I G 2 s 30 2 4□ SA 2SAxfm+□ƒm = □ ma 2 SA×gn+Ogn=one 6 OSA 2SA x bp + □ bp□py, &c. 1. In thefe Equations the □ SA, SA, SA being a Series of Equals, and AB the Number of all the Terms; therefore it will SAX AB the Sum of the Series, by Lemma 1. be 2. Because fm, gn, hp, &c. are as a Series of Squares wherein SA is the greatest Term, and AB the Number of the Terms; 2SA SAXAB therefore 2□ SAXAB = will be the Sum of all that 3 3. And the fm, ogn, hp, &c. will be a Series of Terms in the Ratio of Biquadrates, as above; dB = □ SA being the greatest Term, and AB the Number of all the Terms; therefore it will be. Lemma.5. Whence it follows, that SA × AB 2□SA× AB 3 + SAXAB the Sum of all the Series of□ma, ne, py, of all the Series of ma, One, py, &c. which do constitute the Solidity of half the Spindle, viz. of SAB. Therefore putting D= 2SA, and H2AB, (viz. bAB) it will be 0,41888 DDH 8 the Solidity of the whole Parabolick Spindle b SB, being of = 0,7854DDH the Solidity of its circumfcribing Cylinder. Q. E. D. From hence we may alfo raise a Theorem for finding the Fruftum SApy of the laft Figure. For SA being the greatest Term, Opy the leaft Term, and Ay the Number of all the Terms or Circles included between A and y, of all the Series □ SA, □ ma, one, opr 2 Ay 33 But 4 SA—2S A × hp + =3 30hp 32 5 Ay SA- 2 SA × hp = □py-hp, by 6th Step. 30bp = 32 3 – 4 5/2 DS4+ 5+ &c. 5 Ay 2 Ay Confeq. 72 SA + O P Y — } © b p x } A y =z, the Sum of all the Series of O SA, Oma, One, Opy, which do conftitute the Solidity of the Fruftum S Apy. Therefore putting D 2 SA, as before, C=2py, x = 2 hp, and H= Ay, it will be 1,5708 DD +0,7854 CC—9,31416 xxx H = the Fruftrm SA py. And if we make L 2 H. Then 1,5708 DD +0,7854 CC -0,31416 xx x L Double of that Fruftum, being the middle Zone. And by turning these Factors into one common Divifor, as in the Fruftum of the Conoid at Theorem 25, Page 430, there will arife this following Theorem. = It may be here expected that I should now proceed to fhew how the Area of any Hyperbola, and the Contents of fuch Solids as may be form'd by the Rotation of that Figure about its Axis, &c. may be found; but because thofe Things cannot be exactly perform'd by any certain or fettled Theorem, as thefe of the Circle, Ellipfis, and Parabola have been, I have therefore omitted them, and refer the Reader to Dr. Wallis's Algebra, Chap. 20, &c. or to the Philofoph. Tranfalt. Numb. 34, wherein he may find the Method of forming infinite Series relating to the fquaring of an Hyperbola, &c. which are too tedious to be fully explain'd and demonstrated in this small Tract, it being only intended as an Introduction, the which I fhall here conclude. ΑΝ AN 433 APPENDIX O F 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 forts of Veffels ufed 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 easily understand, without any farther Directions. However, becaufe '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 ufeful 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 should understand the principal Rules very well, especially 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 Num-. -bers 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, whose 'Radius of the double Line of Numbers is not fix Inches; and therefore the Graduations or Divifions of thefe Lines are so very close, that they cannot be well diftinguifh'd. 'Tis true, when the Ru'es 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 Dimensions, and for that Purpose they are very convenient. II. In Geometry the Gauger fhould understand not only how to take Dimensions (which is best learnt by Practice) but also how to divide any irregular Figure or Superficies, as Brewers Backs or Coolers, &c. into the eafieft and fewest 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 Dimenfions 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 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, (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 Kirid, 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 Superficial or Solid Content of any Veffel, as Back, Tun, Cafk, &c. be once computed in Cubick Inches, 'twill be eafy 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 feem 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 understood, the young Gauger will be fitly prepar'd to understand the following Problems, which are fuch as have (moft 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 Theosems and Problems: Sect. 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. Rule. Multiply the given Length or Breadth (being here equal) 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 fquare 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. S 54,96 &c. Then 2827 { Ale Gallons. Wine Gallons. Corn 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.) Thus 282 And 231 1,000000 (0,003546 Ale Gallons. 0,004329 the Multiplica. for W. Gallons. 0,003722 = C. Gallons. Confequently 15500,25 x 0,003546 54,96 &c. the Area in Ale Gallons; as before; and fo on for the rest. 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 Galions. 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 × 85,6=18618. Then 282) 18618 (66,02, &c, Or 18618 x 0,003546-66,02&c. the Area requir'd, &c. |