1with their Diameters; as alſo, the Angles at Y and G being equall;
Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.]
Let the Line A Q cut the Diameter D B in γ, and let it cut A O in δ. Now becauſe that in
283[Figure 283]
the equall and like Portions A P O L & A M Q L,
from the Extremities of their Baſes, A O and
A Q are drawn, that contain equall Angles with
thoſe Baſes; and ſince the Angles at D, are both
Right; Therefore, the Remaining Angles A δ D
and A γ D ſhall be equall to one another: But
the Line P G is parallel unto the Line A O; alſo
M Y is parallel to A que and P S and M C to
A D: Therefore the Triangles P G S and M Y C,
as alſo the Triangles A δ D and A γ D, are all
alike to each other: (b) And as A D is to A δ,
ſo is A D to A γ: and, Permutando, the Lines
A D and A D are equall to each other: Therefore,
A δ and A γ are alſo equall: But A O and
A Q are equall to each other; as alſo their halves
A T and A N: Therefore the Remainders T δ and N γ; that is, TG and MY, are alſo
284[Figure 284]
equall. And, as (c) P G is to G S, ſo is M Y to
Y C: and Permutando, as P G is to M Y, ſo is
G S to Y C: And, therefore, G S and Y C are
equall; as alſo their halves B S and B C: From
whence it followeth, that the Remainders S R and C R
are alſo equall: And, conſequently, that P Z and
M V, and V N and Z T, are lkiewiſe equall to one
another.
Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.]
Let the Line A Q cut the Diameter D B in γ, and let it cut A O in δ. Now becauſe that in
283[Figure 283]the equall and like Portions A P O L & A M Q L,
from the Extremities of their Baſes, A O and
A Q are drawn, that contain equall Angles with
thoſe Baſes; and ſince the Angles at D, are both
Right; Therefore, the Remaining Angles A δ D
and A γ D ſhall be equall to one another: But
the Line P G is parallel unto the Line A O; alſo
M Y is parallel to A que and P S and M C to
A D: Therefore the Triangles P G S and M Y C,
as alſo the Triangles A δ D and A γ D, are all
alike to each other: (b) And as A D is to A δ,
ſo is A D to A γ: and, Permutando, the Lines
A D and A D are equall to each other: Therefore,
A δ and A γ are alſo equall: But A O and
A Q are equall to each other; as alſo their halves
A T and A N: Therefore the Remainders T δ and N γ; that is, TG and MY, are alſo
284[Figure 284]equall. And, as (c) P G is to G S, ſo is M Y to
Y C: and Permutando, as P G is to M Y, ſo is
G S to Y C: And, therefore, G S and Y C are
equall; as alſo their halves B S and B C: From
whence it followeth, that the Remainders S R and C R
are alſo equall: And, conſequently, that P Z and
M V, and V N and Z T, are lkiewiſe equall to one
another.
H
(b) By 4. of the
ſixth.
ſixth.
(c) By 34 of the
firſt,
firſt,
Since, therefore, that N V is leſſer
than double of V N.] For M H is double of
H N, and M V is leſſer than M H: Therefore, M V
is leſſer than double of H N, and much leſſer than
double of V N.
than double of V N.] For M H is double of
H N, and M V is leſſer than M H: Therefore, M V
is leſſer than double of H N, and much leſſer than
double of V N.
K
Therefore, the Portion ſhall not abide, but ſhall turn about,
ſo, as that its Baſe do not in the leaſt touch the Surface of
the Liquid; in regard that now when it toucheth in but one Point
only, it moveth upwards on the part towards A.] Tartaglia's his Tranſla
tion hath it thus, Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum
unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina
tur: Which we have thought fit in this manner to correct, from other Places of
Archimedes, that the ſenſe might be the more perſpicuous. For in the ſixth Propoſition of this,
he thus writeth (as we alſo have it in the Tranſlation,) The Solid A P O L, therefore, ſhall
turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid. Again,
in the ſeventh Propoſition; From whence it is manifeſt, that its Baſe ſhall turn about in
ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For
that now when it toucheth but in one Point only, it moveth downwards on the part
towards L. And that the Portion moveth upwards, on the part towards A, doth plainly ap
pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow ω, de
fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary
that the Centre ω do move upwards, and the Centre E downwards.
ſo, as that its Baſe do not in the leaſt touch the Surface of
the Liquid; in regard that now when it toucheth in but one Point
only, it moveth upwards on the part towards A.] Tartaglia's his Tranſla
tion hath it thus, Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum
unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina
tur: Which we have thought fit in this manner to correct, from other Places of
Archimedes, that the ſenſe might be the more perſpicuous. For in the ſixth Propoſition of this,
he thus writeth (as we alſo have it in the Tranſlation,) The Solid A P O L, therefore, ſhall
turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid. Again,
in the ſeventh Propoſition; From whence it is manifeſt, that its Baſe ſhall turn about in
ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For
that now when it toucheth but in one Point only, it moveth downwards on the part
towards L. And that the Portion moveth upwards, on the part towards A, doth plainly ap
pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow ω, de
fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary
that the Centre ω do move upwards, and the Centre E downwards.
L
It is therefore perſpicuous, that the Portion ſhall conſiſt, ſo, as that
its Axis ſhall make an Angle with the Liquids Surface greater than
the Angle X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter
its Axis ſhall make an Angle with the Liquids Surface greater than
the Angle X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter