zj 6 Mr. Atwood's Disquisition on 
tions and restrictions, will approximate to the measures of 
areas, to a degree of exactness fully sufficient for naval men- 
surations ; and further, will be useful in determining by ap- 
proximation the integral values of fluxional quantities in ge- 
neral, especially those which occur in the investigation of 
practical subjects. 
CASE XII. 
Still supposing the vertical sections of a vessel to be equal 
and similar figures, let BOA (fig. 27.) represent one of these 
sections ; the figure being either a curve of the higher dimen- 
sions, or a curve not formed according to any geometrical law, 
of which the lengths of the ordinates, and of any other lines 
given in position, are supposed to be measurable, and given in 
quantity : the angle at which the vessel is inclined from the 
upright, and the other necessary conditions being known, it is 
required to find, by geometrical construction, a line which 
shall approximate nearly to the measure of the vessel's sta- 
bility. 
1st Method . 
BA represents the intersection of the water's surface when 
the vessel floats upright: bisect BA in the point D; and, 
through D, draw the line NDM inclined to the line BA at an 
angle ADM, equal to the given angle of the vessel's inclina- 
tion : let the area of the figure ADM£, also the area of the 
figure BDN c, be found by means of the rules which have been 
described : suppose the area ADM£ to be greater than the area 
BDN c } and let E represent the difference between them : from 
r " 
