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the depth 22 instead of 2.2 inches. A similar mode of using the double lines S S and SL, as laid down on the pocket slide-rules, must be observed whenever the wet or the dry inches fall, as it is technically said, " off the rule.”
Large ullage-rule.-A rule of much greater length than the ordinary excise sliderule, but containing nothing more than the necessary lines for the ullaging of lying casks, is furnished to each distillery and warehouse station. On this rule, the scale S L is laid down in one continuous progression, instead of being divided, as in the case of the shorter instruments, into two portions. The openness of the graduation and the clearness of the marking render the ullage-rule especially available for the accurate determination of small vacuities in lying casks.
It is needless to give any examples of the manner of using this rule, as the line of segments on it has merely to be read off in one unbroken series corresponding to the figures actually marked upon the slide. It may be useful, however, to inexperienced officers to be enabled to cheok by some other means their estimations of small ullages on the rule, and for this purpose the following short table may be referred to, as coinciding almost exactly with the segments laid down on the large ullage-rules.
TABLE OP ULLAGES FOR CASKS LYING.
Quotient of dry
inches divided by bung-diam. Segment. by bung-diam, Segment. by bung diam.
Segment .005 . . . .000 .065
.032 .150 . , 076 .035 . . .
.080 040 .007 .100 . . .037
165 . . . .088 .011 .110 . . .044
170 . . .094 055 . . . •014 .115 . . . .047 .175 . . . .098 .060 . . .016 1 .120 . . .051 .180 . . .103
Example. Suppose a lying cask, bung-diam. 18.7 inches, dry inches 2.5, content 54 galls. What is the vacuity in galls. ? 18.7)2.5(-134. Opposite .135 (the nearest number to .134) in table, is .063.
Segment .063 multiplied by 54 content = 3.4 galls. vacuity. At various points on the excise slide-rule are inserted brass pins or dots, with distinctive letters attached, which serve to indicate what are termed «gauge points." On some rules also there are short lines marked “X 1st V," " X 2nd V," &c.; and along the edges of most rules are certain other scales for special purposes ; but an explanation of any of these cannot properly be introduced in the present chapter. A sufficient account of the points and lines in question (very few of which are applied in the modern practice of the Excise) will be given in the next chapter.
For a general history of the slide-rule in its original form, and under its latest modifications, officers are referred to the article SLIDE-RULE, in the Penny Cyclopædia, or the English Cyclopædia.
Practical Mensuration--Gauging-Application of the Slide-Rule.
(1) GENERAL OBSERVATIONS.-The art of Menguration, as at present practised by the Excise, is of a very simple character, and is confined within a very limited range of objects. At one time the daily business of an officer was almost wholly made up of the execution of various tasks and problems in “ gauging ;" a great number of different articles was then subject to duty, and from the irregular form and awkward position of many of the vessels employed by traders for the reception of the raw materials or the products of their manufacture, a considerable degree of skill and ingenuity was required to obtain correct accounts of the quantities of goods in stock or in operation. Now, however, owing to the extensive repeals that have taken place, and to changes in the mode of securing the revenue on the few remaining heads of taxation, the business of gauging has lost much of its former difficulty and importance, and, except as regards the survey of maltsters, constitutes but a small portion of the services which officers are called upon to render. A knowledge of the first principles of Mensuration, together with a short course of technical instruction under an experienced officer, will enable any person who exercises ordinary care and intelligence, to perform, in a competent manner, all the necessary details of the existing system.
"Gauging" is that branch of practical Mensuration which especially applies to certain expeditious methods of finding the total or partial capacities of vessels, such as casks, vats, tuns, &c., and expressing the contents of these vessels in standard denominations of measure. The magnitude of bodies consisting of loose particles, such as masses of grain, malt, &c., laid together in regular forms, may also be conveniently determined by the process of gauging, when at least one of the dimensions admits of being taken internally.
In the common mensuration of solids, results are usually determined and expressed in cubic inches, cubic feet, &c., but in all operations where gauging is employed, the results are reduced, either in the course of the calculation, or, as a final step, to their equivalents in gallons, bushels, or other appropriate expressions of dry or liquid measure.
