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.

* Or according

to the Perpendi

cular.

to the Perpendi

cular.

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.

A

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.

B

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:

239[Figure 239]

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:

239[Figure 239]

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