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A P P E N D I X
Practical Gauging. T
HE Art of Gauging is that Branch of the Mathematicks call'd Stereometry, or the Measuring of Solids, because the Capacities or Contents of all sorts of Vessels used for
Liquors, &c. are computed as tho' they were really folid Bodies; which any one that hath made himself Master of the 'foregoing Parts of this Treatise may easily understand, without any farther Directions.
However, because 'is not to be suppos'd that every one, who designs to undertake the Office or Imployment of a Gauger, hath made so great a Progress in Mathematical Learning, I have therefore presented the young Gauger with this Appendix, wherein I have only inserted such Rules as are useful in Gauging, and have been already demonstrated in this Treatise. But berein, I prefuppose that he hath acquir'd (or it not, 'tis very necessary he should acquire) a competent Knowledge both in Arithmetick and Geometry: That is,
I. In Arithmetick he should understand the principal Rules very well, especially Multiplication and Division, both in whole Numbers and Decimal Parts, (which may be easily learnt out of the 2d, 3d, and 5th Chapters of Part 1.) that so he may be ready at computing the Contents of any Vessel, and casting up his Gauges by the Pen only, viz. without the Help of those Lines of Numbers upon Sliding Rules, so much applauded, and but too much practis’d, which at best do but help to guess at the Truth; I mean fuch Pocket Rules as are but nine Inches (or a Foot) long, whose Radius of the double Line of Numbers is not fix Inches; and therefore the Graduations or Divisions of those Lines are so very close, that they cannot be well distinguish'd. 'Tis true, when the Rules are made two or three Foot long (I had one of fix Foot) there they may be of some Use, especially in small Numbers; altho' even then the Operations may be much better and almost as soon) done by the Pen: For, indeed, the chief Use of Sliding Rules is only in taking of Dimensions, and for that Purpose they are very convenient.
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II. In Geometry the Gauger should understand not only how to take Dimensions (which is belt learnt by Practice) but also how to divide any irregular Figure or Superficies, as Brewers Backs or Coolers, &c. into the easiest and fewest regular Figures they will admit of, that fo their Area's may be truly computed with the least Trouble. And this may be learn’d (with a little Care and Diligence) out of the ift, 2d, and 5th Chapters of Part III, which the Gauger should be well acquainted with. Also he ought to have to much Skill in Solids, as to be able, even at sight (but this must be acquir’d by Experience) to determine what Sort of Figure any Veľsel is of (viz. any Tun, or close Calk) or what Figures it may be best reduced to, so that its Dimensions may be truly taken, and the Content thereof computed with the least Error. I say, with the leaft Error, because 'tis very difficult, if not impossible, to do it exactly; for there is not any Tun, or Calk, &c. fo regularly made, as by the Rules of Art 'is requir'd to be.
III. Besides the aforemention'd, the young Gauger must know, that all Dimenfions useful in Gauging are to be taken in Inches, and Decimal Parts of an Inch; and if they are taken in any other Measures, as Feet, Yards, &c. those Measures must be reduced to Inches, (see Sect. 4. Pag. 42.) because the Contents of all Soris of Vessels (taken notice of in Gauging) are computed by the Standard Gallon of its kind, whose Content is known to be a certain Num. ber of Cubick Inches : That is, the Beer or Ale Gallon contains 282, the Wine 231, and the Corn Gallon 268, 8 Cubick Inches, [See the five Tables, &c. in Pages 34, 35, 36, which I here suppose the Gauger to have learnt perfectly, by heart.] Consequently, if either the superficial or Solid Content of any Vessel, as Back, Tun, Caik, &c. be once computed in Cubick Inches, 'twill be easy to know how many Gallons, either of Ale, Wine, or Corn, that Vessel will hold.
Note, I have here faid, the Superficial Content in Cubick Inches, which may seem to be very improper, according to the Definition given of a Superficies in Page 279; but you must know, that, in the Business of Gauging, all Superficies or Area's are always understood to be one Inch deep, otherwise it could not be said (as in the Gaugers Language it is) that the Area of such a Back, or of fuch a Circle, &c. is so many Gallons.
These Things being very well understood, the young Gauger will be fidly prepar’d to understand the following Problems, which are such as have (most of them) been already propos'd in the 'foregoing Parts of this Treatife, and only are here apply'd to Practice ; and therefore I Mall, for Brevity's Sake, often refer to those Theorems and Problems.
Se&t. 1. To find the Area of any right-lined Superficies in Gallons.
PROBLEM I. To find the Area of any square Tun, Back, or Cooler, &c. either
in Ale, Wine, or Corn Gallons.
Multiply the given Length or Breadth (being here eKule.
qual) into itself, and the Product will be the Area in Inches; then divide that Area by 282, or 231, or
-268,8 and the Quotient will be the Area requir’d. Example. Suppose the Side of a square Tun, Back, or Cooler be 124, 5 Inches, what will its Area be in Gallons ? First 124,5 * 124,5=15500,25 the Area in Inches. And 231_15500,255 56,20 &*:? the Area in Wine Gallons. Or 268,8
Corn Gallons. But if any one would rather work by Multiplication than by Division, he may turn or change any Divisor into a Multiplicator, if he divide Unity, or 1, by that Divisor. (Vide Probl. 3. Pag. 402.) 0,003546
Ale Gallons. 1,000000 0,004329 the Multiplica. for W. Gallons. Or 268,8 0,003722
C. Gallons. Consequently 15500,25 x 0,003546 = 54,96 &c. the Area in Ale Gallons ; as before ; and so on for the rest.
To find the Area of any Tun, Back, or Cooler in the Form of a
Right-angled Parallelogram in Ale Gallons, &c. See the Rule for finding its Area in Inches, at Probl. 1,P.339, then either divide (or multiply) that Area, as above, and you will have the Area in Gallons.
