bung-diameter to the wet inches; and that extent will reach on SL, from 100 to a number, and the extent on the numbers from 1 to the content of the calk, will reach from that number to the quantity of liquor in the cask, in ale-gallons. There are also lines on the rod, and on the inside of the sliders, for shewing the contents of cylinders, at one inch deep, in ale and wine gallons; and belides these, on the fliding rule there are lines marked spheroid or ist variety, 2d variety, 3d variety, and conical variety, on the inside of the slider N, for finding the mean diameters of casks; that is, the number belonging to the variety, and opposite to the difference between the bung and head diameters, added to the head-diameter, gives the mean diameter, or the diameter of a cylinder of the same length and content with the cask. These varieties are the casks mentioned in the sth, 7th, 3d, and 2d rules; but the inean diameters found by the sliding rule are not exact. The most remarkable lines on the red, are those called Diagonals; for if the rod be put into a calk from the bung to the fartheft edge of the head, the numbers on these lines, which are cut by the bung, will thew how many ale and wine gallons the call can hold ; and for common casks, this method seems to give the content more exadly thaa X X 2 I'ART III. than any rule that depends on their forms, and very nearly the fame with the gth rule, which in every case gives the true con- For example, let the depth of a vessel be 56 inches, and the mean diameter 42 inches ; set 56 on C, oppolite to 18.95 on D, then opposite to 42 on D, is 275 on C, which is the content in ale-gallons; or if so on C be set to 17.15 on D, then opposite to 42 on D, is 336 wine-gallons on C. Again, let the length of a cask of the conical variety be 40 inches, and the bung-diameter 32 inches, and the head 24 inches : first set 24 on C, to 24 on D, and opposite to 32 on C, is 27.7 on D, the square root of the product of 32 and 24; therefore, set 40 on C, to 32.8 on D, and then oppofite to 56, and 27.7 on D, are 116.5, and 28.5 on C, of which the diffe. rence is 88 ale-gallons, the content. Whenever the base of a vefsel has two diameters, the square root of their produét is to be found on D, and used instead of them, in the same manner as 27.7 in this example. Again, let the length be 49 inches, the bung-diameter 32, the head 26, and the diameter in the middle between the head and bung 30.4 inches; and working as in the 9th rule, set 40 on C, opposite to 42 on D, and then opposite to 26, and 32, and 60.8 on D, there are found on C, 15.3, and and 84.8, which, added together, give 122.3 wine-gallons for the content: and in the same manner, may any other of the above rules be wrought on the sliding rule. Some of these numbers are marked on the line D, and are called Guage-points, such as W.G at 17.15, and A. G at 18.95, which are the guage-points for wine and ale galloris, in finding the contents of cylindrical veilels ; also M.R at 52.32, being the guage-point for malt-busheis in the same vessels, and M.S aç 46.37, the guage-point for malt-buíbels, in finding the contents of square or rectangular vefiels : and in the same manner, may any any of the other guage-points be marked on the line D. And Part III. . besides the guage-points, there are several other numbers marked on the rule, such as M.B at 2150.4, the inches in a malt-bufhel, A at 282, the inches in an ale-gallon; these are marked on the line A: and on the line B, there are marked W at 231, the inches in a wine-gallon, S.I at 707, and S.e at 886, which are the sides of squares inscribed, and equal to the circle of which the diameter is 1003, and C at 3141.6, the circumference of the same circle: and on the line C are marked OC at .0796, the area of a circle of which the circumference is 1, and Od at .7854, the area of a circle of which the diameter is 1. The use of these is obvious; for if the area of a circle be required of which the diameter is 8, place i on D, opposite to .7854, or Od on C, then, opposite to 8 on D, there will be found 50.3 on C, the