ARCHIMEDES
HIS TRACT
De Incidentibus Humido,
OR OF THE
NATATION OF BODIES VPON,
OR SVBMERSION IN,
THE
WATER
OR OTHER LIQUIDS.
IN TWO BOOKS.
Tran&longs;lated from the Original Greek,
Fir&longs;t into Latine, and afterwards into Italian, by
TARTAGLIA,
&longs;trated by way of Dialogue, with
a Noble Engli&longs;h Gentleman, and his Friend.
Together with the Learned Commentaries of
Commandino,
as, thorow the Injury of Time, were obliterated.
Now compared with the ORIGINAL, and Engli&longs;hed
By
ARCHIMEDES
HIS TRACT
INCIDENTIBUS HUMIDO,
OR OF
The Natation of Bodies upon, or Submer&longs;ion in,
the Water, or other Liquids.
BOOK I.
RICARDO.
in which I find not any thing that will not certainly hold
true; but, truth is, there are many of your Conclu&longs;ions
of which I under&longs;tand uot the Cau&longs;e, and therefore, if it
be not a trouble to you, I would de&longs;ire you to declare them
to me, for, indeed, nothing plea&longs;eth me, if the Cau&longs;e
thereof be hid from me.
NICOLO.
My obligations unto you are &longs;o many and
great,
to be trouble&longs;ome to me, and therefore tell me what tho&longs;e Perticulars are of which
you know not the Cau&longs;e, for I &longs;hall endeavour with the utmo&longs;t of my power and
under&longs;tanding to &longs;atisfie you in all your demands.
RIC.
In the fir&longs;t
you conclude, That it is impo&longs;&longs;ible that the Water &longs;hould wholly receive into it
any material Solid Body that is lighter than it &longs;eif (as to
there will alwaies a part of the Body &longs;tay 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 &longs;hall be immerged, or received, into the Wa
ter, &longs;o &longs;hall the Gravity of the Water be to the Gravity
material Body: And that tho&longs;e Solid Bodies, that are by nature more Grave than the
Water, being put into the Water, &longs;hall pre&longs;ently make the &longs;aid Water give place;
and, That they do not only wholly enter or &longs;ubmerge in the &longs;ame, but go continu
ally de&longs;cending untill they arrive at
tom &longs;o much fa&longs;ter, by how much they are more Grave than the Water. And,
again, That tho&longs;e which are preci&longs;ely of the &longs;ame Gravity with the Water, being
put into the &longs;ame, are of nece&longs;&longs;ity wholly received into, or immerged by it, but
yet retained in the Surface of the &longs;aid Water, and much le&longs;s will the Water con
&longs;ent that it do de&longs;cend to the Bottom: and, now, albeit that all the&longs;e things are
manife&longs;t to Sen&longs;e and Experience, yet neverthele&longs;s would I be very glad, if it be
po&longs;&longs;ible, that you would demon&longs;trate to me the mo&longs;t apt and proper Cau&longs;e of
the&longs;e Effects.
NIC.
The Cau&longs;e of all the&longs;e Effects is a&longs;&longs;igned by
that Book
your &longs;elf, as I al&longs;o &longs;aid in the beginning of that my
by me
the more Compre
hen&longs;ive word, for
his Doctrine holds
true in all Liquids
as well as in Wa
ter,
Oyl, Milk,
RIC.
I have &longs;een that &longs;ame
two Books in which he treateth
Center of Gravity in Figures plain, or parallel to the Horizon; and likewi&longs;e tho&longs;e
eth of Solids that Swim upon, or &longs;ink in Liquids, is &longs;o ob&longs;cure, that, to &longs;peak the
truth, there are many things in
we proceed any farther, I &longs;hould take it for a favour if you would declare it to me
in your Vulgar Tongue, beginning with his fir&longs;t
manner.
one Book,
lia
ted no more.
SVPPOSITION I.
its parts being equi-jacent and contiguous, the le&longs;s
pre&longs;&longs;ed are repul&longs;ed by the more pre&longs;&longs;ed. And
that each of its parts is pre&longs;&longs;ed or repul&longs;ed by the
Liquor that lyeth over it, perpendicularly, if the
Liquid be de&longs;cending into any place, or pre&longs;&longs;ed any
whither by another.
NIC.
Every Science, Art, or Doctrine (as you know,
hath its fir&longs;t undemon&longs;trable Principles, by which (they being
granted or &longs;uppo&longs;ed) the &longs;aid Science is proved, maintained, or de
mon&longs;trated. And of the&longs;e Principles, &longs;ome are called
and others
of tho&longs;e Material Solids that Swim or Sink in Liquids, hath only two undemon
&longs;trable
with your de&longs;ire I have &longs;et down in our Vulgar Tongue.
RIC.
Before you proceed any farther tell me, how we are to under&longs;tand the
parts of a Liquid to be
NIC.
When they are equidi&longs;tant from the Center of the World, or of the
Earth (which is the &longs;ame, although ^{*} &longs;ome hold that the Centers of the Earth
and Worldare different.)
RIC.
I under&longs;tand you not unle&longs;s you give me &longs;ome Example thereof in
Figure.
cans.
NIC.
To exemplifie this particular, Let us &longs;uppo&longs;e a quantity of Liquor (as
for in&longs;tance 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 &longs;uch a
manner as that the &longs;aid Section may pa&longs;s ^{*} by the Center of the Earth: And let
us &longs;uppo&longs;e 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 de&longs;cribe a Circle upon the
&longs;aid Center K, of &longs;uch a bigne&longs;s as that the Circumference may pa&longs;s 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 &longs;aid Circumference, cutting the
&longs;ame, as KE, KHO, KFQ KLP, KM. Now I &longs;ay, that all the&longs;e parts of
the &longs;aid Water, terminated in that Circumference, are Equijacent, as being all
M L, L F, F H, H E.
RIC.
I under&longs;tand you very well, as to this particular: But tell me a little; he
&longs;aith that each of the parts of the Liquid is pre&longs;&longs;ed or repul&longs;ed 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 &longs;ome Water, each part of the Water that is in that Line he &longs;uppo&longs;eth to
be pre&longs;&longs;ed or repul&longs;ed by the Water that lieth above it in that &longs;ame Line, and that
that repul&longs;e is made according to the &longs;ame Line, (that is, directly towards the
Center of the World) which Line is called a Perpendicular; becau&longs;e 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 under&longs;tand me, let
us imagine
the Line KHO,
and in that
let us imagine
&longs;everal parts,
as &longs;uppo&longs;e RS,
S T, T V, V H,
H O. I &longs;ay,
that he &longs;up
po&longs;eth that
the part V H
is pre&longs;&longs;ed by
that placed a
bove it, H O,
according to
the Line OK;
the which
O K, as hath been &longs;aid above, is called the Perpendicular pa&longs;&longs;ing thorow tho&longs;e two
parts. In like manner, I &longs;ay that the part T V is expul&longs;ed by the part V H, ac
cording to the &longs;aid Line O K: and &longs;o the part S T to be pre&longs;&longs;ed by T V, according
to the &longs;aid Perpendicular O K, and R S by S T. And this you are to under&longs;tand
in all the other Lines that were protracted from the &longs;aid Point K, penetrating the
&longs;aid Water, As for Example, in
like kind.
RIC. Indeed,
tisfaction; for, in my Judgment, it &longs;eemeth that all the difficulty of this Suppo&longs;ition
con&longs;i&longs;ts in the&longs;e two particulars which you have declared to me.
NIC.
It doth &longs;o; for having under&longs;tood that the parts E H, H F, F L, L M, and
MG, determining in the Circumference of the &longs;aid Circle are equijacent, it is an
ea&longs;ie matter to under&longs;tand the fore&longs;aid
&longs;uppo&longs;ed that the Liquid is of &longs;uch a nature, that the part thereof le&longs;s pre&longs;&longs;ed or thrust is re
pul&longs;ed by the more thru&longs;t or pre&longs;&longs;ed.
more thru&longs;t, crowded, or pre&longs;&longs;ed from above downwards by the Liquid, or &longs;ome
other matter that was over it, than the part H F, contiguous to it, it is &longs;uppo&longs;ed
that the &longs;aid part H F, le&longs;s pre&longs;&longs;ed, would be repul&longs;ed by the &longs;aid part E H. And
thus we ought to under&longs;tand of the other parts equijacent, in ca&longs;e that they be
contiguous, and not &longs;evered. That each of the parts thereof is pre&longs;&longs;ed and repul.
&longs;ed by the
&longs;aid above, to wit, that it &longs;hould be repul&longs;ed, in ca&longs;e the
any place, and thru&longs;t, or driven any whither by another.
RIC.
I under&longs;tand this Suppo&longs;ition very well, but yet me thinks that before
the Suppo&longs;ition, the Author ought to have defined tho&longs;e two particulars, which
you fir&longs;t declared to me, that is, how we are to under&longs;tand the parts of the
equijacent, and likewi&longs;e the Perpendicular.
NIC.
You &longs;ay truth.
RIC.
I have another que&longs;tion to aske you, which is this, Why the Author
u&longs;eth the word
NIC.
It may be for two of the&longs;e two Cau&longs;es; the one is, that Water being the
principal of all
the chief Liquid, that is Water: The other, becau&longs;e that all the Propo&longs;itions of
this Book of his, do not only hold true in Water, but al&longs;o in every other
as in Wine, Oyl, and the like: and therefore the Author might have u&longs;ed the word
RIC.
This I under&longs;tand, therefore let us come to the fir&longs;t
you know, in the Original &longs;peaks in this manner.
PROP. I. THEOR. I.
Point, and the Section be alwaies the Circumference
of a Circle, who&longs;e Center is the &longs;aid Point: that Su
perficies &longs;hall be Spherical.
Let any Superficies be cut at plea&longs;ure by a Plane thorow the
Point K; and let the Section alwaies de&longs;cribe the Circumfe
rence of a Circle that hath for its Center the Point K: I &longs;ay,
that that &longs;ame Superficies is Sphærical. For were it po&longs;&longs;ible that the
&longs;aid Superficies were not Sphærical, then all the Lines drawn
through the &longs;aid Point K unto that Superficies would not be equal,
Let therefore A and B be two
Points in the &longs;aid Superficies, &longs;o that
drawing the two Lines K A and
K B, let them, if po&longs;&longs;ible, be une
qual: Then by the&longs;e two Lines let
a Plane be drawn cutting the &longs;aid
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-&longs;uppo&longs;al, a Circle, and
the Center thereof is the Point K, for &longs;uch the &longs;aid Superficies was
&longs;uppo&longs;ed to be. Therefore the two Lines K A and K B are equal:
But they were al&longs;o &longs;uppo&longs;ed to be unequal; which is impo&longs;&longs;ible:
It followeth therefore, of nece&longs;&longs;ity, that the &longs;aid Superficies be
Sphærical, that is, the Superficies of a Sphære.
RIC.
I under&longs;tand you very well; now let us proceed to the &longs;econd
which, you know, runs thus.
PROP. II. THEOR. II.
&longs;etled &longs;hall be of a Sphærical Figure, which Figure
&longs;hall have the &longs;ame Center with the Earth.
