io 6 Mr. Atwood's Propositions determining the Positions 
Then the proposition affirms that the solid will float per- 
manently on the fluid with the surface thereof in contact with 
one extremity of the base, when the specific gravity of the 
solid is to that of the fluid as the square of the line OX is to 
the square of the axis BD, or as the square of the line PF is 
to the square of the axis BD. 
Instead of inserting the geometrical demonstration of this 
construction, it will be more expedient, in the present instance, 
to proceed by a contrary method of argument, i. e. by assum- 
ing the construction as true, and inferring from it the propor- 
tions of the specific gravities in question, and comparing the 
proportions so inferred with those which have been already 
found by analytical investigation. Proceeding, according to 
this method, through the points X and O draw the lines SX, 
OY, parallel to AD ; and since the axis DB = a , and DK = 
— by construction, and KC =tj, it follows that DC = ~,and 
BC =7^; moreover, by the properties of the parabola DA = 
\/ ap, and the triangles ABD, ECB, being similar, EC = 
= - aJ) * - * 6 * * = ZD ; moreover, as DB : ZE or DC : : DA : 
ZA, that is, as a : ^ : : v/ ap : ZA = 9 ; and consequently 
the parameter of the parabola AEI = ~ 
x - £ = 2£. And because RC = EM = KC — KR = 
9 a 3 ° 9 
and ZN = parameter of the parabola AEI x ME, it follows 
that ZN = V /'' ME [ * = v/!EEIIE, and ND = ZD — ZN 
EC — ZN = — ■ £— 
y 
Sap — 1 sp z V ' Sap — VSap — 15 p z 
5 ° 
^50 