Example. Suppose the Length of a Brewer's Tun, Back, or Cooler be 217,5 Inches, and its Breadth 85,6 Inches, what will its Area be in Ale or Beer Gallons, &c? First 217, 5 x 85,6=18648. Then 282) 18648 (66,12, &c. Or 18648 x 0,003546=66, 12 &c. the Area requir’d, &c. Kkk 2
To find the Area oj any Triangular Tun, Back, or Cooler,
in Ale Gallons, &c.
See the Rule for finding its Area in Inches at Prob. 3, p. 340 ; then divide (or multiply) that Area as before, and you will have the Area required.
Example. If the Length of the Base of a Triangular Cooler be 86,4 Inches, and its perpendicular Breadth be 57 Inches, what will its Area be in Ale Gallons ? First, 86, 4 x? = 2462,4. Then 282) 2462,4 (8,73 &c. Or 2462,4 X 0,003546 = 8,73&c. the Area in Ale Gallons.
Proceeding thus, you may eafily find the Area of any Tun, Back, or Cooler, whether it be in the Form of a Rhombus, Rhomboides, Trapezium, or of any other Polygon, either regular or irregular, in Ale or Beer Gallons, &c. if you first divide it into Triangles, and then find the Area's of those Triangles; (as in the 2d, 4th, 5th, and 6th Problems in Chiap. 5, Part III.) the Sum of those Area's being divided (or multiplyöd) by its proper Divi. for (or Multiplicator) as above, will give the Area requir’d.
Now, the Practical Way of dividing any Polygonus Tun, Back, &c. into Triangles, is by help of a chalk'd Line, such as the Carpenters use, and may be thus perform'd.
Suppose any Brewer's Tun, Back, or Cooler, in the Form of the annex'd Figure A B C D FG. Let one End of the chalk'd Line be fastend with a Nail (or otherwife) in any Corner or Angle of the Back, as at A ; then straining it to the Angle at C, strike the
с Diagonal Line Å C upon the Bottom of the Back; and straining it a
D gain to the Angle D, strike another Diagonal Line, as AD, and so on for the Diagonal Line G D, &c. Then
F having mark'd out all the Diagonals, the Perpendiculars may be thus found: Fasten (as before) one End of the chalk'd Line in the Angle B, and then, by moving it to and fro apon the Stretch, find out the nearest Distance between the Angle at B and the Diagonal Line AC; and there Atrike a Line, and it will mark out the Perpendicular from B to the Line, A C; and so on for the other Perpendiculars: Which being all mark'd out upon the Bottom of the Back, measure them and each
Diagonal by a Line of Inches, &c. and then the Area of that Back may be computed ; as directed above.
And here, by the way, it may be observed, that the Number of Triangles will always be lefs by two, and the Number of the Diagonals lefs by three, than the Number of the sides of any Right-lin'd Figure that is so divided.
Having found (as above) the true Area of any Brewer's Back or Cooler (which, according to the Laws of Excise, ought always to be fix'd or immoveable) the next Thing will be to find out the true dipping or gauging Place in that Back, that so the true Quantity of Worts may be computed or (cast up) at any Depth; which may be thus done.
1. When the Bottom of the Back is cover'd all over (of any Depth) either with Worts or Liquor (viz. IVater) then dip it in eight or ten several Places (more or less according to the Largeness of the Back) as remote and equally distant ore from another as you well can, noting down the wet Inches and decimal Parts of every dip.
2. Divide the Sum of all those Dips or wet Inches by the Number of Places you dipp'd in, and the Quotient will be the mean Wet of all those Dips.
3. Laftly, find out such a Place by the Side of the Back (if you can) that just wets the same with that mean Dip, and make a Notch or Mark there, for the true and constant Dipping place of that Back.' Then if any Quantity of Worts (which do cover the whole Back) be dipp'd or gaug'd at that Place, and the wet Inches fo taken be multiply'd into the Area of the Back in Gallons, the Product will thew what Quantity (viz. how many Gallons) of Worts are in that Back at thac Time, provided the sides of the Back do stand at Right Angles with its Bottom.
Sect. 2. To find the Area of any Circular and Elliptical
Superficies in Gallons, .. 1. I have demonstrated in Cap. 6, Part III, and Theorem 3, 5, 6. Part V, that the Periphery of the Circle whose Diameter is Unity, or 1, is 3,14159265 &c. (or for common Use 3,1416) and that its Area is 0,78539816&c. (or 0,7854 fere.)
2. Also, that the Peripheries of all Circles are in Proportion one to another as their Diameters are ; and their Area's are in Proportion to the Squares of the Diameters. That is, as 1: 3,1416 ; : the Diameter of any Circle : to its Periphery. And 1:0,7854 :: the Square of the Diameter : to the Area.