1contingent unto the Section in the Point P: Wherefore it alſo
maketh Right Angles with the Surface of the Liquid: and that
part of the Conoidall Solid which is within the Liquid ſhall move
upwards according to the Perpendicular drawn thorow B parallel
to R T; and that part which is above the Liquid ſhall move down
wards according to that drawn thorow G, parallel to the ſaid R T:
And thus it ſhall continue to do ſo long untill that the Conoid be
reſtored to uprightneſſe, or to ſtand according to the Perpendicular.
maketh Right Angles with the Surface of the Liquid: and that
part of the Conoidall Solid which is within the Liquid ſhall move
upwards according to the Perpendicular drawn thorow B parallel
to R T; and that part which is above the Liquid ſhall move down
wards according to that drawn thorow G, parallel to the ſaid R T:
And thus it ſhall continue to do ſo long untill that the Conoid be
reſtored to uprightneſſe, or to ſtand according to the Perpendicular.
(a) By 10. of the
fifth.
fifth.
A
B
(b) By 19. of the
fifth.
fifth.
C
(c) By 1. of this
ſecond Book.
ſecond Book.
(d) By 6. De Co
noilibus & Sphæ
roidibus of Archi
medes.
noilibus & Sphæ
roidibus of Archi
medes.
D
E
F
(e) By 2. of this
ſecond Book.
ſecond Book.
COMMANDINE.
A
Let therefore R H be equall to the Semi-parameter.] So it is to be
read, and not R M, as Tartaglia's Tranſlation hath is; which may be made appear from
that which followeth.
read, and not R M, as Tartaglia's Tranſlation hath is; which may be made appear from
that which followeth.
B
And O H double to H M.] In the Tranſlation aforenamed it is falſly render
ed, O N double to R M.
ed, O N double to R M.
C
And look what proportion the Submerged Portion hath to the whole
Portion, the ſame hath the Square of P F unto the Square of N O.]
This place we have reſtored in our Tranſlation, at the requeſt of ſome friends: But it is demon
ſtrated by Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. 26.
Portion, the ſame hath the Square of P F unto the Square of N O.]
This place we have reſtored in our Tranſlation, at the requeſt of ſome friends: But it is demon
ſtrated by Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. 26.
D
Wherefore P F is not leſſe than M O.] For by 10 of the fifth it followeth
that the Square of P F is not leſſe than the Square of M O: and therefore neither ſhall the
Line P F be leße than the Line M O, by 22 of the
that the Square of P F is not leſſe than the Square of M O: and therefore neither ſhall the
Line P F be leße than the Line M O, by 22 of the
253[Figure 253]ſixth.
E
(a) By 14. of the
ſixth.
ſixth.
Nor B P than H O,] For as P F is to
P B, ſo is M O to H O: and, by Permutation, as
P B, ſo is M O to H O: and, by Permutation, as
P F is to M O, ſo is B P to H O; But P F is not
leſſe than M O as hath bin proved; (a) Therefore
neither ſhall B P be leſſe than H O.
F
If therefore a Right Line be drawn
from H at Right Angles unto N O, it
ſhall meet with B P, and ſhall fall be
twixt B and P.] This Place was corrupt in the
Tranſlation of Tartaglia: But it is thus demonstra
ted. In regard that P F is not leſſe than O M, nor P B than O H, if we ſuppoſe P F equall to
O M, P B ſhall be likewiſe equall to O H: Wherefore the Line drawn thorow O, parallel to A L
ſhall fall without the Section, by 17 of the firſt of our Treatiſe of Conicks; And in regard that
B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth
alſo meet with the ſame beneath B, and it doth of neceſſity fall betwixt B and P: But the
ſame is much more to follow, if we ſuppoſe P F to be greater than O M.
from H at Right Angles unto N O, it
ſhall meet with B P, and ſhall fall be
twixt B and P.] This Place was corrupt in the
Tranſlation of Tartaglia: But it is thus demonstra
ted. In regard that P F is not leſſe than O M, nor P B than O H, if we ſuppoſe P F equall to
O M, P B ſhall be likewiſe equall to O H: Wherefore the Line drawn thorow O, parallel to A L
ſhall fall without the Section, by 17 of the firſt of our Treatiſe of Conicks; And in regard that
B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth
alſo meet with the ſame beneath B, and it doth of neceſſity fall betwixt B and P: But the
ſame is much more to follow, if we ſuppoſe P F to be greater than O M.