1
that is, H G to N C: and as (d) O H is to H P, ſo is G B to C K; For O H is double
to G B, and H P alſo double to G F; that is, to C K; Therefore H G hath the ſame propor
tion to N C, that G B hath to C K: And Permutando, N C hath to C K the ſame proportion
that H G hath to G B.
that is, H G to N C: and as (d) O H is to H P, ſo is G B to C K; For O H is double
to G B, and H P alſo double to G F; that is, to C K; Therefore H G hath the ſame propor
tion to N C, that G B hath to C K: And Permutando, N C hath to C K the ſame proportion
that H G hath to G B.
(a) By 2. Lemma.
(b) By 4. Lemma.
(b) By 19. of the
fifth.
fifth.
(d) By 15. of the
fifth.
fifth.
Then take ſome other Point at pleaſure in the Section, which
let be S: and thorow S draw two Lines, the one S T paral
lel to D B, and cutting the Diameter in the Point T; the
other S V parallel to A C, and cutting C E in V. I ſay
that V C hath greater proportion to C K, than T G hath
to G B.
let be S: and thorow S draw two Lines, the one S T paral
lel to D B, and cutting the Diameter in the Point T; the
other S V parallel to A C, and cutting C E in V. I ſay
that V C hath greater proportion to C K, than T G hath
to G B.
For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the
Diameter parallel to B D: G T ſhall be leſſe than G Y, in regard that V S is leße than V X:
And, by the firſt Lemma, Y G ſhall be to V C, as H G to N C; that is, as G B to C K, which
was demonſtrated but now: And, Permutando, Y G ſhall be to G B, as V C to C K: But
T G, for that it is leſſe than Y G, hath leſſe proportion to G B, than Y G hath to the ſame;
Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de
monſtrated. Therefore a Poſition given G K, there ſhall be in the Section one only Point, to
wit M, from which two Lines M E H and M N O being drawn, N C ſhall have the ſame pro
portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall
eth betwixt A C, and the Line parallel unto it ſhall alwayes have greater proportion to C K,
than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there
fore, is manifeſt which was affirmed by Archimedes, to wit, that the Line P I hath unto P H,
either the ſame proportion that N ω hath to ω O, or greater.
Diameter parallel to B D: G T ſhall be leſſe than G Y, in regard that V S is leße than V X:
And, by the firſt Lemma, Y G ſhall be to V C, as H G to N C; that is, as G B to C K, which
was demonſtrated but now: And, Permutando, Y G ſhall be to G B, as V C to C K: But
T G, for that it is leſſe than Y G, hath leſſe proportion to G B, than Y G hath to the ſame;
Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de
monſtrated. Therefore a Poſition given G K, there ſhall be in the Section one only Point, to
wit M, from which two Lines M E H and M N O being drawn, N C ſhall have the ſame pro
portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall
eth betwixt A C, and the Line parallel unto it ſhall alwayes have greater proportion to C K,
than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there
fore, is manifeſt which was affirmed by Archimedes, to wit, that the Line P I hath unto P H,
either the ſame proportion that N ω hath to ω O, or greater.
D
Wherefore P H is to H I either double, or leſſe than double.]
If leſſe than double, let P T be double to T I: The Centre of Gravity of that part of the
Portion that is within the Liquid ſhall be the
262[Figure 262]
Point T: But if P H be double to H I, H ſhall
be the Centre of Gravity; And draw H F, and
prolong it unto the Centre of that part of the Por
tion which is above the Liquid, namely, unto G,
and the reſt is demonſtrated as before. And the
ſame is to be underſtood in the Propoſition that
followeth.
If leſſe than double, let P T be double to T I: The Centre of Gravity of that part of the
Portion that is within the Liquid ſhall be the
262[Figure 262]Point T: But if P H be double to H I, H ſhall
be the Centre of Gravity; And draw H F, and
prolong it unto the Centre of that part of the Por
tion which is above the Liquid, namely, unto G,
and the reſt is demonſtrated as before. And the
ſame is to be underſtood in the Propoſition that
followeth.
The Solid A P O L, therefore,
ſhall turn about, and its Baſe ſhall
not in the leaſt touch the Surface
of the Liquid.] In Tartaglia's Tranſlation it is rendered ut Baſis ipſius non tangent
ſuperficiem humidi ſecundum unum ſignum; but we have choſen to read ut Baſis ipſius
nullo modo humidi ſuperficiem contingent, both here, and in the following Propoſitions,
becauſe the Greekes frequently uſe ὡδὲεἶς, ὡδὲ pro ὠδεὶσ & οὐδὶν: ſo that οὐδἔσινουδείς, nullus
eſt; οὐδὑπ̓ἑρὸς à nullo, and ſo of others of the like nature.
ſhall turn about, and its Baſe ſhall
not in the leaſt touch the Surface
of the Liquid.] In Tartaglia's Tranſlation it is rendered ut Baſis ipſius non tangent
ſuperficiem humidi ſecundum unum ſignum; but we have choſen to read ut Baſis ipſius
nullo modo humidi ſuperficiem contingent, both here, and in the following Propoſitions,
becauſe the Greekes frequently uſe ὡδὲεἶς, ὡδὲ pro ὠδεὶσ & οὐδὶν: ſo that οὐδἔσινουδείς, nullus
eſt; οὐδὑπ̓ἑρὸς à nullo, and ſo of others of the like nature.