1Remaining Angle Y V I is equall to the Remaining Angle B E ψ. And therefore the
(e) Line V I hath to the Line I Y the ſame proportion that the Line E ψ hath to ψ B: But
the (f) Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-
to the ſame: Therefore (g) T I ſhall have greater proportion unto I Y, than E ψ hath to ψ B:
And, by the ſame reaſon, the Square T I ſhall have greater proportion to the Square I Y, than
the Square E ψ hath to the Square ψ B.
(e) Line V I hath to the Line I Y the ſame proportion that the Line E ψ hath to ψ B: But
the (f) Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-
to the ſame: Therefore (g) T I ſhall have greater proportion unto I Y, than E ψ hath to ψ B:
And, by the ſame reaſon, the Square T I ſhall have greater proportion to the Square I Y, than
the Square E ψ hath to the Square ψ B.
E
(e) By 4. of the
ſixth.
ſixth.
(f) By 8. of the
fifth.
fifth.
(g) By 13 of the
fifth.
fifth.
F
But as the Square P I is to the Square Y I, ſo is the Line K R unto
the Line I Y] For by 11. of the firſt of our Conicks, the Square P I is equall
to the Rectangle contained under the Line I O, and under the Parameter; which
we ſuppoſed to be eqnall to the Semi-parameter; that is, the double of K R:
the Line I Y] For by 11. of the firſt of our Conicks, the Square P I is equall
to the Rectangle contained under the Line I O, and under the Parameter; which
we ſuppoſed to be eqnall to the Semi-parameter; that is, the double of K R:
But I Y is double of I O, by 33 of the ſame: And, therefore, the (h) Rectangle made of K R
and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;
that is, to the Square P I: But as the (i) Rectangle compounded of K R and I Y is to the
Square I Y, ſo is the Line K R unto the Line I Y: Therefore the Line K R ſhall have unto I
Y, the ſame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I
hath to the Square I Y.
(h) By 26. of the
ſixth.
ſixth.
(i) By Lem. 22 of
the tenth.
the tenth.
G
And as the Square E Ψ is to the Square Ψ B, ſo is half of the
Line K R unto the Line ψ B.] For the Square E ψ having been ſuppoſed equall
to half the Rectangle contained under the Line K R and ψ B; that is, to that contained under
the half of K R and the Line ψ B; and ſeeing that as the (k) Rectangle made of half K R
Line K R unto the Line ψ B.] For the Square E ψ having been ſuppoſed equall
to half the Rectangle contained under the Line K R and ψ B; that is, to that contained under
the half of K R and the Line ψ B; and ſeeing that as the (k) Rectangle made of half K R
and of B ψ is to the Square ψ B, ſo is half K R unto the Line ψ B; the half of K R ſhall have
the ſame proportion to ψ B, as the Square E ψ hath to the Square ψ B.
(k) By Lem. 22 of
the tenth.
the tenth.
H
And, conſequently, I Y is leſſe than the double of ψ B.]
For, as half K R is to ψ B, ſo is K R to another Line: it ſhall be (1) greater than I Y; that
For, as half K R is to ψ B, ſo is K R to another Line: it ſhall be (1) greater than I Y; that
is, than that to which K R hath leſſer proportion; and it ſhall be double of ψ B: Therefore
I Y is leſſe than the double of ψ B.
(l) By 10 of the
fifth.
fifth.
K
And I ω greater than ψ R.] For O having been ſuppoſed equall to B R,
if from B R, ψ B be taken, and from O ω, O I, which is leſſer than B, be taken; the
Remainder I ω ſhall be greater than the Remainder Ψ R.
if from B R, ψ B be taken, and from O ω, O I, which is leſſer than B, be taken; the
Remainder I ω ſhall be greater than the Remainder Ψ R.
L
And, therefore, F Q is equall to P M.] By the fourteenth of the fifth of
Euclids Elements: For the Line O N is equall to B D.
Euclids Elements: For the Line O N is equall to B D.
M
But it hath been demonſtrated that P H is greater than F.]
For it was demonſtrated that I ω is greater than F: And P H is equall to I ω.
For it was demonſtrated that I ω is greater than F: And P H is equall to I ω.
N
In the ſame manner we might demonſtrate the Line T H
to be Perpendicular unto the Surface of the Liquid.] For T α is equall
to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated,
the Line T H ſhall be drawn Perpendicular unto the Liquids Surface.
to be Perpendicular unto the Surface of the Liquid.] For T α is equall
to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated,
the Line T H ſhall be drawn Perpendicular unto the Liquids Surface.
O
Therefore, the Square P I hath leſſer proportion unto the
Square I Y, than the Square E hath to the Square ψ B.]
Theſe, and other particulars of the like nature, that follow both in this and the following
Propoſitions, ſhall be demonſtrated by us no otherwiſe than we have done above.
Square I Y, than the Square E hath to the Square ψ B.]
Theſe, and other particulars of the like nature, that follow both in this and the following
Propoſitions, ſhall be demonſtrated by us no otherwiſe than we have done above.
P
Therefore Perpendiculars being drawn thorow Z and G, unto
the Surface of the Liquid, that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till that its
Axis do make an Angle with the Waters Surface greater than that
which it now maketh.] For in that the Line drawn thorow G, doth fall perpendicu
larly towards thoſe parts which are next to L; but that thorow Z, towards thoſe next to A;
It is neceſſary that the Centre G do move downwards, and Z upwards: and, therefore, the
parts of the Solid next to L ſhall move downwards, and thoſe towards A upwards, that the
Axis may makea greater Angle with the Surface of the Liquid.
the Surface of the Liquid, that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till that its
Axis do make an Angle with the Waters Surface greater than that
which it now maketh.] For in that the Line drawn thorow G, doth fall perpendicu
larly towards thoſe parts which are next to L; but that thorow Z, towards thoſe next to A;
It is neceſſary that the Centre G do move downwards, and Z upwards: and, therefore, the
parts of the Solid next to L ſhall move downwards, and thoſe towards A upwards, that the
Axis may makea greater Angle with the Surface of the Liquid.
Q
For ſo ſhall I O be equall to ψ B; and ω I equall to I R; and
P H equall to F.] This plainly appeareth in the third Figure, which is added by us.
P H equall to F.] This plainly appeareth in the third Figure, which is added by us.