Let us &longs;uppo&longs;e a Liquid that is of &longs;uch a con&longs;i&longs;tance 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 &longs;ay that the Line A B G D is the Circumference of a
Circle, and that the Center
thereof is the Point K And
if it be po&longs;&longs;ible that it may
not be the Circumference
of a Circle, the Right
Lines drawn ^{*} by the Point
K to the &longs;aid Line A B G D
&longs;hall not be equal. There
fore let a Right-Line be
taken greater than &longs;ome of tho&longs;e produced from the Point K unto
the &longs;aid Line A B G D, and le&longs;&longs;er than &longs;ome other; and upon the
Point K let a Circle be de&longs;cribed at the length of that Line,
Now the Circumference of this Circle &longs;hall fall part without the
&longs;aid Line A B G D, and part within: it having been pre&longs;uppo&longs;ed
that its Semidiameter is greater than &longs;ome of tho&longs;e Lines that may
be drawn from the &longs;aid Point K unto the &longs;aid Line A B G D, and
le&longs;&longs;er than &longs;ome other. Let the Circumference of the de&longs;cribed
Circle be R B G H, and from B to K draw the Right-Line B K: and
drawn al&longs;o the two Lines K R, and K E L which make a Right
Angle in the Point K: and upon the Center K de&longs;cribe 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 rea&longs;ons alledged upon the fir&longs;t
jacent, or equipo&longs;ited, and contiguous to each other; and both
the&longs;e parts are pre&longs;t or thru&longs;t, according to the &longs;econd part of the And becau&longs;e the
two Angles E K B and B K R are &longs;uppo&longs;ed equal [
be equal (fora&longs;much as R B G H was a Circle de&longs;cribed for &longs;atis
faction of the Oponent, and K its Center:) And in like manner
the whole Triangle B E K &longs;hall be equal to the whole Triangle
B R K. And becau&longs;e al&longs;o the Triangle O P K for the &longs;ame rea&longs;on
Notion) &longs;ub&longs;tracting tho&longs;e two &longs;mall Triangles O P K and O X K
from the two others B E K and B R K, the two Remainders &longs;hall
be equal: one of which Remainders &longs;hall be the Quadrangle
B E O P, and the other B R X O. And becau&longs;e 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 re&longs;idue 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 &longs;aid
Quadrangle B E O P be more Grave than the Quadrangle
B R X O, much more &longs;hall the Quadrangle B L O P exceed in Gra
vity the &longs;aid Quadrangle B R X O: whence it followeth, that the
part O P is more pre&longs;&longs;ed than the part O X. But, by the fir&longs;t part
of the Suppo&longs;ition, the part le&longs;s pre&longs;&longs;ed &longs;hould be repul&longs;ed by the
part more pre&longs;&longs;ed: Therefore the part O X mu&longs;t be repul&longs;ed by
the part O P: But it was pre&longs;uppo&longs;ed that the Liquid did not
move: Wherefore it would follow that the le&longs;s pre&longs;&longs;ed would not
be repul&longs;ed by the more pre&longs;&longs;ed: And therefore it followeth of
nece&longs;&longs;ity that the Line A
and that the Center of it is the point K. And in like manner &longs;hall
it be demon&longs;trated, if the Surface of the Liquid be cut by a Plane
thorow the Center of the Earth, that the Section &longs;hall be the Cir
cumference of a Circle, and that the Center of the &longs;ame &longs;hall be
that very Point which is Center of the Earth. It is therefore mani
fe&longs;t that the Superficies of a Liquid that is con&longs;i&longs;tant and &longs;etled
&longs;hall have the Figure of a Sphære, the Center of which &longs;hall be
the &longs;ame with that of the Earth, by the fir&longs;t
&longs;uch that being ever cut thorow the &longs;ame Point, the Section or Di
vi&longs;ion de&longs;cribes the Circumference of a Circle which hath for Cen
ter the &longs;elf-&longs;ame Point that is Center of the Earth: Which was to
be demon&longs;trated.
RIC.
I do thorowly under&longs;tand the&longs;e your Rea&longs;ons, and &longs;ince there is in them
no umbrage of Doubting, let us proceed to his third
PROP. III. THEOR. III.
Liquid are al&longs;o equal to it in Gravity, being demit-
&longs;ame as that they lie or appear not at all above the
Surface of the Liquid, nor yet do they &longs;ink to the
Bottom.
&longs;etled, as nece&longs;&longs;ary
in making the Ex
periment.
NIC.
In this
pen to be equal in &longs;pecifical Gravity with the Liquid being lefeat liber
ty in the &longs;aid Liquid do &longs;o &longs;ubmerge in the &longs;ame, as that they lie or ap
pear not at all above the Surface of the Liquid, nor yet do they go or &longs;ink to the
Bottom.
For &longs;uppo&longs;ing, on the contrary, that it were po&longs;&longs;ible for one of
tho&longs;e Solids being placed in the Liquid to lie in part without the
Liquid, that is above its Surface, (alwaies provided that the &longs;aid
Liquid be &longs;etled and undi&longs;turbed,) 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 Super&longs;icies E Z H T, and let us &longs;uppo&longs;e
the Center of the Earth to be the Point K: and let the part of the
&longs;aid Solid &longs;ubmerged in the Liquid be B G H T, and let that above
be B E Z G: and let the Solid Body be &longs;uppo&longs;ed to be comprized in
a Pyramid that hath its Parallelogram Ba&longs;e in the upper Surface of
the Liquid, and its Summity or Vertex in the Center of the Earth:
which Pyramid let us al&longs;o &longs;uppo&longs;e to be cut or divided by the &longs;ame
Plane in which is the Circumference A B G D, and let the Sections
of the Planes of the &longs;aid
Pyramid be K L and
K M: and in the Liquid
about the Center K let
there be de&longs;cribed 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 &longs;uppo&longs;ed equal and like to that which compri&longs;eth the &longs;aid
Solid Body, and contiguous and conjunct with the &longs;ame; and let
the Sections of its Superficies be K M and K N: and let us &longs;uppo&longs;e
another Solid to be taken or imagined, of Liquor, contained in that
&longs;ame Pyramid, which let be R S C Y, equal and like to the partial
Solid B H G T, which is immerged in the &longs;aid Liquid: But the
part of the Liquid which in the fir&longs;t 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 equipo&longs;ited and contiguous, but
are not pre&longs;&longs;ed equally; for that which is under the Superficies
X O is pre&longs;&longs;ed 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 pre&longs;&longs;ed by the Solid R S C Y, and by the Liquid
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, &longs;hall be le&longs;&longs;er
than that which is according to L M X O; becau&longs;e that Solid of
Liquor which proceeds according to R S C Y is le&longs;s than the Solid
E Z H T (having been &longs;uppo&longs;ed to be equal in quantity to only
the part H B G T of that:) And the &longs;aid Solid E Z H T hath been
&longs;uppo&longs;ed to be equally grave with the Liquid: Therefore the Gra
vity of the
cies L M and
Pyramid, together with
the whole Solid EZHT,
&longs;hall exceed the Gravity
of the Liquid compri
&longs;ed 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, &longs;uppo&longs;ed to remain above the Surface of
the Liquid: And therefore it is manife&longs;t that the part which pro
ceedeth according to the Circumference O P is pre&longs;&longs;ed, driven, and
repul&longs;ed, according to the
cording to the Circumference X O, by which means the Liquid
would not be &longs;etled and &longs;till: But we did pre&longs;uppo&longs;e that it was
&longs;etled, namely &longs;o, as to be without motion: It followeth, therefore,
that the &longs;aid Solid cannot in any part of it exceed or lie above the
Superficies of the Liquid: And al&longs;o that being dimerged in the Li
quid it cannot de&longs;cend to the Bottom, for that all the parts of the
Liquid equijacent, or di&longs;po&longs;ed equally, are equally pre&longs;&longs;ed, becau&longs;e
the Solid is equally grave with the Liquid, by what we pre&longs;uppo&longs;ed.