Quantities of exciseable goods, when ascertained by gauging or actual measurement, are generally stated, as regards liquids, in gallons, and in decimal or vulgar fractions of a gallon, the commercial sub-divisions of quarts, pints, &c., being rarely adopted. On the exportation of beer, the amount entitled to drawback is expressed in barrels, firkins, and gallons, agreeably to the usage of the trade. The bulk of brewers' grains, as estimated by gauge, is reduced to quarters, bushels, and
gallons, in order to form a compare with the trader's entry of malt brewed. The amounts of gauges of grain making into malt are set forward in bushels and tenths, and the net charges, in bushels and hundredths of a bushel.
Tables of all the standard weights and measures will be found in the Appendix.
In computing the number of bushels, gallons, &c., which any given vessel contains, or is capable of containing, or which make up the bulk of any given solid, the cubic inches corresponding to the space in question may first be determined by the proper rule, as laid down in the last chapter, under the head of “ Principles of Mensuration," and the result thus obtained be then reduced to its equivalent in the required measure by applying the divisor or factor which represents a unit of that denomination. Considerable labor will, however, be saved in many cases by the use of a modified divisor or factor-that is, where separate multiplications and divisions, or successive processes of the same kind would otherwise be necessary, a single operation combining all these effects may be resorted to with advantage, if a suitable divisor or multiplier be prepared for the purpose.
As an example of the working of this expedient, suppose the diameter of a cylinder (see page 211) to be 40 inches, and the height 120 inchos. To find the solid content in gallons. 1st Method.
2nd Method. Diameter 40
192000, sq. of diam. x height.
Now, to multiply by •7854,
and then divide by 277-274,
is the same as to divide only
by the quotient of
277274=•7854 = 353.04.
Galla 277-274)150796-8(543-8. Answer. 353.04)192000(543-8 galls. Answer. A still greater simplification is to employ the factor formed by dividing unity by 353-04, that is, .0028326.
Thus, 192000 x .0028326 = 543-8 galls., as before. Other examples of the usefulness of this manner of computing contents will appear in the subsequent articles on the gauging of each class of traders' utensils.
In applying the rules prescribed by the theory of mensuration, to the various objects which actually present themselves in the practice of the art, it should be recollected that no vessel, utensil, or self-contained solid, however carefully and skilfully formed, is, at all its points, in strict accordance with the outline of the geometrical figure to which it bears the closest resemblance. There is, in every case, some amount of deviation from the perfect and definite model ; and when this want of conformity is seen to be considerable, it will be necessary, if great exactness be demanded, either to treat the body as composed of several distinct sections, each of which must be measured by appropriate methods, and the results then added together; or else to take a sufficient number of dimensions at equal distances apart, and to compute the total magnitude by the rule laid down on page 215, for the mensuration of irregular bodies in general.
Uniformity and simplicity of procedure, combined with expedition, being of more importance in the assessment of the revenue than extreme accuracy, it is the custom of the Excise to gauge all vessels, &c., as though they were of precisely similar shape to certain regular solids, obtaining, when requisite, a mean of several dimensions in the
Cubic inches 150796-8
same direction, and making such allowances as judgment or experience shows will nearly compensate the errors that are thus committed.
It would obviously be inexpedient to permit each officer, unless in rare and exceptional instances, to pursue his own methods of gauging. A system of general applicability has therefore been devised, which officers are enjoined to carry out on all ordinary occasions, without regard to the fact that a different mode of operation would, in some cases, give more accurate results.
Dimensions are ascertained by means of graduated rods or tapes. Tapes are divided into whole inches only, and are used chiefly to measure the lengths and breadtbs of « malt-floors," or to check the like dimensions of fixed malt utensils. The tape carried by officers, admits of being wound up in a compact form within a circular leathern box or case. " Gauging," " Diameter," or " Dimension" rods, as they are variously termed, are divided into inches and tenths; the pieces are made of equal and convenient lengths, which fit closely into one another at the ends, or are connected together with screws, so as to form, when requisite, one continuous rod of sufficient length. Such rods must always be employed to fix the dimensions of vessels, &c., in which charges of duty arise. « Dipping rods," of similar construction, are used to take the depth of liquor in a vessel, and of grain laid in frames or in loose heaps. Each parcel of the larger “ gauging-rods" consists generally of several blank pieces, and of two graduated joints held together with a clamp, and capable of sliding one by the other. The total distance measured in any instance is found by adding the sum of the known lengths of the blank pieces to the number of inches and tenths exhibited at the point of junction of the two graduated joints, the divisions on these joints being marked in opposite directions. There are other special arrangements of sets of gauging rods, the principle of which will at once be understood on inspection of the instruments.
