1
250[Figure 250]
Angle K H M: Therefore (f) O G and H N are parallel,
and the (g) Angle H N F equall to the Angle O G F; for
that G O being Perpendicular to E F, H N ſhall alſo be per-
pandicnlar to the ſame: Which was to be demon ſtrated.
250[Figure 250]Angle K H M: Therefore (f) O G and H N are parallel,
and the (g) Angle H N F equall to the Angle O G F; for
that G O being Perpendicular to E F, H N ſhall alſo be per-
pandicnlar to the ſame: Which was to be demon ſtrated.
(a) By Cor. of 8. of
6. of Euclide.
6. of Euclide.
(b) By 17. of the
6.
6.
(c) By 14. of the
6.
6.
(d) By 33. of the
1.
1.
(e) By 4. of the 1.
(f) By 28. of the
1.
1.
(g) By 29. of th
1
1
And the part which is within the Liquid
doth move upwards according to the Per
pendicular that is drawn thorow B parallel
to R T.] The reaſon why this moveth upwards, and that
other downwards, along the Perpendicular Line, hath been ſhewn above in the 8 of the firſt
Book of this; ſo that we have judged it needleſſe to repeat it either in this, or in the reſt
that follow.
doth move upwards according to the Per
pendicular that is drawn thorow B parallel
to R T.] The reaſon why this moveth upwards, and that
other downwards, along the Perpendicular Line, hath been ſhewn above in the 8 of the firſt
Book of this; ſo that we have judged it needleſſe to repeat it either in this, or in the reſt
that follow.
G
THE TRANSLATOR.
In the Antient Parabola (namely that aſſumed in a Rightangled
Cone) the Line juxta quam Poſſunt quæ in Sectione ordinatim du
cuntur (which I, following Mydorgius, do call the Parameter) is (a)
double to that quæ ducta eſt à Vertice Sectionis uſque ad Axem, or in
Archimedes phraſe, τᾱς υσ́χρι τοῡ ἄξον<34>; which I for that cauſe, and
for want of a better word, name the Semiparameter: but in Modern
Parabola's it is greater or leſſer then double. Now that throughout this
Book Archimedes ſpeaketh of the Parabola in a Rectangled Cone, is mani
feſt both by the firſt words of each Propoſition, & by this that no Parabola
hath its Parameter double to the Line quæ eſt a Sectione ad Axem, ſave
that which is taken in a Rightangled Cone. And in any other Parabola, for
the Line τᾱς μσ́χριτοῡ ἄεον<34> or quæ uſque ad Axem to uſurpe the Word Se
miparameter would be neither proper nor true: but in this caſe it may paſs
Cone) the Line juxta quam Poſſunt quæ in Sectione ordinatim du
cuntur (which I, following Mydorgius, do call the Parameter) is (a)
double to that quæ ducta eſt à Vertice Sectionis uſque ad Axem, or in
Archimedes phraſe, τᾱς υσ́χρι τοῡ ἄξον<34>; which I for that cauſe, and
for want of a better word, name the Semiparameter: but in Modern
Parabola's it is greater or leſſer then double. Now that throughout this
Book Archimedes ſpeaketh of the Parabola in a Rectangled Cone, is mani
feſt both by the firſt words of each Propoſition, & by this that no Parabola
hath its Parameter double to the Line quæ eſt a Sectione ad Axem, ſave
that which is taken in a Rightangled Cone. And in any other Parabola, for
the Line τᾱς μσ́χριτοῡ ἄεον<34> or quæ uſque ad Axem to uſurpe the Word Se
miparameter would be neither proper nor true: but in this caſe it may paſs
(a) Rîvalt. in Ar
chimed. de Cunoid
& Sphæroid. Prop.
3. Lem. 1.
chimed. de Cunoid
& Sphæroid. Prop.
3. Lem. 1.
PROP. III. THEOR. III.
The Right Portion of a Rightangled Conoid, when it
ſhall have its Axis leſſe than ſeſquialter of the Se
mi-parameter, the Axis having any what ever pro
portion to the Liquid in Gravity, being demitted into
the Liquid ſo as that its Baſe be wholly within the
ſaid Liquid, and being ſet inclining, it ſhall not re
main inclined, but ſhall be ſo reſtored, as that its Ax
is do ſtand upright, or according to the Perpendicular.
ſhall have its Axis leſſe than ſeſquialter of the Se
mi-parameter, the Axis having any what ever pro
portion to the Liquid in Gravity, being demitted into
the Liquid ſo as that its Baſe be wholly within the
ſaid Liquid, and being ſet inclining, it ſhall not re
main inclined, but ſhall be ſo reſtored, as that its Ax
is do ſtand upright, or according to the Perpendicular.
Let any Portion be demitted into the Liquid, as was ſaid; and
let its Baſe be in the Liquid;
251[Figure 251]
and let it be cut thorow the
Axis, by a Plain erect upon the Sur
face of the Liquid, and let the Se
ction be A P O L, the Section of a
Right angled Cone: and let the Axis
of the Portion and Diameter of the
let its Baſe be in the Liquid;
251[Figure 251]and let it be cut thorow the
Axis, by a Plain erect upon the Sur
face of the Liquid, and let the Se
ction be A P O L, the Section of a
Right angled Cone: and let the Axis
of the Portion and Diameter of the