RIC.
I do under&longs;tand your Argumentation, but I under&longs;tand not that Phra&longs;e
NIC.
I will declare this Term unto you.
re&longs;pecteth all the Species of Continual Quantity; and the Species of Continual
Quantity are three, that is, the
is al&longs;o called a Solid, as having in it &longs;elf
and therefore that none might equivocate or take that Term
meant of
did &longs;pecifie it by that manner of expre&longs;&longs;ion, as was &longs;aid. The truth is, that he
might have expre&longs;t that
of equal Gravity with an equal Ma&longs;s of the Liquid,And this
been more cleer and intelligible, for it is as &longs;ignificant to &longs;ay, a
to &longs;ay, a
kinds of words indifferently.
RIC.
You have &longs;ufficiently &longs;atisfied me, wherefore that we may lo&longs;e no time
let us go forwards to the fourth
PROP. IV. THEOR. IV.
being demitted into the &longs;etled Liquid, will not total
ly &longs;ubmerge in the &longs;ame, but &longs;ome part thereof will
lie or &longs;tay above the Surface of the Liquid.
NIC.
In this fourth
lighter (as to Specifical Gravity) than the
will &longs;tay and appear without the
For &longs;uppo&longs;ing, on the contrary, that it were po&longs;&longs;ible for a Solid
more light than the Liquid, being demitted in the Liquid to &longs;ub
merge totally in the &longs;ame, that is, &longs;o as that no part thereof re
maineth above, or without the &longs;aid Liquid, (evermore &longs;uppo&longs;ing
that the Liquid be &longs;o con&longs;tituted 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
Center of the Earth be K. And let there be imagined a Pyramid
that compri&longs;eth 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
and K And let us ima
gine another Pyramid equal and like to this, and let its Superficies
be cut by the Superficies A
the Superficies of another Sphære be de&longs;cribed in the Liquid, upon
the Center K, and beneath the Solid R; and let that be cut by the
&longs;ame Plane according to
ther Solid taken ^{*} from the Liquid, in this &longs;econd 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
ramid, and that which is under the Spherical Superficies that pro
ceeds according to the Circumference O P, in the &longs;econd Pyramid,
are equijacent, and contiguous, but are not pre&longs;&longs;ed equally; for
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 pre&longs;&longs;ed by the Solid H, &longs;uppo&longs;ed to be of the &longs;ame Li
quid, and by the Liquid which that containeth, that is, that which
is in the place of the &longs;aid Pyramid according to P O B G: and the
Gravity of the Solid R is le&longs;s than the Gravity of the Liquid
H, for that the&longs;e two Magnitudes were &longs;uppo&longs;ed to be equal in
Ma&longs;s, and the Solid R was &longs;uppo&longs;ed to be lighter than the Liquid:
and the Ma&longs;&longs;es of the two Pyramids of Liquor that containeth the&longs;e
two Solids R and H are equal ^{*} by what was pre&longs;uppo&longs;ed: There
fore the part of the Liquid that is under the Superficies that pro
ceeds according to the Circumference O P is more pre&longs;&longs;ed; and,
therefore, by the
pre&longs;&longs;ed, whereby the &longs;aid Liquid will not be &longs;etled: But it was be
fore &longs;uppo&longs;ed that it was &longs;etled: Therefore that Solid R &longs;hall not
totally &longs;ubmerge, but &longs;ome part thereof will remain without the
Liquid, that is, above its Surface, Which was the
the Liquid.
ramids were &longs;uppo
&longs;ed equal.
RIC.
I have very well under&longs;tood you, therefore let us come to the fifth
po&longs;ition,
PROP. V. THEOR. V.
being demitted in the (&longs;etled) Liquid, will &longs;o far
&longs;ubmerge, till that a Ma&longs;s of Liquor, equal to the
Part &longs;ubmerged, doth in Gravity equalize the
whole Magnitude.
NIC.
It having, in the precedent, been demon&longs;trared that Solids lighter than
the Liquid, being demitted in the
without the
a&longs;&longs;erted, that &longs;o much of &longs;uch a Solid &longs;hall &longs;ubmerge, as that a Ma&longs;s of the
Solid.
And to demon&longs;trate this, let us a&longs;&longs;ume all the &longs;ame Schemes
as before, in
led, and let the Solid E Z H T be lighter than the Liquid. Now if the &longs;aid Liquid be &longs;etled, the parts of it that are equija
cent are equally pre&longs;&longs;ed: Therefore the Liquid that is beneath
and P O are equally pre&longs;&longs;ed; whereby the Gravity pre&longs;&longs;ed is equal.
But the Gravity of the
Liquid which is in the
fir&longs;t 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 manife&longs;t, 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 manife&longs;t that a Ma&longs;s of Liquor
equal in Ma&longs;s to the part of the Solid &longs;ubmerged is equal in Gra
vity to the whole Solid.
being deducted.
RIC.
This was a pretty Demon&longs;tration, and becau&longs;e I very well under&longs;tand
it, let us lo&longs;e no time, but proceed to the &longs;ixth
PROP. VI. THEOR. VI.
into the Liquid, are repul&longs;ed upwards with a Force
as great as is the exce&longs;s of the Gravity of a Ma&longs;s
of Liquor equal to the Magnitude above the Gra
vity of the &longs;aid Magnitude.
NIC.
This &longs;ixth
demitted, thru&longs;t, or trodden by Force underneath the Liquids Sur
face, are returned or driven upwards with &longs;o much Force, by
how much a quantity of the Liquid equal to the. Solid &longs;hall
exceed the &longs;aid Solid in Gravity.
