Demonftration. If any Semi-Parabola (as BSA) be turn'd or mov'd about its Axis (SA) 'twill form a folid Parabolick Conoid, constituted of an infinite Series of Circles, viz. O ba, fe, og y, &c. by Defini tion 17. Now, according to the Property of every Parabola, it will be, SA: AB::AB: ОАВ L, the Latus Rectum. Then Sex L=0 fe Syx L□gy, &c. B e Here Sax L, Sex L, Sy x L, &c. are a Series of Terms in Arithmetick Progreffion : therefore ba, fe, ☐gy, &c. are also a Series of Terms in the fame Progreffion, beginning at the Point S; wherein AB is the greatest Term, and 8 A the Number of all the Terms. Therefore ABSA the Sum of all the Series, by Lemma 2. Confequently, o AB SA= the Sum of all the Series ba, ofe, ogy, &c. which do conftitute the Solidity of the Conoid. And putting D 2 AB, and H=S A. Then 0,7854 D D x H = 0,3927 DDH will be the folid Content of the Conoid, which is juft half the Cylinder whofe Bafe = D and Height H. [See Theorem 11.] = = X Q. E. D. This being understood, 'twill be easy to raise a Theorem for finding the lower Fruftum of any Parabolick Conoid. For fuppofing ba A the Height of the Fruftum, and p = S a the Height of the Part 6 Sb cut off; then h+p=SA, the Height of the whole Conoid. Confequently, ABxh+. AB‹p the whole Conoid. And dity of the Part cut off. Ergo I ba+p 2 2 = the Soli © ABxb+ ABxp—baxp 2 the Solidity of the Fruftum. But 2 b+p:□ AB::p:ba Confeq. 3h+p: ☺ AB : :p : 3 ba · · · √ 40 1Bxp=oba x h+o baxp 4-0 ba XP IX 2 510 AB Xp - Oba x p = bax b 6 0 ABXh+o ABX po ba xp=2F 70 9 O AB+Oba 2 xbF the Fruftum's Solidity. Let D-AB, as before, and d=2 b a the Diameter of the Part cut off; then we fhall have this following DD +0,3927dd X b = the THEOREM XXV. {Solidity of the Fruftum requir’d. Or { DD + dd 2,5404 Xh=the Fruftum; for,3927) 1,0000 (=2,5464 and becaufe 2,5464 + 2,5464 2 3,8196; therefore it may be made 3,8196) DD+ddxb (the fame Fruftum, &c. Note, The Reason why I have reduced this Theorem to have the fame Divifor with thofe at the Fruftums of Pyramids, &c. will beft appear farther on, viz. when they all come to be apply'd to Practice in Gauging. THEOREM XXVI, Every Parabolick Spindle (or Pyramidoid) is equalto eight Fifteenths of its circumfcribing Cylinder. Demonftration. If any acute Parabola, as bS B, be turn'd or mov'd about its greatest Ordinate b AB, it will form a Solid call'd a Parabolick Spindle, conftituted of an infinite Series of Om a, One, &c. by Definition 18. py, Let us fuppofe the Line S d, parallel to A B, &c. (as at Theorem 23) then it hath already been prov'd, that the Lines fm, gn, hp, &c. are a Series of Squares whofe Roots are in Arithmetick Progreffion: confequently their Squares, viz. □ƒ m, gn, bp. &c. will be a Series of Biquadrates, whofe Roots will be in Arithmetick Progreffion which being premis'd, we may proceed thus. [1]SA—fm=ma First, 2S A-g n = ne S f g h d m Aaey 12 3 4□ S4-28Axƒm+□ƒm= 5SA ma SA 2SAXgn+Ogn=One 6 SA-2SA x bp + Ob p□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 be SAXA B the Sum of the Series, by Lemma 1. 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 2 SAXAB will be the Sum of all that therefore 3 Series, by Lemma 3. = 3 SA 3. And the fm, □ gn, □ hp, &c. will be a Series of Terms in the Ratio of Biquadrates, as above; dB = being the greatest Term, and AB the Number of all the Terms; OSA AB therefore it will be the Sum of all that Series, by Whence it follows, that SAXAB 20SAXAB + 3 the Sum of all the Series of ma, ne, □py's the Sum of all the Series of ma, One, Ohp, Od B. &c. confequently, of all the Series of Oma, One, Opy, &c. which do conftitute. the Solidity of half the Spindle, viz. of SAB. Therefore putting D=2SA, and H=24B, (viz. bAB) it will be 0,41888DDH the Solidity of the whole Parabolick Spindle b S B, being of 0,7854DDH the Solidity of its circumfcribing Cylinder. Q. E. D. From hence we may also 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, 2-Ay 330SA 2SA Xhp 30hp = 3 ༧ Ay 5 But 4 SA- 2 SA × hp = □py-hp, by 6th Step. ·30hp. 3% 3 4 520SA+: 5 = Ay 5+&c. 62 Spy bp= = 3% A y Confeq. 720S A + O р y → O bp x 1⁄2 A y = z12 ś Zy the Sum of all the Series of SA, Oma, One, Opy, which do conftitute the Solidity of the Fruftum S Apy. Therefore putting D 2SA, as before, C2 pv, x=2hp, and H= Ay, it will be 1,5708 DD +0,7854 CC — 0,31416××× ¦ H= the Fruftum S Apy. And if we make L 2 H. Then 1,5708 DD+ 0,7854 CC 0,31410xx XL 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 Thes rem 25, Page 430, there will arife this following Theorem. THEOREM XXVII. 3,8196) 2 DD + CC — 0,4xx × L (= It may be here expected that I fhould now proceed to shew how the Area of any Hyperbola, and the Contents of such Solids as may be form'd by the Rotation of that Figure about its Axis, &c. may be found; but becaufe 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. 90, &c. or to the Philofoph. Tranfa&t. 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 demonftrated in this small Tract, it being only intended as an Introduction, the which I shall here conclude. ΑΝ AN 433 APPENDIX OF 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 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, 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 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 neceffary 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 thofe 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 almost as foon) done by the Pen: For, indeed, the chief Ufe of Sliding Rules is only in taking of Dimenfions, and for that Purpofe they are very convenient. |