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. in Figure to fucb à Solid, yet the Officer has no manner of Rule to afif him in afcertaining the fame ; so that it is but mere guef fing at best.

But by the Method I bere propose, 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 Dimenhon required among the Data; and the Operation will be very easy 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 Illustration of what was delivered in the Second Part: The Measure here being estimated by Gallons or Bushels, which, there were Cubical Inches. - Here is hoewn the Manner of Cask-Gauging ; First, on the Supposition of a Cask having a known Form : Secondly, Without any Regard to the Form, by the Help of a Fourth Dimenhon taken ; fo that every one may follow tpat wbich feems to him beft.

Then the Practice of Ullaging Casks standing or lying is delivered in a Metbod, 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 hgnifies Segments standing, or Segments lying) on the SlidingRuler; which certainly can serve but one fort of Cafks, and that must be fimilar to the

Cask

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Cask from whence the Lines themselves were made.

Afterwards follows the several Rules and Precepts for Gauging Tuns, Coppers, Stills, Cifterns, &c. with Examples at large to eack. And that nothing might be wanting on my Part to render this Treatise as compleat as possibly I could, I have carefully consulted all that bas been wrote hitherto upon the Subječt, and particularly J. Kepler, P. Guildin, J. Wallis; W. Jones,

Sharp, and J. Mat. Hasius; being the most considerable Authors who have delivered any thing to the purpose about this Affair, and at the same time bave demonstrated the Rules they gave.

From these I have selected whatever was thought might conduce either to the Improvement of the Practice of Gauging, or that might make the Reason of the Methods already given more easy to be compréhended.

And as to the Way taken in finding out the several 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 easy. In this I have followed the Example of one of the ingenious Authors above-mentioned, whoje Yudgment in these Matters will never be doubted, be having deduced the Solutions of several

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Propositions in bis. Synopsis after the same manner.

Į hope the Reader will favourably pass over any Inaccuracy of Style be may posibly meet with, and excuse fuch Errors as too often oct fur in Subjects of this Nature,

October 30, 1740. Portland-street, Corner

of Mortimer-street, near Oxford-market

THE

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* 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 there-
of, in the Middle betwixt the extreme ones MN,mn,
continue nm, and throm draw a Line perpendicu.
lar to mn,MN, let it meet the first in R, the other
in q, and the Circumference in P; also put mn=y,
MN=b, 0=1, Rq=1, P=v, and m'n'=m the Mid.
Diam. then by Corol. 2. Page 191. the Measure of
MmN=y+4mo+bex Now 'tis manifest
from 3. Eucl. 36. Rnx Rm=RP x m'P ; also by the
35 of the fame Elements, m'q xqP = MaxğN ; but
Rx=y+ = Rm=

RP=1 to,
Rm' = 5, also m'q=, qp=v, Mq =
9 N=m+ 5* = het, hence the above e-
quations give to the =l+ox
and

the latter taken from the for.
2m2_62---2
mer leaves

therefore 4m2=41 t.
2bo+y?, put this for 4m” in the above Expression,
then we have the Measure of Mmn N =

.62+72 5+y? +-41% +262 +2y2 x

+ xal in cubic Inches, or

62 -2 272

x?

-xlx,0034. 2. E. O.

2

X

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4

212

al 6

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3

2

TH

Hose that are inclined to have the Sliding-Rule,

as constructed Page 240. may have it accurately made by the ingenious Mathematical Instru. ment-makers, Mr.Jobn Coggs and Mr.William Wyeth, near St. Dunftan's Church in Fleetstreet,

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