1ly. The like ſhall alſo hold true in the Portion of the Sphære
leſs than an Hemiſphere that lieth with its whole Baſe above the
Liquid.
leſs than an Hemiſphere that lieth with its whole Baſe above the
Liquid.
COMMANDINE.
The Demonſtration of this Propoſition is defaced by the Injury of Time, which we have re
ſtored, ſo far as by the Figures that remain, one may collect the Meaning of Archimedes,
for we thought it not good to alter them: and what was wanting to their declaration and ex
planation we have ſupplyed in our Commentaries, as we have alſo determined to do in the ſe
cond Propoſition of the ſecond Book.
ſtored, ſo far as by the Figures that remain, one may collect the Meaning of Archimedes,
for we thought it not good to alter them: and what was wanting to their declaration and ex
planation we have ſupplyed in our Commentaries, as we have alſo determined to do in the ſe
cond Propoſition of the ſecond Book.
If any Solid Magnitude lighter than the Liquid.] Theſe words, light-
er than the Liquid, are added by us, and are not to be found in the Tranſiation; for of theſe
kind of Magnitudes doth Archimedes ſpeak in this Propoſition.
er than the Liquid, are added by us, and are not to be found in the Tranſiation; for of theſe
kind of Magnitudes doth Archimedes ſpeak in this Propoſition.
Shall be demitted into the Liquid in ſuch a manner as that the
Baſe of the Portion touch not the Liquid.] That is, ſhall be ſo demitted into
the Liquid as that the Baſe ſhall be upwards, and the Vertex downwards, which he oppoſeth
to that which he ſaith in the Propoſition following; Be demitted into the Liquid, ſo, as
that its Baſe be wholly within the Liquid; For theſe words ſignifie the Portion demit
ted the contrary way, as namely, with the Vertex upwards and the Baſe downwards. The
ſame manner of ſpeech is frequently uſed in the ſecond Book; which treateth of the Portions
of Rectangle Conoids.
Baſe of the Portion touch not the Liquid.] That is, ſhall be ſo demitted into
the Liquid as that the Baſe ſhall be upwards, and the Vertex downwards, which he oppoſeth
to that which he ſaith in the Propoſition following; Be demitted into the Liquid, ſo, as
that its Baſe be wholly within the Liquid; For theſe words ſignifie the Portion demit
ted the contrary way, as namely, with the Vertex upwards and the Baſe downwards. The
ſame manner of ſpeech is frequently uſed in the ſecond Book; which treateth of the Portions
of Rectangle Conoids.
Now becauſe every Portion of a Sphære hath its Axis in the Line
that from the Center of the Sphære is drawn perpendicular to its
Baſe.] For draw a Line from B to C, and let K L cut the Circumference A B C D in the
Point G, and the Right Line B C in M:
13[Figure 13]
and becauſe the two Circles A B C D, and
E F H do cut one another in the Points
B and C, the Right Line that conjoyneth
their Centers, namely, K L, doth cut the
Line B C in two equall parts, and at
Right Angles; as in our Commentaries
upon Prolomeys Planiſphære we do
prove: But of the Portion of the Circle
B N C the Diameter is M N; and of the
Portion B G C the Diameter is M G;
for the (a) Right Lines which are drawn
on both ſides parallel to B C do make
Right Angles with N G; and (b) for
that cauſe are thereby cut in two equall
parts: Therefore the Axis of the Portion
of the Sphære B N C is N M; and the
Axis of the Portion B G C is M G:
from whence it followeth that the Axis of
the Portion demerged in the Liquid is
in the Line K L, namely N G. And ſince the Center of Gravity of any Portion of a Sphære is
in the Axis, as we have demonstrated in our Book De Centro Gravitatis Solidorum, the
Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is,
of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra
vity of thoſe Portions of Sphæres. For ſuppoſe, if poſſible, that it be out of the Line N G, as
in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V que Becauſe
therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha
ving the ſame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the
Portion B G C ſhall, by the 8 of the firſt Book of Archimedes, De Centro Gravitatis
that from the Center of the Sphære is drawn perpendicular to its
Baſe.] For draw a Line from B to C, and let K L cut the Circumference A B C D in the
Point G, and the Right Line B C in M:
13[Figure 13]and becauſe the two Circles A B C D, and
E F H do cut one another in the Points
B and C, the Right Line that conjoyneth
their Centers, namely, K L, doth cut the
Line B C in two equall parts, and at
Right Angles; as in our Commentaries
upon Prolomeys Planiſphære we do
prove: But of the Portion of the Circle
B N C the Diameter is M N; and of the
Portion B G C the Diameter is M G;
for the (a) Right Lines which are drawn
on both ſides parallel to B C do make
Right Angles with N G; and (b) for
that cauſe are thereby cut in two equall
parts: Therefore the Axis of the Portion
of the Sphære B N C is N M; and the
Axis of the Portion B G C is M G:
from whence it followeth that the Axis of
the Portion demerged in the Liquid is
in the Line K L, namely N G. And ſince the Center of Gravity of any Portion of a Sphære is
in the Axis, as we have demonstrated in our Book De Centro Gravitatis Solidorum, the
Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is,
of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra
vity of thoſe Portions of Sphæres. For ſuppoſe, if poſſible, that it be out of the Line N G, as
in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V que Becauſe
therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha
ving the ſame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the
Portion B G C ſhall, by the 8 of the firſt Book of Archimedes, De Centro Gravitatis