88 Mr. Atwood's Propositions determining the Positions 
angle is given, the specific gravity will be known. If it should 
be required to find the sine of the angle of inclination from 
having given the specific gravity, it is evident from the nature 
of the equation, that such determination would require analy- 
tical operations extremely complex and troublesome, which 
may be avoided by having recourse to well known methods of 
approximation. By assuming the quantities s and t by estima- 
tion, let the value of v / n be calculated from the equation, which 
being compared with the given value of s/ n, the difference will 
be the error arising from the error in the assumed values of s and 
t, which are therefore to be corrected, and the operation re- 
peated until the value of s/ n, deduced from calculation, coin- 
cides with its true value ; from which method of proceeding, 
the angle of inclination from the original position of equili- 
brium will be known. 
This solution is evidently applicable to all cases in which 
the specific gravity of the solid is between the limits ^ and 
and by an investigation entirely similar, an equation is de- 
duced expressing the relation of the specific gravity of the 
solid and the sine or tangent of the angle of inclination from 
the perpendicular, when the specific gravity of the solid is be- 
tween ^and in which case the solid will float permanently 
with the diagonal line IC obliquely upward, being inclined to 
the vertical at some angle between the limits o and 18 0 2 6' 8 ", 6 . 
These determinations comprehend all the positions in which 
a square parallelopiped can be placed on the surface of a fluid 
in a position of equilibrium, provided the solid is moveable only 
round one axis, namely, that which passes through the centre 