And to delucidate this
than the
Solid A is B: and let the Gravity of a
be B G. I &longs;ay, that the Solid A depre&longs;&longs;ed or demitted with Force
into the &longs;aid
a Force equal to the Gravity G. And to demon&longs;trate this
&longs;ition,Now
the Solid compounded of the two Solids A and D will be lighter
than the
them both is BG, and the Gravity of as much Liquor as equal
leth in greatne&longs;s the Solid A, is greater than the &longs;aid Gravity BG,
Therefore the Solid compounded of tho&longs;e two Solids A and D
being dimerged, it &longs;hall, by the precedent, &longs;o much of it &longs;ubmerge,
as that a quantity of the Liquid equal to the &longs;aid &longs;ubmerged part
&longs;hall have equal Gravity with the &longs;aid compounded Solid. And
for an example of that
perficies of any Liquid be that which pro
ceedeth according to the Circumference
A B G D: Becau&longs;e now a Ma&longs;s or quantity
of Liquor as big as the Ma&longs;s A hath equal
Gravity with the whole compounded Solid
A D: It is manife&longs;t that the &longs;ubmerged part
thereof &longs;hall be the Ma&longs;s A: and the remain
der, namely, the part D, &longs;hall be wholly a
top, that is, above the Surface of the Liquid. It is therefore evident, that the part A hath &longs;o much virtue or
Force to return upwards, that is, to ri&longs;e from below above the Li
quid, as that which is upon it, to wit, the part D, hath to pre&longs;s it
downwards, for that neither part is repul&longs;ed by the other: But D
pre&longs;&longs;eth downwards with a Gravity equal to G, it having been &longs;up
po&longs;ed that the Gravity of that part D was equal to G: Therefore
that is manife&longs;t which was to be demon&longs;trated.
RIC.
This was a fine Demon&longs;tration, and from this I perceive that you colle
cted your
the fir&longs;t Book for the recovering of a Ship &longs;unk: and, indeed, I have many Que
&longs;tions to ask you about that, but I will not now interrupt the Di&longs;cour&longs;e in hand, but
de&longs;ire that we may go on to the &longs;eventh
PROP. VII. THEOR. VII.
mitted into the [&longs;etled] Liquid, are boren down
wards as far as they can de&longs;cend: and &longs;hall be lighter
in the Liquid by the Gravity of a Liquid Ma&longs;s of
the &longs;ame bigne&longs;s with the Solid Magnitude.
NIC.
This &longs;eventh
The fir&longs;t is, That all Solids heavier than the Liquid, being demit
ted into the
as they can de&longs;cend, that is untill they arrive at the Bottom. Which
fir&longs;t part is manife&longs;t, becau&longs;e the Parts of the
under that Solid, are more pre&longs;&longs;ed than the others equijacent,
becau&longs;e that that Solid is &longs;uppo&longs;ed more grave than the Liquid.
is affirmed in the &longs;econd part, &longs;hall be demon&longs;trated in this man
ner. Take a Solid, as &longs;uppo&longs;e A, that is more grave than the Li
quid, and &longs;uppo&longs;e the Gravity of that &longs;ame Solid A to be BG. And of a Ma&longs;s of
po&longs;e the Gravity to be B: It is to be demon&longs;trated that the Solid
A, immerged in the Liquid, &longs;hall have a Gravity equal to G. And
to demon&longs;trate this, let us imagine another Solid, as &longs;uppo&longs;e D,
more light than the Liquid, but of &longs;uch a quality as that its Gravi
ty is equal to B: and let this D be of &longs;uch a Magnitude, that a
Ma&longs;s of
B G. Now the&longs;e two Solids D and A being compounded toge
ther, all that Solid compounded of the&longs;e two &longs;hall be equally
Grave with the Water: becau&longs;e the Gravity of the&longs;e two Solids
together &longs;hall be equal to the&longs;e two Gravities, that is, to B G, and
to B; and the Gravity of a Liquid that hath its
Ma&longs;s equal to the&longs;e two Solids A and D, &longs;hall be
equal to the&longs;e two Gravities B G and B.
the&longs;e two Solids, therefore, be put in the
and they &longs;hall ^{*} remain in the Surface of that
quid, (that is, they &longs;hall not be drawn or driven
upwards, nor yet downwards:) For if the Solid
A be more grave than the Liquid, it &longs;hall be
drawn or born by its Gravity downwards to
wards the Bottom, with as much Force as by the Solid D it is thru&longs;t
upwards: And becau&longs;e the Solid D is lighter than the
&longs;hall rai&longs;e it upward with a Force as great as the Gravity G: Be
cau&longs;e it hath been demon&longs;trated, in the &longs;ixth
lid Magnitudes that are lighter than the Water, being demitted in
the &longs;ame, are repul&longs;ed or driven upwards with a Force &longs;o much the
greater by how much a
Grave than the &longs;aid Solid: But the
with the Solid D, is more grave than the &longs;aid Solid D, by the Gra
vity G: Therefore it is manife&longs;t, that the Solid A is pre&longs;&longs;ed or
born downwards towards the Centre of the World, with a Force
as great as the Gravity G: Which was to be demon&longs;trated.
be equall in Gravi
ty to the Liquid,
neither moving up
wards or down
wards.
RIC.
This hath been an ingenuous Demon&longs;tration; and in regard I do &longs;uffici
ently under&longs;tand it, that we may lo&longs;e no time, we will proceed to the &longs;econd
&longs;ition,
SVPPOSITION II.
wards, do all a&longs;cend according to the Perpendicular
which is produced thorow their Centre of Gravity.
COMMANDINE.
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 under&longs;tanding of this &longs;econd
that every Solid that is lighter than the Liquid being by violence, or by &longs;ome other
occa&longs;ion, &longs;ubmerged in the Liquid, and then left at liberty, it &longs;hall, by that which
hath been proved in the &longs;ixth
and that impul&longs;e or thru&longs;ting is &longs;uppo&longs;ed 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 &longs;heweth in that Book, intituled
ponderantibus
it to you in this place would cau&longs;e very great confu&longs;ion.
RIC.
I under&longs;tand you: &longs;ome other time we will talk of this, becau&longs;e I have
a mind at pre&longs;ent to proceed to the la&longs;t
to me very confu&longs;ed, and, as I conceive, the Author hath not therein &longs;hewn all
the Subject of that
&longs;peaketh, as you know, in this form.
PROP. VIII. THEOR. VIII.
hath the Figure of a Portion of a Sphære, &longs;hall be
the Ba&longs;e of the Portion touch not the Liquid, the
Figure &longs;hall &longs;tand erectly, &longs;o, as that the Axis of
the &longs;aid Portion &longs;hall be according to the Perpen
dicular. And if the Figure &longs;hall be inclined to any
&longs;ide, &longs;o, as that the Ba&longs;e of the Portion touch the
Liquid, it &longs;hall not continue &longs;o inclined as it was de
mitted, but &longs;hall return to its uprightne&longs;s.
