1made of the Exceſs by which the Axis is greater than Seſquialter
of the Semi-parameter, hath to the Square made of the Axis, being
demitted into the Liquid, ſo as hath been ſaid, it ſhall ſtand erect,
or Perpendicular.
of the Semi-parameter, hath to the Square made of the Axis, being
demitted into the Liquid, ſo as hath been ſaid, it ſhall ſtand erect,
or Perpendicular.
F
G
H
K
L
M
N
O
P
Q
R
COMMANDINE.
The particulars contained in this Tenth Propoſition, are divided by Archimedes
into five Parts and Concluſions, each of which he proveth by a diſtinct Demonſtration.
into five Parts and Concluſions, each of which he proveth by a diſtinct Demonſtration.
A
It ſhall ſometimes ſtand perpendicular.] This is the firſt Concluſion, the
Demonstration of which he hath ſubjoyned to the Propoſition.
Demonstration of which he hath ſubjoyned to the Propoſition.
B
And ſometimes ſo inclined, as that its Baſe touch the Surface
of the Liquid, in one Point only.] This is demonſtrated in the third Con
cluſion.
of the Liquid, in one Point only.] This is demonſtrated in the third Con
cluſion.
Sometimes, ſo that its Baſe be moſt ſubmerged in the Liquid.]
This pertaineth unto the fourth Concluſion.
C
And, ſometimes, ſo as that it doth not in the leaſt touch the Sur
face of the Liquid.] This it doth hold true two wayes, one of which is explained is
the ſecond, and the other in the fifth Concluſion.
face of the Liquid.] This it doth hold true two wayes, one of which is explained is
the ſecond, and the other in the fifth Concluſion.
D
According to the proportion, that it hath to the Liquid in Gra
vity. Every one of which Caſes ſhall be anon demonſtrated.]
In Tartaglia's Verſion it is rendered, to the confuſion of the ſence, Quam autem pro
portionem habeant ad humidum in Gravitate fingula horum demonſtrabuntur.
vity. Every one of which Caſes ſhall be anon demonſtrated.]
In Tartaglia's Verſion it is rendered, to the confuſion of the ſence, Quam autem pro
portionem habeant ad humidum in Gravitate fingula horum demonſtrabuntur.
E
It is manifeſt, therefore, that K C is greater than the Semi
parameter] For, ſince B D hath to K C the ſame proportion, as fifteen to four, and
hath unto the Semi-parameter greater proportion; (a) the Semi-parameter ſhall be leſs
than K C.
parameter] For, ſince B D hath to K C the ſame proportion, as fifteen to four, and
hath unto the Semi-parameter greater proportion; (a) the Semi-parameter ſhall be leſs
than K C.
F
(a) By 10. of the
fifth.
fifth.
Let the Semi-parameter be equall to KR.] We have added theſe words,
which are not to be found in Tartaglia.
which are not to be found in Tartaglia.
G
But S B is alſo Seſquialter of BR.] For, D B is ſuppoſed Seſquialter of
B K; and D S alſo is Seſquialter of K R: Wherefore as (b) the whole D B, is to the whole
B K, ſo is the part D S to the part K R. Therefore, the Remainder S B, is alſo to the
Remainder B R, as D B is to B K.
B K; and D S alſo is Seſquialter of K R: Wherefore as (b) the whole D B, is to the whole
B K, ſo is the part D S to the part K R. Therefore, the Remainder S B, is alſo to the
Remainder B R, as D B is to B K.
H
(b) By 19 of the
fifth.
fifth.
And let them be like to the Portion A B L.] Apollonius thus defineth
like Portions of the Sections of a Cone, in Lib. 6. Conicornm, as Eutocius writeth ^{*};
ὄν οἱ̄ς ἀχ δεισω̄ν ὄν ἑχάσῳ ωαραλλήλων τη̄ <35>ὰσει, ἵσων τὸ πλη̄ο<34>, αἱ παράλληλος, καὶ ἁι <35>άσεις ωρὸς τάς αποτρμ
νομένας ἀπὸ διαμέτσων τω̄ς κορυφαῑς ἐν τοῑς ἀντοῑς λόγοις εἰσι, καὶ αἱ ἀποτεμνόμεναι ωρὸς τὰς ἀ τεμνομίνασ
that is, In both of which an equall number of Lines being drawn parallel to the
Baſe; the parallel and the Baſes have to the parts of the Diameters, cut off from
the Vertex, the ſameproportion: as alſo, the parts cut off, to the parts cut off.
Now the Lines parallel to the Baſes are drawn, as I ſuppoſe, by making a Rectilineall Figure (cal-
led) Signally inſcribed [χη̄μα γιωρίμως ἐγν̀<36>ρόμενον] in both portions, having an equall num
ber of Sides in both. Therefore, like Portions are cut off from like Sections of a Cone; and
their Diameters, whether they be perpendicular to their Baſes, or making equall Angles with their
Baſes, have the ſame proportion unto their Baſes.
like Portions of the Sections of a Cone, in Lib. 6. Conicornm, as Eutocius writeth ^{*};
ὄν οἱ̄ς ἀχ δεισω̄ν ὄν ἑχάσῳ ωαραλλήλων τη̄ <35>ὰσει, ἵσων τὸ πλη̄ο<34>, αἱ παράλληλος, καὶ ἁι <35>άσεις ωρὸς τάς αποτρμ
νομένας ἀπὸ διαμέτσων τω̄ς κορυφαῑς ἐν τοῑς ἀντοῑς λόγοις εἰσι, καὶ αἱ ἀποτεμνόμεναι ωρὸς τὰς ἀ τεμνομίνασ
that is, In both of which an equall number of Lines being drawn parallel to the
Baſe; the parallel and the Baſes have to the parts of the Diameters, cut off from
the Vertex, the ſameproportion: as alſo, the parts cut off, to the parts cut off.
Now the Lines parallel to the Baſes are drawn, as I ſuppoſe, by making a Rectilineall Figure (cal-
led) Signally inſcribed [χη̄μα γιωρίμως ἐγν̀<36>ρόμενον] in both portions, having an equall num
ber of Sides in both. Therefore, like Portions are cut off from like Sections of a Cone; and
their Diameters, whether they be perpendicular to their Baſes, or making equall Angles with their
Baſes, have the ſame proportion unto their Baſes.
K
* Upon prop.
3 lib. 2
Archim. Æqui
pond.
Archim. Æqui
pond.
Vide Archim, ante
prop. 2. lib. 2.
Æquipond.
prop. 2. lib. 2.
Æquipond.
L
Now the Section of the Cone A E I ſhall paſs thorow K.]
For, if it be poſſible, let it not paſs thorow K, but thorow ſome other Point of the Line D B, as
thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, whoſe
Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth
both A E and A I; A E in B, and A I in D; D B ſhall have to B V, the ſame proportion
For, if it be poſſible, let it not paſs thorow K, but thorow ſome other Point of the Line D B, as
thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, whoſe
Diameter is E Z, A E is drawn and prolonged; and D B parallel unto the Diameter, cutteth
both A E and A I; A E in B, and A I in D; D B ſhall have to B V, the ſame proportion