340 
turning freely on fixed supports at the surface of the water 
MN. Draw ce at right angles to pc, and equal to it. 
Suppose pe divided into any number of indefinitely small 
parts; the pressure on any part, as at c, will be as its depth 
from p, which is equal to the perpendicular ce; and similar 
perpendiculars, drawn from any other part, will be equal to 
the depth of such part, and the whole pressure on Dc is 
represented by the triangle pce. Bisect ce in F, and join 
pF; the centre of gravity of the triangle DcE is at two 
thirds of pr, from p at the point o, through which draw 
GH parallel to ce. The sum of all the perpendiculars, mul- 
tiplied by their respective forces, is equal to the sum ofall the 
forces multiplied by their mean distance, which is co; and, 
therefore, the pressure may be considered as concentrated at 0, 
and acting along the line Go, or at G, which is at two-thirds 
of pc from p, and, therefore, the force of the water pressing 
against the line pe, is expressed by two-thirds of pc, mul- 
tiplied by the weight ofa quantity of water represented by, or 
equal to, the triangle pce. Now, it is evident that an equal 
pressure or weight acting perpendicularly at P, two-thirds of 
DB, from D, will balance the pressure at c. 
Suppose the figure to have revolved about the axis p, till 
pc became pc’, and ps, DB’. The angles snc’ and B’DA are 
equal, and their sines also equal; but the rotating power of 
any weight acting at any given point in BD is to its power at 
B'D as the sine of the angle made by Bp with pa, and the pres- 
sure of the water on pc’ is as the sine of the equal angle spc’, 
for the pressure on each of the parts into which pc was sup- 
posed to be divided is as their depths, or the perpendiculars 
let fall on them from the line Bp, that is, as the sine of the 
angle spc’; therefore, the pressure of the sum is likewise as the 
sine of spc’, and consequently equal to the power of the weight 
at P on BD, and they will balance each other in all positions, 
as the same may be proved of any position of the float. 
Now, if the water sink below the level of Bp, the pressure 