For the declaration of this
that hath the Figure of a portion of a Sphære, as hath been &longs;aid,
be imagined to be de
mitted into the Liquid; and
al&longs;o, let a Plain be &longs;uppo&longs;ed
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 po&longs;&longs;ible, for &longs;atisfact
ion of the Adver&longs;ary, Let
it be &longs;uppo&longs;ed that the &longs;aid Axis were not according to the
pendicular; we are then to demon&longs;trate, that the Figure will not
continue as it was con&longs;tituted by the Adver&longs;ary, but that it will re
turn, as hath been &longs;aid, unto its former po&longs;ition, that is, that the
Axis F T &longs;hall be according to the Perpendicular. It is manife&longs;t, by
the
is in the Line F T, fora&longs;much as that is the Axis of that Figure. And in regard that the Por
tion of a Sphære, may be
greater or le&longs;&longs;er than an He
mi&longs;phære, and may al&longs;o be
an Hemi&longs;phære, let the Cen
tre of the Sphære, in the He
mi&longs;phære, be the Point T,
and in the le&longs;&longs;er Portion the
Point P, and in the greater,
the Point K, and let the Cen
tre of the Earth be the Point
L. And &longs;peaking, fir&longs;t, of
that greater Portion which
hath its Ba&longs;e 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, becau&longs;e
every Portion of a Sphære, hath its Axis in the Line, that from the
Centre of the Sphære is drawn perpendicular unto its Ba&longs;e, 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 Por
drawn through the point K; and its Centre of Gravity, for the &longs;ame
rea&longs;on, &longs;hall be in the Line N K: let us &longs;uppo&longs;e 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
the Point F; let us &longs;uppo&longs;e
it to be the Point
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 &longs;o,
that the part prolonged may
have the &longs;ame 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 &longs;uppo&longs;e
that ^{*} that Centre of the &longs;aid Figure be the Point S: and thorow that
&longs;ame Centre S draw the Perpendicular L S. Now the Gravity of the Fi
gure that is above the Liquid &longs;hall pre&longs;&longs;e from above downwards ac
cording to the Perpendicular S L; & the Gravity of the Portion that
is &longs;ubmerged in the Liquid, &longs;hall pre&longs;&longs;e from below upwards, accor
ding to the Perpendicular R L. Therefore that Figure will not conti
nue according to our Adver&longs;aries Propo&longs;all, but tho&longs;e parts of the
&longs;aid Figure which are towards E, &longs;hall be born or drawn downwards,
& tho&longs;e which are towards H &longs;hall be born or driven upwards, and
this &longs;hall be &longs;o long untill that the Axis F T comes to be according
to the 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.
e,
of Gravity.
And this &longs;ame Demon&longs;tration is in the &longs;ame manner verified in
the other As, fir&longs;t, in the Hæmi&longs;phere that lieth with its
whole Ba&longs;e above or without the
hath been &longs;uppo&longs;ed to be the
to be in the place, in which, in the other above mentioned, the
find that the Figure which is above the
above downwards according to the
wards according to the
follow, as in the other, namely, that the parts of the whole Figure
which are towards E, &longs;hall be born or pre&longs;&longs;ed downwards, and tho&longs;e
that are towards H, &longs;hall be born or driven upwards: and this &longs;hall
be &longs;o long untill that the Axis F T come to &longs;tand ^{*} The like &longs;hall al&longs;o hold true in the
le&longs;s than an Hemi&longs;phere that lieth with its whole Ba&longs;e above the
Liquid.
to the Perpendi
cular.
COMMANDINE.
&longs;tored, &longs;o far as by the Figures that remain, one may collect the Meaning of
planation we have &longs;upplyed in our Commentaries, as we have al&longs;o determined to do in the &longs;e
cond Propo&longs;ition of the &longs;econd Book.
If any Solid Magnitude lighter than the Liquid.]
kind of Magnitudes doth
Shall be demitted into the Liquid in &longs;uch a manner as that the
Ba&longs;e of the Portion touch not the Liquid.]
the Liquid as that the Ba&longs;e &longs;hall be upwards, and the
to that which he &longs;aith in the Propo&longs;ition following
that its Ba&longs;e be wholly within the Liquid;
ted the contrary way, as namely, with theThe
&longs;ame manner of &longs;peech is frequently u&longs;ed in the &longs;econd Book; which treateth of the Portions
of Rectangle Conoids.
Now becau&longs;e every Portion of a Sphære hath its Axis in the Line
that from the Center of the Sphære is drawn perpendicular to its
Ba&longs;e.]
Point G, and the Right Line B C in M
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
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;
on both &longs;ides parallel to B C do make
that cau&longs;e 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 &longs;ince the Center of Gravity of any Portion of a Sphære is
in the Axis, as we have demonstrated in our Book
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 tho&longs;e Portions of Sphæres. For &longs;uppo&longs;e, if po&longs;&longs;ible, 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
therefore from the Portion demerged in the Liquid the Portion of the Sphære B N C, not ha
ving the &longs;ame Center of Gravity, is cut off, the Center of Gravity of the Remainder of the
Portion B G C &longs;hall, by the 8 of the fir&longs;t Book of
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.
fir&longs;t of
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 &longs;uppo&longs;e it to be
the Point X.]
the Axis T Y, and the Center of Gravity Z. And becau&longs;e that from the whole Sphære,
who&longs;e Centre of Gravity is K, as we have al&longs;o demon&longs;trated 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
fall betwixt K and F.
fir&longs;t
The remainder, therefore, of the Figure, elevated above the Sur
face of the Liquid, hath its Center of Gravity in the
prolonged.]
tis Planorum.
Now the Gravity of the Figure that is above the
pre&longs;s from above downwards according to S L; and the Gravit
of the Portion that is &longs;ubmerged in the
low upwards, according to the Perpendicular R L.]
po&longs;ition 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 &longs;hewn by Propo&longs;ition 6. of this. And becau&longs;e they are moved along &longs;everall other Lines,
neither cau&longs;eth the others being le&longs;s moved; the which it continually doth when the Portion
is &longs;et according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes
do concur in one and the &longs;ame Perpendicular, namely, in the Axis of the Portion: and look
with what force or
doth that which is above or without the Liquid tend downwards along the &longs;ame Line: And
but &longs;hall &longs;tay and re&longs;t allwayes in one and the &longs;ame Po&longs;ition, unle&longs;s &longs;ome extrin&longs;ick Cau&longs;e
chance to intervene.
