ΑΝ 433 APPENDIX OF Practical Gauging. T HE Art of Gauging is that Branch of the Mathematicks call'd Stereometry, or the Meafuring of Solids, because the Capacities or Contents of all forts of Veffels 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 Treatife may eafily understand, without any farther Directions. However, because 'tis not to be fuppos'd that every one, who defigns to undertake the Office or Imployment of a Gauger, hath made fo great a Progrefs in Mathematical Learning, I have therefore prefented the young Gauger with this Appendix, wherein I have only inferted fuch Rules as are useful in Gauging, and have been already demonftrated in this Treatife. But herein, I prefuppofe that he hath acquir'd (or if not, 'tis very necessary he should acquire) a competent Knowledge both in Arithmetick and Geometry: That is, I. In Arithmetick he fhould underftand the principal Rules very well, efpecially Multiplication and Divifion, both in whole Numbers and Decimal Parts, (which may be eafily learnt out of the 2d, 3d, and 5th Chapters of Part 1.) that fo he may be ready at computing the Contents of any Veffel, and cafting up his Gauges by the Pen only, viz. without the Help of thofe Lines of Numbers upon Sliding Rules, fo much applauded, and but too much practis'd, which at beft do but help to guefs at the Truth; I mean fuch Pocket-Rules as are but nine Inches (or a Foot) long, whofe Radius of the double Line of Numbers is not fix Inches; and therefore the Graduations or Divifions of those Lines are so very close, that they cannot be well diftinguifh'd. 'Tis true, when the Rules are made two or three Foot long (I had one of fix Foot) there they may be of fome Ufe, efpecially in fmall Numbers; altho' even then the Operations may be much better (and almoft as foon) done by the Pen: For, indeed, the chief Ufe of Sliding Rules is only in taking of Dimenfions, and for that Purpose they are very convenient. II. In Geometry the Gauger fhould understand not only how to take Dimenfions (which is beft learnt by Practice) but alfo how to divide any irregular Figure or Superficies, as Brewers Backs or Coolers, &c. into the eafieft and feweft regular Figures they will admit of, that fo their Area's may be truly computed with the leaft Trouble. And this may be learn'd (with a little Care and Diligence) out of the 1ft, 2d, and 5th Chapters of Part III, which the Gauger fhould be well acquainted with. Alfo he ought to have fo much Skill in Solids, as to be able, even at fight (but this must be acquir'd by Experience) to determine what Sort of Figure any Veffel is of (viz. any Tun, or clofe Cafk) or what Figures it may be beft reduced to, fo that its Dimensions may be truly taken, and the Content thereof computed with the leaft Error. I fay, with the leaft Error, becaufe 'tis very difficult, if not impoffible, to do it exactly; for there is not any Tun, or Cafk, &c. fo regularly made, as by the Rules of Art 'tis requir'd to be. III. Befides the aforemention'd, the young Gauger muft 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. thofe Measures must be reduced to Inches, (fee Sect. 4. Pag. 42.) because the Contents of all Sorts of Veffels (taken notice of in Gauging) are computed by the Standard Gallon of its Kind, whofe Content is known to be a certain Number 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 fuppofe the Gauger to have learnt perfectly, by heart.] Confequently, if either the fuperficial or Solid Content of any Veffel, as Back, Tun, Cafk, &c. be once computed in Cubick Inches, 'twill be easy to know how many Gallons, either of Ale, Wine, or Corn, that Veffel 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 Bufinefs of Gauging, all Superficies or Area's are always underflood to be one Inch deep, otherwise it could not be faid (as in the Gaugers Language it is) that the Area of fuch a Back, or of fuch a Circle, &c. is fo many Gallons. Thefe Things being very well underflood, the young Gauger will be ficly prepar'd to understand the following Problems, which are fuch 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 fhall, for Brevity's Sake, often refer to thofe Theorems and Problems. Se&t. E 1 Sec. 1. To find the Area of any right-lined Superficies in Gallons. PROBLEM I. To find the Area of any fquare Tun, Back, or Cooler, &c. either in Ale, Wine, or Corn Gallons. Kule. Multiply the given Length or Breadth (being here equal) into itfelf, 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. Suppofe the Side of a fquare Tun, Back, or Cooler be 124,5 Inches, what will its Area be in Gallons? Firft 124,5 × 124,5=15500,25 the Arca in Inches. 54,96 &c. Then 282 Or 268,8 57,66 &c. Ale Gallons. Wine Gallons. But if any one would rather work by Multiplication than by Divifion, he may turn or change any Divifor into a Multiplicator, if he divide Unity, or 1, by that Divifor. (Vide Probl. 3. Pag. 402.) Confequently 15500,25 x 0,00354654,96 &c. the Area in Ale Gallons; as before; and fo on for the reft. PROBLEM II. 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. Suppofe 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. K kk 2 PRO PROBLEM III. To find the Area of 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 272462.4. Then 282) 2462,4 (8,73 E Or 2462,4 X o,c035468,73 &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 of irregular, in Ale or Beer Gallons, &c. if you first divide it into Triangles, and then find the Area's of thofe Triangles; (as in the 2d, 4th, 5th, and 6th Problems in Chap. 5, Part III.) the Sum of thofe Area's being divided (or multiply'd) 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, fuch as the Carpenters ufe, and may be thus perform'd. B Suppofe any Brewer's Tun, Back, or Cooler, in the Form of the annex'd Figure A B C D F G. Let one End of the chalk'd Line be faften'd with a Nail (or otherwife) in any Corner or Angle of the Back, as at A; then straining it to the Angle at C, ftrike the Diagonal Line AC upon the Bottom of the Back; and ftraining it again to the Angle D, ftrike another Diagonal Line, as AD, and fo on for the Diagonal Line GD, &c. Then having mark'd out all the Diagonals, C D F the Perpendiculars may be thus found: Faften (as before) one End of the chalk'd Line in the Angle B, and then, by moving it to and fro upon the Stretch, find out the nearest Distance between the Angle at B and the Diagonal Line AC; and there ftrike a Line, and it will mark out the Perpendicular from B to the Line, AC; and fo on for the other Perpendiculars: Which being all mark'd out upon the Bottom of the Back, measure them and each Diagonal T Diagonal by a Line of Inches, &c. and then the Area of that And here, by the Way, it may be obferved, that the Number Having found (as above) the true Area of any Brewer's Back or Cooler (which, according to the Laws of Excife, 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 fo the true Quantity of Worts may be computed or (caft 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. Water) then dip it in eight or ten feveral Places (more or lefs according to the Largenefs of the Back) as remote and equally diftant one from another as you well can, noting down the wet Inches and decimal Parts of every dip. 2. Divide the Sum of all thofe Dips or wet Inches by the Number of Places you dipp'd in, and the Quotient will be the mean Wet of all thofe Dips. 3. Laftly, find out fuch a Place by the Side of the Back (if you can) that juft wets the fame with that mean Dip, and make a Notch or Mark there, for the true and conftant 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 fhew what Quantity (viz. how many Gallons) of Worts are in that Back at that 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 I. I have demonftrated in Cap. 6, Part III, and Theorem 3, 5, 6. Part V, that the Periphery of the Circle whofe Diameter is Unity, or 1, is 3,14159265 c. (or for common Ufe 3,1416) and that its Area is 0,78539816 &c. (or 0,7854 fere.) 2. Alfo, 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 I: 0,7854: the Square of the Diameter: to the Area. Upon |