To being divided into 1000 equal parts, the number 2 is placed PART III. at 301 of thefe parts, and the number 3 at 477 of them; and fo on; because thefe 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 I 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 thofe oppofite to them on D, and the numbers on the other lines are the fquares of those on D. Any proportion may be wrought on the fliding rule, by fetting the first term on the rule oppofite to the third on the flider; and then, oppofite to the second term on the rule, will be found the fourth term, or answer on the flider: Or it may be wrought on the rod, by extending the compaffes from the first to the third term, and that extent, laid the fame way from the fecond term, will reach to the fourth term. The lines, marked SS, and SL, are ufed for ullaging ftanding and lying casks; fet the length of a standing cafk, or the bung-diameter of a lying caik on N, oppofite to 100 on the other lines, and oppofite to the wet inches on N, take off the number on SS, if it be a ftanding cafk, or on SL, if it be a lying cask; then having placed the content of the cafk on A, oppofite to 100 on B, look on B, for the number taken off, and oppofite to it, on A, is the quantity of ale-gallons in the cafk. In like manner, on the rod, extend the compaffes on the line of numbers, from 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 cafk, will reach from that number to the quantity of liquor in the cafk, in ale-gallons. There are alfo lines on the rod, and on the infide of the fliders, for fhewing the contents of cylinders, at one inch deep, in ale and wine gallons; and befides thefe, on the fliding rule there are lines marked fpheroid or ift variety, 2d variety, 3d variety, and conical variety, on the infide of the flider N, for finding the mean diameters of cafks; that is, the number belonging to the variety, and oppofite 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 fame length and content with the cafk. Thefe varieties are the cafks mentioned in the 5th, 7th, 3d, and 2d rules; but the mean diameters found by the fliding rule are not exact. The moft remarkable lines on the rod, are thofe called Diagonals; for if the rod be put into a cafk from the bung to the fartheft edge of the head, the numbers on these lines, which are cut by the bung, will fhew how many ale and wine gallons the cafk can hold; and for common cafks, this method feems to give the content more exactly X X 2 than PART III than any rule that depends on their forms, and very nearly the fame with the 9th rule, which in every cafe gives the true con tent. Since on thefe inftruments every thing is performed by proportions, the multipliers mentioned in the first rule must be converted into divifors, by annexing cyphers to I, and then dividing it by thefe multipliers, and the quotients will be divifors, anfwering the fame purpose; but in ufing the line D, the square roots of thefe divifors are used in the first rule, or the square roots of 3 times the divisors, in working any of the varieties, er of 6 times the fame divifors, in ufing the 9th rule; these are called Guage-points on D, and for cylinders, they are 18.95 for ale-gallons, and 17.15 for wine; and for any of the varieties, except the third, which is the fame as for the cylinder, they are 32.8 for ale, and 29.7 for wine: and for the 9th rule, they are 46.4 for ale, and 42 for wine gallons. For example, let the depth of a veffel be 56 inches, and the mean diameter 42 inches; fet 56 on C, oppofite to 18.95 on D, then oppofite to 42 on D, is 275 on C, which is the content in ale-gallons; or if 56 on C be fet to 17.15 on D, then oppofite to 42 on D, is 336 wine-gallons on C. Again, let the length of a cafk of the conical variety be 40 inches, and the bung-diameter 32 inches, and the head 24 inches first fet 24 on C, to 24 on D, and oppofite to 32 on C, is 27.7 on D, the fquare root of the product of 32 and 24; therefore, fet 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 bafe of a veffel has two diameters, the fquare root of their product is to be found on D, and ufed inftead of them, in the fame manner as 27.7 in this example. Again, let the length be 40 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, fet 40 on C, oppofite to 42 on D, and then oppofite to 26, and 32, and 60.8 on D, there are found on C, 15.3, and 23.2, and 84.8, which, added together, give 122.3 wine-gallons for the content: and in the fame manner, may any other of the above rules be wrought on the fliding rule. Some of these numbers are marked on the line D, and are called Guage-points, fuch as W.G at 17.15, and A.G at 18.95, which are the guage-points for wine and ale gallons, in finding the contents of cylindrical veffels; alfo M.R at 52.32, being the guage-point for malt-bufhels in the fame veffels, and M.S at 46.37, the guage-point for malt-bufbels, in finding the contents of fquare or rectangular veffels: and in the fame manner, may any : any of the other guage-points be marked on the line D. And PART III. befides the guage-points, there are feveral other numbers marked on the rule, fuch 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 fides of fquares infcribed, and equal to the circle of which the diameter is 1000, and C at 3141.6, the circumference of the fame 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, oppofite to .7854, or Od on C, then, oppofite to 8 on D, there will be found 50.3 on C, the area required. The ufe of the line MD, is for guaging rectangular veffels, of floors of malt; fet the length on B, oppofite to the breadth on MD, and then oppofite to the depth on A, there will be found the content in malt-bufhels 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 oppofite to 5 on A, there is found 35 on B, the number of bushels in the floor. By the line E, the contents of fimilar veffels may be found from one another, and the dimenfions of fimilar veffels may be found from their contents. Suppofe the depth of a veffel containing 100 gallons to be 40 inches; what is the content of a fimilar veffel, of which the depth is 36 inches? fet 40 on D, against 100 on E, then, oppofite to 36 on D, there will be found 72.9 on E, which is the content. Again, let the depth of a veffel be 30 inches, and that it is required to find the depth of a fimilar veffel that fhall contain twice as much; fet 30 on D, to I on E, and againft 2 on E, there will be found 37.8 on D, which is the depth required. Suppofe, again, that the diagonal, a cak containing a hogfhead, is 31 inches, and that it is required to find the diagonal of a fimilar cafk 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 fame 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 fame manner with those for English meafures. For example, the fextier of Paris contains 384 cubical Paris inches; therefore, if .785398 be divided by 3842 PART III. 384, the quotient .0020453 is a multiplier for giving the contents in fextiers; or if 384 be divided by .785398, the quotient 488.924, will be a divifor for the fame purpose, and the fquare root 22.1 of this divifor will give the guage-point for cylinders to be used on the line D: Thefe numbers fuppofe that the veffel is guaged in Paris inches, but if the veffel be guaged in English inches, the fextier contains 464.8 fuch inches; and therefore the multiplier in this cafe will be .00169, and the guage-point on D will be 24.3. In like manner, there being 1040 cubical English inches in a ftechan of Amfterdam, or in an almuda of Portugal, numbers may be found for finding the contents in the measures of these countries; and fo on. When a cafk, not full, is lying with its axis parallel to the horizon, it was fhown in Prob. g. of this Part, how to find the content of the full or empty part; but it required the area of the fegment of a circle to be found, which being tedious, is ufually done by means of a table of circular fegments, of which the following is a specimen. A TABLE Of the Areas of Segments. [V. S. Segm. V. S. Segm. V. S. Segm. ||V. S. | Sexm. ||V. S. | Segm. To ullage a lying cafk by the table of fegments, 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 fide, is to be taken out of the table, and multiplied by the whole content of the cafk; divide the product by 8, and from the quotient cut off four figures for a decimal, the rest of the figures fhew the quantity of ale-gallons in the cask. Suppofe the content of a cafk to be 92 gallons, the bung-diameter 32 inches, and the wet inches 8; divide 8.co by 32, the quotient quotient is .25, oppofite 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 cafk be above half full, divide the dry inches inftead of the wet inches, and thus find the content of the empty part, which, fubtracted from the whole content, gives the quantity of liquor in it. |