706 
MECHANIC S. 
Ad— ; whence we find BS-f-B'S 
—Br-fB's, orB :B':: S— s' : s—S, or laftly, B : B'+B' 
:: S —s' : s— s', \ 
Cor. j. The analogy B : B' :: S—s' : s —S, may be thus 
exprefted in words: “As the part or the folid within the 
heavier fluid, to the part within the lighter; fo is the dif¬ 
ference between the fpecific gravities of the folid and lighter 
fluid to the difference between the fpecific gravities of the 
heavier fluid and the folid.” 
Cor. 2. When the fpecific gravity of the lighter fluid is 
very fmali, compared with that of the heavier or that of the 
folid, then we may, without any fenfible error, ufe the 
proportion B : B-f-B' :: S : s. Thus, if the lighter fluid 
be air, the heavier water, and the folid elm-wood, their 
fpecific gravities being 1000, and 600, refpeftively ; 
then would the ratio of 600—1-| ; 100—1-|, or 588-2. ; 988-Z. 
be very nearly equal to that of 600 : 1000. 
Cor. 3. Hence alfo it appears, that the common ride for 
nfeertaining the fpecific gravities of a fluid and a lighter 
folid by the ratio of the part immerfed to the whole, is not 
accurately, though nearly, true; becaufe the air is a heavy 
fluid, and therefore every folid floating on a fluid and in 
air is in faff a folid of intermediate fpecific gravity float¬ 
ing between two immifciable fluids. We may, however, 
render the rule accurate, by fubdu&ing the number ex- 
p re fling the fpecific gravity of air from the two numbers 
exprefling the fpecific gravity of the folid and the fluid on 
which it floats ; the remainders will exprefs the aflual ratio 
between thofe fpecific gravities, and may be reduced to 
the ufual (tandard by a Ample and obvious analogy. 
Prop. XIV. To find the fpecific gravity of a body .—This 
may be done generally by means of the hydrolfatic balance, 
which is a kind of feales contrived for the exafif and eafy 
determination of the weight of bodies, either in the air, or 
when immerfed in water. This inftrument, and others 
for the fame purpofe, will be deferibed farther on. The 
problem may be divided iqto three cafes, as below. 
1. When the body is heavier than water. —Weigh it both 
in water and out of water; and the difference of thefe 
weights will exprefs the weight loft in water. Then, if B 
reprefent the weight of the body out of water, B' its weight 
In water, S its fpecific gravity, and s the fpecific gravity 
of water, the firft equation in Cor. 9. Prop. XII. will be- 
jB s 
come (B—B') S=B s, whence we find S=:-— - v confe- 
.B ■—-B' 
quently the general rule in words at length is this: 
As the weight loft in water 
Is to the whole or abfolute weight; 
So is the fpecific gravity of water 
To the fpecific gravity of the body. 
2. When the body will not fink in water, being fpecifically 
lighter. —In this cale attach to it a piece of another body 
heavier than water, fo that the mafs compounded of the two 
may fink together. Weigh the denfer body and the com¬ 
pound body feparately, both out of the water and in it; 
and find how much each lofes in water, by lubtratiling its 
weight in water from its weight in air; and fubtraft the 
lefs of thefe remainders from the greater. Then ufe this 
proportion : 
As the laft remainder 
Is to the weight of the light body in air ; 
So is the fpecific gravity of water 
To the fpecific gravity of the body. 
This alfo follows from the fame Cor. 9. where it was 
3. When the fpecific gravity of a fiuid is required. —Take a 
piece of fome body of known fpecific gravity ; weigh it 
both in and out of the fluid, and find the lofs of weight 
by taking the difference of thefe two: then fay, 
As the whole or abfolute weight 
Is to the lofs of weight; 
So is the fpecific gravity of the folid 
To the fpecific gravity of the fluid. 
