of floating Bodies, and the Stability of Ships. 115 
distance Xx will be the solid contents of this segment, to a 
degree of exactness fully sufficient for the purposes of this 
approximation. In the same manner the solid contents of 
all the segments which are elevated above the surface are to 
be obtained by making XI = AB — NX, X W = AB - PX, 
and proceeding as in the former case. If the aggregate of the 
segments NXP representing the part immersed, in consequence 
of the vessel's inclination, should not be equal to the aggregate 
of the segments I AW, (fig. 2.) which are elevated above the 
surface, the position of the point X, or rather of the line which 
that point denotes, must be altered, and the same operations 
repeated till the sums of the segments on each side of the said 
line of inclination are precisely equal. 
This having been effected, the magnitude of the volume 
immersed, denoted by A in the expression W x ^ — ds, will 
be known ; and the magnitude of each of the individual seg- 
ments NXPwx/> and IXWfrw, &c. will also be known ; the 
quantity b A will be found in the following manner. The 
area PXNTP and its centre of gravity d are to be deter- 
mined by methods of approximation. Through d draw dc 
perpendicular to the horizontal line PX ; Xc* will be the 
• The solution of problems by geometrical construction has been little practised 
since methods of calculation have been so much improved by the invention of 
logarithms and other facilities : the solutions of difficult cases are, however, some- 
times obtained with sufficient exactness by construction, which would be more trouble- 
some by any other method: in the present instance, after the area PTNP and its 
centre of gravity have been determined, the position of the centre of gravity d, of the 
entire area XNTP, and the length of the line Xc, may be most easily ascertained by 
the method of construction. If the line PN is bisected in the point C, the centre of 
gravity of the triangle PXN will be situated at the distance of i CX from the point C : 
the centre of gravity of the triangle PXN being thus constructed with geometrical ex- 
actness, it follows, that the centre of gravity of the entire area PXNTP, which is the 
O 2 
