nearly in Figure to fuch a Solid, yet the Officer bas no manner of Rule to affift him in afcertaining the fame; fo that it is but mere guef fing at beft. But by the Method I here propofe, there is always a Certainty, either to have the Content exactly true, or exceedingly near the Truth: neither is there any Difficulty in taking the fourth Dimenfion required among the Data; and the Operation will be very eafy by the SlidingRuler The third Part of this Book is wholly taken up with the Practice of Gauging; and is only a more full and particular Illuftration of what was delivered in the Second Part: The Meafure here being estimated by Gallons or Bufhels, which, there, were Cubical Inches. -Here is fhewn the Manner of Cafk-Gauging; Firft, on the Suppofition of a Cafk having a known Form: Secondly, Without any Regard to the Form, by the Help of a Fourth Dimenfon taken; fo that every one may follow that which feems to him beft. Then the Practice of Ullaging Cafks ftanding or lying is delivered in a Method, which will be found more universal and exact, than what is given for that Purpose, by the Line mark'd Seg. fta. or Seg. ly. (which fignifies Segments standing, or Segments lying) on the SlidingRuler which certainly can ferve but one fort of Cafks, and that must be fimilar to the Cafk Cafk from whence the Lines themselves were made. Afterwards follows the feveral Rules and Precepts for Gauging Tuns, Coppers, Stills, Cifterns, &c. with Examples at large to each. And that nothing might be wanting on my Part to render this Treatife as compleat as poffibly I could, I have carefully confulted all that has been wrote hitherto upon the Subject, and particularly J. Kepler, P. Guildin, J. Wallis, W. Jones, Sharp, and J. Mat. Hafius; being the most confiderable Authors who have delivered any thing to the purpose about this Affair, and at the fame time have demonftrated the Rules they gave. From thefe I have felected whatever was thought might conduce either to the Improvement of the Practice of Gauging, or that might make the Reafon of the Methods already given more eafy to be comprehended. And as to the Way taken in finding out the feveral Rules herein delivered, I have not fcrupled to make use of either the Arithmetic of Infinites, or the Method of Increments, according as I imagined the one or the other would render the Investigation the most eafy In this I have followed the Example of one of the ingenious Authors above-mentioned, whofe Judgment in thefe Matters will never be doubted, he having deduced the Solutions of feveral Pro Propofitions in his Synopfis after the fame manner. I hope the Reader will favourably pass over any Inaccuracy of Style he may pafibly meet with, and excufe fuch Errors as too often oc cur in Subjects of this Nature. October 30, 1740. Portland-freet, Corner of Mortimer-ftreet, near Oxford-market. THE xiii * The Theorem, p. 272. is deduced from Cor. 2. Page 191. thus, let MNnm (Fig. 74.) be any Part of the Fruftum of a Sphere, and m'n' a Diameter thereof, in the Middle betwixt the extreme ones MN,mn, continue nm, and thro' m' draw a Line perpendicu lar to mn, MN, let it meet the firft in R, the other in q, and the Circumference in P; alfo put mn=y, MN=b, Ool, Rq=1, qP=v, and m'n'm the Mid. Diam. then by Corol. 2. Page 191. the Measure of MmnN=y2+4m*+b2 x. Now 'tis manifest al X from Rm Ra =y+ "==", R m = 2 my RP=1+0, b-m Rm' ==—, also m'q=2, qp=v, Mq= q N = m + b = m b+m 2 2b2+y, put this for 4m2 in the above Expreffion, b3+y2+412+2b2+2y2 x 2/ 62ty2 al 2/2 = + xal in 2 3 cubic Inches, or 622 2/2 -x/x,0034. 2, E. 0. 3 TH Hofe that are inclined to have the Sliding-Rule, as conftructed Page 240. may have it accurately made by the ingenious Mathematical Inftrument-makers, Mr. John Coggs and Mr.William Wyeth, near St. Dunstan's Church in Fleetftreet, And by The Cooke in the Old Jury |