This rule flows evidently from Cor. 4.. Prop. XII, 
Prop. XV. To find the refpeBivt weights of two known in¬ 
gredients iil a given compound .—If we adopt the notation in 
Cor. 9, Prop.XII.and make ufe of the 4th and 6th equations 
H L C 
there given, namely, H + L=C, and we fliall 
O S j 
thence find Hr 
(/—SOS 
C,andL: 
(S—/)S' 
■ From whence 
( s ~S')/ (S—S')/ 
we deduce the following rule in words at length : Take 
the three differences of every pair of the three fpecific gra¬ 
vities, viz. the fpecific gravities of the compound, and of 
each ingredient; and multiply each fpecific gravity by the 
difference of the ocher two : then fay, 
As. the greateft product 
Is to the whole weight of tire compound; 
So is each of the other two products 
To each refpeCtive weight of the two ingred ents. 
Cor. If, inftead of finding the weights, we were to. find 
the magnitudes, M and M', of the two ingredients, the 
fpecific gravities being as above; we fliould have the 
weight of M=SM, and the weight of while 
the weight of the compound would be /X(MxM'). 
Hence we fliould have SM-4-S'M'=/ M-j-/ . M', an 
equation fimilar to that in Prop. XIII. and, confequently, 
fimilar analogies, viz. M : M' :; J — S' : S— fi, and M : 
M-f-M* : : /—S' : S—S'. 
It is here fuppofed that the magnitude M + M' of the 
compound is equal to the fum of the magnitudes of the 
two parts when feparate. But it very frequently happens 
that the magnitude of the mixture is lefs than this fum ; a 
circumftance which is probably occafloned by tw’o caufes, 
the different magnitudes of the conftituent particles of the 
two bodies, and rheirchemical affinity. Butthis rule is,not- 
withflanding, of ufe in determining the quantity of penetra¬ 
tion or rarefaction, by comparing the computed magnitudes 
or denfities with thofe which are difeovered by obfervation, 
Tofave trouble upon many occafions, Tables of Specific 
Gravities have been formed, which include the metals and 
other bodies moft commonly ufed in phiiofophicai experi¬ 
ments. A Table of this kind has been given under the 
article Gravity, vol. viii. p. 809, 10. upon which it is 
now proper to obferve, that the fpecific gravity of rain-water 
being there reprefented by 1000, and a cubic foot of rain¬ 
water weighing 1000 ounces avoirdupois, the numbers 
againft each fubftance reprefent the weight of a cubic foot 
thereof in avoirdupois ounces. Upon the above Tabieare 
founded the following Propofition and Examples. 
Prop. XVI. Given the weight, W, of a body whofe fpecific 
gravity is known by the Table 5 to find its magnitude, M.—- 
Let a — the fpecific gravity of the body found in the 
Table, which reprefents the weight of a cubic foot in 
avoirdupois ounces ; then, as the weight is in proportion 
to the magnitude, when the fpecific gravity is given, \ve 
have a : W :: 1 foot : M the magnitude in cubic feet. 
Ex. If a piece of heart of oak weigh 1720 ounces, what 
is its magnitude?—Here, a — 1170, W 1720 5 hence, 
J 7 2 ° , ,. - 
1170: 1720 :: 1 : -= 1 461 cubic feet. 
1170 
Cor. As W=mM, if the magnitude be given we can 
find the weight. 
Ex. If the magnitude of a piece of iron be i’3S cubic 
feet, what is its weight ?— Here, a— 7207, M = i’35; 
hence, W =: 1 ‘35 X 7207 9729°45 ounces. 
Instruments for discovering Specific Gravities. 
Plate XVIII. and XIX. 
The Hydrofiatic Balance is an inftrument contrived to 
determine accurately the fpecific gravity of both folid and 
fluid bodies. One of the moft ingenious forms of this ba¬ 
lance is exhibited at fig. 10. where V C G is the ftand, or 
pillar, which is to be fixed in a table. From the top A 
hangs by two filk firings' the horizontal bar B B, from 
which is fulpended by a ring i the fine beam of a ba¬ 
lance b 5 which is prevented from defeending too low on 
either fide by the gently-fpringing piece ty z, fixed on the 
fupport M. The harnefs is annulated at o, to fhow dif- 
4 ti.nftly 
