5 8 Mr. Atwood's Propositions determining the Positions 
vated above it ; consequently, whatever may be the position 
of the point of intersection X, the volume I X W must be 
equal to the volume P X N. Suppose a to be the centre of 
gravity of the space I X W, and let d be the centre of gravity 
of the space NXP; then, the part immersed WRMP, is 
equal to the space IRMN, diminished by the space IWX, 
considered as concentered in the point a, and increased by 
the equal space NXP, concentered in the point d ; conse- 
quently the centre of gravity Q of the space WRMP will 
be at such a distance from E, the centre of gravity of the 
space IRMN, as corresponds to the alteration occasioned by 
removing the volume IWX, concentered in the point a , to 
the point d. These are the data from which the perpendi- 
cular distance G Z, of the two vertical lines K O, SO, pass- 
ing through the centres of gravity G and O, is to be obtained 
in the manner following : through the centres of gravity a 
and by draw the lines ab , dc, perpendicular to the horizontal 
line AB ; through E draw the indefinite line EY parallel 
to AB, and in the line EY, take a part ET, so that ET shall 
be to the line be as the volume IWX, or its equal NXP, is 
to the whole volume immersed, WRMP or ADHB : through 
the point T thus found, draw the line F T S parallel to the 
vertical line G O ; the centre of gravity Q, of the immersed 
part, will be somewhere in the line FS ; and because ER 
is to E G, as the sine of the given angle of inclination is to 
radius, the line GO = EG being supposed given, the line ER 
will therefore be known, and being subtracted from the line 
ET before found, will leave RT or GZ the perpendicular dis- 
tance between the -two vertical lines, which it was required to 
determine by geometrical construction, and which has been 
accordingly determined. 