area required. The use of the line MD, is for guaging re&angular vessels, of floors of malt: set the length on B, opposite to the breadth on MD, and then opposite to the depth on A, there will be found the content in malt-bushels on B. Thus, if the length of a floor of malt be 270 inches, its breadth 56 inches, and its mean depth 5 inches ; fet 270 on B, oppofite to 56 on MD, counted towards the left hand, then opposite to 5 on A, there is found 35 on B, the number of bushels in the floor. By the line E, the contents of similar vessels may be found from one another, and the dimensions of similar vessels may found from their contents. Suppose the depth of a vessel containing 160 gallons to be 40 inches ; what is the content of a similar vessel, of which the depth is 36 inches ? set 40 on D, against 100 on E, then, opposite to 36 on D, there will be found 72.9 on E, which is the content. Again, let the depth of a vefsel be 30 inches, and that it is required to find the depth of a fimilar vefsel that shall contain twice as much; set 30 on D, to I on E, and against 2 on E, there will be found 37.8 on D, which is the depth required. Suppose, again, that the diagonal, a cak containing a hogshead, is 31 inches, and that it is required to find the diagonal of a similar cask that shall hold a puncheon; fet 31 on D, to 63 gallons on E, and against 84 gallons on E, there will be found 34.1 inches on D, which is the diagonal required. It is evident, that the same rules will give the contents of veffels in the measures of foreign countries, if proper multipliers or guage-points be found for these measures; and these multipliers and guage-points are found from the number of inches in any measure exactly in the same manner with those for English measures. For example, the sextier of Paris contains 384 çubical Paris inches; therefore, if .785398 be divided by 3842 be PART I'I. 384, the quotient .0020453 is a multiplier for giving the con tents in fextiers; or if 384 be divided by :785398, the quotient When a cajk, not full, is lying with its axis parallel to the A TABLE Of the Areas of Segments. V. S. Segm. V. S. Segm. V. S. Segm. IV. S. Segm. IV. S. Segm. To ullage a lying calk by the table of segments, divide the wet inches by the bung-diameter, and find the quotient in the table, in one of the columns marked V. S. and the area of the fegment nearest to it on the right side, is to be taken out of the table, and multiplied by the whole content of the cak; divide the produet by 8, and from the quotient cut off four figures for a decis mal, the rest of the figures shew the quantity of ale-gallons in the cak Suppose the content of a cask to be 92 gallons, the bung-diameter 32 inches, and the wet inches 8; divide 8.co by 32, the quotient quotient is .25, opposite to which in the table is found 15355, PART III. which, multiplied by 92, and four figures cut off from the product, gives 141.3660; and this, divided by 8, gives 18 ale-gallons nearly for the quantity of liquor in it. If the cask be above half full, divide the dry inches instead of the wet inches, and thus find the content of the empty part, which, subtracted from the whole content, gives the quantity of liquor in it. Τ Α Β Ι Ε aune aune cane auiie Inches, 46.680 47.604 77.220 27.1 201 27.228 27.276 27.260 22.860 22.896 25.008 23.466 22.836 21.9721 $ Inches. Paris foot 12.792 Lyons 13.458 Leyden 12.396 Amsterdam 11.304 Antwerp 11.352 Bruffels 10.828 Hamburgh 11.376 Lubeck 11.448 Denmark 12.504. Sweden 11.733 Dantzick 11.328 Riga 10.986 Ruslia arsheen 9.090 pal. merc. Rome 9.791 pal. arch. 8.779 Venice foot 13.944 Leghorn palm Genoa 9.900 Naples palm 10.332 Madrid S palm 9.012 12.012 Toledo foot 10.788 Gibraltar 11.039 Lisbon cavido 20. 122 Constantinople sh. pike 25.576 Persia Old Greek foot 12.087 lold Roman foot 11.604 Scripture palm 3.648 9.182 palon cane brace mer. 34.270 brace arch. 30.739 cane. 81.900 brace 26.460 brace 22.95 cane 88.290 brace 25.200 82.5601 vara 35.0401 vara 32.220 33.1201 33.000 27.220 arish 38.364 cubit 18.132 cubit 17.490 cubit 21.8831 vara vara gr. pike FINI S. |