Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 13, MAT-MIC

MENS U
• and HB X h. BrrDH*—DB1 (Geometry, Seft. IV.
Theor. 12.), therefore APzzDH2—DB1, and DB*
—HI)1—DG*. Hence it appears, that if the figure
be conceived to revolve about CA as an axis, fo that
the hyperbolic arc AB may generate a hyperboloid,
the. triangle i)CH a eone, and the re&angle HAFG a
cylinder, any fection of the firft of thefe folids by a
plane H perpendicular to the axis, will be equal to
the difterence of the feftions of the other two by the
fame plane. Therefore the hyperboloid B A b is equal
to the difference between the conic fruftutn FH h f and
the cylinder FG gj. Let A a the tranfverfe axis be
denoted by p, F/hr its conjugate axis by AD the
height of the folid by h,l&> b its bafe by b. Then,
becaufe by fimilar triangles, &c.
CA : CD :: Yf: H h :: F'f* : F/xH/z,
therefore F/xH^ = 3?xF/2= (j P+h) ? , ,
CA J ip ~/ t-
2J1?
Now Yand H Ji (zrB b' + Yf1)—^ -j-y1, there¬
fore putting n for .7854, we have (by Prob. 6.) the
content of the conic fruflum FH h f equal to
from this fubtraft n k q*, the expreffion for the content
of the cylinder FG ^y', and there will remain
3 V T /
for the content of the hyperboloid. But from the na¬
ture of the hyperbola
A a* : F/2 AD x D a : BD1,
that is,/.* : qx :: (p+h)h : j
2 h q* __ p bl
therefore
P. 2(i0+^),
the hyperboloid is alfo equal to
n h(pb1 \ n hb
2(p+/i)J
and hence the content of
y-+
= — X
p-\-\h
3 \ ,2(T+/0/ 2 ~ p+/i ‘
Now if it be confidered that the quantity » b b* is the
expreflion for the content of a cylinder whofe bafe is b
and height fi, it will appear evident, that this laft for¬
mula is the fame as would refult from the foregoing
rule.
Ex. Suppofe the height of the hyperboloid to be 10,
the radius of its bafe 12, and its tranfverfe axis 30.
What is its content ?
1. Becaufe a cylinder of the fame bafe and altitude
24* X-7854 X !0> therefore, we have the proportion,
40 : — 24*X.7854X 10,
^ 3 " ^
24*X.7854x10xr 10 „
“o x 3 x 2 —2073.456, the content
of the folid as required.
Of GAUGING.
Gauging treats of the meafuring of calks, and other
things falling under the cognizance of the officers of
3
RATION,
the excife, and it has received its name from a gauge
or rod ufed by the praftitioners of the art.
From the way in which calks are conitrufled, they
are evidently folids of no determinate geometrical fi¬
gure. It is, however, ufual to conlider them as having
one or other of the four following forms :
1. The middle frultum of a fpheroid.
2. d he middle frultum of a parabolic fpindle.
3. The two equal frultums of a paraboloid.
4. The two equal frultums of a cone.
We have already given rules by w hich the content of
each of thefe folids may be found in cubit feet, inches,
&c. But as it is ufual to exprefs the contents of calks
in gallons, we ffiall give the rules again in a form
fuited to that mode of eltimating capacity. Obferving
that in each cafe the lineal dimenfions of the calk ■ ar*
fuppofed to be taken in inches.
Problem I,
To find the content of a calk of the firft, or fphe-
roidal variety.
Rule.
To the fquare of the head diameter add double the
fquare of the bung diameter, and multiply the fum by
the length of the calk. Then let the produift be mul¬
tiplied by..0009:3;, or divided by 1077 for ale gallons,
or multiplied by .001 ly or divided by 882 for wine
gallons.
The truth of this rule may be proved thus. Put B j
for FG, the bung diameter, H for AH the head dia¬
meter, and L for i\.D, the length of the calk, then
(by Prob. 14.) the content of the calk is (2B24-H*)L
X which being divided by 282 (the cubic inches
in an ale gallon) gives (2 B1-]-H*)L X-000928371,
or (2 B*-J-H*) X X k.) for the content in ale
. 1077.157
gallons. And being divided by 231, (the cubic inches
in a wine gallon) gives (2B*-f-Hz) x -OOl 13333 k, or
(2B2-f-IT) Xett— Xk, for the content in wine
882.355
gallons. .
Ex. Suppofe the bung and head diameters to be 32
and 24, and the length 40 inches. Required the con¬
tent ?
Here (2 X 5 2s+ 24*) X 40 X .00093=97.44 ale gallons,
is the content required.
And (2X32Z-ffi24*) X40X -00113=118.95 wine gal¬
lons is the fame content.
Problem II.
To find the content of a calk of the fecond, or pa¬
rabolic fpindle form.
Rule.
To the fquare of the head diameter add double that
of the bung diameter, and from the fum take |, or T%-
of