q6o Mr. Atwood's Disquisition on 
bola*, the stability-)- is the same as when the sides are plane 
surfaces, parallel to the plane of the masts. It is inferred that 
if the sides of a vessel are formed to coincide with a parabola 
of the lowest, and the sides of another vessel with a parabolic 
curve of the highest dimension, all the other conditions being 
the same, the stabilities of the two vessels will be equal in 
these two extreme cases. 
In proceeding to ascertain the stability of vessels, the verti- 
cal sections of which coincide -with any parabolic curve, the 
rigid strictness of geometrical inference cannot be well pre- 
served, when the oblique segments are objects of consideration, 
on account of the complicated properties of the figures. J But, 
in these and similar cases, methods of approximation may be 
employed, by which the stability corresponding to any given 
figure of the sides may be inferred, to a degree of exactness 
exceeding any that can be necessary in practice. These me- 
thods of approximation are either such as are required for the 
mensuration of curvilinear areas, or geometrical constructions 
which exhibit the lineal measures of stability not strictly and 
rigidly true, but approaching, as nearly as may be desired, to 
the true and correct measures. 
The methods of approximation to be used for the quadra- 
ture of curvilinear spaces, are founded on Sir Isaac Newton's 
discovery of a theorem, by which, from having given any 
* Case xx. pages 255, 256. 
f The comparative stability, in this and similar observations, is understood to im- 
ply, that the vessels are inclined at equal angles of inclination from the upright, all 
the other conditions (the shape of the sides excepted) being the same. 
j The areas of any parabolic segments, either direct or oblique, are geometrically 
quadrable, but, in the oblique segments, the positions of the centres of gravity are not 
determinable generally by direct methods. 