PROP. IX. THEOR. IX.
Coppies this is no
di&longs;tinct Propo&longs;i
tion, but all
Commentators,
do divide it
from the Prece
dent, as having a
di&longs;tinct demon
&longs;tration in the
Originall.
^{*}
ted into the Liquid, &longs;o, as that its Ba&longs;e be wholly
within the &longs;aid Liquid, it &longs;hall continue in &longs;uch
manner erect, as that its Axis &longs;hall &longs;tand according
to the Perpendicular.
For &longs;uppo&longs;e, &longs;uch 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
cumference A B C D, and of the Figure the Circumference E F
And let E H be a Right Line, and F T the Axis of the Portion. If
now it were po&longs;&longs;ible, for &longs;atisfaction of the Adver&longs;ary, let it be
&longs;uppo&longs;ed that the &longs;aid Axis were not according to the Perpendicu
lar: we are now to demon&longs;trate that the Figure will not &longs;o conti
Perpendieular. It is manife&longs;t that the Gen
tre of the Sphære is in the Line F T. And
again, fora&longs;much as the Portion of a Sphære
may be greater or le&longs;&longs;er than an Hemi&longs;
phære, and may al&longs;o be an Hemi&longs;phære, let
the Centre of the Sphære in the Hemi&longs;
phære be the Point T, & in the le&longs;&longs;er Por
tion the Point P, and in the Greater the
Point R. And &longs;peaking fir&longs;t of that greater
Portion which hath its Ba&longs;e within the
Liquid, thorow R and L, the Earths Cen
tre, draw the line RL. The Portion that is
above the Liquid, hath its Axis in the Per
pendicular pa&longs;&longs;ing thorow R; and by
what hath been &longs;aid before, its Centre of
Gravity &longs;hall 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 &longs;hall have its Centre in
the Right Line R
towards
longed may have the &longs;ame 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 &longs;ame Figure:
and thorow O draw the Perpendicular L
O. Now the Gravity of the Portion that
is above the Liquid &longs;hall pre&longs;s 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
&longs;hall not continue; but the parts of it towards H &longs;hall move down
wards, and tho&longs;e towards E upwards: &
this &longs;hall ever be, &longs;o long as F T is accord
ing to the Perpendicular.
COMMANDINE.
The Portion that is above the Liquid
hath its Axis in the Perpendicular pa&longs;&longs;ing
thorow K.]
M; and let N K out the CircumferenceIn
the &longs;ame manner as before me will demon&longs;trate, that the Axis
the Centre of Gravity of them both &longs;hall be in the Line N M: And becau&longs;e that from the Por
tion B N C the Portion B G C, not having the &longs;ame Centre of Gravity, is cut off, the Centre
of Gravity of the remainder of the Magnitude that is above the Surface of the Liquid &longs;hall be
in the Line N K; namely, in the Line which conjoyneth the Centres of Gravity of the &longs;aid
Portions by the fore&longs;aid 8 of
NIC.
Truth is, that in &longs;ome of the&longs;e Figures C is put for X, and &longs;o it was in
the Greek Copy that I followed.
RIC.
This Demo&longs;tration is very difficult, to my thinking; but I believe that
it is becau&longs;e I have not in memory the Propo&longs;itions of that Book entituled
tris Gravium.
NIC.
It is &longs;o.
RIC.
We will take a more convenient time to di&longs;cour&longs;e of that, and now return
to &longs;peak of the two la&longs;t Propo&longs;itions. And I &longs;ay that the Figures incerted in the
demon&longs;tration would in my opinion, have been better and more intelligble unto
me, drawing the Axis according to its proper Po&longs;ition; that is in the half Arch of
the&longs;e Figures, and then, to &longs;econd the Objection of the Adver&longs;ary, to &longs;uppo&longs;e
that the &longs;aid Figures &longs;tood &longs;omewhat Obliquely, to the end that the &longs;aid Axis, if it
were po&longs;&longs;ible, did not &longs;tand according to the Perpendicular &longs;o often mentioned,
which doing, the Propo&longs;ition would be proved in the &longs;ame manner as before:
and this way would be more naturall and clear.
NIC.
You are in the right, but becau&longs;e thus they were in the Greek Copy,
I thought not fit to alter them, although unto the better.
RIC. Companion, you have thorowly &longs;atisfied me in all that in the beginning
of our Di&longs;cour&longs;e I asked of you, to morrow, God permitting, we will treat of
&longs;ome other ingenious Novelties.
THE TRANSLATOR.
I &longs;ay that the Figures, &c.
would have been more intelligible to
me, drawing the Axis Z T according to its proper Po&longs;ition, that
is in the half Arch of the&longs;e Figures.]
the Schemes of
Propo&longs;itions, hath well con&longs;idered what
&longs;aid Axis (which in the Manu&longs;cripts that he had by him is lettered F T, and not as in that of
Tartaylia
But becau&longs;e thus they were in the Greek Copy, I thought not
fit to alter them although unto the better.]
in Commandines Copies; but for variety &longs;ake, take here one
of
to an Hemi&longs;phære, with its Axis oblique, and its Ba&longs;e dimitted
into the Liquid, and Lettered as in this Edition.
great Ob&longs;curity of the Originall in the Demon&longs;trations of the&longs;e
two la&longs;t Propo&longs;itions, be able from the joynt light of the&longs;e two Famous Commentators of our
more famous Author, to di&longs;cern the truth of the Doctrine affirmed, namely, That Solids of the
Figure of Portions of Sphæres demitted into the Liquid with their Ba&longs;es upwards &longs;hall &longs;tand
erectly, that is, with their Axis according to the Perpendicular drawn from the Centre of the
Earth unto its Circumference: And that if the &longs;aid Portions be demitted with their Ba&longs;es
oblique and touching the Liquid in one Point, they &longs;hall not rest in that Obliquity, but &longs;hall
return to Rectitude: And that la&longs;tly, if the&longs;e Portions be demitted with their Ba&longs;es downwards,
they &longs;hall continue erect with their Axis according to the Perpendicular afore&longs;aid: &longs;o that no
more remains to be done, but that we&longs;et before you the 2 Books of this our Admirable Author.
ARCHIMEDES,
HIS TRACT
INSIDENTIBUS HUMIDO,
OR,
Of the NATATION of BODIES Upon, or
Submer&longs;ion In the WATER, or other LIQUIDS.
