GAUGING. 
rectilinear bases. The following very easy 
mode of ascertaining the contents of a co- 
nic frnstrum is given by the ingenious New- 
ton. Multiply each diameter (i. e. of top 
and bottom) by itself; then the one by the 
other, and the aggregate of those products 
by the altitude ; multiply also the last pro- 
duct by 78539, (the superficial content of 
a circle whose diameter is 1000) ; a third 
part of the product is the measure of the 
frnstrum. \ 
Therefore, when vessels have their sides 
composed of straight ribs, proceeding in 
right lines from one to the other end of the 
conic frustrum, the measurement is easily 
'made ; thus we may, without difficulty, as- 
certain the contents of great coppers, mash- 
ing-tubs, corn-binns, and a great variety of 
similar vessels. But we rarely see casks of 
any description formed by the union of two 
frustrated cones ; their usual shape is more 
spheroidal; that is, they have an arched 
or swelling course from the buhg to the 
chine or end : conspquently these contain 
more than such as are truly conical. This 
occasions the necessity for allowing some- 
thing for the bulge or swell, and of taking 
the diameter at the centre, between the 
bung and the chine, which diameter will 
give a true medinm. The thickness of the 
cask may easily be ascertained by aid of 
calibre compasses applied to the proper 
part. The length of the cask may be mea- 
sured internally, by putting a rod or wand 
in at the tap hole, and the internal diame- 
ter may be taken in a similar way at the 
bung ; but such can only be done when the 
cask is empty, or, at least, opened for the 
purpose : whereas casks that are filled and 
sealed must often be measured; for this 
purpose the calibre compasses are extremely 
useful, since they embrace the outside mea- 
sure. To correct the computation, we must 
usually allow an inch anc! a half in the whole 
length, and the same in the whole diameters 
at the bung and chine, thus exteriorly taken, 
for the thickness of the cask itself. This de- 
duction being made, we must compute ac- 
cording to the form or swell of the staves. If 
they be much raised, we multiply the differ- 
ence between the diameter at the bung, and 
at the end, by .7 ; if less raised, or swelling, 
we multiply the difference by .65 ; if nearly 
straight, by .6, and if rectilinear, or truly 
conical, by .55 : the product added to the 
diameter at the end, or head, will give a 
mean diameter. Suppose the diameter with 
in the bung to be 32 inches, at the head 
24, and that the length within be 40 ; the 
difference between 32 and 24 is 8, which, 
multiplied by .7, gives 5.6 ; add thereto 
the diameter at the head, 24, and the me- 
dium will be 29.6 ; multiply by the length 
40, and divide by 359.05, and the quotient 
will be ale gallons 97.4. And thus, with 
the other multipliers, according to the ap- 
parent bulge or swell between the bung 
and the chine, and according to wine or ale 
measure. 
To find the ullage, or quantity of liquor 
deficient in a cask; we have the following 
rule. Take the diameter at the bung, and 
ascertain the number of inches and parts 
that are dry, say that of 29 inches 13 be 
dry ; also that the whole cask measures 80 
gallons. Divide the dry inches 13 by 29, 
the bung diameter ;■ the' quotient will be 
.448 ; find the two first figures, .44, under 
V. S. in the annexed table, and its sequent 
will be .4238, to which add a proportional 
part for the 8, and the whole sequent will 
be .4343, which multiplied by the contents 
of the cask, will shew a deficiency of 
34,661 gallons. This measurement, how- 
ever, applies to cylinders only ; if the cask 
be conical, you must find the mean diame- 
ter, which should be deducted from that at 
the bung ; and noting half the difference, 
which is to be deducted from the wet inches 
and reserved. Then, as the mean diameter 
is to 100, so is the reserved difference to a 
versed line in the table ; and if the seg- 
ment (to be found in the table) be multi- 
plied, as before shewn, into the contents, 
the product will be the quantity of liquor 
in the cask. 
Example. Let the bung-diameter be 32, 
the mean-diameter 29.6, and the whole 
measure 97.4 gallons : say there be 19 
inches wet : 
From 32.0 From 90 
deduct 29.6 take 1.2 
remain 2.4 remain 17.8 reserved. 
its half is 1.2 
Now as 29.6 is to 100, so is 17.8 to 
.60, the versed sine. The segment to 60 is 
.6265 ; which* multiplied by 97.4, the whole 
contents, the product gives 61 gallons of 
liquor remaining. By working upon the 
dry inches, you would have found the ul- 
lage, or deficiency. 
