238 Mr. Atwood’s Disquisition on 
are drawn through these points perpendicular to CH and ch. 
The line SL will therefore coincide with the line si, and is equal 
to it. In the same manner, it is proved that the line SK is 
equal to the line s k ; consequently, KL is equal to k l. And 
since, by construction, the area ASH is equal to the area 
BSC, and the area ash equal to the area bsc; and, on appli- 
cation of the figure AWB to the figure awb, the triangle ASH 
coincides with the triangle a s h, it follows, that the four areas 
ASH, ash, BSC, bsc, are all equal. 
But ET* 
KL x area ASH 
total volume 
a /ion aild C t 4- « * X volume as o 
immersed’ ' total volume immersed ’ 
and, since KL = k l, and the volume ASH = the volume ash, 
KL x volume ASH = kl x volume ash; and the entire volume 
immersed being the same in both vessels, by the supposition, it 
follows that ET — et. 
This equality between the lines ET, e t, is independent of 
the position of the centres of gravity of the vessels, G, g, and 
also of the position of the centres of gravity, E, e, in the lines 
WD, wd. If the distances of GE, ge, should be equal, since the 
angles of inclination from the upright, or EGR, egr, are equal 
by the supposition, it follows that the sines of those angles to 
equal radii must be equal, or ER = er. Subtracting, therefore, 
ER from ET, and e r from e t, the remaining lines RT, r t , 
must be equal, or G Z = g z. The stability, therefore, of a 
vessel, the sides of which are inclined to an angle under the 
water’s surface, is equal to the stability of the vessel of which 
the sides are inclined to an angle which is above the water’s 
surface : the breadth at the water-line, and the other condi- 
tions, being the same in both vessels. 
This proposition is not confined to the case here demon - 
* Fig. 9. Seepage 212. 
t Fig. 12. See page 212. 
