1For if, the Line B R being prolonged unto G, G R hath the ſame proportion to R B as the Por
tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the
Liquid, the Toine G ſhall be its Centre of Gravity; by the 8 of the ſecond of Archimedes
de Centro Gravitatis Planorum, vel de Æquiponderantibus.
tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the
Liquid, the Toine G ſhall be its Centre of Gravity; by the 8 of the ſecond of Archimedes
de Centro Gravitatis Planorum, vel de Æquiponderantibus.
R O ſhall be leſs than quæ uſque ad Axem (or than the Semi
parameter.] By the 10 Propofit. of Euclids fifth Book of Elements. The Line quæ
uſque ad Axem, (or the Semi-parameter) according to Archimedes, is the half of that
juxta quam poſſunt, quæ á Sectione ducuntur, (or of the Parameter;) as appeareth
by the 4 Propoſit of his Book De Conoidibus & Shpæroidibus: and for what reaſon it is
ſo called, we have declared in the Commentaries upon him by us publiſhed.
parameter.] By the 10 Propofit. of Euclids fifth Book of Elements. The Line quæ
uſque ad Axem, (or the Semi-parameter) according to Archimedes, is the half of that
juxta quam poſſunt, quæ á Sectione ducuntur, (or of the Parameter;) as appeareth
by the 4 Propoſit of his Book De Conoidibus & Shpæroidibus: and for what reaſon it is
ſo called, we have declared in the Commentaries upon him by us publiſhed.
Whereupon the Angle R P ω ſhall be acute.] Let the Line N O be
continued out to H, that ſo RH may be equall to
the Semi-parameter. If now from the Point H
248[Figure 248]
a Line be drawn at Right Angles to N H, it ſhall
meet with FP without the Section; for being
drawn thorow O parallel to A L, it ſhall fall
without the Section, by the 17 of our ſirst Book of
Conicks; Therefore let it meet in V: and
becauſe F P is parallel to the Diameter, and H
V perpendicular to the ſame Diameter, and R H
equall to the Semi-parameter, the Line drawn
from the Point R to V ſhall make Right Angles
with that Line which the Section toucheth in the Point P: that is with K ω, as ſhall anon be
demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and ω; and the Argle R
P ω ſhall be an Acute Angle.
continued out to H, that ſo RH may be equall to
the Semi-parameter. If now from the Point H
248[Figure 248]a Line be drawn at Right Angles to N H, it ſhall
meet with FP without the Section; for being
drawn thorow O parallel to A L, it ſhall fall
without the Section, by the 17 of our ſirst Book of
Conicks; Therefore let it meet in V: and
becauſe F P is parallel to the Diameter, and H
V perpendicular to the ſame Diameter, and R H
equall to the Semi-parameter, the Line drawn
from the Point R to V ſhall make Right Angles
with that Line which the Section toucheth in the Point P: that is with K ω, as ſhall anon be
demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and ω; and the Argle R
P ω ſhall be an Acute Angle.
Let A B C be the Section of a Rightangled Cone, or a Parabola,
and its Diameter B D; and let the Line E F touch the
ſame in the Point G: and in the Diameter B D take the Line
H K equall to the Semi-parameter: and thorow G, G L be
ing drawn parallel to the Diameter, draw KM from the
Point K at Right Angles to B D cutting G L in M: I ſay
that the Line prolonged thorow Hand Mis perpendicular to
E F, which it cutteth in N.
and its Diameter B D; and let the Line E F touch the
ſame in the Point G: and in the Diameter B D take the Line
H K equall to the Semi-parameter: and thorow G, G L be
ing drawn parallel to the Diameter, draw KM from the
Point K at Right Angles to B D cutting G L in M: I ſay
that the Line prolonged thorow Hand Mis perpendicular to
E F, which it cutteth in N.
For from the Point G draw the Line G O at Right Angles to E F cutting the Diameter in
O: and again from the ſame Point draw G P perpendicular to the Diameter: and let the
ſaid Diameter prolonged cut the Line E F in que P B ſhall be equall to B Q, by the 35 of
O: and again from the ſame Point draw G P perpendicular to the Diameter: and let the
ſaid Diameter prolonged cut the Line E F in que P B ſhall be equall to B Q, by the 35 of
our firſt Book of Conick Sections, (a) and G
249[Figure 249]P a Mean-proportion all betmixt Q P and PO;
(b) and therefore the Square of G P ſhall be e
quall to the Rectangle of O P Q: But it is alſo
equall to the Rectangle comprehended under P B
and the Line juxta quam poſſunt, or the Par
ameter, by the 11 of our firſt Book of Conicks:
(c) Therefore, look what proportion Q P hath to
P B, and the ſame hath the Parameter unto P O:
But Q P is double unto P B, for that P B and B
Q are equall, as hath been ſaid: And therefore
the Parameter ſhall be double to the ſaid P O:
and by the ſame Reaſon P O is equall to that which we call the Semi-parameter, that is, to K H:
But (d) P G is equall to K M, and (e) the Angle O P G to the Angle H K M; for they are both
Right Angles: And therefore O G alſo is equall to H M, and the Angle P O G unto the