the Stability of Ships. 253 
NW is a tangent to the curve in the point V : through the 
point V, draw VK parallel and equal to DL ; and, through the 
point K, draw CH parallel to NW : let DL be divided into five 
equal parts, and let LE be taken equal to three of those parts : 
make VQ equal to LE ; and through Q draw Pr perpendicular 
to NW ; through G draw GZ perpendicular to rP : GZ is the 
measure of the vessel’s stability, when inclined from the upright 
through the given angle MOL The demonstration follows. 
Through E, draw ET perpendicular to rP ; and, through G, 
draw GR parallel to rP ; let the parameter of the curve be de- 
noted by^>. 
By the construction, LX : LI : : LI : LF : : tang. MOI to rad. 
therefore - LX : LF : : tang. 2 MOI : rad. 2 and 
and - LX : 4LF : : tang. 2 MOI : 41’ad. 2 
By the properties of the curve, 
LX: XV:: XV : 4LF 
LX : 4LF : : LX 2 : XV 2 . 
LX : 4LF : : tang. 2 MOI : 4 rad. 2 
LX 2 : XV 2 : : tang. 2 MOI : 4rad. 2 
LX : XV : : tang. MOI : 2 rad. 
or, since LX = ±XN 
^XN: XV:: tang. MOI: 2 rad. 
XN : XV : : tang. MOI : rad. but, by 
wherefore 
But 
therefore 
and 
XN : XV : : tang. XVN : rad. 
or 
the construction, 
consequently tang. XVN is equal to the tangent of MOI to the 
same radius; and therefore the angle XVN is equal to the 
angle MOI, or the given. angle of the vessel’s inclination from 
the upright. Moreover, since it appears from the construction, 
that the angle XVN is equal to the angle NrP, NrP is equal 
to the vessel’s inclination from the upright, and because the 
