1the Angles at X and N are equall. And, therefore, if drawing HK,
it be prolonged to ω, the Centre of Gravity of the whole Portion ſhall
be K; of the part which is within the Liquid H; and of the part which
is above the Liquid in K ὠ as ſuppoſe in ω; and H K perpendicular to
63[Figure 63]
the Surface of the Liquid. Therfore
along the ſame Right Lines ſhall the
part which is within the Liquid move
upwards, and the part above it down
wards: And therfore the Portion
ſhall reſt with one of its Points
touching the Surface of the Liquid,
and its Axis ſhall make with the
ſame an Angle equall to X. It is
to be demonſtrated in the ſame
manner that the Portion that hath
the ſame proportion in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid, ſo, as that its
Baſe touch not the Liquid, it ſhall ſtand inclined, ſo, as that its Baſe
touch the Surface of the Liquid in one Point only; and its Axis ſhall
make therwith an Angle equall to the Angle φ.
it be prolonged to ω, the Centre of Gravity of the whole Portion ſhall
be K; of the part which is within the Liquid H; and of the part which
is above the Liquid in K ὠ as ſuppoſe in ω; and H K perpendicular to
63[Figure 63]the Surface of the Liquid. Therfore
along the ſame Right Lines ſhall the
part which is within the Liquid move
upwards, and the part above it down
wards: And therfore the Portion
ſhall reſt with one of its Points
touching the Surface of the Liquid,
and its Axis ſhall make with the
ſame an Angle equall to X. It is
to be demonſtrated in the ſame
manner that the Portion that hath
the ſame proportion in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid, ſo, as that its
Baſe touch not the Liquid, it ſhall ſtand inclined, ſo, as that its Baſe
touch the Surface of the Liquid in one Point only; and its Axis ſhall
make therwith an Angle equall to the Angle φ.
A
B
C
D
E
F
COMMANDINE.
A
That is the Square T P to the Square B D.] By the twenty ſixth of the Book
of Archimedes, De Conoidibus & Sphæroidibus: Therefore, (a) the Square T P
ſhall be equall to the Square X O: And for that reaſon, the Line T P equall to the
Line X O.
(a) By 9 of the
fifth.
fifth.
B
The Portions ſhall alſo be equall.] By the twenty fifth of the ſame Book.
C
Again, becauſe that in the Equall and Like Portions, A O Q L
and A P M L.] For, in the Portion A P M L, deſcribe the Portion A O Q equall
to the Portion I P M: The Point Q falleth beneath M; for otherwiſe, the Whole would be
equall to the Part. Then draw I V parallel to A Q, and cutting the Diameter is ψ; and
let I M cut the ſame ς; and A Q in ς. I ſay
that the Angle A υ D, is leſſer than the Angle
64[Figure 64]
I σ D. For the Angle I ψ D is equall to the
Angle A υ D: (b) But the interiour Angle
and A P M L.] For, in the Portion A P M L, deſcribe the Portion A O Q equall
to the Portion I P M: The Point Q falleth beneath M; for otherwiſe, the Whole would be
equall to the Part. Then draw I V parallel to A Q, and cutting the Diameter is ψ; and
let I M cut the ſame ς; and A Q in ς. I ſay
that the Angle A υ D, is leſſer than the Angle
64[Figure 64]I σ D. For the Angle I ψ D is equall to the
Angle A υ D: (b) But the interiour Angle
I ψ D is leſſer than the exteriour I σ D: There-
fore, (c) A υ D ſhall alſo be lefter than I σ D.
(b) By 29 of the
firſt.
firſt.
(c) By 16 of the
firſt.
firſt.
D
And the Angle at X, being leſſe
than the Angle at N.] Thorow O draw twe
Lines, O C perpendicular to the Diameter B D, and
O X touching the Section in the Point O, and cutting
than the Angle at N.] Thorow O draw twe
Lines, O C perpendicular to the Diameter B D, and
O X touching the Section in the Point O, and cutting
the Diameter in X: (d) O X ſhall be parallel
to A que and the (e) Angle at X, ſhall be equall to
that at υ: Therefore, the (f) Angle at X,
ſhall be leſſer than the Angle at ς; that is, to
that at N: And, conſequently, X ſhall fall beneath N: Therefore, the Line X B is greater than
N B. And, ſince B C is equall to X B, and B S equall to N B; B C ſhall be greater than B S.