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.

D

E

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.

F

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