When, from the largeness of a vessel it becomes necessary to apply several connected lengths of a gauging rod, supports should be placed beneath so as to keep the rods extended in a straight line between the sides of the vessel. The length, breadth, or diameter of a vessel, is not taken by the Excise to the nearest tenth, but only to the last completed tenth, a slight advantage being thus afforded to traders.
In the practice of gauging, it is found convenient, as was remarked on page 171, to extend the original and legitimate meaning of the word area, and to include under it the content of a regularly formed vessel or solid, at one inch of its altitude. The number which expresses the area (properly so called) of any surface in square inches, must evidently be the same as that which expresses the content in cubic inches of a solid having this surface as its base, of equal boundary throughout its extent, and having the depth or thickness of one inch. To compute the magnitude of such a solid, we multiply the area of the base by 1, which produces no change of numerical value, but bas simply the effect of transforming superficial units into solid units. Thus, & cylinder measuring 16 square inches at its base or at its upper surface, and possessing the depth of one inch, contains 16 x 1, or 16 cubic inches of space or matter in its entire volume. If the depth of this cylinder were increased, say to 20 inches, its total content would be found by multiplying 16 by 20; accordingly, it will be understood that the term “ area” may be used, without confusion, to denote the content at one inch deep, of any cylindrical, rectangular, or other solid of uni. form dimensions, if, only, it be borne in mind that an expression of solidity is here meant, and not one of superficial magnitude, as would be implied by the proper signification of the word.
The total or partial content of conical, tapering, or irregularly-shaped bodies, in which the horizontal sections differ considerably from each other, cannot, it is obvious, be estimated by means of an « area” representing the solidity at one inch deep, since the space comprised between every two successive inches of the depth must, in these cases, be greater or less than the space answering to the first inch. But by taking dimensions at small distances apart, and computing a separato « area" for each such fraction of the depth, the magnitude of the whole vessel, &c., or of any given part of it, may be obtained without great error, by multiplying each area into the depth to which it applies, and adding the results together. An illustration of this point will be furnished under the head of " Browers' Mash Tuns."
Areas, in the practice of gauging, are reduced from cubio inches into bushels, gallons, or other appropriate measures, and are generally inserted for convenience of reference at the top of the column in the officer's survey book, showing the state of the vessel or utensil in which gauges are required to be taken.
Fixed Divisors and Factors. It has already been remarked, that in order to convert any number of cubic inches into its equivalent in bushels, gallons, &c., it is necessary, as a general rule, either to divide by the standard unit of the required measure, or to multiply by the reciprocal of that unit. The same effect, for instance, is produced, whether we use 2218.192 as a divisor, or .00045082, the decimal answering to 1 ; 2218-192, as & multiplier. Any constant multiplier thus employed is called a « factor.” The divisors and factors most commonly needed in excise calculations are the following :
By the several numbers under the title of “divisors" in the preceding table is meant
1st. As regards squares and straight lined figures generally, or more correctly, all classes of solid bodies enclosed by plane surfaces. If the cubic inches in the content bo divided by 277-274, the result will be an expression of the content in gallons; if, by 2218-192, in bushels. As a gallon is the one-eighth of a bushel, so 277.274 is the eighth part of 2218-192.
2nd. As regards circles, or rather all classes of cylindrical and elliptical solids. If the cubic inches in the content be divided by 353-036, the result will be tho content in gallons; if by 2824.29 (= 353.036 X 8), in bushels.
By the several numbers under the title of "factors" is meant
1st. With respect to squares, &c.-as taken in the sense above explained-if the cubio inches in the content be multiplied by .0036065, the result will express the content in gallons; if by .00045082 in bushels.
2nd. With respect to circles—as taken in the sense above explained-if the cubic inches in the content be multiplied by .0028326, the result will express the content in gallons; if by .00035407, in bushols.
The formation of the divisor 853.036 was pointed out at the beginning of this chapter ; all the numbers in the columns of “Factors are merely the reciprocals of those in the corresponding columns of “Divisors."