1the Portion hath to the Liquid of equall Maſſe, the ſame hath the
Magnitude of the Portion ſubmerged unto the whole Portion; as
hath been demonſtrated in the firſt Propoſition; The Magnitude
ſubmerged, therefore, ſhall not have greater proportion to the
whole (b) Portion, than that which hath been mentioned: ^{*}And
therefore the whole Portion hath not greater proportion unto that
which is above the Liquid, than the Square N O hath to the Square
M O: But the (c) whole Portion hath the ſame proportion unto
that which is above the Liquid that the Square N O hath to the
Square P F: Therefore the Square N O hath not greater propor
tion unto the Square P F, than it hath unto the Square M O: ^{*}And
hence it followeth that P F is not leſſe than O M, nor P B than O
H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
ſhall meet with B P betwixt P and B: Let it be in T: And be
cauſe that in the Section of the Rectangled Cone P F is parallel unto
the Diameter N O; and H T perpendicular unto the ſaid Diame
ter; and R H equall to the Semi-parameter: It is manifeſt that
R T prolonged doth make Right Angles with K P ω: And there
fore doth alſo make Right Angles with I S: Therefore R T is per
pendicular unto the Surface of the Liquid; And if thorow the
Points B and G Lines be drawn parallel unto R T, they ſhall be
perpendicular unto the Liquids Surface. The Portion, therefore,
which is above the Liquid ſhall move downwards in the Liquid ac
cording to the Perpendicular drawn thorow B; and that part
which is within the Liquid ſhall move upwards according to the
Perpendicular drawn thorow G; and the Solid Portion A P O L
ſhall not continue ſo inclined, [as it was at its demerſion], but ſhall
move within the Liquid untill ſuch time that N O do ſtand accor
ding to the Perpendicular.
Magnitude of the Portion ſubmerged unto the whole Portion; as
hath been demonſtrated in the firſt Propoſition; The Magnitude
ſubmerged, therefore, ſhall not have greater proportion to the
whole (b) Portion, than that which hath been mentioned: ^{*}And
therefore the whole Portion hath not greater proportion unto that
which is above the Liquid, than the Square N O hath to the Square
M O: But the (c) whole Portion hath the ſame proportion unto
that which is above the Liquid that the Square N O hath to the
Square P F: Therefore the Square N O hath not greater propor
tion unto the Square P F, than it hath unto the Square M O: ^{*}And
hence it followeth that P F is not leſſe than O M, nor P B than O
H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
ſhall meet with B P betwixt P and B: Let it be in T: And be
cauſe that in the Section of the Rectangled Cone P F is parallel unto
the Diameter N O; and H T perpendicular unto the ſaid Diame
ter; and R H equall to the Semi-parameter: It is manifeſt that
R T prolonged doth make Right Angles with K P ω: And there
fore doth alſo make Right Angles with I S: Therefore R T is per
pendicular unto the Surface of the Liquid; And if thorow the
Points B and G Lines be drawn parallel unto R T, they ſhall be
perpendicular unto the Liquids Surface. The Portion, therefore,
which is above the Liquid ſhall move downwards in the Liquid ac
cording to the Perpendicular drawn thorow B; and that part
which is within the Liquid ſhall move upwards according to the
Perpendicular drawn thorow G; and the Solid Portion A P O L
ſhall not continue ſo inclined, [as it was at its demerſion], but ſhall
move within the Liquid untill ſuch time that N O do ſtand accor
ding to the Perpendicular.
COMMANDINE.
And therefore the whole Portion hath not greater proportion
unto that which is above the Liquid, than the Square N O hath to
the Square M O.] For in regard that the Magnitude of the Portion demerged
within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which
the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of
the Proportion, by the 26. of the fifth of Euclid, of Campanus his Tranſlation, the whole
Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O
hath unto the Exceſſe by which N O is greater than the Square M O. Let a Portion be taken;
and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which
is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe
by which the Square N O is greater than the Square M O be the fourth. Now of theſe Mag
nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third &
fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square
N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi
on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the
firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame
unto that which is above the Liquid, than the Square N O hath to
the Square M O.] For in regard that the Magnitude of the Portion demerged
within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which
the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of
the Proportion, by the 26. of the fifth of Euclid, of Campanus his Tranſlation, the whole
Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O
hath unto the Exceſſe by which N O is greater than the Square M O. Let a Portion be taken;
and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which
is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe
by which the Square N O is greater than the Square M O be the fourth. Now of theſe Mag
nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third &
fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square
N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi
on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the
firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame