Archimedes, Natation of bodies, 1662

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Author: Archimedes
Title: Natation of bodies
Date: 1662

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1
ARCHIMEDES
HIS TRACT
De Incidentibus Humido,
OR OF THE
NATATION OF BODIES VPON,
OR SVBMERSION IN,
THE
WATER
OR OTHER LIQUIDS.
IN TWO BOOKS.
Tranſlated from the Original Greek,
Firſt into Latine, and afterwards into Italian, by NICOLO
TARTAGLIA, and by him familiarly demon­
ſtrated by way of Dialogue, with Richard Wentworth,
a Noble Engliſh Gentleman, and his Friend.
Together with the Learned Commentaries of Federico
Commandino, who hath Reſtored ſuch of the Demonſtrations
as, thorow the Injury of Time, were obliterated.
Now compared with the ORIGINAL, and Engliſhed
By THOMAS SALVSBVRY, Eſque
LONDON, Printed by W. Leybourn, 1662.
1
[Empty page]
1
ARCHIMEDES
HIS TRACT
De
INCIDENTIBUS HUMIDO,
OR OF
The Natation of Bodies upon, or Submerſion in,
the Water, or other Liquids.
1
BOOK I.
RICARDO.
Dear Companion, I have peruſed your Induſtrious Invention,
in which I find not any thing that will not certainly hold
true; but, truth is, there are many of your Concluſions
of which I underſtand uot the Cauſe, and therefore, if it
be not a trouble to you, I would deſire you to declare them
to me, for, indeed, nothing pleaſeth me, if the Cauſe
thereof be hid from me.
NICOLO. My obligations unto you are ſo many and
great, Honoured Campanion, that no requeſt of yours ought
to be troubleſome to me, and therefore tell me what thoſe Perticulars are of which
you know not the Cauſe, for I ſhall endeavour with the utmoſt of my power and
underſtanding to ſatisfie you in all your demands.
RIC. In the firſt Direction of the firſt Book of that your Induſtrious Invention
you conclude, That it is impoſſible that the Water ſhould wholly receive into it
any material Solid Body that is lighter than it ſeif (as to ſpeciæ) nay, you ſay, That
there will alwaies a part of the Body ſtay or remain above the Waters Surface
(that is uncovered by it;) and, That as the whole Solid Body put into the Water
is in proportion to that part of it that ſhall be immerged, or received, into the Wa­
ter, ſo ſhall the Gravity of the Water be to the Gravity (in ſpeciæ) of that ſame
material Body: And that thoſe Solid Bodies, that are by nature more Grave than the
Water, being put into the Water, ſhall preſently make the ſaid Water give place;
and, That they do not only wholly enter or ſubmerge in the ſame, but go continu­
ally deſcending untill they arrive at the Bottom; and, That they ſink to the Bot­
tom ſo much faſter, by how much they are more Grave than the Water. And,
again, That thoſe which are preciſely of the ſame Gravity with the Water, being
put into the ſame, are of neceſſity wholly received into, or immerged by it, but
yet retained in the Surface of the ſaid Water, and much leſs will the Water con­
ſent that it do deſcend to the Bottom: and, now, albeit that all theſe things are
manifeſt to Senſe and Experience, yet nevertheleſs would I be very glad, if it be
poſſible, that you would demonſtrate to me the moſt apt and proper Cauſe of
theſe Effects.
1
NIC. The Cauſe of all theſe Effects is aſſigned by Archimedes, the Siracuſan, in

that Book De Incidentibus (^{*}) Aquæ, by me publiſhed in Latine, and dedicated to
your ſelf, as I alſo ſaid in the beginning of that my Induſtrions Invention.
* Aquæ, tanſlated
by me Humido, as
the more Compre­
henſive word, for
his Doctrine holds
true in all Liquids
as well as in Wa­
ter, ſoil. in Wine,
Oyl, Milk, &c.
RIC. I have ſeen that ſame Archimedes, and have very well underſtood thoſe
two Books in which he treateth De Centro Gravitatis æquerepentibus, or of the
Center of Gravity in Figures plain, or parallel to the Horizon; and likewiſe thoſe
De Quadratura Parabolæ, or, of Squaring the Parabola; but ^{*}that in which he treat­
eth of Solids that Swim upon, or ſink in Liquids, is ſo obſcure, that, to ſpeak the
truth, there are many things in it which I do not underſtand, and therefore before

we proceed any farther, I ſhould take it for a favour if you would declare it to me
in your Vulgar Tongue, beginning with his firſt Suppoſition, which ſpeaketh in this
manner.
* He ſpeaks of but
one Book, Tartag­
lia having tranſla­
ted no more.
SVPPOSITION I.
It is ſuppoſed that the Liquid is of ſuch a nature, that
its parts being equi-jacent and contiguous, the leſs
preſſed are repulſed by the more preſſed. And
that each of its parts is preſſed or repulſed by the
Liquor that lyeth over it, perpendicularly, if the
Liquid be deſcending into any place, or preſſed any
whither by another.
NIC. Every Science, Art, or Doctrine (as you know, Honoured Companion,)
hath its firſt undemonſtrable Principles, by which (they being
granted or ſuppoſed) the ſaid Science is proved, maintained, or de­
monſtrated. And of theſe Principles, ſome are called Petitions,
and others Demands, or Suppoſitions. I ſay, therefore, that the Science or Doctrine
of thoſe Material Solids that Swim or Sink in Liquids, hath only two undemon­
ſtrable Suppoſitions, one of which is that above alledged, the which in compliance
with your deſire I have ſet down in our Vulgar Tongue.
RIC. Before you proceed any farther tell me, how we are to underſtand the
parts of a Liquid to be Equijacent.
NIC. When they are equidiſtant from the Center of the World, or of the
Earth (which is the ſame, although ^{*} ſome hold that the Centers of the Earth
and Worldare different.)
RIC. I underſtand you not unleſs you give me ſome Example thereof in
Figure.
* The Coperni­
cans.
NIC. To exemplifie this particular, Let us ſuppoſe a quantity of Liquor (as
for inſtance of Water) to be upon the Earth; then let us with the Imagination
cut the whole Earth together with that Water into two equal parts, in ſuch a
manner as that the ſaid Section may paſs ^{*} by the Center of the Earth: And let
us ſuppoſe that one part of the Superficies of that Section, as well of the Water
as of the Earth, be the Superficies A B, and that the Center of the Earth be the
point K. This being done, let us in our Imagination deſcribe a Circle upon the

ſaid Center K, of ſuch a bigneſs as that the Circumference may paſs by the Super­
ficies of the Section of the Water: Now let this Circumference be E F G: and
let many Lines be drawn from the point K to the ſaid Circumference, cutting the
ſame, as KE, KHO, KFQ KLP, KM. Now I ſay, that all theſe parts of
the ſaid Water, terminated in that Circumference, are Equijacent, as being all
1equidiſtant from the point K, the Center of the World, which parts are G M,
M L, L F, F H, H E.
* Or through.
RIC. I underſtand you very well, as to this particular: But tell me a little; he
ſaith that each of the parts of the Liquid is preſſed or repulſed by the Liquid that
is above it, according to the Perpendicular: I know not what that Liquid is that
lieth upon a part of another Perpendicularly.
NIC. Imagining a Line that cometh from the Center of the Earth penetrating
thorow ſome Water, each part of the Water that is in that Line he ſuppoſeth to
be preſſed or repulſed by the Water that lieth above it in that ſame Line, and that
that repulſe is made according to the ſame Line, (that is, directly towards the
Center of the World) which Line is called a Perpendicular; becauſe every
Right-Line that departeth from any point, and goeth directly towards the Worlds
Center is called a Perpendicular. And that you may the better underſtand me, let
1[Figure 1]
us imagine
the Line KHO,
and in that
let us imagine
ſeveral parts,
as ſuppoſe RS,
S T, T V, V H,
H O. I ſay,
that he ſup­
poſeth that
the part V H
is preſſed by
that placed a­
bove it, H O,
according to
the Line OK;
the which
O K, as hath been ſaid above, is called the Perpendicular paſſing thorow thoſe two
parts. In like manner, I ſay that the part T V is expulſed by the part V H, ac­
cording to the ſaid Line O K: and ſo the part S T to be preſſed by T V, according
to the ſaid Perpendicular O K, and R S by S T. And this you are to underſtand
in all the other Lines that were protracted from the ſaid Point K, penetrating the
ſaid Water, As for Example, in K G, K M, K L, K F, K E, and infinite others of the
like kind.
RIC. Indeed, Dear Companion, this your Explanation hath given megreat ſa­
tisfaction; for, in my Judgment, it ſeemeth that all the difficulty of this Suppoſition
conſiſts in theſe two particulars which you have declared to me.
NIC. It doth ſo; for having underſtood that the parts E H, H F, F L, L M, and
MG, determining in the Circumference of the ſaid Circle are equijacent, it is an
eaſie matter to underſtand the foreſaid Suppoſition in Order, which ſaith, That it is
ſuppoſed that the Liquid is of ſuch a nature, that the part thereof leſs preſſed or thrust is re­
pulſed by the more thruſt or preſſed. As for example, if the part E H were by chance
more thruſt, crowded, or preſſed from above downwards by the Liquid, or ſome
other matter that was over it, than the part H F, contiguous to it, it is ſuppoſed
that the ſaid part H F, leſs preſſed, would be repulſed by the ſaid part E H. And
thus we ought to underſtand of the other parts equijacent, in caſe that they be
contiguous, and not ſevered. That each of the parts thereof is preſſed and repul.
ſed by the Liquid that lieth over it Perpendicularly, is manifeſt by that which was
ſaid above, to wit, that it ſhould be repulſed, in caſe the Liquid be deſcending into
any place, and thruſt, or driven any whither by another.
RIC. I underſtand this Suppoſition very well, but yet me thinks that before
the Suppoſition, the Author ought to have defined thoſe two particulars, which
you firſt declared to me, that is, how we are to underſtand the parts of the Liquid
equijacent, and likewiſe the Perpendicular.
1
NIC. You ſay truth.
RIC. I have another queſtion to aske you, which is this, Why the Author
uſeth the word Liquid, or Humid, inſtead of Water.
NIC. It may be for two of theſe two Cauſes; the one is, that Water being the
principal of all Liquids, therefore ſaying Humidum he is to be underſtood to mean
the chief Liquid, that is Water: The other, becauſe that all the Propoſitions of
this Book of his, do not only hold true in Water, but alſo in every other Liquid,
as in Wine, Oyl, and the like: and therefore the Author might have uſed the word
Humidum, as being a word more general than Aqua.
RIC. This I underſtand, therefore let us come to the firſt Propoſition, which, as
you know, in the Original ſpeaks in this manner.
PROP. I. THEOR. I.
If any Superficies ſhall be cut by a Plane thorough any
Point, and the Section be alwaies the Circumference
of a Circle, whoſe Center is the ſaid Point: that Su­
perficies ſhall be Spherical.
Let any Superficies be cut at pleaſure by a Plane thorow the
Point K; and let the Section alwaies deſcribe the Circumfe­
rence of a Circle that hath for its Center the Point K: I ſay,
that that ſame Superficies is Sphærical. For were it poſſible that the
ſaid Superficies were not Sphærical, then all the Lines drawn
through the ſaid Point K unto that Superficies would not be equal,
Let therefore A and B be two
Points in the ſaid Superficies, ſo that
2[Figure 2]
drawing the two Lines K A and
K B, let them, if poſſible, be une­
qual: Then by theſe two Lines let
a Plane be drawn cutting the ſaid
Superficies, and let the Section in
the Superficies make the Line
D A B G: Now this Line D A B G
is, by our pre-ſuppoſal, a Circle, and
the Center thereof is the Point K, for ſuch the ſaid Superficies was
ſuppoſed to be. Therefore the two Lines K A and K B are equal:
But they were alſo ſuppoſed to be unequal; which is impoſſible:
It followeth therefore, of neceſſity, that the ſaid Superficies be
Sphærical, that is, the Superficies of a Sphære.
RIC. I underſtand you very well; now let us proceed to the ſecond Propoſition,
which, you know, runs thus.
1
PROP. II. THEOR. II.
The Superficies of every Liquid that is conſiſtant and
ſetled ſhall be of a Sphærical Figure, which Figure
ſhall have the ſame Center with the Earth.
Let us ſuppoſe a Liquid that is of ſuch a conſiſtance as that it
is not moved, and that its Superficies be cut by a Plane along
by the Center of the Earth, and let the Center of the Earth
be the Point K: and let the Section of the Superficies be the Line
A B G D. I ſay that the Line A B G D is the Circumference of a
3[Figure 3]
Circle, and that the Center
thereof is the Point K And
if it be poſſible that it may
not be the Circumference
of a Circle, the Right­

Lines drawn ^{*} by the Point
K to the ſaid Line A B G D
ſhall not be equal. There­
fore let a Right-Line be
taken greater than ſome of thoſe produced from the Point K unto
the ſaid Line A B G D, and leſſer than ſome other; and upon the
Point K let a Circle be deſcribed at the length of that Line,
Now the Circumference of this Circle ſhall fall part without the
ſaid Line A B G D, and part within: it having been preſuppoſed
that its Semidiameter is greater than ſome of thoſe Lines that may
be drawn from the ſaid Point K unto the ſaid Line A B G D, and
leſſer than ſome other. Let the Circumference of the deſcribed
Circle be R B G H, and from B to K draw the Right-Line B K: and
drawn alſo the two Lines K R, and K E L which make a Right­
Angle in the Point K: and upon the Center K deſcribe the Circum­
ference X O P in the Plane and in the Liquid. The parts, there­
fore, of the Liquid that are ^{*} according to the Circumference

X O P, for the reaſons alledged upon the firſt Suppoſition, are equi­
jacent, or equipoſited, and contiguous to each other; and both
theſe parts are preſt or thruſt, according to the ſecond part of the
Suppoſition, by the Liquor which is above them. And becauſe the
two Angles E K B and B K R are ſuppoſed equal [by the 26. of 3.
of Euclid,] the two Circumferences or Arches B E and B R ſhall
be equal (foraſmuch as R B G H was a Circle deſcribed for ſatis­
faction of the Oponent, and K its Center:) And in like manner
the whole Triangle B E K ſhall be equal to the whole Triangle
B R K. And becauſe alſo the Triangle O P K for the ſame reaſon
1ſhall be equal to the Triangle O X K; Therefore (by common
Notion) ſubſtracting thoſe two ſmall Triangles O P K and O X K
from the two others B E K and B R K, the two Remainders ſhall
be equal: one of which Remainders ſhall be the Quadrangle
B E O P, and the other B R X O. And becauſe the whole Quadran­
gle B E O P is full of Liquor, and of the Quadrangle B R X O,
the part B A X O only is full, and the reſidue B R A is wholly void
of Water: It followeth, therefore, that the Quadrangle B E O P
is more ponderous than the Quadrangle B R X O. And if the ſaid
Quadrangle B E O P be more Grave than the Quadrangle
B R X O, much more ſhall the Quadrangle B L O P exceed in Gra­
vity the ſaid Quadrangle B R X O: whence it followeth, that the
part O P is more preſſed than the part O X. But, by the firſt part
of the Suppoſition, the part leſs preſſed ſhould be repulſed by the
part more preſſed: Therefore the part O X muſt be repulſed by
the part O P: But it was preſuppoſed that the Liquid did not
move: Wherefore it would follow that the leſs preſſed would not
be repulſed by the more preſſed: And therefore it followeth of
neceſſity that the Line A B G D is the Circumference of a Circle,
and that the Center of it is the point K. And in like manner ſhall
it be demonſtrated, if the Surface of the Liquid be cut by a Plane
thorow the Center of the Earth, that the Section ſhall be the Cir­
cumference of a Circle, and that the Center of the ſame ſhall be
that very Point which is Center of the Earth. It is therefore mani­
feſt that the Superficies of a Liquid that is conſiſtant and ſetled
ſhall have the Figure of a Sphære, the Center of which ſhall be
the ſame with that of the Earth, by the firſt Propoſition; for it is
ſuch that being ever cut thorow the ſame Point, the Section or Di­
viſion deſcribes the Circumference of a Circle which hath for Cen­
ter the ſelf-ſame Point that is Center of the Earth: Which was to
be demonſtrated.
* O: through.
* i.e. Parallel.
RIC. I do thorowly underſtand theſe your Reaſons, and ſince there is in them
no umbrage of Doubting, let us proceed to his third Propoſition.
PROP. III. THEOR. III.
Solid Magnitudes that being of equal Maſs with the
Liquid are alſo equal to it in Gravity, being demit-

ted into the [^{*} ſetled] Liquid do ſo ſubmerge in the
ſame as that they lie or appear not at all above the
Surface of the Liquid, nor yet do they ſink to the
Bottom.
1
* I add the word
ſetled, as neceſſary
in making the Ex­
periment.
NIC. In this Propoſition it is affirmed that thoſe Solid Magnitules that hap­
pen to be equal in ſpecifical Gravity with the Liquid being lefeat liber­
ty in the ſaid Liquid do ſo ſubmerge in the ſame, as that they lie or ap­
pear not at all above the Surface of the Liquid, nor yet do they go or ſink to the
Bottom.
For ſuppoſing, on the contrary, that it were poſſible for one of
thoſe Solids being placed in the Liquid to lie in part without the
Liquid, that is above its Surface, (alwaies provided that the ſaid
Liquid be ſetled and undiſturbed,) let us imagine any Plane pro­
duced thorow the Center of the Earth, thorow the Liquid, and
thorow that Solid Body: and let us imagine that the Section of the
Liquid is the Superficies A B G D, and the Section of the Solid
Body that is within it the Superſicies E Z H T, and let us ſuppoſe
the Center of the Earth to be the Point K: and let the part of the
ſaid Solid ſubmerged in the Liquid be B G H T, and let that above
be B E Z G: and let the Solid Body be ſuppoſed to be comprized in
a Pyramid that hath its Parallelogram Baſe in the upper Surface of
the Liquid, and its Summity or Vertex in the Center of the Earth:
which Pyramid let us alſo ſuppoſe to be cut or divided by the ſame
Plane in which is the Circumference A B G D, and let the Sections
4[Figure 4]
of the Planes of the ſaid
Pyramid be K L and
K M: and in the Liquid
about the Center K let
there be deſcribed a Su­
perficies of another
Sphære below E Z H T,
which let be X O P;
and let this be cut by
the Superficies of the Plane: And let there be another Pyramid ta­
ken or ſuppoſed equal and like to that which compriſeth the ſaid
Solid Body, and contiguous and conjunct with the ſame; and let
the Sections of its Superficies be K M and K N: and let us ſuppoſe
another Solid to be taken or imagined, of Liquor, contained in that
ſame Pyramid, which let be R S C Y, equal and like to the partial
Solid B H G T, which is immerged in the ſaid Liquid: But the
part of the Liquid which in the firſt Pyramid is under the Super­
ficies X O, and that, which in the other Pyramid is under the Su­
perficies O P, are equijacent or equipoſited and contiguous, but
are not preſſed equally; for that which is under the Superficies
X O is preſſed by the Solid T H E Z, and by the Liquor that is
contained between the two Spherical Superficies X O and L M
and the Planes of the Pyramid, but that which proceeds accord­
ing to F O is preſſed by the Solid R S C Y, and by the Liquid
1contained between the Sphærical Superficies that proceed accord­
ing to P O and M N and the Planes of the Pyramid; and the Gra­
vity of the Liquid, which is according to M N O P, ſhall be leſſer
than that which is according to L M X O; becauſe that Solid of
Liquor which proceeds according to R S C Y is leſs than the Solid
E Z H T (having been ſuppoſed to be equal in quantity to only
the part H B G T of that:) And the ſaid Solid E Z H T hath been
ſuppoſed to be equally grave with the Liquid: Therefore the Gra­
vity of the Liquid compriſed betwixt the two Sphærical Superfi­
cies L M and X O, and betwixt the ſides L X and M O of the
5[Figure 5]
Pyramid, together with
the whole Solid EZHT,
ſhall exceed the Gravity
of the Liquid compri­
ſed betwixt the other
two Sphærical Superfi­
cies M N and O P, and
the Sides M O and N P
of the Pyramid, toge­
ther with the Solid of Liquor R S C Y by the quantity of the Gra­
vity of the part E B Z G, ſuppoſed to remain above the Surface of
the Liquid: And therefore it is manifeſt that the part which pro­
ceedeth according to the Circumference O P is preſſed, driven, and
repulſed, according to the Suppoſition, by that which proceeds ac­
cording to the Circumference X O, by which means the Liquid
would not be ſetled and ſtill: But we did preſuppoſe that it was
ſetled, namely ſo, as to be without motion: It followeth, therefore,
that the ſaid Solid cannot in any part of it exceed or lie above the
Superficies of the Liquid: And alſo that being dimerged in the Li­
quid it cannot deſcend to the Bottom, for that all the parts of the
Liquid equijacent, or diſpoſed equally, are equally preſſed, becauſe
the Solid is equally grave with the Liquid, by what we preſuppoſed.
RIC. I do underſtand your Argumentation, but I underſtand not that Phraſe
Solid Magnitudes.
NIC. I will declare this Term unto you. Magnitude is a general Word that
reſpecteth all the Species of Continual Quantity; and the Species of Continual
Quantity are three, that is, the Line, the Superficies, and the Body; which Body
is alſo called a Solid, as having in it ſelf Length, Breadth, and Thickneſs, or Depth:
and therefore that none might equivocate or take that Term Magnitudes to be
meant of Lines, or Superficies, but only of Solid Magnitudes, that is, Bodies, he
did ſpecifie it by that manner of expreſſion, as was ſaid. The truth is, that he
might have expreſt that Propoſition in this manner: Solids (or Bodies) which being
of equal Gravity with an equal Maſs of the Liquid, &c. And this Propoſition would have
been more cleer and intelligible, for it is as ſignificant to ſay, a Solid, or, a Body, as
to ſay, a Solid Magnitude: therefore wonder not if for the future I uſe theſe three
kinds of words indifferently.
RIC. You have ſufficiently ſatisfied me, wherefore that we may loſe no time
let us go forwards to the fourth Propoſition.
1
PROP. IV. THEOR. IV.
Solid Magnitudes that are lighter than the Liquid,
being demitted into the ſetled Liquid, will not total­
ly ſubmerge in the ſame, but ſome part thereof will
lie or ſtay above the Surface of the Liquid.
NIC. In this fourth Propoſition it is concluded, that every Body or Solid that is
lighter (as to Specifical Gravity) than the Liquid, being put into the
Liquid, will not totally ſubmerge in the ſame, but that ſome part of it
will ſtay and appear without the Liquid, that is above its Surface.
For ſuppoſing, on the contrary, that it were poſſible for a Solid
more light than the Liquid, being demitted in the Liquid to ſub­
merge totally in the ſame, that is, ſo as that no part thereof re­
maineth above, or without the ſaid Liquid, (evermore ſuppoſing
that the Liquid be ſo conſtituted as that it be not moved,) let us
imagine any Plane produced thorow the Center of the Earth, tho­
row the Liquid, and thorow that Solid Body: and that the Surface
of the Liquid is cut by this Plane according to the Circumference
A B G, and the Solid Body according to the Figure R; and let the
Center of the Earth be K. And let there be imagined a Pyramid
6[Figure 6]
that compriſeth the Figure
R, as was done in the pre.
cedent, that hath its Ver­
tex in the Point K, and let
the Superficies of that
Pyramid be cut by the
Superficies of the Plane
A B G, according to A K
and K B. And let us ima­
gine another Pyramid equal and like to this, and let its Superficies
be cut by the Superficies A B G according to K B and K G; and let
the Superficies of another Sphære be deſcribed in the Liquid, upon
the Center K, and beneath the Solid R; and let that be cut by the
ſame Plane according to X O P. And, laſtly, let us ſuppoſe ano­
ther Solid taken ^{*} from the Liquid, in this ſecond Pyramid, which

let be H, equal to the Solid R. Now the parts of the Liquid, name­
ly, that which is under the Spherical Superficies that proceeds ac­
cording to the Superficies or Circumference X O, in the firſt Py­
ramid, and that which is under the Spherical Superficies that pro­
ceeds according to the Circumference O P, in the ſecond Pyramid,
are equijacent, and contiguous, but are not preſſed equally; for
1that of the firſt Pyramid is preſſed by the Solid R, and by the Liquid
which that containeth, that is, that which is in the place of the Py­
ramid according to A B O X: but that part which, in the other Py­
ramid, is preſſed by the Solid H, ſuppoſed to be of the ſame Li­
quid, and by the Liquid which that containeth, that is, that which
is in the place of the ſaid Pyramid according to P O B G: and the
Gravity of the Solid R is leſs than the Gravity of the Liquid
H, for that theſe two Magnitudes were ſuppoſed to be equal in
Maſs, and the Solid R was ſuppoſed to be lighter than the Liquid:
and the Maſſes of the two Pyramids of Liquor that containeth theſe

two Solids R and H are equal ^{*} by what was preſuppoſed: There­
fore the part of the Liquid that is under the Superficies that pro­
ceeds according to the Circumference O P is more preſſed; and,
therefore, by the Suppoſition, it ſhall repulſe that part which is leſs
preſſed, whereby the ſaid Liquid will not be ſetled: But it was be­
fore ſuppoſed that it was ſetled: Therefore that Solid R ſhall not
totally ſubmerge, but ſome part thereof will remain without the
Liquid, that is, above its Surface, Which was the Propoſition.
* That is a Maſs of
the Liquid.
* For that the Py­
ramids were ſuppo­
ſed equal.
RIC. I have very well underſtood you, therefore let us come to the fifth Pro­
poſition, which, as you know, doth thus ſpeak.
PROP. V. THEOR. V.
Solid Magnitudes that are lighter than the Liquid,
being demitted in the (ſetled) Liquid, will ſo far
ſubmerge, till that a Maſs of Liquor, equal to the
Part ſubmerged, doth in Gravity equalize the
whole Magnitude.
NIC. It having, in the precedent, been demonſtrared that Solids lighter than
the Liquid, being demitted in the Liquid, alwaies a part of them remains
without the Liquid, that is above its Surface; In this fifth Propoſition it is
aſſerted, that ſo much of ſuch a Solid ſhall ſubmerge, as that a Maſs of the
Liquid equal to the part ſubmerged, ſhall have equal Gravity with the whole
Solid.
And to demonſtrate this, let us aſſume all the ſame Schemes
as before, in Propoſition 3. and likewiſe let the Liquid be ſet­
led, and let the Solid E Z H T be lighter than the Liquid.
Now if the ſaid Liquid be ſetled, the parts of it that are equija­
cent are equally preſſed: Therefore the Liquid that is beneath
1the Superficies that proceed according to the Circumferences X O
and P O are equally preſſed; whereby the Gravity preſſed is equal.
7[Figure 7]
But the Gravity of the
Liquid which is in the

firſt Pyramid ^{*} without
the Solid B H T G, is
equal to the Gravity of
the Liquid which is in
the other Pyramid with­
out the Liquid R S C Y:
It is manifeſt, therefore,
that the Gravity of the Solid E Z H T, is equal to the Gravity of
the Liquid R S C Y: Therefore it is manifeſt that a Maſs of Liquor
equal in Maſs to the part of the Solid ſubmerged is equal in Gra­
vity to the whole Solid.
* Without, i.e. that
being deducted.
RIC. This was a pretty Demonſtration, and becauſe I very well underſtand
it, let us loſe no time, but proceed to the ſixth Propoſition, ſpeaking thus.
PROP. VI. THEOR. VI.
Solid Magnitudes lighter than the Liquid being thruſt
into the Liquid, are repulſed upwards with a Force
as great as is the exceſs of the Gravity of a Maſs
of Liquor equal to the Magnitude above the Gra­
vity of the ſaid Magnitude.
NIC. This ſixth Propoſition ſaith, that the Solids lighter than the Liquid
demitted, thruſt, or trodden by Force underneath the Liquids Sur­
face, are returned or driven upwards with ſo much Force, by
how much a quantity of the Liquid equal to the. Solid ſhall
exceed the ſaid Solid in Gravity.
And to delucidate this Propoſition, let the Solid A be lighter
than the Liquid, and let us ſuppoſe that the Gravity of the ſaid
Solid A is B: and let the Gravity of a Liquid, equal in Maſs to A,
be B G. I ſay, that the Solid A depreſſed or demitted with Force
into the ſaid Liquid, ſhall be returned and repulſed upwards with
a Force equal to the Gravity G. And to demonſtrate this Propo­
ſition, take the Solid D, equal in Gravity to the ſaid G. Now
the Solid compounded of the two Solids A and D will be lighter
than the Liquid: for the Gravity of the Solid compounded of
them both is BG, and the Gravity of as much Liquor as equal­
leth in greatneſs the Solid A, is greater than the ſaid Gravity BG,
1for that B G is the Gravity of the Liquid equal in Maſs unto it:
Therefore the Solid compounded of thoſe two Solids A and D
being dimerged, it ſhall, by the precedent, ſo much of it ſubmerge,
as that a quantity of the Liquid equal to the ſaid ſubmerged part
ſhall have equal Gravity with the ſaid compounded Solid. And
8[Figure 8]
for an example of that Propoſition let the Su­
perficies of any Liquid be that which pro­
ceedeth according to the Circumference
A B G D: Becauſe now a Maſs or quantity
of Liquor as big as the Maſs A hath equal
Gravity with the whole compounded Solid
A D: It is manifeſt that the ſubmerged part
thereof ſhall be the Maſs A: and the remain­
der, namely, the part D, ſhall be wholly a­
top, that is, above the Surface of the Liquid.
It is therefore evident, that the part A hath ſo much virtue or
Force to return upwards, that is, to riſe from below above the Li­
quid, as that which is upon it, to wit, the part D, hath to preſs it
downwards, for that neither part is repulſed by the other: But D
preſſeth downwards with a Gravity equal to G, it having been ſup­
poſed that the Gravity of that part D was equal to G: Therefore
that is manifeſt which was to be demonſtrated.
RIC. This was a fine Demonſtration, and from this I perceive that you colle­
cted your Induſtrious Invention; and eſpecially that part of it which you inſert in
the firſt Book for the recovering of a Ship ſunk: and, indeed, I have many Que­
ſtions to ask you about that, but I will not now interrupt the Diſcourſe in hand, but
deſire that we may go on to the ſeventh Propoſition, the purport whereof is this.
PROP. VII. THEOR. VII.
Solid Magnitudes beavier than the Liquid, being de­
mitted into the [ſetled] Liquid, are boren down­
wards as far as they can deſcend: and ſhall be lighter
in the Liquid by the Gravity of a Liquid Maſs of
the ſame bigneſs with the Solid Magnitude.
NIC. This ſeventh Propoſition hath two parts to be demonſtrated.
The firſt is, That all Solids heavier than the Liquid, being demit­
ted into the Liquid, are boren by their Gravities downwards as far
as they can deſcend, that is untill they arrive at the Bottom. Which
firſt part is manifeſt, becauſe the Parts of the Liquid, which ſtill lie
under that Solid, are more preſſed than the others equijacent,
becauſe that that Solid is ſuppoſed more grave than the Liquid.
1But now that that Solid is lighter in the Liquid than out of it, as
is affirmed in the ſecond part, ſhall be demonſtrated in this man­
ner. Take a Solid, as ſuppoſe A, that is more grave than the Li­
quid, and ſuppoſe the Gravity of that ſame Solid A to be BG.
And of a Maſs of Liquor of the ſame bigneſs with the Solid A, ſup­
poſe the Gravity to be B: It is to be demonſtrated that the Solid
A, immerged in the Liquid, ſhall have a Gravity equal to G. And
to demonſtrate this, let us imagine another Solid, as ſuppoſe D,
more light than the Liquid, but of ſuch a quality as that its Gravi­
ty is equal to B: and let this D be of ſuch a Magnitude, that a
Maſs of Liquor equal to it hath its Gravity equal to the Gravity
B G. Now theſe two Solids D and A being compounded toge­
ther, all that Solid compounded of theſe two ſhall be equally
Grave with the Water: becauſe the Gravity of theſe two Solids
together ſhall be equal to theſe two Gravities, that is, to B G, and
9[Figure 9]
to B; and the Gravity of a Liquid that hath its
Maſs equal to theſe two Solids A and D, ſhall be
equal to theſe two Gravities B G and B. Let
theſe two Solids, therefore, be put in the Liquid,

and they ſhall ^{*} remain in the Surface of that Li­
quid, (that is, they ſhall not be drawn or driven
upwards, nor yet downwards:) For if the Solid
A be more grave than the Liquid, it ſhall be
drawn or born by its Gravity downwards to­
wards the Bottom, with as much Force as by the Solid D it is thruſt
upwards: And becauſe the Solid D is lighter than the Liquid, it
ſhall raiſe it upward with a Force as great as the Gravity G: Be­
cauſe it hath been demonſtrated, in the ſixth Propoſition, That So­
lid Magnitudes that are lighter than the Water, being demitted in
the ſame, are repulſed or driven upwards with a Force ſo much the
greater by how much a Liquid of equal Maſs with the Solid is more
Grave than the ſaid Solid: But the Liquid which is equal in Maſs
with the Solid D, is more grave than the ſaid Solid D, by the Gra­
vity G: Therefore it is manifeſt, that the Solid A is preſſed or
born downwards towards the Centre of the World, with a Force
as great as the Gravity G: Which was to be demonſtrated.
* Or, according to
Commandine, ſhall
be equall in Gravi­
ty to the Liquid,
neither moving up­
wards or down­
wards.
RIC. This hath been an ingenuous Demonſtration; and in regard I do ſuffici­
ently underſtand it, that we may loſe no time, we will proceed to the ſecond Suppo­
ſition, which, as I need not tell you, ſpeaks thus.
1
SVPPOSITION II.
It is ſuppoſed that thoſe Solids which are moved up­
wards, do all aſcend according to the Perpendicular
which is produced thorow their Centre of Gravity.
COMMANDINE.
And thoſe which are moved downwards, deſcend, likewiſe, according to the Perpendicular
that is produced thorow their Centre of Gravity, which he pretermitted either as known,
or as to be collected from what went before.
NIC. For underſtanding of this ſecond Suppoſition, it is requiſite to take notice
that every Solid that is lighter than the Liquid being by violence, or by ſome other
occaſion, ſubmerged in the Liquid, and then left at liberty, it ſhall, by that which
hath been proved in the ſixth Propoſition, be thruſt or born up wards by the Liquid,
and that impulſe or thruſting is ſuppoſed to be directly according to the Perpendi­
cular that is produced thorow the Centre of Gravity of that Solid; which Per­
pendicular, if you well remember, is that which is drawn in the Imagination
from the Centre of the World, or of the Earth, unto the Centre of Gravity of
that Body, or Solid.
RIC. How may one find the Centre of Gravity of a Solid?
NIC. This he ſheweth in that Book, intituled De Centris Gravium, vel de Æqui­
ponderantibus; and therefore repair thither and you ſhall be ſatisfied, for to declare
it to you in this place would cauſe very great confuſion.
RIC. I underſtand you: ſome other time we will talk of this, becauſe I have
a mind at preſent to proceed to the laſt Propoſition, the Expoſition of which ſeemeth
to me very confuſed, and, as I conceive, the Author hath not therein ſhewn all
the Subject of that Propoſition in general, but only a part: which Propoſition
ſpeaketh, as you know, in this form.
PROP. VIII. THEOR. VIII.
A
If any Solid Magnitude, lighter than the Liquid, that
hath the Figure of a Portion of a Sphære, ſhall be

demitted into the Liquid in ſuch a manner as that
the Baſe of the Portion touch not the Liquid, the
Figure ſhall ſtand erectly, ſo, as that the Axis of
the ſaid Portion ſhall be according to the Perpen­
dicular. And if the Figure ſhall be inclined to any
ſide, ſo, as that the Baſe of the Portion touch the
Liquid, it ſhall not continue ſo inclined as it was de­
mitted, but ſhall return to its uprightneſs.
1
B
For the declaration of this Propoſition, let a Solid Magnitude
that hath the Figure of a portion of a Sphære, as hath been ſaid,
be imagined to be de­
10[Figure 10]
mitted into the Liquid; and
alſo, let a Plain be ſuppoſed
to be produced thorow the
Axis of that portion, and
thorow the Center of the
Earth: and let the Section
of the Surface of the Liquid
be the Circumference A B
C D, and of the Figure, the
Circumference E F H, & let
E H be a right line, and F T
the Axis of the Portion. If now
it were poſſible, for ſatisfact­
ion of the Adverſary, Let
it be ſuppoſed that the ſaid Axis were not according to the (a) Per­

pendicular; we are then to demonſtrate, that the Figure will not
continue as it was conſtituted by the Adverſary, but that it will re­
turn, as hath been ſaid, unto its former poſition, that is, that the
Axis F T ſhall be according to the Perpendicular. It is manifeſt, by
the Corollary of the 1. of 3. Euclide, that the Center of the Sphære
is in the Line F T, foraſmuch as that is the Axis of that Figure.
And in regard that the Por­
11[Figure 11]
tion of a Sphære, may be
greater or leſſer than an He­
miſphære, and may alſo be
an Hemiſphære, let the Cen­
tre of the Sphære, in the He­
miſphære, be the Point T,
and in the leſſer Portion the
Point P, and in the greater,
the Point K, and let the Cen­
tre of the Earth be the Point
L. And ſpeaking, firſt, of
that greater Portion which
hath its Baſe out of, or a­
bove, the Liquid, thorew the Points K and L, draw the Line KL
cutting the Circumference E F H in the Point N, Now, becauſe

every Portion of a Sphære, hath its Axis in the Line, that from the
Centre of the Sphære is drawn perpendicular unto its Baſe, and hath
its Centre of Gravity in the Axis; therefore that Portion of the Fi­
gure which is within the Liquid, which is compounded of two
1tions of a Sphære, ſhall have its Axis in the Perpendicular, that is
drawn through the point K; and its Centre of Gravity, for the ſame
reaſon, ſhall be in the Line N K: let us ſuppoſe it to be the Point R:

But the Centre of Gravity of the whole Portion is in the Line F T,
betwixt the Point R and
12[Figure 12]
the Point F; let us ſuppoſe
it to be the Point X: The re­
mainder, therefore, of that

Figure elivated above the
Surface of the Liquid, hath
its Centre of Gravity in
the Line R X produced or
continued right out in the
Part towards X, taken ſo,
that the part prolonged may
have the ſame proportion to
X R, that the Gravity of
that Portion that is demer­
ged in the Liquid hath to
the Gravity of that Figure which is above the Liquid; let us ſuppoſe

that ^{*} that Centre of the ſaid Figure be the Point S: and thorow that

ſame Centre S draw the Perpendicular L S. Now the Gravity of the Fi­
gure that is above the Liquid ſhall preſſe from above downwards ac­
cording to the Perpendicular S L; & the Gravity of the Portion that
is ſubmerged in the Liquid, ſhall preſſe from below upwards, accor­
ding to the Perpendicular R L. Therefore that Figure will not conti­
nue according to our Adverſaries Propoſall, but thoſe parts of the
ſaid Figure which are towards E, ſhall be born or drawn downwards,
& thoſe which are towards H ſhall be born or driven upwards, and
this ſhall be ſo long untill that the Axis F T comes to be according
to the Perpendicular.
(a) Perpendicular
is taken kere, as
in all other places,
by this Author for
the Line K L
drawn thorow the
Centre and Cir­
cumference of the
Earth.
C
D
E
* i. e, The Center
of Gravity.
F
And this ſame Demonſtration is in the ſame manner verified in
the other Portions. As, firſt, in the Hæmiſphere that lieth with its
whole Baſe above or without the Liquid, the Centre of the Sphære
hath been ſuppoſed to be the Point T; and therefore, imagining T
to be in the place, in which, in the other above mentioned, the
Point R was, arguing in all things elſe as you did in that, you ſhall
find that the Figure which is above the Liquid ſhall preſs from
above downwards according to the Perpendicular S L; and the
Portion that is ſubmerged in the Liquid ſhall preſs from below up­
wards according to the Perpendicular R L. And therefore it ſhall
follow, as in the other, namely, that the parts of the whole Figure
which are towards E, ſhall be born or preſſed downwards, and thoſe

that are towards H, ſhall be born or driven upwards: and this ſhall
be ſo long untill that the Axis F T come to ſtand ^{*} P
1ly. The like ſhall alſo hold true in the Portion of the Sphære
leſs than an Hemiſphere that lieth with its whole Baſe above the
Liquid.
* Or according
to the Perpendi­
cular.
COMMANDINE.
The Demonſtration of this Propoſition is defaced by the Injury of Time, which we have re­
ſtored, ſo far as by the Figures that remain, one may collect the Meaning of Archimedes,
for we thought it not good to alter them: and what was wanting to their declaration and ex­
planation we have ſupplyed in our Commentaries, as we have alſo determined to do in the ſe­
cond Propoſition of the ſecond Book.
If any Solid Magnitude lighter than the Liquid.] Theſe words, light-

er than the Liquid, are added by us, and are not to be found in the Tranſiation; for of theſe
kind of Magnitudes doth Archimedes ſpeak in this Propoſition.
A
Shall be demitted into the Liquid in ſuch a manner as that the

Baſe of the Portion touch not the Liquid.] That is, ſhall be ſo demitted into
the Liquid as that the Baſe ſhall be upwards, and the Vertex downwards, which he oppoſeth
to that which he ſaith in the Propoſition following; Be demitted into the Liquid, ſo, as
that its Baſe be wholly within the Liquid; For theſe words ſignifie the Portion demit­
ted the contrary way, as namely, with the Vertex upwards and the Baſe downwards. The
ſame manner of ſpeech is frequently uſed in the ſecond Book; which treateth of the Portions
of Rectangle Conoids.
B
Now becauſe every Portion of a Sphære hath its Axis in the Line

that from the Center of the Sphære is drawn perpendicular to its
Baſe.] For draw a Line from B to C, and let K L cut the Circumference A B C D in the
Point G, and the Right Line B C in M:
13[Figure 13]
and becauſe the two Circles A B C D, and
E F H do cut one another in the Points
B and C, the Right Line that conjoyneth
their Centers, namely, K L, doth cut the
Line B C in two equall parts, and at
Right Angles; as in our Commentaries
upon Prolomeys Planiſphære we do
prove: But of the Portion of the Circle
B N C the Diameter is M N; and of the
Portion B G C the Diameter is M G;

for the (a) Right Lines which are drawn
on both ſides parallel to B C do make

Right Angles with N G; and (b) for
that cauſe are thereby cut in two equall
parts: Therefore the Axis of the Portion
of the Sphære B N C is N M; and the
Axis of the Portion B G C is M G:
from whence it followeth that the Axis of
the Portion demerged in the Liquid is
in the Line K L, namely N G. And ſince the Center of Gravity of any Portion of a Sphære is
in the Axis, as we have demonstrated in our Book De Centro Gravitatis Solidorum, the
Centre of Gravity of the Magnitude compounded of both the Portions B N C & B G C, that is,
of the Portion demerged in the Water, is in the Line N G that doth conjoyn the Centers of Gra­
vity of thoſe Portions of Sphæres. For ſuppoſe, if poſſible, that it be out of the Line N G, as
in Q, and let the Center of the Gravity of the Portion B N C, be V, and draw V que Becauſe
therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha­
ving the ſame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the
Portion B G C ſhall, by the 8 of the firſt Book of Archimedes, De Centro Gravitatis
1Planotum, be in the Line V Q prolonged: But that is impoſſible; for it is in the Axis
G: It followeth, therefore, that the Center of Gravity of the Portion demerged in
Liquid be in the Line N K: which we propounded to be proved.
C
(a) By 29. of the
firſt of Encl.
(b) By 3. of the
third.
But the Centre of Gravity of the whole Portion is in the Line

T, betwixt the Point R and the Point F; let us ſuppoſe it to be
the Point X.] Let the Sphære becompleated, ſo as that there be added of that Portion
the Axis T Y, and the Center of Gravity Z. And becauſe that from the whole Sphære,
whoſe Centre of Gravity is K, as we have alſo demonſtrated in the (c) Book before named, the
is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind

of the Portion E F H ſhall be in the Line Z K prolonged: And therefore it muſt of neceſſity
fall betwixt K and F.
D
(c) By 8 of the
firſt of Archimedes.
E
The remainder, therefore, of the Figure, elevated above the Sur­
face of the Liquid, hath its Center of Gravity in the Line R X
prolonged.] By the ſame 8 of the firſt Book of Archimedes, de Centro Gravita­
tis Planorum.
Now the Gravity of the Figure that is above the Liquid ſhall
preſs from above downwards according to S L; and the Gravit
of the Portion that is ſubmerged in the Liquid ſhall preſs from be
low upwards, according to the Perpendicular R L.] By the ſecond Sup­
poſition of this. For the Magnitude that is demerged in the Liquid is moved upwards with as
much Force along R L, as that which is above the Liquid is moved downwards along S L; as
may be ſhewn by Propoſition 6. of this. And becauſe they are moved along ſeverall other Lines,
neither cauſeth the others being leſs moved; the which it continually doth when the Portion
is ſet according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes
do concur in one and the ſame Perpendicular, namely, in the Axis of the Portion: and look
with what force or Impetus that which is in the Lipuid tendeth upwards, and with the like
doth that which is above or without the Liquid tend downwards along the ſame Line: And

therefore, in regard that the one doth not ^{*} exceed the other, the Portion ſhall no longer move
but ſhall ſtay and reſt allwayes in one and the ſame Poſition, unleſs ſome extrinſick Cauſe
chance to intervene.
F
* Or overcome.
PROP. IX. THEOR. IX.
* In ſome Greek
Coppies this is no
diſtinct Propoſi­
tion, but all
Commentators,
do divide it
from the Prece­
dent, as having a
diſtinct demon­
ſtration in the
Originall.
^{*} But if the Figure, lighter than the Liquid, be demit­
ted into the Liquid, ſo, as that its Baſe be wholly
within the ſaid Liquid, it ſhall continue in ſuch
manner erect, as that its Axis ſhall ſtand according
to the Perpendicular.
For ſuppoſe, ſuch a Magnitude as that aforenamed to be de
mitted into the Liquid; and imagine a Plane to be produced
thorow the Axis of the Portion, and thorow the Center of the
Earth: And let the Section of the Surface of the Liquid, be the Cir­
cumference A B C D, and of the Figure the Circumference E F H
And let E H be a Right Line, and F T the Axis of the Portion. If
now it were poſſible, for ſatisfaction of the Adverſary, let it be
ſuppoſed that the ſaid Axis were not according to the Perpendicu­
lar: we are now to demonſtrate that the Figure will not ſo
1nue, but will return to be according to the
14[Figure 14]
Perpendieular. It is manifeſt that the Gen­
tre of the Sphære is in the Line F T. And
again, foraſmuch as the Portion of a Sphære
may be greater or leſſer than an Hemiſ­
phære, and may alſo be an Hemiſphære, let
the Centre of the Sphære in the Hemiſ­
phære be the Point T, & in the leſſer Por­
tion the Point P, and in the Greater the

Point R. And ſpeaking firſt of that greater
Portion which hath its Baſe within the
Liquid, thorow R and L, the Earths Cen­
15[Figure 15]
tre, draw the line RL. The Portion that is
above the Liquid, hath its Axis in the Per­
pendicular paſſing thorow R; and by
what hath been ſaid before, its Centre of
Gravity ſhall be in the Line N R; let it
be the Point R: But the Centre of Gra­
vity of the whole Portion is in the line F
T, betwixt R and F; let it be X: The re­
mainder therefore of that Figure, which is
within the Liquid ſhall have its Centre in
the Right Line R X prolonged in the part
16[Figure 16]
towards X, taken ſo, that the part pro­
longed may have the ſame Proportion to
X R, that the Gravity of the Portion that
is above the Liquid hath to the Gravity
of the Figure that is within the Liquid.
Let O be the Centre of that ſame Figure:
and thorow O draw the Perpendicular L
O. Now the Gravity of the Portion that
is above the Liquid ſhall preſs according
to the Right Line R L downwards; and
the Gravity of the Figure that is in the
Liquid according to the Right Line O L upwards: There the Figure
ſhall not continue; but the parts of it towards H ſhall move down­
wards, and thoſe towards E upwards: &
17[Figure 17]
this ſhall ever be, ſo long as F T is accord­
ing to the Perpendicular.
A
COMMANDINE.
The Portion that is above the Liquid

hath its Axis in the Perpendicular paſſing
thorow K.] For draw B C cutting the Line N K in
M; and let N K out the Circumference A B C D in G. In
the ſame manner as before me will demonſtrate, that the Axis
1of the Portion of the Sphære is N M; and of the Portion B G C the Axis is G M: Wherefore
the Centre of Gravity of them both ſhall be in the Line N M: And becauſe that from the Por­
tion B N C the Portion B G C, not having the ſame Centre of Gravity, is cut off, the Centre
of Gravity of the remainder of the Magnitude that is above the Surface of the Liquid ſhall be
in the Line N K; namely, in the Line which conjoyneth the Centres of Gravity of the ſaid
Portions by the foreſaid 8 of Archimedis de Centro Gravitatis Planorum.
A
NIC. Truth is, that in ſome of theſe Figures C is put for X, and ſo it was in
the Greek Copy that I followed.
RIC. This Demoſtration is very difficult, to my thinking; but I believe that
it is becauſe I have not in memory the Propoſitions of that Book entituled De Cen­
tris Gravium.
NIC. It is ſo.
RIC. We will take a more convenient time to diſcourſe of that, and now return

to ſpeak of the two laſt Propoſitions. And I ſay that the Figures incerted in the
demonſtration would in my opinion, have been better and more intelligble unto
me, drawing the Axis according to its proper Poſition; that is in the half Arch of
theſe Figures, and then, to ſecond the Objection of the Adverſary, to ſuppoſe
that the ſaid Figures ſtood ſomewhat Obliquely, to the end that the ſaid Axis, if it
were poſſible, did not ſtand according to the Perpendicular ſo often mentioned,
which doing, the Propoſition would be proved in the ſame manner as before:
and this way would be more naturall and clear.
A
B
NIC. You are in the right, but becauſe thus they were in the Greek Copy,
I thought not fit to alter them, although unto the better.
RIC. Companion, you have thorowly ſatisfied me in all that in the beginning
of our Diſcourſe I asked of you, to morrow, God permitting, we will treat of
ſome other ingenious Novelties.
THE TRANSLATOR.
I ſay that the Figures, &c. would have been more intelligible to

me, drawing the Axis Z T according to its proper Poſition, that
is in the half Arch of theſe Figures.] And in this conſideration I have followed
the Schemes of Commandine, who being the Reſtorer of the Demonſtrations of theſe two laſt
Propoſitions, hath well conſidered what Ricardo here propoſeth, and therefore hath drawn the
ſaid Axis (which in the Manuſcripts that he had by him is lettered F T, and not as in that of
Tartaylia Z T,) according to that its proper Poſition.
A
But becauſe thus they were in the Greek Copy, I thought not

fit to alter them although unto the better.] The Schemes of thoſe Manu-
18[Figure 18]
ſcripts that Tartaylia had ſeen were more imperfect then thoſe
in Commandines Copies; but for variety ſake, take here one
of Tartaylia, it being that of the Portion of a Sphære, equall
to an Hemiſphære, with its Axis oblique, and its Baſe dimitted
into the Liquid, and Lettered as in this Edition.
B
Now Courteous Readers, I hope that you may, amidſt the
great Obſcurity of the Originall in the Demonſtrations of theſe
two laſt Propoſitions, be able from the joynt light of theſe two Famous Commentators of our
more famous Author, to diſcern the truth of the Doctrine affirmed, namely, That Solids of the
Figure of Portions of Sphæres demitted into the Liquid with their Baſes upwards ſhall ſtand
erectly, that is, with their Axis according to the Perpendicular drawn from the Centre of the
Earth unto its Circumference: And that if the ſaid Portions be demitted with their Baſes
oblique and touching the Liquid in one Point, they ſhall not rest in that Obliquity, but ſhall
return to Rectitude: And that laſtly, if theſe Portions be demitted with their Baſes downwards,
they ſhall continue erect with their Axis according to the Perpendicular aforeſaid: ſo that no
more remains to be done, but that weſet before you the 2 Books of this our Admirable Author.
1
ARCHIMEDES,
HIS TRACT
DE
INSIDENTIBUS HUMIDO,
OR,
Of the NATATION of BODIES Upon, or
Submerſion In the WATER, or other LIQUIDS.
BOOK II.
PROP. I. THEOR. I.
If any Magnitude lighter than the Liquid be demitted
into the ſaid Liquid, it ſhall have the ſame proporti­
on in Gravity to a Liquid of equal Maſſe, that the
part of the Magnitude demerged hath unto the
whole Magnitude.
For let any Solid Magnitude, as for in­
ſtance F A, lighter than the Liquid, be de­
merged in the Liquid, which let be F A:
And let the part thereof immerged be A,
and the part above the Liquid F, It is to
be demonſtrated that the Magnitude F A
hath the ſame proportion in Gravity to a
Liquid of Equall Maſſe that A hath to F
A. Take any Liquid Magnitude, as ſup­
poſe N I, of equall Maſſe with F A; and let F be equall to N, and
A to I: and let the Gravity of the whole Magnitude F A be B, and
let that of the Magnitude N I be O,
and let that of I be R. Now the
19[Figure 19]
Magnitude F A hath the ſame pro­
portion unto N I that the Gravity B
hath to the Gravity O R: But for
aſmuch as the Magnitude F A demit­
ted into the Liquid is lighter than
the ſaid Liquid, it is manifeſt that a Maſſe of the Liquid, I, equall
to the part of the Magnitude demerged, A, hath equall Gravity

with the whole Magnitnde, F A: For this was (a) above demon­
ſtrated: But B is the Gravity of the Magnitude F A, and R of I:
1Therefore B and R are equall. And becauſe that of the Magni­
tude FA the Gravity is B: Therefore of the Liquid Body N I the
Gravity is O R. As F A is to N I, ſo is B to O R, or, ſo is R to
O R: But as R is to O R, ſo is I to N I, and A to F A: Therefore

I is to N I, as F A to N I: And as I to N I ſo is (b) A to F A.
Therefore F A is to N I, as A is to F A: Which was to be demon­
ſtrated.
(a) By 5. of the
firſt of this.
(b) By 11. of the
fifth of Eucl.
PROP. II. THEOR. II.
A
^{*} The Right Portion of a Right angled Conoide, when it
ſhall have its Axis leſſe than ſeſquialter ejus quæ ad
Axem (or of its Semi-parameter) having any what
ever proportion to the Liquid in Gravity, being de­
mitted into the Liquid ſo as that its Baſe touch not
the ſaid Liquid, and being ſet ſtooping, it ſhall not
remain ſtooping, but ſhall be restored to uprightneſſe.
I ſay that the ſaid Portion ſhall ſtand upright when
the Plane that cuts it ſhall be parallel unto the Sur­
face of the Liquid.
Let there be a Portion of a Rightangled Conoid, as hath been
ſaid; and let it lye ſtooping or inclining: It is to be demon­
ſtrated that it will not ſo continue but ſhall be reſtored to re­
ctitude. For let it be cut through the Axis by a plane erect upon
the Surface of the Liquid, and let the Section of the Portion be
A PO L, the Section of a Rightangled Cone, and let the Axis
20[Figure 20]
of the Portion and Diameter of the
Section be N O: And let the Sect­
ion of the Surface of the Liquid be
I S. If now the Portion be not
erect, then neither ſhall A L be Pa­
rallel to I S: Wherefore N O will
not be at Right Angles with I S.

Draw therefore K ω, touching the Section of the Cone I, in the
Point P [that is parallel to I S: and from the Point P unto I S

draw P F parallel unto O N, ^{*} which ſhall be the Diameter of the
Section I P O S, and the Axis of the Portion demerged in the Li­

quid. In the next place take the Centres of Gravity: ^{*} and of
the Solid Magnitude A P O L, let the Centre of Gravity be R; and

of I P O S let the Centre be B: ^{*} and draw a Line from B to R
prolonged unto G; which let be the Centre of Gravity of the
1remaining Figure I S L A. Becauſe now that N O is Seſquialter
of R O, but leſs than Seſquialter ejus quæ uſque ad Axem (or of its
Semi-parameter;) ^{*} R O ſhall be leſſe than quæ uſque ad Axem (or

than the Semi-parameter;) ^{*} whereupon the Angle R P ω ſhall be

acute. For ſince the Line quæ uſque ad Axem (or Semi-parameter)
is greater than R O, that Line which is drawn from the Point R,
and perpendicular to K ω, namely RT, meeteth with the line F P
without the Section, and for that cauſe muſt of neceſſity fall be­
tween the Points P and ω; Therefore if Lines be drawn through
B and G, parallel unto R T, they ſhall contain Right Angles with
the Surface of the Liquid: ^{*} and the part that is within the Li­

quid ſhall move upwards according to the Perpendicular that is
drawn thorow B, parallel to R T, and the part that is above the Li­
quid ſhall move downwards according to that which is drawn tho­
row G; and the Solid A P O L ſhall not abide in this Poſition; for
that the parts towards A will move upwards, and thoſe towards
B downwards; Wherefore N O ſhall be conſtituted according to
the Perpendicular.]
* Supplied by Fe­
derico Comman­
dino.
B
C
D
E
F
G
COMMANDINE.
The Demonſtration of this propoſition hath been much deſired; which we have (in like man­
ner as the 8 Prop. of the firſt Book) reſtored according to Archimedes his own Schemes, and
illustrated it with Commentaries.
The Right Portion of a Rightangled Conoid, when it ſhall

have its Axis leſſe than Seſquialter ejus quæ uſque ad Axem (or of
its Semi-parameter] In the Tranſlation of Nicolo Tartaglia it is falſlyread great­
er then Seſquialter, and ſo its rendered in the following Propoſition; but it is the Right
Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay
that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the
Baſe, ſhall be parallel to the Surface of the Liquid.
A
Which ſhall be the Diameter of the Section I P O S, and the

Axis of the Portion demerged in the Liquid.] By the 46 of the firſt of
the Conicks of Apollonious, or by the Corol­
lary of the 51 of the ſame.
B
21[Figure 21]
And of the Solid Magnitude A P

O L, let the Centre of Gravity be R;
and of I P O S let the Centre be B.]
For the Centre of Gravity of the Portion of a Right­
angled Conoid is in its Axis, which it ſo divideth
as that the part thereof terminating in the vertex,
be double to the other part terminating in the Baſe; as
in our Book De Centro Gravitatis Solidorum Propo. 29. we have demonſtrated. And
ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there­
fore N O ſhall be Seſquialter of O R. And for the ſame reaſon, B the Centre of Gravity of the Por­
tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;
C
And draw a Line from B to R prolonged unto G; which let

be the Centre of Gravity of the remaining Eigure I S L A.]
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.
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.
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
22[Figure 22]
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.
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

our firſt Book of Conick Sections, (a) and G
23[Figure 23]
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
124[Figure 24]
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.
(b) By 17. of the
6.
(c) By 14. of the
6.
(d) By 33. of the
1.
(e) By 4. of the 1.
(f) By 28. of the
1.
(g) By 29. of th
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.
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
(a) Rîvalt. in Ar­
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.
Let any Portion be demitted into the Liquid, as was ſaid; and
let its Baſe be in the Liquid;
25[Figure 25]
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
1Section of the Portion be A P O L, the Section of a Rightangled
Cone; and let the Axis of the Portion and Diameter of the Section
be N O, and the Section of the Surface of the Liquid I S. If now
the Portion be not erect, then N O ſhall not be at equall Angles with
I S. Draw R ω touching the Section of the Rightangled Conoid
in P, and parallel to I S: and from the Point P and parall to O N
draw P F: and take the Centers of Gravity; and of the Solid A
P O L let the Centre be R; and of that which lyeth within the
Liquid let the Centre be B; and draw a Line from B to R pro­
longing it to G, that G may be the Centre of Gravity of the Solid
that is above the Liquid. And becauſe N O is ſeſquialter of R
O, and is greater than ſeſquialter of the Semi-Parameter; it is ma­

nifeſt that (a) R O is greater than the
26[Figure 26]
Semi-parameter. ^{*}Let therefore R

H be equall to the Semi-Parameter,

^{*} and O H double to H M. Foraſ­
much therefore as N O is ſeſquialter

of R O, and M O of O H, (b) the
Remainder N M ſhall be ſeſquialter
of the Remainder R H: Therefore
the Axis is greater than ſeſquialter
of the Semi parameter by the quan­
tity of the Line M O. And let it be
ſuppoſed that the Portion hath not leſſe proportion in Gravity unto
the Liquid of equall Maſſe, than the Square that is made of the
Exceſſe by which the Axis is greater than ſeſquialter of the Semi­
parameter hath to the Square made of the Axis: It is therefore ma­
nifeſt that the Portion hath not leſſe proportion in Gravity to the
Liquid than the Square of the Line M O hath to the Square of N
O: But look what proportion the Portion hath to the Liquid in
Gravity, the ſame hath the Portion ſubmerged to the whole Solid:
for this hath been demonſtrated (c) above: ^{*}And look what pro­

portion the ſubmerged Portion hath to the whole Portion, the

ſame hath the Square of P F unto the Square of N O: For it hath
been demonſtrated in (d) Lib. de Conoidibus, that if from a Right­

angled Conoid two Portions be cut by Planes in any faſhion pro­
duced, theſe Portions ſhall have the ſame Proportion to each
other as the Squares of their Axes: The Square of P F, therefore,
hath not leſſe proportion to the Square of N O than the Square of
M O hath to the Square of N O: ^{*}Wherefore P F is not leſſe than

M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn

from H at Right Angles unto N O, it ſhall meet with B P, and ſhall

fall betwixt B and P; let it fall in T: (e) And becauſe P F is

parallel to the Diameter, and H T is perpendicular unto the ſame
Diameter, and R H equall to the Semi-parameter; a Line drawn
from R to T and prolonged, maketh Right Angles with the Line
1contingent unto the Section in the Point P: Wherefore it alſo
maketh Right Angles with the Surface of the Liquid: and that
part of the Conoidall Solid which is within the Liquid ſhall move
upwards according to the Perpendicular drawn thorow B parallel
to R T; and that part which is above the Liquid ſhall move down­
wards according to that drawn thorow G, parallel to the ſaid R T:
And thus it ſhall continue to do ſo long untill that the Conoid be
reſtored to uprightneſſe, or to ſtand according to the Perpendicular.
(a) By 10. of the
fifth.
A
B
(b) By 19. of the
fifth.
C
(c) By 1. of this
ſecond Book.
(d) By 6. De Co­
noilibus & Sphæ­
roidibus of Archi­
medes.
D
E
F
(e) By 2. of this
ſecond Book.
COMMANDINE.
A
Let therefore R H be equall to the Semi-parameter.] So it is to be
read, and not R M, as Tartaglia's Tranſlation hath is; which may be made appear from
that which followeth.
B
And O H double to H M.] In the Tranſlation aforenamed it is falſly render­
ed, O N double to R M.
C
And look what proportion the Submerged Portion hath to the whole
Portion, the ſame hath the Square of P F unto the Square of N O.]
This place we have reſtored in our Tranſlation, at the requeſt of ſome friends: But it is demon­
ſtrated by Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. 26.
D
Wherefore P F is not leſſe than M O.] For by 10 of the fifth it followeth
that the Square of P F is not leſſe than the Square of M O: and therefore neither ſhall the
Line P F be leße than the Line M O, by 22 of the
27[Figure 27]

ſixth.
E
(a) By 14. of the
ſixth.
Nor B P than H O,] For as P F is to
P B, ſo is M O to H O: and, by Permutation, as

P F is to M O, ſo is B P to H O; But P F is not
leſſe than M O as hath bin proved; (a) Therefore
neither ſhall B P be leſſe than H O.
F
If therefore a Right Line be drawn
from H at Right Angles unto N O, it
ſhall meet with B P, and ſhall fall be­
twixt B and P.] This Place was corrupt in the
Tranſlation of Tartaglia: But it is thus demonstra­
ted. In regard that P F is not leſſe than O M, nor P B than O H, if we ſuppoſe P F equall to
O M, P B ſhall be likewiſe equall to O H: Wherefore the Line drawn thorow O, parallel to A L
ſhall fall without the Section, by 17 of the firſt of our Treatiſe of Conicks; And in regard that
B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth
alſo meet with the ſame beneath B, and it doth of neceſſity fall betwixt B and P: But the
ſame is much more to follow, if we ſuppoſe P F to be greater than O M.
1
PROP. V. THEOR. V.
The Right Portion of a Right-Angled Conoid lighter
than the Liquid, when it ſhall have its Axis great­
er than Seſquialter of the Semi-parameter, if it have
not greater proportion in Gravity to the Liquid [of
equal Maſs] than the Exceſſe by which the Square
made of the Axis is greater than the Square made
of the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-Parameter hath to the
Square made of the Axis being demitted into the Li­
quid, ſo as that its Baſe be wholly within the Liquid,
and being ſet inclining, it ſhall not remain ſo inclined,
but ſhall turn about till that its Axis ſhall be accor­
ding to the Perpendicular.
For let any Portion be demitted into the Liquid, as hath been
ſaid; and let its Baſe be wholly within the Liquid, And being
cut thorow its Axis by a Plain erect upon the Surface of the
Liquid; its Section ſhall be the Section
28[Figure 28]
of a Rightangled Cone: Let it be
A P O L, and let the Axis of the Por­
tion and Diameter of the Section be
N O; and the Section of the Surface of
the Liquid I S. And becauſe the Axis
is not according to the Perpendicu­
lar, N O will not be at equall angles
with I S. Draw K ω touching the Se­
ction A P O L in P, and parallel unto
I S: and thorow P, draw P F parallel unto N O: and take the
Centres of Gravity; and of the Solid A P O L let the Centre be
R; and of that which lyeth above the Liquid let the Centre be B;
and draw a Line from B to R, prolonging it to G; which let be the
Centre of Gravity of the Solid demerged within the Liquid: and
moreover, take R H equall to the Semi-parameter, and let O H be
double to H M; and do in the reſt as hath been ſaid (a) above.

Now foraſmuch as it was ſuppoſed that the Portion hath not greater
proportion in Gravity to the Liquid, than the Exceſſe by which
the Square N O is greater than the Square M O, hath to the ſaid
Square N O: And in regard that whatever proportion in Gravity
1the Portion hath to the Liquid of equall Maſſe, the ſame hath the
Magnitude of the Portion ſubmerged unto the whole Portion; as
hath been demonſtrated in the firſt Propoſition; The Magnitude
ſubmerged, therefore, ſhall not have greater proportion to the

whole (b) Portion, than that which hath been mentioned: ^{*}And
therefore the whole Portion hath not greater proportion unto that

which is above the Liquid, than the Square N O hath to the Square

M O: But the (c) whole Portion hath the ſame proportion unto
that which is above the Liquid that the Square N O hath to the
Square P F: Therefore the Square N O hath not greater propor­

tion unto the Square P F, than it hath unto the Square M O: ^{*}And
hence it followeth that P F is not leſſe than O M, nor P B than O

H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
ſhall meet with B P betwixt P and B: Let it be in T: And be­
cauſe that in the Section of the Rectangled Cone P F is parallel unto
the Diameter N O; and H T perpendicular unto the ſaid Diame­
ter; and R H equall to the Semi-parameter: It is manifeſt that
R T prolonged doth make Right Angles with K P ω: And there­
fore doth alſo make Right Angles with I S: Therefore R T is per­
pendicular unto the Surface of the Liquid; And if thorow the
Points B and G Lines be drawn parallel unto R T, they ſhall be
perpendicular unto the Liquids Surface. The Portion, therefore,
which is above the Liquid ſhall move downwards in the Liquid ac­
cording to the Perpendicular drawn thorow B; and that part
which is within the Liquid ſhall move upwards according to the
Perpendicular drawn thorow G; and the Solid Portion A P O L
ſhall not continue ſo inclined, [as it was at its demerſion], but ſhall
move within the Liquid untill ſuch time that N O do ſtand accor­
ding to the Perpendicular.
(a) In 4. Prop. of
this.
(a) By 11. of the
fifth.
A
(b) By 26. of the
Book De Conoid.
& Sphæroid.
B
C
COMMANDINE.
A
And therefore the whole Portion hath not greater proportion
unto that which is above the Liquid, than the Square N O hath to
the Square M O.] For in regard that the Magnitude of the Portion demerged
within the Liquid hath not greater proportion unto the whole Portion than the Exceſſe by which
the Square N O is greater than the Square M O hath to the ſaid Square N O; Converting of
the Proportion, by the 26. of the fifth of Euclid, of Campanus his Tranſlation, the whole
Portion ſhall not have leſſer proportion unto the Magnitude ſubmerged, than the Square N O
hath unto the Exceſſe by which N O is greater than the Square M O. Let a Portion be taken;
and let that part of it which is above the Liquid be the firſt Magnitude; the part of it which
is ſubmerged the ſecond: and let the third Magnitude be the Square M O; and let the Exceſſe
by which the Square N O is greater than the Square M O be the fourth. Now of theſe Mag­
nitudes, the proportion of the firſt and ſecond, unto the ſecond, is not leſſe than that of the third &
fourth unto the fourth: For the Square M O together with the Exceſſe by which the Square
N O exceedeth the Square M O is equall unto the ſaid Square N O: Wherefore, by Converſi­
on of Proportion, by 30 of the ſaid fifth Book, the proportion of the firſt and ſecond unto the
firſt, ſhall not be greater than that of the third and fourth unto the third: And, for the ſame
1the proportion of the whole Portion unto that part thereof which is above the Liquid ſhall not be
greater than that of the Square N O unto the Square M O: Which was to be demonſtrated.
And hence it followeth that P F is not leſſe than O M, nor P B

than O H.] This followeth by the 10 and 14 of the fifth, and by the 22 of the ſixth of
Euclid, as hath been ſaid above.
B
A Line, therefore, drawn from Hat Right Angles unto N O ſhall

meet with P B betwixt P and B.] Why this ſo falleth out, we will ſhew in the
next.
C
PROP. VI. THEOR. VI.
The Right Portion of a Rightangled Conoid lighter
than the Liquid, when it ſhall have its Axis greater
than ſeſquialter of the Semi-parameter, but leſſe than
to be unto the Semi-parameter in proportion as fifteen
to fower, being demitted into the Liquid ſo as that
its Baſe do touch the Liquid, it ſhall never stand ſo
enclined as that its Baſe toucheth the Liquid in one
Point only.
Let there be a Portion, as was ſaid; and demit it into the Li­
quid in ſuch faſhion as that its Baſe do touch the Liquid in
one only Point: It is to be demonſtrated that the ſaid Portion

ſhall not continue ſo, but ſhall turn about in ſuch manner as that
its Baſe do in no wiſe touch the Surface of the Liquid. For let it be
cut thorow its Axis by a Plane erect
29[Figure 29]
upon the Liquids Surface: and let
the Section of the Superficies of the
Portion be A P O L, the Section of
a Rightangled Cone; and the Sect­
ion of the Surface of the Liquid be
A S; and the Axis of the Portion
and Diameter of the Section N O:
and let it be cut in F, ſo as that O
F be double to F N; and in ω ſo, as that N O may be to F ω in the
ſame proportion as fifteen to four; and at Right Angles to N O
draw ω Now becauſe N O hath greater proportion unto F ω than
unto the Semi-parameter, let the Semi-parameter be equall to F B:

and draw P C parallel unto A S, and touching the Section A P O L
in P; and P I parallel unto N O; and firſt let P I cut Kω in H. For­

aſmuch, therefore, as in the Portion A P O L, contained betwixt
the Right Line and the Section of the Rightangled Cone, K ω is
parallel to A L, and P I parallel unto the Diameter, and cut by the
1ſaid K ω in H, and A S is parallel unto the Line that toucheth in
P; It is neceſſary that P I hath unto P H either the ſame proportion
that N ω hath to ω O, or greater; for this hath already been de­
monſtrated: But N ω is ſeſquialter of ω O; and P I, therefore, is
either Seſquialter of H P, or more than ſeſquialter: Wherefore

P H is to H I either double, or leſſe than double. Let P T be
double to T I: the Centre of Gravity of the part which is within
the Liquid ſhall be the Point T. Therefore draw a Line from T
to F prolonging it; and let the Centre of
30[Figure 30]
Gravity of the part which is above the Liquid
be G: and from the Point B at Right Angles
unto N O draw B R. And ſeeing that P I is
parallel unto the Diameter N O, and B R
perpendicular unto the ſaid Diameter, and F
B equall to the Semi-parameter; It is mani­
feſt that the Line drawn thorow the Points
F and R being prolonged, maketh equall
Angles with that which toucheth the Section
A P O L in the Point P: and therefore doth alſo make Right An­
gles with A S, and with the Surface of the Liquid: and the Lines
drawn thorow T and G parallel unto F R ſhall be alſo perpendicu­
lar to the Surface of the Liquid: and of the Solid Magnitude A P
O L, the part which is within the Liquid moveth upwards according
to the Perpendicular drawn thorow T; and the part which is above
the Liquid moveth downwards according to that drawn thorow G:

The Solid A P O L, therefore, ſhall turn about, and its Baſe ſhall
not in the leaſt touch the Surface of the Liquid, And if P I do not
cut the Line K ω, as in the ſecond Figure, it is manifeſt that the
Point T, which is the Centre of Gravity of the ſubmerged Portion,
falleth betwixt P and I: And for the other particulars remaining,
they are demonſtrated like as before.
A
B
C
D
E
COMMANDINE.
A
It is to be demonſtrated that the ſaid Portion ſhall not continue
ſo, but ſhall turn about in ſuch manner as that its Baſe do in no wiſe
touch the Surface of the Liquid.] Theſe words are added by us, as having been
omitted by Tartaglia.
Now becauſe N O hath greater proportion to F ω than unto

the Semi parameter.] For the Diameter of the Portion N O hath unto F ω the
ſame proportion as fifteen to fower: But it was ſuppoſed to have leſſe proportion unto the
Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F ω
than unto the Semi-parameter: And therefore (a) the Semi-parameter ſhall be greater

than the ſaid F ω.
B
(a) By 10. of the
fifth.
Foraſmuch, therefore, as in the Portion A P O L, contained, be­

twixt the Right Line and the Section of the Rightangled Cone K
ω is parallel to A L, and P I parallel unto the Diameter, and cut by
1the ſaid K ω in H, and A S is parallel unto the Line that toucheth
in P; It is neceſſary that P I hath unto P H either the ſame propor­
tion that N ω hath to ω O, or greater; for this hath already been
demonſtrated.] Where this is demonſtrated either by Archimedes himſelf, or by
any other, doth not appear; touching which we will here inſert a Demonſtration, after that
we have explained ſome things that pertaine thereto.
C
LEMMA I.
Let the Lines A B and A C contain the Angle B A C; and from
the point D, taken in the Line A C, draw D E and D F at
pleaſure unto A B: and in the ſame Line any Points G and L
being taken, draw G H & L M parallel to D E, & G K and
L N parallel unto F D: Then from the Points D & G as farre
as to the Line M L draw D O P, cutting G H in O, and G Q
parallel unto B A. I ſay that the Lines that lye betwixt the Pa­
rallels unto F D have unto thoſe that lye betwixt the Par­
allels unto D E (namely K N to G Q or to O P; F K to D O;
and F N to D P) the ſame mutuall proportion: that is to ſay,
the ſame that A F hath to A E.
For in regard that the Triangles A F D, A K G, and A N L
31[Figure 31]
are alike, and E F D, H K G, and M N L are alſo alike: There-

fore, (a) as A F is to F D, ſo ſhall A K be to K G; and as F D is to
F E, ſo ſhall K G be to K H: Wherefore, ex equali, as A F is to F
E, ſo ſhall A K be to K H: And, by Converſion of proportion, as
A F is to A E, ſo ſhall A K be to K H. It is in the ſame manner
proved that, as A F is to A E, ſo ſhall A N be to A M. Now A

N being to A M, as A K is to A H; The (b) Remainder K N ſhall
be unto the Remainder H M, that is unto G Q, or unto O P, as
A N is to A M; that is, as A F is to A E: Again, A K is to
A H, as A F is to A E; Therefore the Remainder F K ſhall be to
the Remainder E H, namely to D O, as A F is to A E. We might in
like manner demonstrate that ſo is F N to D P: Which is that that
was required to be demonstrated.
(a) By 4. of the
ſixth.
(b) By 5. of the
fifth.
LEMMA II.
In the ſame Line A B let there be two Points R and S, ſo diſpo­
ſed, that A S may have the ſame Proportion to A R that
A F hath to A E; and thorow R draw R T parallel to E D,
and thorow S draw S T parallel to F D, ſo, as that it may
meet with R T in the Point T. I ſay that the Point T fall­
eth in the Line A C.
132[Figure 32]
For if it be poſſible, let it fall ſhort of it: and let R T be pro­
longed as farre as to A C in V: and then thorow V draw V X pa­
rallel to F D. Now, by the thing we have last demonſtrated, A X
ſhall have the ſame proportion unto A R, as A F hath to A E.
But A S hath alſo the ſame proportion to A R: Wherefore (a)

A S is equall to A X, the part to the whole, which is impoſſi­
ble. The ſame abſurdity will follow if we ſuppoſe the Toint
T to fall beyond the Line A C: It is therefore neceſſary that
it do fall in the ſaid A C. Which we propounded to be demonstrated.
(a) By 9. of the
fifth.
LEMMA III.
Let there be a Parabola, whoſe Diameter

let be A B; and let the Right Lines A C and B D be ^{*} con­
tingent to it, A C in the Point C, and B D in B: And two
Lines being drawn thorow C, the one C E, parallel unto
the Diameter; the other C F, parallel to B D; take any
Point in the Diameter, as G; and as F B is to B G, ſo let B
G be to B H: and thorow G and H draw G K L, and H E
M, parallel unto B D; and thorow M draw M N O parallel
to A C, and cutting the Diameter in O: and the Line N P
being drawn thorow N unto the Diameter let it be parallel
to B D. I ſay that H O is double to G B.
* Or touch it.
For the Line M N O cutteth the Diameter either in G, or in other Points: and if it do
cut it in G, one and the ſame Point ſhall be noted by the two letters G and O. Therfore F C,
P N, and H E M being Parallels, and A C being Parallels to M N O, they ſhall make the
33[Figure 33]
Triangles A F C, O P N and O H M like to

each other: Wherefore (a) O H ſhall be to
H M, as A F to FC: and ^{*} Permutando,

O H ſhall be to A F, as H M to F C: But
the Square H M is to the Square G L as the Line
H B is to the Line B G, by 20. of our firſt Book
of Conicks; and the Square G L is unto the
Square F C, as the Line G B is to the Line B F:
and the Lines H B, B G and B F are thereupon

Proportionals: Therefore the (b) Squares
H M, G L and F C and there Sides, ſhall alſo be
Proportionals: And, therefore, as the (c)
Square H M is to the Square G L, ſo is the Line

H M to the Line F C: But as H M is to F C, ſo
is O H to A F; and as the Square H M is to
the Square G L, ſo is the Line H B to B G; that
is, B G to B F: From whence it followeth that
O H is to A F, as B G to B F: And Permu­
tando, O H is to B G, as A F to F B; But A F is double to F B: Therefore A B and B F
are equall, by 35. of our firſt Book of Conicks: And therefore N O is double to G B:
Which was to be demonſtrated.
1
(a) By 4. of the
ſixth.
* Or permitting.
(b) By 22. of the
ſixth.
(c) By Cor. of 20.
of the ſixth.
LEMMA IV.
The ſame things aſſumed again, and M Q being drawn from the
Point M unto the Diameter, let it touch the Section in the
Point M. I ſay that H Q hath to Q O, the ſame proportion
that G H hath to C N.
For make H R equall to G F; and ſeeing that
34[Figure 34]
the Triangles A F C and O P N are alike, and
P N equall to F C, we might in like manner de­
monſtrate P O and F A to be equall to each other:
Wherefore P O ſhall be double to F B: But H O
is double to G B: Therefore the Remainder P H
is alſo double to the Remainder F G; that is, to
R H: And therefore is followeth that P R, R H
and F G are equall to one another; as alſo that
R G and P F are equall: For P G is common to
both R P and G F. Since therefore, that H B is
to B G, as G B is to B F, by Converſion of Pro­
portion, B H ſhall be to H G, as B G is to G F:
But Q H is to H B, as H O to B G. For by 35
of our firſt Book of Conicks, in regard that Q
M toucheth the Section in the Point M, H B and
B Q ſhall be equall, and Q H double to H B:
Therefore, ex æquali, Q H ſhall be to H G, as
H O to G F; that is, to H R: and, Permu­
tando, Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q
O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de­
monſtrated.
Theſe things therefore being explained, we come now to that
which was propounded. I ſay, therefore, firſt that N C hath
to C K the ſame proportion that H G hath to G B.
For ſince that H Q is to Q O, as H G to C N;
35[Figure 35]
that is, to A O, equall to the ſaid C N: The Re­
mainder G Q ſhall be to the Remainder Q A, as
H Q to Q O: and, for the ſame cauſe, the Lines
A C and G L prolonged, by the things that wee
have above demonstrated, ſhall interſect or meet
in the Line Q M. Again, G Q is to Q A,
as H Q to Q O: that is, as H G to F P; as

(a) was bnt now demonstrated, But unto (b) G

Q two Lines taken together, Q B that is H B, and
B G are equall: and to Q A H F is equall; for
if from the equall Magnitudes H B and B Q there
be taken the equall Magnitudes F B and B A, the
Re mainder ſhall be equall; Therefore taking H
G from the two Lines H B and B G, there ſhall re­
main a Magnitude double to B G; that is, O H:
and P F taken from F H, the Remainder is H P:
Wherefore (c) O H is to H P, as G Q to Q A:

But as G Q is to Q A, ſo is H Q to Q O;
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.
(a) By 2. Lemma.
(b) By 4. Lemma.
(b) By 19. of the
fifth.
(d) By 15. of the
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.
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.
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
36[Figure 36]
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.
1
PROP. VII. THE OR. VII.
The Right Portion of a Rightangled Conoid lighter
than the Liquid, when it ſhall have its Axis greater
than Seſquialter of the Semi-parameter, but leſſe
than to be unto the ſaid Semi-parameter in proportion
as fiſteen to fower, being demitted into the Liquid ſo
as that its Baſe be wholly within the Liquid, it ſhall
never ſtand ſo as that its Baſe do touch the Surface
of the Liquid, but ſo, that it be wholly within the
Liquid, and ſhall not in the leaſt touch its Surface.
Let there be a Portion as hath been ſaid; and let it be de­
mitted into the Liquid, as we have ſuppoſed, ſo as that its
Baſe do touch the Surface in one Point only: It is to be de­
monſtrated that the ſame ſhall not ſo
37[Figure 37]
continue, but ſhall turn about in
ſuch manner as that its Baſe do in no
wiſe touch the Surface of the Liquid.
For let it be cut thorow its Axis by
a Plane erect upon the Liquids Sur­
face: and let the Section be A P O L,
the Section of a Rightangled
Cone; the Section of the Liquids
Surface S L; and the Axis of the
Portion and Diameter of the Section P F: and let P F be cut in
R, ſo, as that R P may be double to R F, and in ω ſo as that P F
may be to R ω as fifteen to fower: and draw ω K at Right Angles

to P F: (a) R ω ſhall be leſſe than the Semi-parameter. There­
fore let R H be ſuppoſed equall to the Semi-parameter: and
draw C O touching the Section in O and parallel unto S L; and
let N O be parallel unto P F; and firſt let N O cut K ω in the Point
I, as in the former Schemes: It ſhall be demonſtrated that N O is
to O I either ſeſquialter, or greater than ſeſquialter. Let O I be
leſſe than double to I N; and let O B be double to B N: and let
them be diſpoſed like as before. We might likewiſe demonſtrate
that if a Line be drawn thorow R and T it will make Right Angles
with the Line C O, and with the Surface of the Liquid: Where­
fore Lines being drawn from the Points B and G parallels unto
R T, they alſo ſhall be Perpendiculars to the Surface of the Liquid:
The Portion therefore which is above the Liquid ſhall move
138[Figure 38]
wards according to that ſame Perpendicular
which paſſeth thorow B; and the Portion
which is within the Liquid ſhall move up­
wards acording to that paſſing thorow G:
From whence it is manifeſt that the Solid
ſhall turn about in ſuch manner, as that
its Baſe ſhall in no wiſe touch the Surface
of the Liquid; for that now when it touch­
eth but in one Point only, it moveth down­
wards on the part towards L. And though
N O ſhould not cut ω K, yet ſhall the ſame hold true.
(a) By 10 of the
fifth.
PROP. VIII. THE OR. VIII.
The Right Portion of a Rightangled Conoid, when it
ſhall have its Axis greater than ſeſquialter of the Se­
mi-parameter, but leſſe than to be unto the ſaid Semi­
parameter, in proportion as fifteen to fower, if it
have a leſſer proportion in Gravity to the Liquid, than
the Square made of the Exceſſe 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 that its Baſe touch not the Liquid,
it ſhall neither return to Perpendicularity, nor conti­
nue inclined, ſave only when the Axis makes an
Angle with the Surface of the Liquid, equall to that
which we ſhall preſently ſpeak of.
Let there be a Portion as hath been ſaid; and let B D be equall
to the Axis: and let B K be double to K D; and R K equall

to the Semi-parameter: and let C B be Seſquialter of B R:
C D ſhall be alſo Sefquialter of K R. And as the Portion is to the
Liquid in Gravity, ſo let the Square F Q be to the Square D B;
and let F be double to Q: It is manifeſt, therefore, that F Q hath
to D B, leſs proportion than C B hath to B D; For C B is the
Exceſs by which the Axis is greater than Seſquialter of the Semi­

parameter: And, therefore, F Q is leſs than B C; and, for the

ſame reaſon, F is leſs than B R. Let R ψ be equall to F; and draw
ψ E perpendicular to B D; which let be in power or contence the
half of that which the Lines K R and ψ B containeth; and
draw a Line from B to E: It is to be demonſtrated, that the
1Portion demitted into the Liquid, like as hath been ſaid, ſhall ſtand
enclined ſo as that its Axis do make an Angle with the Surface of
the Liquid equall unto the Angle E B Ψ. For demit any Portion
into the Liquid ſo as that its Baſe
39[Figure 39]
touch not the Liquids Surface;
and, if it can be done, let the
Axis not make an Angle with the
Liquids Surface equall to the
Angle E B Ψ; but firſt, let it be
greater: and the Portion being
cut thorow the Axis by a Plane e­
rect unto [or upon] the Surface of
the Liquid, let the Section be A P
O L the Section of a Rightangled
Cone; the Section of the Surface of the Liquid X S; and let the
Axis of the Portion and Diameter of the Section be N O: and
draw P Y parallel to X S, and touching the Section A P O L in P;
and P M parallel to N O; and P I perpendicular to N O: and
moreover, let B R be equall to O ω, and R K to T ω; and let ω H
be perpendicular to the Axis. Now becauſe it hath been ſuppoſed

that the Axis of the Portion doth make an Angle with the Surface
of the Liquid greater than the Angle B, the Angle P Y I ſhall be
greater than the Angle B: Therefore the Square P I hath greater

proportion to the Square Y I, than the Square E Ψ hath to the
Square Ψ B: But as the Square P I is to the Square Y I, ſo is the

Line K R unto the Line I Y; and as the Square E Ψ is to the Square

Ψ B, ſo is half of the Line K R unto the Line Ψ B: Wherefore
(a) K R hath greater proportion to I Y, than the half of K R hath

to Ψ B: And, conſequently, I Y isleſſe than the double of Ψ B,
and is the double of O I: Therefore O I is leſſe than Ψ B; and I ω

greater than Ψ R: but Ψ R is equall to F: Therefore I ω is greater

than F. And becauſe that the Portion is ſuppoſed to be in Gra­
vity unto the Liquid, as the Square F Q is to the Square B D; and
ſince that as the Portion is to the Liquid in Gravity, ſo is the part
thereof ſubmerged unto the whole Portion; and in regard that as
the part thereof ſubmerged is to the whole, ſo is the Square P M to
the Square O N; It followeth, that the Square P M is to the Square
N O, as the Square F Q is to the Square B D: And therefore F

Q is equall to P M: But it hath been demonſtrated that P H is

greater than F: It is manifeſt, therefore, that P M is leſſe than
ſeſquialter of P H: And conſequently that P H is greater than
the double of H M. Let P Z be double to Z M: T ſhall be the Cen­
tre of Gravity of the whole Solid; the Centre of that part of it
which is within the Liquid, the Point Z; and of the remaining

part the Centre ſhall be in the Line Z T prolonged unto G. In
1the ſame manner we might demon­
40[Figure 40]
ſtrate the Line T H to be perpendi­
cular unto the Surface of the Liquid:
and that the Portion demerged with­
in the Liquid moveth or aſcend­
eth out of the Liquid according to
the Perpendicular that ſhall be
drawn thorow Z unto the Surface
of the Liquid; and that the part
that is above the Liquid deſcendeth
into the Liquid according to that
drawn thorow G: therefore the Portion will not continue ſo inclined
as was ſuppoſed: But neither ſhall it return to Rectitude or Per­
pendicularity; For that of the Perpendiculars drawn thorow Z and
G, that paſſing thorow Z doth fall on thoſe parts which are to­
wards L; and that that paſſeth thorow G on thoſe towards A:
Wherefore it followeth that the Centre Z do move upwards,
and G downwards: Therefore the parts of the whole Solid which
are towards A ſhall move downwards, and thoſe towards L up­
wards. Again let the Propoſition run in other termes; and let
the Axis of the Portion make an Angle with the Surface of the

Liquid leſſe than that which is at B. Therefore the Square P I
hath leſſer Proportion unto the Square
41[Figure 41]
I Y, than the Square E Ψ hath to the
Square Ψ B: Wherefore K R hath
leſſer proportion to I Y, than the half
of K R hath to Ψ B: And, for the
ſame reaſon, I Y is greater than dou­
ble of Ψ B: but it is double of O I:
Therefore O I ſhall be greater than
Ψ B: But the Totall O ω is equall
to R B, and the Remainder ω I leſſe
than ψ R: Wherefore P H ſhall alſo
be leſſe than F. And, in regard that
M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui­
alter of P H; and that P H is leſſe than double of H M. Let P Z
be double to Z M. The Centre of Gravity of the whole Solid ſhall
again be T; that of the part which is within the Liquid Z; and
drawing a Line from Z to T, the Centre of Gravity of that which
is above the Liquid ſhall be found in that Line portracted, that is
in G: Therefore, Perpendiculars being drawn thorow Z and G

unto the Surface of the Liquid that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till
that its Axis do make an Angle with the Waters Surface greater than
that which it now maketh. And becauſe that when before we
1did ſuppoſe that it made an Angle greater than the Angle B, the
Poriton did not reſt then neither; It is manifeſt that it ſhall ſtay

or reſt when it ſhall make an Angle eqnall to B. For ſo ſhall I O
be equall to Ψ B; and ω I equall to
42[Figure 42]
Ψ R; and P H equall to F: There­
fore M P ſhall be ſeſquialter of P H,
and P H double of H M: And there­
fore ſince H is the Centre of Gravity
of that part of it which is within the
Liquid, it ſhall move upwards along
the ſame Perpendicular according to
which the whole Portion moveth;
and along the ſame alſo ſhall the part
which is above move downwards:
The Portion therefore ſhall reſt; for­
aſmuch as the parts are not repulſed by each other.
A
B
C
D
E
F
G
(a) By 13. of the
fifth.
H
K
L
M
N
O
P
Q
COMMANDINE.
And let C B be ſeſquialter of B R: C D ſhall alſo be ſeſquialter

of K R.] In the Tranſlation it is read thus: Sit autem & CB quidem hemeolia
ipſius B R: C D autem ipſius K R. But we at the reading of this paſſage have thought
fit thus to correctit; for it is not ſuppoſed ſo to be, but from the things ſuppoſed is proved to
be ſo. For if B ψ be double of ψ D, D B ſhall be ſeſquialter of B ψ. And becauſe E B is
ſeſquialter of B R, it followeth that the (a) Remainder C D is ſeſquialter of ψ R; that is, of

the Semi-parameter: Wherefore B C ſhall be the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-parameter.
A
(a) By 19. of the
fifth.
And therefore F Q is leſſe than B C.] For in regard that the Portion hath

the ſame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B;
and hath leſſer proportion than the Square made of the Exceſſe by which the Axis
is greater than Seſquialter of the Semi parameter, hath to the Square made of the Axis; that
is, leßer than the Square C B hath to the Square B D; for the Line B D was ſuppoſed to be
equall unto the Axis: Therefore the Square F Q ſhall have to the Square D B leſſer proporti­
on than the Sqnare C B to the ſame Square B D: And therefore the Square (b) F Q ſhall be

leße than the Square C B: And, for that reaſon, the Line F Q ſhall be leße than B C.
B
(b) By 8 of the
fifth.
And, for the ſame reaſon, F is leſſe than B R.] For C B being ſeſqui-

alter of B R, and F Q ſeſquialter of F: (c) F Q ſhall be likewiſe leſſe than B C; and F

leße than B R.
C
(c) By 14 of the
fifth.
Now becauſe it hath been ſuppoſed that the Axis of the Portion

doth make an Angle with the Surface of the Liquid greater than
the Angle B, the Angle P Y I ſhall be greater than the Angle B.]
For the Line P Y being parallel to the Surface of the Liquid, that is, to XS; (d) the Angle

P Y I ſhall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the
Line X S: And therefore ſhall be greater than the Angle B.
D
(d) By 29 of the
firſt.
Therefore the Square P I hath greater proportion to the Square

Y I, than the Square E Ψ hath to the Square Ψ B] Let the Triangles P I Y
and E ψ B, be deſcribed apart: And ſeeing that the Angle P Y I is greater
than the Angle E B ψ, unto the Line I Y, and at the Point Y aſſigned in
43[Figure 43]
the ſame, make the Angle V Y I equall to the Angle E B ψ; But
the Right Angle at I, is equall unto the Right Angle at ψ; therefore the
1Remaining Angle Y V I is equall to the Remaining Angle B E ψ. And therefore the

(e) Line V I hath to the Line I Y the ſame proportion that the Line E ψ hath to ψ B: But
the (f) Line P I, which is greater than V I, hath unto I Y greater proportion than V I hath un-

to the ſame: Therefore (g) T I ſhall have greater proportion unto I Y, than E ψ hath to ψ B:
And, by the ſame reaſon, the Square T I ſhall have greater proportion to the Square I Y, than

the Square E ψ hath to the Square ψ B.
E
(e) By 4. of the
ſixth.
(f) By 8. of the
fifth.
(g) By 13 of the
fifth.
F
But as the Square P I is to the Square Y I, ſo is the Line K R unto
the Line I Y] For by 11. of the firſt of our Conicks, the Square P I is equall
to the Rectangle contained under the Line I O, and under the Parameter; which
we ſuppoſed to be eqnall to the Semi-parameter; that is, the double of K R:

But I Y is double of I O, by 33 of the ſame: And, therefore, the (h) Rectangle made of K R
and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;

that is, to the Square P I: But as the (i) Rectangle compounded of K R and I Y is to the
Square I Y, ſo is the Line K R unto the Line I Y: Therefore the Line K R ſhall have unto I
Y, the ſame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I
hath to the Square I Y.
(h) By 26. of the
ſixth.
(i) By Lem. 22 of
the tenth.
G
And as the Square E Ψ is to the Square Ψ B, ſo is half of the
Line K R unto the Line ψ B.] For the Square E ψ having been ſuppoſed equall
to half the Rectangle contained under the Line K R and ψ B; that is, to that contained under
the half of K R and the Line ψ B; and ſeeing that as the (k) Rectangle made of half K R

and of B ψ is to the Square ψ B, ſo is half K R unto the Line ψ B; the half of K R ſhall have
the ſame proportion to ψ B, as the Square E ψ hath to the Square ψ B.
(k) By Lem. 22 of
the tenth.
H
And, conſequently, I Y is leſſe than the double of ψ B.]
For, as half K R is to ψ B, ſo is K R to another Line: it ſhall be (1) greater than I Y; that

is, than that to which K R hath leſſer proportion; and it ſhall be double of ψ B: Therefore
I Y is leſſe than the double of ψ B.
(l) By 10 of the
fifth.
K
And I ω greater than ψ R.] For O having been ſuppoſed equall to B R,
if from B R, ψ B be taken, and from O ω, O I, which is leſſer than B, be taken; the
Remainder I ω ſhall be greater than the Remainder Ψ R.
L
And, therefore, F Q is equall to P M.] By the fourteenth of the fifth of
Euclids Elements: For the Line O N is equall to B D.
M
But it hath been demonſtrated that P H is greater than F.]
For it was demonſtrated that I ω is greater than F: And P H is equall to I ω.
N
In the ſame manner we might demonſtrate the Line T H
to be Perpendicular unto the Surface of the Liquid.] For T α is equall
to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated,
the Line T H ſhall be drawn Perpendicular unto the Liquids Surface.
O
Therefore, the Square P I hath leſſer proportion unto the
Square I Y, than the Square E hath to the Square ψ B.]
Theſe, and other particulars of the like nature, that follow both in this and the following
Propoſitions, ſhall be demonſtrated by us no otherwiſe than we have done above.
P
Therefore Perpendiculars being drawn thorow Z and G, unto
the Surface of the Liquid, that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till that its
Axis do make an Angle with the Waters Surface greater than that
which it now maketh.] For in that the Line drawn thorow G, doth fall perpendicu­
larly towards thoſe parts which are next to L; but that thorow Z, towards thoſe next to A;
It is neceſſary that the Centre G do move downwards, and Z upwards: and, therefore, the
parts of the Solid next to L ſhall move downwards, and thoſe towards A upwards, that the
Axis may makea greater Angle with the Surface of the Liquid.
Q
For ſo ſhall I O be equall to ψ B; and ω I equall to I R; and
P H equall to F.] This plainly appeareth in the third Figure, which is added by us.
1
PROP. IX. THE OR. IX.
The Right Portion of a Rightangled Conoid, when it
ſhall have its Axis greater than Seſquialter of the
Semi-parameter, but leſſer than to be unto the ſaid
Semi-parameter in proportion as fifteen to four, and
hath greater proportion in Gravity to the Liquid, than
the exceſs by which the Square made of the Axis is
greater than the Square made 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 that its Baſe
be wholly within the Liquid, and being ſet inclining
it ſhall neither turn about, ſo as that its Axis ſtand
according to the Perpendicular, nor remain inclined,
ſave only when the Axis makes an Angle with
the Surface of the Liquid, equall to that aßigned
as before.
Let there be a Portion as was ſaid; and ſuppoſe D B equall to
the Axis of the Portion: and let B K be double to K D; and
K R equall to the Semi-parameter: and C B Seſquialter of
B R. And as the Portion is to the Liquid in Gravity, ſo let the Ex­
ceſſe by which the Square B D exceeds the Square F Q be to the
Square B D: and let F be double to Q: It is manifeſt, therefore,
that the Exceſſe by which the
44[Figure 44]
Square B D is greater than the
Square B C hath leſser proportion
to the Square B D, than the Exceſs
by which the Square B D is greater
than the Square F Q hath to the
Square B D; for B C is the Exceſs
by which the Axis of the Portion is
greater than Seſquialter of the
Semi-parameter: And, therefore,

the Square B D doth more exceed
the Square F Q, than doth the
Square B C: And, conſequently, the Line F Q is leſs than B C;
1and F leſs than B R. Let R Ψ be equall to F; and draw Ψ E
perpendicular to B D; which let be in power the half of that
which the Lines K R and Ψ B containeth; and draw a Line from
B to E: I ſay that the Portion demitted into the Liquid, ſo as that
its Baſe be wholly within the Liquid, ſhall ſo ſtand, as that its Axis
do make an Angle with the Liquids Surface, equall to the Angle B.
For let the Portion be demitted into the Liquid, as hath been ſaid;
and let the Axis not make an Angle with the Liquids Surface, equall
to B, but firſt a greater: and the ſame being cut thorow the Axis
by a Plane erect unto the Surface of the Liquid, let the Section of
the Portion be A P O L, the Section of a Rightangled Cone; the
Section of the Surface of the Liquid Γ I; and the Axis of the
Portion and Diameter of the Section N O; which let be cut in
the Points ω and T, as before: and draw Y P, parallelto Γ I, and
touching the Section in P, and MP parallel to N O, and P S perpen­
dicular to the Axis. And becauſe now that the Axis of the Portion
maketh an Angle with the Liquids Surface greater than the Angle
B, the Angle S Y P ſhall alſo be greater than the Angle B: And,
therefore, the Square P S hath greater proportion to the Square

S Y, than the Square Ψ E hath to the Square Ψ B: And, for that
cauſe, K R hath greater proportion to S Y, than the half of K R
hath to Ψ B: Therefore, S Y is leſs than the double of Ψ B; and

S O leſs than ψ B: And, therefore, S ω is greater than R ψ; and

P H greater than F. And, becauſe that the Portion hath the
ſame proportion in Gravity unto the Liquid, that the Exceſs by
which the Square B D, is greater than the Square F Q, hath unto
the Square B D; and that as the Portion is in proportion to the
Liquid in Gravity, ſo is the part thereof ſubmerged unto the whole
Portion; It followeth that the part ſubmerged, hath the ſame
proportion to the whole Portion, that the Exceſs by which the
Square B D is greater than the Square F Q hath unto the Square
B D: And, therefore, the whole Portion ſhall have the ſame propor­

tion to that part which is above the
45[Figure 45]
Liquid, that the Square B D hath to
the Square F Q: But as the whole
Portion is to that part which is above
the Liquid, ſo is the Square N O unto
the Square P M: Therefore, P M
ſhall be equall to F Q: But it
hath been demonſtrated, that P H is
greater than F. And, therefore,
MH ſhall be leſs than que and P H
greater than double of H M. Let
therefore, P Z be double to Z M:
1and drawing a Line from Z to T pro­
46[Figure 46]
long it unto G. The Centre of
Gravity of the whole Portion ſhall
be T; of that part which is above
the Liquid Z; and of the Remain­
der which is within the Liquid, the
Centre ſhall be in the Line Z T pro­
longed; let it be in G: It ſhall be
demonſtrated, as before, that T H
is perpendicular to the Surface of
the Liquid, and that the Lines
drawn thorow Z and G parallel to the ſaid T H, are alſo perpen­
diculars unto the ſame: Therefore, the Part which is above the
Liquid ſhall move downwards, along that which paſseth thorow Z;
and that which is within it, ſhall move upwards, along that which
paſseth thorow G: And, therefore, the Portion ſhall not remain
ſo inclined, nor ſhall ſo turn about, as that its Axis be perpendicular

unto the Surface of the Liquid; for the parts towards L ſhall move
downwards, and thoſe towards A upwards; as may appear by
the things already demonſtrated. And, if the Axis ſhould make
an Angle with the Surface of the Liquid, leſs than the Angle B;
it ſhall in like manner be demonſtrated, that the Portion will not

reſt, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.
A
B
C
D
E
F
G
COMMANDINE.
And, therefore, the Square B D doth more exceed the Square

F Q, than doth the Square B C: And, conſequently, the Line
F Q, is leſs than B C; and F leſs than B R.] Becauſe the Exceſs by
which the Square B D exceedeth the Square B C; having leſs proportion unto the Square B D,
than the Exceſs by which the Square B D exceedeth the Square F Q, hath to the ſaid Square;
(a) the Exceſs by which the Square B D exceedeth the Square B C ſhall be leſs than the Exceſs

by which it exceedeth the Square F Q: Therefore, the Square F Q is leſs than the Square B C:
and, conſquently, the Line F Q leſs than the Line BC: But F Q hath the ſameproportion
to F, that B C hath to B R; for the Antecedents are each Seſquialter of their conſequents:
And (b) F Q being leſs than B C, F ſhall alſo be leſs than B R.
A
(a) By 8. of the
fifth.
(b) By 14. of the
fifth.
And, for that cauſe, K R hath greater proportion to S Y, than
the half of K R hath to ψ B.] For K R is to S Y, as the Square P S is to the Square

S Y: and the half of the Line K R is to the Line ψ B, as the Square E ψ is to the Square ψ B.
B
And S O leſs than ψ B.] For S Y is double of S O.
C
And P H greater than F.] For P H is equall to S ω, and R ψ equall to F.
D
And, therefore, the whole Portion ſhall have the ſame propor­

tion to that part which is above the Liquid, that the Square B D
hath to the Square F Q] Becauſe that the part ſubmerged, being to the whole Portion
as the Exceſs by which the Square B D is greater than the Square F Q, is to the Square B D;
the whole Portion, Converting, ſhall be to the part thereof ſubmerged, as the Square B D is to
1the Exceſs by which it exceedeth the Square F Q: And, therefore, by Converſion of Proportion,
the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square,
F que for the Square B D is ſo much greater than the Exceſs by which it exceedeth the Squar,
F Q as is the ſaid Square F que
E
F
For the parts towards L ſhall move downwards, and thoſe to­
wards A upwards.] We thus carrect theſe words, for in Tartaglia's Tranſlation it
is falſly, as I conceive, read Quoniam quæ ex parte L ad ſuperiora ferentur, becauſe
the Line thàt paſſeth thorow Z falls perpendicularly on the parts towards L, and that thorow
G falleth perpendicularly on the parts towards A: Whereupon the Centre Z, together with thoſe
parts which are towards L ſhall move downwards; and the Centre G, together with the parts
which are towards A upwards.
G
It ſhall in like manner be demonſtrated that the Portion ſhall not
reſt, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.] This may be eaſily demon­
ſtratred, as nell from what hath been ſaid in the precedent Propoſition, as alſo from the two
latter Figures, by us inſerted
PROP. X. THEOR. X.
The Right Portion of a Rightangled Conoid, lighter
than the Liquid, when it ſhall have its Axis greater
than to be unto the Semiparameter, in proportion as
fifteen to four, being demitted into the Liquid, ſo as

that its Baſe touch not the ſame, it ſhall ſometimes

ſtand perpendicular; ſometimes inclined; and ſome­
times ſo inclined, as that its Baſe touch the Surface
of the Liquid in one Point only, and that in two Po-

ſitions; ſometimes ſo that its Baſe be more ſubmer-

ged in the Liquid; and ſometimes ſo as that it doth
not in the leaſt touch the Surface of the Liquid;

according to the proportion that it hath to the Liquid
in Gravity. Every one of which Caſes ſhall be anon
demonſtrated.
A
B
C
D
E
Let there be a Portion, as hath been ſaid; and it being cut
thorow its Axis, by a Plane erect unto the Superficies of the
Liquid, let the Section be A P O L, the Section of a Right
angled Cone; and the Axis of the Portion and Diameter of the
Section B D: and let B D be cut in the Point K, ſo as that B K
be double of K D; and in C, ſo as that B D may have the ſame

proportion to K C, as fifteen to four: It is manifeſt, therefore,

that K C is greater than the Semi-parameter: Let the
1parameter be equall to K R: and
47[Figure 47]

let D S be Seſquialter of K R: but
S B is alſo Seſquialter of B R:
Therefore, draw a Line from A to
B; and thorow C draw C E Per­
pendicular to B D, cutting the Line
A B in the Point E; and thorow E
draw E Z parallel unto B D. Again,
A B being divided into two equall
parts in T, draw T H parallel to the
ſame B D: and let Sections of
Rightangled Cones be deſcribed, A E I about the Diameter E Z;
and A T D about the Diameter T H; and let them be like to the

Portion A B L: Now the Section of the Cone A E I, ſhall paſs

thorow K; and the Line drawn from R perpendicular unto B D,
ſhall cut the ſaid A E I; let it cut it in the Points Y G: and
thorow Y and G draw P Y Q and O G N parallels unto B D, and
cutting A T D in the Points F and X: laſtly, draw P Φ and O X
touching the Section A P O L in the Points P and O. In regard,

therefore, that the three Portions A P O L, A E I, and A T D are
contained betwixt Right Lines, and the Sections of Rightangled
Cones, and are right alike and unequall, touching one another, upon
one and the ſame Baſe; and N X G O being drawn from the
Point N upwards, and Q F Y P from Q: O G ſhall have to G X
a proportion compounded of the proportion, that I L hath to L A,
and of the proportion that A D hath to DI: But I L is to L A,
as two to five: And C B is to B D, as ſix to fifteen; that is, as two

to five: And as C B is to B D, ſo is E B to B A; and D Z to

D A: And of D Z and D A, L I and L A are double: and A D

is to D I, as five to one: But the proportion compounded of the
proportion of two to five, and of the proportion of five to one, is

the ſame with that of two to one: and two is to one, in double
proportion: Therefore, O G is double of GX: and, in the ſame
manner is P Y proved to be double of Y F: Therefore, ſince that
D S is Seſquialter of K R; B S ſhall be the Exceſs by which the
Axis is greater than Seſquialter of the Semi-parameter. If there­
fore, the Portion have the ſame proportion in Gravity unto the
Liquid, as the Square made of the Line B S, hath to the Square
made of B D, or greater, being demitted into the Liquid, ſo as hat
its Baſe touch not the Liquid, it ſhall ſtand erect, or perpendicular:
For it hath been demonſtrated above, that the Portion whoſe

Axis is greater than Seſquialter of the Semi-parameter, if it have
not leſser proportion in Gravity unto the Liquid, than the Square
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.
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.
A
It ſhall ſometimes ſtand perpendicular.] This is the firſt Concluſion, the
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.
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.
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.
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.
F
(a) By 10. of the
fifth.
Let the Semi-parameter be equall to KR.] We have added theſe words,

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.
H
(b) By 19 of the
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.
K
* Upon prop. 3 lib. 2
Archim. Æqui­
pond.
Vide Archim, ante
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
1that A Z hath to Z D; by the fourth Propoſition of Archimedes, De quadratura Para­
bolæ: But A Z is Seſquialter of Z D; for it is as three to two, as we ſhallanon demon-

ſtrate: Therefore D B is Seſquialter of B V; but D B and B K are Seſquialter:
And, therefore, the Lines (c) B V and B K are equall: Which is imposſible:
Therefore the Section of the Right-angled Cone A E I, ſhall paſs thorow the Point K; which
we would demonstrate.
(c) By 9 of the
fifth,
In regard, therefore, that the three Portions A P O L, A E I

and A T D are contained betwixt Right Lines and the Sections
of Right-angled Cones, and are Right, alike and unequall,
touching one another, upon one and the ſame Baſe.] After theſe words,
upon one and the ſame Baſe, we may ſee that ſomething is obliterated, that is to be
deſired: and for the Demonſtration of theſe particulars, it is requiſite in this place to
premiſe ſome things: which will alſo be neceſſary unto the things that follow.
M
LEMMA. I.
Let there be a Right Line A B; and let it be cut by two Lines,
parallel to one another, A C and D E, ſo, that as A B is to
B D. ſo A C may be to D E. I ſay that the Line that con­
joyneth the Points C and B ſhall likewiſe paſs by E.
48[Figure 48]
For, if poſſible, let it not paſs by E, but either
above or below it. Let it first paſs below it,
as by F. The Triangles A B C and D B F ſhall
be alike: And, therefore, as (a) A B is to B D,

ſo is A C to D F: But as A B is to B D, ſo was
A C to D E: Therefore (b) D F ſhall be equall to

D E: that is, the part to the whole: Which is
abſurd. The ſame abſurditie will follow, if the
Line C B be ſuppoſed to paſs above the Point E:
And, therefore, C B muſt of necesſity paſs thorow
E: Which was required to be demonſtrated.
(a) By 4. of the
ſixth.
(b) By 9. of the
fifth.
LEMMA. II.
Let there be two like Portions, contained betwixt Right Lines,
and the Sections of Right-angled Cones; A B C the great­
er, whoſe Diameter let be B D; and E F C the leſser, whoſe
Diameter let be F G: and, let them be ſo applyed to one
another, that the greater include the leſser; and let their
Baſes A C and E C be in the ſame Right Line, that the ſame
Point C, may be the term or bound of them both: And,
then in the Section A B C, take any Point, as H; and draw
a Line from H to C. I ſay, that the Line H C, hath to that
part of it ſelf, that lyeth betwixt C and the Section E F C, the
ſame proportion that A C hath to C E.
Draw B C, which ſhall paſs thorow F, For, in regard, that the Portions are alike, the
Diameters with the Baſes contain equall Angles: And, therefore, B D and F G are parallel
to one another: and B D is to A C, as F G it to E C: and, Permutando, B D is to F G, as
A C is to C E; that is, (a) as their halfes D C to C G; therefore, it followeth, by the

preceding Lemma, that the Line B C ſhall paſs by the Point F. Moreover, from the Point
H unto the Diameter B D, draw the Line H K, parallel to the Baſe A C: and, draw a Line
149[Figure 49]
from K to C, cutting the Diameter F G in L:
and, thorow L, unto the Section E F. G, on the
part E, draw the Line L M, parallel unto the
ſame Baſe A C. And, of the Section A B C,
let the Line B N be the Parameter; and, of the
Section E F C, let F O be the Parameter. And,
becauſe the Triangles C B D and C F G are alike;
(b) therefore, as B C is to C F, ſo ſhall D C be

to C G, and B D to F G. Again, becauſe the
Triangles C K B and C L F, are alſo alike to
one another; therefore, as B C is to C F, that is,
as B D is to F G, ſo ſhall K C be to C L, and B K to F L: Wherefore, K C to C L, and,

B K to F L, are as D C to C G; that is, (c) as their duplicates A C and C E: But as
B D is to F G, ſo is D C to C G; that is, A D to E G: And, Permutando, as B D is to
A D, ſo is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11
of our firſt of Conicks: Therefore, the (d) three Lines B D, A D and B N are

Proportionalls. By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle
G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is
to A D, ſo is F G to E G: And, therefore, as A D is to B N, ſo is E G to F O: Ex equali,
therefore, as D B is to B N, ſo is G F to F O: And, Permutando, as D B is to G F, ſo is
B N to F O: But as D B is to G F, ſo is B K to F L: Therefore, B K is to F L, as
B N is to F O: And, Permutando, as B K is to B N, ſo is F L to F O. Again,
becauſe the (e) Square H K is equall to the Rectangle B N; and the Square M L, equall

to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals:
and F L, L M, and F O ſhall alſo be Proportionals: And, therefore, (f) as the Line

B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the
Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M:
Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square

B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,
(g) as the Line B K is to the Line K H, ſo is the Line F L to L M: And, Permutando,
as B K is to F L, ſo is K H to L M: But B K was to F L, as K C to C L: Therefore,
K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that
the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is,
as A C to C E, ſo is H C to C M; that is, to the ſame part of it ſelf, that lyeth betwixt C and
the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth
in other Lines, that are produced from the Point C, and the Sections E B C. And, that
B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G;
that is, as their Duplicates A C to C E.
(a) By 15. of the
fifth.
(b) By 4. of the
ſixth.
(c) By 15. of the
fifth.
(d) By 17. of the
ſixth.
(e) By 11 of our
firſt of Conicks.
(f) By Cor. of 20.
of the ſixth.
(g) By 23. of the
ſixth.
From whence it is manifeſt, that all Lines ſo drawn, ſhall be cut by the
ſaid Section in the ſame proportion. For, by Diviſion and Converſion,
C M is to M H, and C F to F B, as C E to E A.
LEMMA. III.
And, hence it may alſo be proved, that the Lines which are
drawn in like Portions, ſo, as that with the Baſes, they con­
tain equall Angles, ſhall alſo cut off like Portions; that is,
as in the foregoing Figure, the Portions H B C and M F C,
which the Lines C H and C M do cut off, are alſo alike to
each other.
For let C H and C M be divided in the midst in the Points P and que and thorow thoſe
Points draw the Lines R P S and T Q V parallel to the Diameters. Of the Portion
H S C the Diameter ſhall be P S, and of the Portion M V C the Diameter ſhall be
1Q V. And, ſuppoſe that as the Square C R is to the Square C P, ſo is the Line B N unto
another Line; which let be S X: And, as the Square C T is to the Square C Q ſo let F O
be to V Y. Now it is manifeſt, by the things which we have demonſtrated, in our Commentaries,
upon the fourth Propoſition of Archimedes, De Conoidibus & Spheæroidibus, that the
Square C P is equall to the Rectangle P S X; and alſo, that the Square C Q is equall to
the Rectangle Q V Y; that is, the Lines S X and V Y, are the Parameters of the Sections H S C
and M V C: But ſince the Triangles C P R and C Q T are alike; C R ſhall have to C P, the
ſame Proportion that C T hath to C Q: And, therefore, the (a) Square C R ſhall have

to the Square C P, the ſame proportion that the
50[Figure 50]
Square C T hath to the Square C Q: There­
fore, alſo, the Line B N ſhall be to the Line
S X, as the Line F O is to V Y: But H C was
to C M, as A C to C E: And, therefore, alſo,
their halves C P and C Q, are alſo to one
another, as A D and E G: And. Permu­
tando, C P is to A D, as C Q is to E G:
But it hath been proved, that A D is to B N,
as E G to F O; and B N to S X, as F O to
V Y: Therefore, exæquali, C P ſhall be
to S X, as C Q is to V Y. And, ſince the
Square C P is equall to the Rectangle P S X, and the Square C Q to the Rectangle Q V Y,
the three Lines S P, PC and S X ſhall be proportionalls, and V Q, Q C and V Y ſhal be
Proportionalls alſo: And therefore alſo S P ſhall be to P C as V Q to Q C And as P C
is to C H, ſo ſhall Q C. be to C M: Therefore, ex æquali, as S P the Diameter of the
Portion H S C is to its Baſe C H, ſo is V Q the Diameter of the portion M V S the
Baſe C M; and the Angles which the Diameter with the Baſes do contain, are equall; and the
Lines S P and V Q are parallel: Therefore the Portions, alſo, H S C and M V C ſhall be alike:
Which was propoſed to be demonſtrated
(a) By 22. of the
ſixth.
LEMMA. IV.
Let there be two Lines A B and C D; and let them be cut in the
Points E and F, ſo that as A E is to E B, C F may be to F D:
and let them be cut again in two other Points G and H; and
let C H be to H D, as A G is to G B. I ſay that C F ſhall be to
F H as A E is E G.
For in regard that as A E is to E B, ſo is C F to F D; it followeth that, by Compounding,
as A B is to E B, ſo ſhall C D be to F D. Again, ſince that as A G is to G B, ſo is C H, to
H D; it followeth that, by Compounding and Converting, as G B is to A B, ſo ſhall H D be
51[Figure 51]
C D: Therefore, ex æquali, and Converting as E B
is to G B, ſo ſhall F D be to H D; And, by Conver­
ſion of Propoſition, as E B is to E G, ſo ſhall F D
be to F H: But as A E is to E B, ſo is C F to F D:
Ex æquali, therefore, as A E is to E G, ſo
ſhall CF be to F H. Again, another way. Let
the Lines A B and C D be applyed to one another,
ſo as that they doe make an Angle at the parts A and C;
and let A and C be in one and the ſame Point: then
draw Lines from D to B, from H to G, and from F to E. And ſince that as A E is to E B,
ſo is C F, that is A F to F D; therefore F E ſhall be parallel to D B; (a) and likewiſe

H G ſhall be parallel to D B; for that A H is to H D, as A G to G B: (b) Therefore F E
and H G are parallel to each other: And conſequently, as A E is to E G, ſo is A H, that is,

C F to F H: Which was to be demonſtrated.
1
(a) By 2. of the
ſixth.
(b) By 30 of the
firſt.
LEMMA. V.
Again, let there be two like Portions, contained betwixt Right
Lines and the Sections of Right-angled Cones, as in the fore­
going figure, A B C, whoſe Diameter is B D; and E F C,
whoſe Diameter is F G; and from the Point E, draw the
Line E H parallel to the Diameters B D and F G; and let it
cut the Section A B C in K: and from the Point C draw C H
touching the Section A B C in C, and meeting with the Line
E H in H; which alſo toucheth the Section E F C in the ſame
Point C, as ſhall be demonſtrated: I ſay that the Line drawn
from C H unto the Section E F C ſo as that it be parallel to
the Line E H, ſhall be divided in the ſame proportion by the
Section A B C, in which the Line C A is divided by the Section
E F C; and the part of the Line C A which is betwixt the
two Sections, ſhall anſwer in proportion to the part of the Line
drawn, which alſo falleth betwixt the ſame Sections: that is,
as in the foregoing Figure, if D B be produced untill it meet
with C H in L, that it may interſect the Section E F C in the
Point M, the Line L B ſhall have to B M the ſame proportion
that C E hath to E A.
For let G F be prolonged untill it meet the ſame Line C H in N, cutting the Section A B C
in O; and drawing a Line from B to C, which ſhall paſſe by F, as hath been ſhewn, the
52[Figure 52]
Triangles C G F and C D B ſhall be alike; as
alſo the Triangles C F N and C B L: Wherefore
(a) as G F is to D B, ſo ſhall C F b to C B:

And as (b) C F is to C B, ſo ſhall F N be
to B L: Therefore G F ſhall be to D B, as F N

to B L: And, Permutando, G F ſhall be to
F N, as D B to B L: But D B is equall to
B L, by 35 of our Firſt Book of Conicks:
Therefore (c) G F alſo ſhall be equall to F N:

And by 33 of the ſame, the Line C H touch­
eth the Section E F C in the ſame Point. There­
fore, drawing a Line from C to M, prolong it
untill it meet with the Section A B C in P; and
from P unto A C draw P Q parallel to B D.
Becauſe, now, that the Line C H toucheth the
Section E F C in the Point C; L M ſhall have
the ſame proportion to M D that C D hath to D E,
by the Fifth Propoſition of Archimedes in his
Book De Quadratura Patabolæ: And by
reaſon of the Similitude of the Triangles C M D
and C P Q, as C M is to C D, ſo ſhall C P
be to C Q: And, Permutando, as C M is to
C P, ſo ſhall C D be to C Q: But as C M is to C P, ſo is C E to C A,; as we have but
even now demonſtrated: And therefore, as C E is to C A, ſo is C D to C que that is as the
whole is to the whole, ſo is the part to the part: The remainder, therefore, D E is to the
Remainder Q A, as C E is to C A; that is, as C D is to C Q: And, Permutando, C D
is to D E, as C Q is to Q A: And L M is alſo to M D, as C D to D E: Therefore L M is
1to M D, as C Q to Q A: But L B is to B D, by 5 of Archimedes, before recited, as C D
to D A: It is manifeſt therefore, by the precedent Lemma, that C D is to D Q, as L B is to
B M: But as C D is to D Q, ſo is C M to M P: Therefore L B is to B M, as C M to M P:

And it haveing been demonſtrated, that C M is to M P, as C E to E A; L B ſhall be to B M,
as C E to E A. And in like manner it ſhall be demonstrated that ſo is N O to O F; as alſo the
Remainders. And that alſo H K is to K E, as C E to E A, doth plainly appeare by the ſame
5. of Archimedes: Which is that that we propounded to be demonſtrated.
(a) By 4. of the
ſixth.
(b) By 11 of the
fifth,
(c) By 14 of the
fifth.
By 2. of the ſixth
LEMMA. VI.
And, therefore, let the things ſtand as above; and deſcribe
yet another like Portion, contained betwixt a Right Line, and
the Section of the Rightangled Cone D R C, whoſe Diameter
is R S, that it may cut the Line F G in T; and prolong S R
unto the Line C H in V, which meeteth the Section A B C in
X, and E F C in Y. I ſay, that B M hath to M D, a propor­
tion compounded of the proportion that E A hath to A C;
and of that which C D hath to D E.
For, we ſhall firſt demonſtrate, that the Line C H toucheth the Section D R C in the
Point C; and that L M is to M D, as alſo N F to F T, and V Y to Y R, as C D is to E D.
And, becauſe now that L B is to B M, as C E is to E A; therefore, Compounding and Conver­
ting, B M ſhall be to L M, as E A to A C: And, as L M is to M D, ſo ſhall C D be to
D E: The proportion, therefore, of B M to M D, is compounded of the proportion that
B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion
of B M to M D, ſhall alſo be compounded of the proportion that E A hath to A C, and of
that which C D hath to D E. In the ſame manner it ſhal be demonſtrated, that O F hath to
F T, and alſo X Y to Y R, a proportion compounded of thoſe ſame proportions; and ſo in
the reſt: Which was to be demonstrated.
By which it appeareth that the Lines ſo drawn; which fall betwixt
the Sections A B C and D R C, ſhall be divided by the Section E F C
in the ſame Proportion.
And C B is to B D, as ſix to fifteen.] For we have ſuppoſed that B K is

double of K D: Wherefore, by Compoſition B D ſhall be to K D as three to one; that is, as
fifteen to five: But B D was to K C as fifteen to four; Therefore B D is to D C as fifteen to nine:
And, by Converſion of proportion and Convert­
ing, C B is to B D, as ſix to ſifteen.
N
53[Figure 53]
And as C B is to B D, ſo is

E B to B A; and D Z to D A.]
For the Triangles C B E and D B A being
alike; As C B is to B E, ſo ſhall D B be to B A:
And, Permutando, as C B is to B D, ſo ſhall
E B be to B A: Againe, as B C is to C E ſo
ſhall B D be to D A, And, Permutando, as
C B is to B D, ſo ſhall C E, that is, D Z
equall to it, be to D A.
O
And of D Z and D A, L I and

L A are double.] That the Line L A is
double of D A, is manifeſt, for that B D is the Diameter of the Portion. And that L I is
dovble to D Z ſhall be thus demonſtrated. For as much as ZD is to D A, as two to five:
therefore, Converting and Dividing, A Z, that is, I Z, ſhall be to Z D, as three to two:
1Again, by dividing, I D ſhall be to D Z, as one to two: But Z D was to D A, that is, to D L,
as two to five: Therefore, ex equali, and Converting, L D is to D I, as five to one: and, by
Converſion of Proportion, D L is to D I, as five to four: But D Z was to D L, as two to
five: Therefore, again, ex equali, D Z is to L I, as two to four: Therefort L I is double
of D Z: Which was to be demonſtrated.
P
Q
And, A D is to D I, as five to one.] This we have but juſt now demon­
ſtrated.
R
For it hath been demonſtrated, above, that the Portion whoſe
Axis is greater than Seſquialter of the Semi-parameter, if it have
not leſſer proportion in Gravity to the Liquid, &c.] He hath demonstra­
ted this in the fourth Propoſition of this Book.
CONCLVSION II.
If the Portion have leſſer proportion in Gravity to the

Liquid, than the Square S B hath to the Square
B D, but greater than the Square X O hath to the
Square B D, being demitted into the Liquid, ſo in­
clined, as that its Baſe touch not the Liquid, it ſhall
continue inclined, ſo, as that its Baſe ſhall not in the
leaſt touch the Surface of the Liquid, and its Axis
ſhall make an Angle with the Liquids Surface, greater
than the Angle X.
A
Therfore repeating the firſt figure, let the Portion have unto
the Liquid in Gravitie a proportion greater than the Square
X O hath to the ſquare B D, but leſſer than the Square made of
the Exceſſe by which the Axis is greater than Seſquialter of the Semi­
54[Figure 54]
Parameter, that is, of S B, hath to
the Square B D: and as the Portion
is to the Liquid in Gravity, ſo let
the Square made of the Line ψ be
to the Square B D: ψ ſhall be great­

er than X O, but leſſer than the
Exceſſe by which the Axis is grea­
ter than Seſquialter of the Semi­
parameter, that is, than S B. Let
a Right Line M N be applyed to
fall between the Conick-Sections
A M Q L and A X D, [parallel to
B D falling betwixt O X and B D,] and equall to the Line ψ: and let
it cut the remaining Conick Section A H I in the point H, and the

Right Line R G in V. It ſhall be demonſtrated that M H is double to
H N, like as it was demonſtrated that O G is double to G X.
155[Figure 55]
And from the Point M draw M Y
touching the Section A M Q L in M;
and M C perpendicular to B D: and
laſtly, having drawn A N & prolong­
ed it to Q, the Lines A N & N Q ſhall
be equall to each other. For in
regard that in the Like Portions

A M Q L and A X D the Lines A Q
and A N are drawn from the Baſes
unto the Portions, which Lines
contain equall Angles with the ſaid
Baſes, Q A ſhall have the ſame proportion to A M that L A hath
to A D: Therefore A N is equall to N Q, and A Q parallel to M Y.

It is to be demonſtrated that the Portion being demitted into the
Liquid, and ſo inclined as that its Baſe touch not the Liquid, it
ſhall continue inclined ſo as that its Baſe ſhall not in the leaſt touch
the Surface of the Liquid, and its Axis ſhall make an Angle with
the Liquids Surface greater than the Angle X. Let it be demitted
into the Liquid, and let it ſtand, ſo, as that its Baſe do touch the
Surface of the Liquid in one Point only; and let the Portion be cut
thorow the Axis by a Plane erect unto the Surface of the Liquid,
56[Figure 56]
and Let the Section of the Super­
ficies of the Portion be A P O L,
the Section of a Rightangled Cone,
and let the Section of the Liquids
Surface be A O; And let the Axis
of the Portion and Diameter of the
Section be B D: and let B D be

cut in the Points K and R as hath
been ſaid; alſo draw P G Parallel to
A O and touching the Section
A P O L in P; and from that Point
draw P T Parallel to B D, and P S perpendicular to the ſame B D.
Now, foraſmuch as the Portion is unto the Liquid in Gravity, as
the Square made of the Line ψ is to the Square B D; and ſince that
as the portion is unto the Liquid in Gravitie, ſo is the part thereof
ſubmerged unto the whole Portion; and that as the part ſubmerged
is to the whole, ſo is the Square T P to the Square B D; It follow­
eth that the Line ψ ſhall be equall to T P: And therefore the Lines
M N and P T, as alſo the Portions A M Q and A P O ſhall like­
wiſe be equall to each other. And ſeeing that in the Equall and
Like Portions A P O L and A M Q L the Lines A O and A Q

are drawn from the extremites of their Baſes, ſo, as that the Portions
cut off do make Equall Angles with their Diameters; as alſo the
1Angles at Y and G being equall; therefore the Lines Y B and G B,
and B C and B S ſhall alſo be equall: And therefore C R and S R,
and M V and P Z, and V N and Z T, ſhall be equall likewiſe.

Since therefore M V is Leſſer than double of V N, it is manifeſt that
P Z is leſſer than double of Z T. Let P ω be double of ω T; and
drawing a Line from ω to K, prolong it to E. Now the Centre of
Gravity of the whole Portion ſhall be the point K; and the Centre
of that part which is in the Liquid ſhall be ω, and of that which is
above the Liquid ſhall be in the Line K E, which let be E: But the
Line K Z ſhall be perpendicular unto the Surface of the Liquid:
And therefore alſo the Lines drawn thorow the Points E and ω parall­

lell unto K Z, ſhall be perpendicular sunto the ſame: Therefore the
Portion ſhall not abide, but ſhall turn about ſo, as that its Baſe
do not in the leaſt touch the Surface of the Liquid; in regard that
now when it toucheth in but one Point only, it moveth upwards, on

the part towards A: It is therefore perſpicuous, that the Portion
ſhall conſiſt ſo, as that its Axis ſhall make an Angle with the Liquids
Surface greater than the Angle X.
B
C
D
E F
G
H
K
L
M
COMMANDINE.
A
If the Portion have leſſer proportion in Gravity to the Liquid,
than the Square S B hath to the Square B D, but greater than the
Square X O hath to the Square B D.] This is the ſecond part of the Tenth
propoſition; and the other pat is with their Demonſtrations, ſhall hereafter follow in the ſame Order.
Ψ ſhall be greater than X O, but leſſer than the Exceſs by

which the Axis is greater than Seſquialter of the Semi-parameter,
that is than S B.] This followeth from the 10 of the fifth Book of Euclids Elements.
B
C
It ſhall be demonſtrated, that M H is double to H N, like as it
was demonſtrated, that O G is double to G X.] As in the firſt Concluſion
of this Propoſition, and from what we have but even now written, thereupon appeareth:
D
For in regard that in the like Portions A M Q L and A X D, the
Lines A Q and A N are drawn from the Baſes unto the Portions,
which Lines contain equall Angles with the ſaid Baſes, Q A ſhall
have the ſame proportion to A N, that L A hath to A D.]
This we have demonstrated above.
E
Therefore A N is equall to N Q] For ſince that Q A is to A N, as L A to
A D; Dividing and Converting, A N ſhall be to N Q as A D to D L: But A D
is equall to D L; for that D B is ſuppoſed to be the Diameter of the Portion: Therefore

alſo (a) A N is equall to N que
(a) By 14 of the
fifth.
And A Q parallel to M Y.] By the fifth of the ſecond Book of Apollonius his Conicks.
F
And let B D be cut in the Points K and R as hath been ſaid.]

In the firſt Conciuſion of this Propoſition: And let it be cut in K, ſo, as that B K be double to
K D, and in R ſo, as that K R may be equall to the Semi-parameter.
G
And, ſeeing that in the Equall and Like Portions A P O L and

A M Q L, the Lines A O and A Q are drawn from the Extremities
of their Baſes, ſo, as that the Portions cut off, do make equall Angles
1with their Diameters; as alſo, the Angles at Y and G being equall;
Therefore, the Lines Y B and G B, & B C & B S, ſhall alſo be equall.]
Let the Line A Q cut the Diameter D B in γ, and let it cut A O in δ. Now becauſe that in
57[Figure 57]
the equall and like Portions A P O L & A M Q L,
from the Extremities of their Baſes, A O and
A Q are drawn, that contain equall Angles with
thoſe Baſes; and ſince the Angles at D, are both
Right; Therefore, the Remaining Angles A δ D
and A γ D ſhall be equall to one another: But
the Line P G is parallel unto the Line A O; alſo
M Y is parallel to A que and P S and M C to
A D: Therefore the Triangles P G S and M Y C,
as alſo the Triangles A δ D and A γ D, are all
alike to each other: (b) And as A D is to A δ,

ſo is A D to A γ: and, Permutando, the Lines
A D and A D are equall to each other: Therefore,
A δ and A γ are alſo equall: But A O and
A Q are equall to each other; as alſo their halves
A T and A N: Therefore the Remainders T δ and N γ; that is, TG and MY, are alſo

58[Figure 58]
equall. And, as (c) P G is to G S, ſo is M Y to
Y C: and Permutando, as P G is to M Y, ſo is
G S to Y C: And, therefore, G S and Y C are
equall; as alſo their halves B S and B C: From
whence it followeth, that the Remainders S R and C R
are alſo equall: And, conſequently, that P Z and
M V, and V N and Z T, are lkiewiſe equall to one
another.
H
(b) By 4. of the
ſixth.
(c) By 34 of the
firſt,
Since, therefore, that N V is leſſer

than double of V N.] For M H is double of
H N, and M V is leſſer than M H: Therefore, M V
is leſſer than double of H N, and much leſſer than
double of V N.
K
Therefore, the Portion ſhall not abide, but ſhall turn about,

ſo, as that its Baſe do not in the leaſt touch the Surface of
the Liquid; in regard that now when it toucheth in but one Point
only, it moveth upwards on the part towards A.] Tartaglia's his Tranſla­
tion hath it thus, Non ergo manet Portio ſed inclinabitur ut Baſis ipſius, nec ſecundum
unum tangat Superficiem Humidi, quon am nunc ſecundum unum tacta ipſa reclina­
tur: Which we have thought fit in this manner to correct, from other Places of
Archimedes, that the ſenſe might be the more perſpicuous. For in the ſixth Propoſition of this,
he thus writeth (as we alſo have it in the Tranſlation,) The Solid A P O L, therefore, ſhall
turn about, and its Baſe ſhall not in the leaſt touch the Surface of the Liquid. Again,
in the ſeventh Propoſition; From whence it is manifeſt, that its Baſe ſhall turn about in
ſuch manner, a that its Baſe doth in no wiſe touch the Surface of the Liquid; For
that now when it toucheth but in one Point only, it moveth downwards on the part
towards L. And that the Portion moveth upwards, on the part towards A, doth plainly ap­
pear: For ſince that the Perpendiculars unto the Surface of the Liquid, that paſs thorow ω, de
fall on the part towards A, and thoſe that paſs thorow E, on the part towards L; it is neceſſary
that the Centre ω do move upwards, and the Centre E downwards.
L
It is therefore perſpicuous, that the Portion ſhall conſiſt, ſo, as that
its Axis ſhall make an Angle with the Liquids Surface greater than
the Angle X.] For dræwing a Line from A to X, prolong it untill it do cut the Diamter
159[Figure 59]
B D in λ; and from the Point O, and parallel to
A λ, draw O X; and let it touch the Section in O,
as in the first Figure: And the (d) Angle at X,

ſhall be equall alſo to the angle λ: But the angle at Y
is equall to the Angle at γ; and the (e) Angle

A Γ D greater than the Angle A λ D, which falleth
without it: Therefore the Angle at Y ſhall be great­
er than that at X. And becauſe now the Portion
turneth about, ſo, as that the Baſe doth not touch
the Liquid, the Axis ſhall make an Angle with its
Surface greater than the Angle G; that is, than the
Angle Y: And, for that reaſon, much greater than
the Angle X.
(d) By 29 of the
firſt.
(e) By 16. of the
firſt.
CONCLUSION III.
If the Portion have the ſame proportion in Gravity to the
Liquid, that the Square X O hath to the Square
BD, being demitted into the Liquid, ſo inclined, as that
its Baſe touch not the Liquid, it ſhall ſtand and
continue inclined, ſo, as that its Baſe touch the Sur­
face of the Liquid, in one Point only, and its Axis ſhall
make an Angle with the Liquids Surface equall to the
Angle X. And, if the Portion have the ſame proportion
in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid,
& ſet ſo inclined, as that its Baſe touch not the Liquid,
it ſhall ſtand inclined, ſo, as that its Baſe touch the
Surface of the Liquid in one Point only, & its Axis ſhall
make an Angle with it, equall to the Angle Φ.
Let the Portion have the ſame proportion in Gravity to tho
Liquid that the Square XO hath to the Square B D; and let
it be demitted into the Liquid ſo inclined, as that its Baſe touch
60[Figure 60]
not the Liquid. And cutting it by
a Plane thorow the Axis, erect unto
the Surface of the Liquid, let the
Section of the Solid, be the Section
of a Right-angled Cone, A P M L;
let the Section of the Surface of the
Liquid be I M; and the Axis of the
Portion and Diameter of the Section
B D; and let B D be divided as be­
fore; and draw PN parallel to IM
1and touching the Section in P, and T P parallel to B D; and P S perpen­
dicular unto B D. It is to be demonſtrated that the Portion ſhall
61[Figure 61]
not ſtand ſo, but ſhall encline until
that the Baſe touch the Surface of
the Liquid, in one Point only, for let
the ſuperior figure ſtand as it was,
and draw O C, Perpendicular to B D;
and drawing a Line from A to X,
prolong it to Q: A X ſhalbe equall
to X que Then draw O X parallel
to A que And becauſe the Portion
is ſuppoſed to have the ſame pro­
portion in Gravity to the Liquid
that the ſquare X O hath to the
Square B D; the part thereof ſubmerged ſhall alſo have the ſame
proportion to the whole; that is, the Square T P to the Square

B D; and ſo T P ſhall be equal to X O: And ſince that of the Portions
I P M and A O Q the Diameters are equall, the portions ſhall alſo be

equall. Again, becauſe that in the Equall and Like Portions A O Q L

and AP ML the Lines A Q and I M, which cut off equall Por­
tions, are drawn, that, from the Extremity of the Baſe, and this
not from the Extremity; it appeareth that that which is drawn from
the end or Extremity of the Baſe, ſhall make the Acute Angle with
the Diameter of the whole Portion leſset. And the Angle at X

being leſſe than the Angle at N, B C ſhall be greater than B S; and
C R leſſer than S R: And, therfore O G ſhall be leſſer than P Z;
and G X greater than Z T: Therfore P Z is greater than double of
Z T; being that O G is double of G X. Let P H be double to H T;
and drawing a Line from H to K, prolong it to ω. The Center of
Gravity of the whole Portion ſhall be K; the Center of the part
which is within the Liquid H, and that of the part which is above
the Liquid in the Line K ω; which ſuppoſed to be ω. Therefore it
ſhall be demonſtrated, both, that K H is perpendicular to the Surface
of the Liquid, and thoſe Lines alſo that are drawn thorow the Points
Hand ω parallel to K H: And therfore the Portion ſhall not reſt, but
ſhall encline untill that its Baſe do touch the Surface of the Liquid
in one Point; and ſo it ſhall continue. For in the Equall Portions
A O Q L and A P M L, the
62[Figure 62]
Lines A Q and A M, that cut off
equall Portions, ſhall be dawn
from the Ends or Terms of the Baſes;
and A O Q and A P M ſhall be
demonſtrated, as in the former, to

be equall: Therfore A Q and A M,
do make equall Acute Angles with
the Diameters of the Portions; and
1the Angles at X and N are equall. And, therefore, if drawing HK,
it be prolonged to ω, the Centre of Gravity of the whole Portion ſhall
be K; of the part which is within the Liquid H; and of the part which
is above the Liquid in K ὠ as ſuppoſe in ω; and H K perpendicular to
63[Figure 63]
the Surface of the Liquid. Therfore
along the ſame Right Lines ſhall the
part which is within the Liquid move
upwards, and the part above it down­
wards: And therfore the Portion
ſhall reſt with one of its Points
touching the Surface of the Liquid,
and its Axis ſhall make with the

ſame an Angle equall to X. It is
to be demonſtrated in the ſame
manner that the Portion that hath
the ſame proportion in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid, ſo, as that its
Baſe touch not the Liquid, it ſhall ſtand inclined, ſo, as that its Baſe
touch the Surface of the Liquid in one Point only; and its Axis ſhall
make therwith an Angle equall to the Angle φ.
A
B
C
D
E
F
COMMANDINE.
A
That is the Square T P to the Square B D.] By the twenty ſixth of the Book

of Archimedes, De Conoidibus & Sphæroidibus: Therefore, (a) the Square T P
ſhall be equall to the Square X O: And for that reaſon, the Line T P equall to the
Line X O.
(a) By 9 of the
fifth.
B
The Portions ſhall alſo be equall.] By the twenty fifth of the ſame Book.
C
Again, becauſe that in the Equall and Like Portions, A O Q L
and A P M L.] For, in the Portion A P M L, deſcribe the Portion A O Q equall
to the Portion I P M: The Point Q falleth beneath M; for otherwiſe, the Whole would be
equall to the Part. Then draw I V parallel to A Q, and cutting the Diameter is ψ; and
let I M cut the ſame ς; and A Q in ς. I ſay
that the Angle A υ D, is leſſer than the Angle
64[Figure 64]
I σ D. For the Angle I ψ D is equall to the
Angle A υ D: (b) But the interiour Angle

I ψ D is leſſer than the exteriour I σ D: There-

fore, (c) A υ D ſhall alſo be lefter than I σ D.
(b) By 29 of the
firſt.
(c) By 16 of the
firſt.
D
And the Angle at X, being leſſe
than the Angle at N.] Thorow O draw twe
Lines, O C perpendicular to the Diameter B D, and
O X touching the Section in the Point O, and cutting

the Diameter in X: (d) O X ſhall be parallel
to A que and the (e) Angle at X, ſhall be equall to

that at υ: Therefore, the (f) Angle at X,

ſhall be leſſer than the Angle at ς; that is, to
that at N: And, conſequently, X ſhall fall beneath N: Therefore, the Line X B is greater than
N B. And, ſince B C is equall to X B, and B S equall to N B; B C ſhall be greater than B S.
1
(d) By 5 of our ſe­
cond of Conicks.
(e) By 29 of the
firſt.
(f) By 39 of our
firſt of Conicks.
Therefore, A Q and A M do make equall Acute Angles with

the Diameters of the Portions.] We demonſtrate this as in the Commentaries
upon the ſecond Concluſion.
E
It is to be demonſtrated in the ſame manner, that the Portion

that hath the ſame proportion in Gravity to the Liquid, that the
Square P F hath to the Square B D,
being demitted into the Liquid, ſo,
65[Figure 65]
as that its Baſe touch not the Li­
quid, it ſhall ſtand inclined, ſo, as
that its Baſe touch the Surface of the
Liquid in one point only; and its Axis
ſhall make therewith an angle equall
to the Angle φ.] Let the Portion be to the
Liquid in Gravity, as the Square P F to the
Square B D: and being demitted into the
Liquid, ſo inclined, as that its Baſe touch not
the Liquid, let it be cut thorow the Axis by a
Plane erect to the Surface of the Liquid, that
that the Section may be A M O L, the Section
of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit
of the Portion and Diameter of the Section B D; which let be cut into the ſame parts as
we ſaid before, and draw M N parallel to I O, that it may touch the Section in the Point
M; and M T parallel to B D, and P M S perpe ndicular to the ſame. It is to be demon­
strated, that the Portion ſhall not reſt, but ſhall incline, ſo, as that it touch the Liquids
Surface, in one Point of its Baſe only. For,
66[Figure 66]
draw P C perpendicular to B D; and drawing
a Line from A to F, prolong it till it meet with
the Section in que and thorow P draw P φ pa­
rallel to A Q: Now, by the things allready de­
monſtrated by us, A F and F Q ſhall be equall
to one another. And being that the Portion hath
the ſame proportion in Gravity unto the Liquid,
that the Square P F hath to the Square B D; and
ſeeing that the part ſubmerged, hath the ſame pro-

partion to the whole Portion; that is, the Squàre
M T to the Square B D; (g) the Square M T
ſhall be equall to the Square P F; and, by the
ſame reaſon, the Line M T equall to the Line
P F. So that there being drawn in the equall & like
portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the
firſt from the Extreme term of the Baſe, the laſt not from the Extremity; it followeth, that
A Q drawn from the Extremity, containeth a leſſer Acute Angle with the Diameter of the
Portion, than I O: But the Line P φ is parallel to the Line A Q, and M N to I O: There­
fore, the Angle at φ ſhall be leſſer than the Angle at N; but the Line B C greater than B S;
and S R, that is, M X, greater than C R, that is, than P Y: and, by the ſame reaſon, X T
leſſer than Y F. And, ſince P Y is double to Y F, M X ſhall be greater than double to
Y F, and much greater than double of X T. Let M H be double to H T, and draw a
Line from H to K, prolonging it. Now, the Centre of Gravity of the whole Portion
ſhall be the Point K; of the part within the Liquid H; and of the Remaining part above
the Liquid in the Line H K produced, as ſuppoſe in ω It ſhall be demonſtrated in the ſame
manner, as before, that both the Line K H and thoſe that are drawn thorow the Points H
and ω parallel to the ſaid K H, are perpendicular to the Surface of the Liquid: The
Portion therefore, ſhall not reſt; but when it ſhall be enclined ſo far as to touch the Sur­
face of the Liquid in one Point and no more, then it ſhall ſtay. For the Angle at N
167[Figure 67]
ſhall be equall to the Angle at φ; and the Line B S
equall to the Line B C; and S R to C R: Where­
fore, M H ſhall be likewiſe equall to P Y. There­
fore, having drawn HK and prolonged it; the
Centre of Gravity of the whole Portion ſhall be
K; of that which is in the Liquid H; and of
that which is above it, the Centre ſhall be in
the Line prolonged: let it be in ω. There­
fore, along that ſame Line K H, which is per­
pendicular to the Surface of the Liquid, ſhall
the part which is within the Liquid move up­
wards, and that which is above the Liquld
downwards: And, for this cauſe, the Portion,
ſhall be no longer moved, but ſhall ſtay, and
reſt, ſo, as that its Baſe do touch the Liquids Surface in but one Point; and its Axis
maketh an Angle therewith equall to the Angle φ; And, this is that which we were to
demonſtrate.
F
(g) By 9 of t
fifth.
CONCLVSION IV.
If the Portion have greater proportion in Gravity
to the Liquid, than the Square F P to the Square
B D, but leſſer than that of the Square X O to the
Square B D, being demitted into the Liquid,
and inclined, ſo, as that its Baſe touch not the
Liquid, it ſhall ſtand and reſt, ſo, as that its Baſe
ſhall be more ſubmerged in the Liquid.
Again, let the Portion have greater proportion in
Gravity to the Liquid, than the Square F P to the
Square B D, but leſſer than that of the Square X O to
the Square B D; and as the Portion is in Gravity to the Liquid,
ſo let the Square made of the Line ψ be to the Square B D. Ψ
ſhall be greater than F P, and leſſer than X O. Apply, therefore,
the right Line I V to fall betwixt the Portions A V Q L and A X D;
and let it be equall to ψ, and parallel to B D; and let it meet
the Remaining Section in Y: V Y ſhall alſo be proved double
to Y I, like as it hath been demonſtrated, that O G is double off
G X. And, draw from V, the Line V ω, touching the Section
A V Q L in V; and drawing a Line from A to I, prolong it unto
que We prove in the ſame manner, that the Line A I is equall
to I que and that A Q is parallel to V ω. It is to be demonſtrated,
that the Portion being demitted into the Liquid, and ſo inclined,
as that its Baſe touch not the Liquid, ſhall ſtand, ſo, that its Baſe
ſhall be more ſubmerged in the Liquid, than to touch it Surface in
1but one Point only. For let it be de­
68[Figure 68]
mitted into the Liquid, as hath been
ſaid; and let it firſt be ſo inclined, as
that its Baſe do not in the leaſt
touch the Surface of the Liquid. And
then it being cut thorow the Axis,
by a Plane erect unto the Surface of
the Liquid, let the Section of the
Portion be A N Z G; that of the
Liquids Surface E Z; the Axis of
the Portion and Diameter of the
Section B D; and let B D be cut in
the Points K and R, as before; and
draw N L parallel to E Z, and touching the Section A N Z G
in N, and N S perpendicular to
69[Figure 69]
B D. Now, ſeeing that the Por­
tion is in Gravity unto the Liquid,
as the Square made of the Line
is to the Square B D; ψ ſhall
be equall to N T: Which is to
be demonſtrated as above: And,
therefore, N T is alſo equall to
V I: The Portions, therefore,
A V Q and E N Z are equall to
one another. And, ſince that in
the Equall and like Portions A V
Q L and A N Z G, there are drawn A Q and E Z, cutting off
equall Portions, that from the
70[Figure 70]
Extremity of the Baſe, this not
from the Extreme, that which is
drawn from the Extremity of the
Baſe, ſhall make the Acute Angle
with the Diameter of the Portion
leſſer: and in the Triangles N L S
and V ω C, the Angle at L is
greater than the Angle at ω:
Therefore, B S ſhall be leſſer
than B C; and S R leſſer than
C R: and, conſequently, N X
greater than V H; and X T leſſer than H I. Seeing, therefore,
that V Y is double to Y I; It is manifeſt, that N X is greater than
double to X T. Let N M be double to M T: It is manifeſt, from what
hath been ſaid, that the Portion ſhall not reſt, but will incline, untill
that its Bafe do touch the Surface of the Liquid: and it toucheth it in
one Point only, as appeareth in the Figure: And other things
171[Figure 71]
ſtanding as before, we will again
demonſtrate, that N T is equall to
V I; and that the Portions A V Q
and A N Z are equall to each other.
Therefore, in regard, that in the
Equall and Like Portions A V Q L
and A N Z G, there are drawn
A Q and A Z cutting off equall Por­
tions, they ſhall with the Diameters
of the Portions, contain equall
Angles. Therefore, in the Triangles
N L S and V ω C, the Angles at
the Points L and ω are equall; and the Right Line B S equall to
B C; S R to C R; N X to V H; and X T to H I: And, ſince
V Y is double to Y I, N X ſhall be greater than double of X T.
Let therefore, N M be double to M T. It is hence again manifeſt,
that the Portion will not remain, but ſhall incline on the part
towards A: But it was ſuppoſed, that the ſaid Portion did
touch the Surface of the Liquid in one ſole Point: Therefore,
its Baſe muſt of neceſſity ſubmerge farther into the Liquid.
CONCLVSION V.
If the Portion have leſſer proportion in Gravity to
the Liquid, than the Square F P to the Square
B D, being demitted into the Liquid, and in­
clined, ſo, as that its Baſe touch not the Liquid,
it ſhall ſtand ſo inclined, as that its Axis ſhall
make an Angle with the Surface of the Liquid,
leſſe than the Angle ψ; And its Baſe ſhall
not in the leaſt touch the Liquids Surface.
Finally, let the Portion have leſſer proportion to the Liquid
in Gravity, than the Square F P hath to the Square B D; and
as the Portion is in Gravity to the Liquid, ſo let the
Square made of the Line ψ be to the Square B D. ψ ſhall be
leſſer than P F. Again, apply any Right Line as G I, falling
betwixt the Sections A G Q L and A X D, and parallel to B D;
and let it cut the Middle Conick Section in the Point H, and
1the Right Line R Y in Y. We
72[Figure 72]
ſhall demonſtrate G H to be double
to H I, as it hathbeen demonſtra­
ted, that O G is double to G X.
Then draw G ω touching the Section
A G Q L in G; and G C perpen di­
cular to B D; and drawing a Line
from A to I, prolong it to que Now
A I ſhall be equall to I que and
A Q parallel to G ω. It is to be
demonſtrated, that the Portion being
demitted into the Liquid, and inclined, ſo, as that its Baſe touch
the Liquid, it ſhall ſtand ſo incli­
73[Figure 73]
ned, as that its Axis ſhall make
an Angle with the Surface of the
Liquid leſſe than the Angle φ;
and its Baſe ſhall not in the leaſt
touch the Liquids Surface. For
let it be demitted into the Liquid,
and let it ſtand, ſo, as that its Baſe
do touch the Surface of the Liquid
in one Point only: and the Portion
being cut thorow the Axis by a
Plane erect unto the Surface of the Liquid, let the Section of
74[Figure 74]
the Portion be A N Z L, the Section
of a Rightangled Cone; that of
the Surface of the Liquid A Z; and
the Axis of the Portion and Dia­
meter of the Section B D; and let
B D be cut in the Points K and R
as hath been ſaid above; and draw
N F parallel to A Z, and touching
the Section of the Cone in the Point
N; and N T parallel to B D; and
N S perpendicular to the ſame. Be­
cauſe, now, that the Portion is in Gravity to the Liquid, as
the Square made of ψ is to the Square B D; and ſince that as the
Portion is to the Liquid in Gravity, ſo is the Square N T to the
Square B D, by the things that have been ſaid; it is plain, that
N T is equall to the Line ψ: And, therefore, alſo, the Portions
A N Z and A G Q are equall. And, ſeeing that in the Equall and
Like Portions A G Q L and A N Z L; there are drawn from the
Extremities of their Baſes, A Q and A Z which cut off equall Porti­
ons: It is obvious, that with the Diameters of the Portions they
1make equall Angles; and that in the Triangles N F S and G ω C
the Angles at F and ω are equall; as alſo, that S B and B C, and
S R and C R are equall to one another: And, therefore, N X and
G Y are alſo equall; and X T and Y I. And ſince G H is double
to H I, N X ſhall be leſſer than double of X T. Let N M therefore
be double to M T; and drawing a Line from M to K, prolong it
unto E. Now the Centre of Gravity of the whole ſhall be the
Point K; of the part which is in the Liquid the Point M; and
that of the part which is above the Liquid in the Line prolonged
as ſuppoſe in E. Therefore, by what was even now demonſtrated
it is manifeſt that the Portion ſhall not ſtay thus, but ſhall incline, ſo
as that its Baſe do in no wiſe touch the Surface of the Liquid
And that the Portion will ſtand, ſo, as to make an Angle with the
Surface of the Liquid leſſer than
75[Figure 75]
the Angle φ, ſhall thus be demon
ſtrated. Let it, if poſſible, ſtand,
ſo, as that it do not make an Angle
leſſer than the Angle φ; and diſpoſe
all things elſe in the ſame manner a
before; as is done in the preſet
Figure. We are to demonſtrat
in the ſame method, that N T is e­
quall to ψ; and by the ſame reaſor
equall alſo to G I. And ſince that in
the Triangles P φ C and N F S, the Angle F is not leſſer than the
Angle φ, B F ſhall not be greater than B C: And, therefore, neither
ſhall S R be leſſer than C R; nor N X than P Y: But ſince P F is
greater than N T, let P F be Seſquialter of P Y: N T ſhall be leſſer
than Seſquialter of N X: And, therefore, N X ſhall be greate
than double of X T. Let N M be double of M T; and drawing
Line from M to K prolong it. It is manifeſt, now, by what hath
been ſaid, that the Portion ſhall not continue in this poſition, but ſhall
turn about, ſo, as that its Axis do make an Angle with the Surface
of the Liquid, leſſer than the Angle φ.