104 Mr. Atwood's Propositions determining the Positions 
specific gravity n — 
26 a 
+ 6 x ^ 2a X V' Sa — 1 ?/> 
50 a 
or n 
26—15 — 6 X V 7 2,/ X v' 8a 
5 oa 
l sp 
, the specific gravity of the 
fluid being == 1, the solid will float with the extremity of 
the base in contact with the fluid's surface. If the specific 
. , 26a — x ^ 8a — icfil , 
gravity is greater than — — — , or less 
than 26 - a ~ l<i L r 6 .- * - ^ ~ ^ , the solid will float with 
50 a 
the base wholly above the surface. If the specific gravity 
of the solid is to that of the fluid in any proportion be- 
tween the limits 
26 a — 15 p + 6 x v 7 2a X v 8./ _ 15/) 
_ 
to a \ and 
I 6u .. JJ.i L . 6 - . 1 5 ^ | to a, the solid will float with 
the base partly immersed beneath the fluid's surface. 
These limits are determined by geometrical construction 
in the treatise before quoted (lib. 11. prop. x. et seq .) to 
which construction the preceding investigation may serve as 
a comment and analysis ; and some elucidation of this kind 
may perhaps be deemed the more requisite, since no traces 
are to be found in the work referred to of the method of 
investigation or train of reasoning, by which a problem of so 
much difficulty was solved, without assistance from analytical 
operations, at least from any that would seem competent to 
such an inquiry.* 
* Before any proposition can be demonstrated synthetically, it must have been 
investigated or discovered by some previous train of reasoning : it has been supposed 
that the ancient geometricians purposely concealed the analysis of their propositions ; 
but as no satisfactory evidence is produced to support this conjecture, it is probable 
that the supposed concealment arose from the want of a proper notation, by which 
analytical investigations might be conveniently expressed. 