PROP. I. THEOR. I.
into the &longs;aid Liquid, it &longs;hall have the &longs;ame proporti
on in Gravity to a Liquid of equal Ma&longs;&longs;e, that the
part of the Magnitude demerged hath unto the
whole Magnitude.
For let any Solid Magnitude, as for in
&longs;tance 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 demon&longs;trated that the Magnitude F A
hath the &longs;ame proportion in Gravity to a
Liquid of Equall Ma&longs;&longs;e that A hath to F
A. Take any Liquid Magnitude, as &longs;up
po&longs;e N I, of equall Ma&longs;&longs;e 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
Magnitude F A hath the &longs;ame pro
portion unto N I that the Gravity B
hath to the Gravity O R: But for
a&longs;much as the Magnitude F A demit
ted into the Liquid is lighter than
the &longs;aid Liquid, it is manife&longs;t that a Ma&longs;&longs;e 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
&longs;trated: But B is the Gravity of the Magnitude F A, and R of I:
And becau&longs;e that of the Magni
tude FA the
Gravity is O R. As F A is to N I, &longs;o is B to O R, or, &longs;o is R to
O R: But as R is to O R, &longs;o 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 &longs;o is Therefore F A is to N I, as A is to F A: Which was to be demon
&longs;trated.
fir&longs;t of this.
fifth of
PROP. II. THEOR. II.
^{*}
&longs;hall have its Axis le&longs;&longs;e than
Axem (
ever proportion to the Liquid in Gravity, being de
mitted into the Liquid &longs;o as that its Ba&longs;e touch not
the &longs;aid Liquid, and being &longs;et &longs;tooping, it &longs;hall not
remain &longs;tooping, but &longs;hall be restored to uprightne&longs;&longs;e. I &longs;ay that the &longs;aid Portion &longs;hall &longs;tand upright when
the Plane that cuts it &longs;hall be parallel unto the Sur
face of the Liquid.
Let there be a Portion of a Rightangled Conoid, as hath been
&longs;aid; and let it lye &longs;tooping or inclining: It is to be demon
&longs;trated that it will not &longs;o continue but &longs;hall be re&longs;tored 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
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 &longs;hall A L be Pa
rallel to I S: Wherefore N O will
not be at Right Angles with I S.
Draw therefore K
Point P [that is parallel to I S: and from the Point P unto I S
draw P F parallel unto O N, ^{*} which &longs;hall be the Diameter of the
Section I P O S, and the Axis of the Portion demerged in the
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
Becau&longs;e now that N O is
of R O, but le&longs;s than
than the
acute. For &longs;ince the Line
is greater than R O, that Line which is drawn from the Point R,
and perpendicular to K
without the Section, and for that cau&longs;e mu&longs;t of nece&longs;&longs;ity fall be
tween the Points
B and G, parallel unto R T, they &longs;hall contain Right Angles with
the Surface of the Liquid: ^{*} and the part that is within the Li
quid &longs;hall move upwards according to the Perpendicular that is
drawn thorow B, parallel to R T, and the part that is above the Li
quid &longs;hall move downwards according to that which is drawn tho
row G; and the Solid A P O L &longs;hall not abide in this Po&longs;ition; for
that the parts towards A will move upwards, and tho&longs;e towards
B downwards; Wherefore N O &longs;hall be con&longs;tituted according to
the Perpendicular.]
derico Comman
dino.
COMMANDINE.
ner as the 8 Prop. of the fir&longs;t Book) re&longs;tored according to
illustrated it with Commentaries.
The Right Portion of a Rightangled Conoid, when it &longs;hall
have its Axis le&longs;&longs;e than
its
er then Se&longs;quialter,
Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we &longs;ay
that Conoids are then con&longs;tituted erect when the cutting Plane, that is to &longs;ay, the Plane of the
Ba&longs;e, &longs;hall be parallel to the Surface of the Liquid.
Which &longs;hall be the Diameter of the Section I P O S, and the
Axis of the Portion demerged in the Liquid.]
the Conicks of
lary of the 51 of the &longs;ame.
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.]
angled Conoid is in its Axis, which it &longs;o divideth
as that the part thereof terminating in the vertex,
be double to the other part terminating in the Ba&longs;e; as
in our Book29.
And
&longs;ince the Centre of Gravity of the Portion A P O L is R, O R &longs;hall be double to RN and there
fore N O &longs;hall be Se&longs;quialter of O R. And for the &longs;ame rea&longs;on, B the Centre of Gravity of the Por
tion I P O S is in the Axis P F, &longs;o dividing it as that P B is double to B F;
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.]
tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the
Liquid, the Toine G &longs;hall be its Centre of Gravity; by the 8 of the &longs;econd of
de Centro Gravitatis Planorum, vel de
R O &longs;hall be le&longs;s than
parameter.] of
The Line
u&longs;que ad Axem,
juxta quam po&longs;&longs;unt, quæ á Sectione ducuntur, (
by the 4 Propo&longs;it of his Book
&longs;o called, we have declared in the Commentaries upon him by us publi&longs;hed.
Whereupon the Angle R P
continued out to H, that &longs;o RH may be equall to
the Semi-parameter. If now from the Point H
meet with FP without the Section; for being
drawn thorow O parallel to A L, it &longs;hall fall
without the Section, by the 17 of our &longs;irst Book of
Conicks;
becau&longs;e F P is parallel to the Diameter, and H
V perpendicular to the &longs;ame Diameter, and R H
equall to the Semi-parameter, the Line drawn
from the Point R to V &longs;hall make Right Angles
with that Line which the Section toucheth in the Point P: that is with K
demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and
P
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
&longs;ame 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
that the Line prolonged thorow Hand Mis perpendicular to
E F, which it cutteth in N.
O: and again from the &longs;ame Point draw G P perpendicular to the Diameter: and let the
&longs;aid Diameter prolonged cut the Line E F in
(b)
quall to the Rectangle of O P Q: But it is al&longs;o
equall to the Rectangle comprehended under P B
and the Line
ameter, by the 11 of our fir&longs;t Book of
(c)
P B, and the &longs;ame hath the Parameter unto P O:
But Q P is double unto
Q are equall, as hath been &longs;aid: And therefore
the Parameter &longs;hall be double to the &longs;aid P O:
and by the &longs;ame Rea&longs;on P O is equall to that which we call the Semi-parameter, that is, to K H
that G O being Perpendicular to E F, H N &longs;hall al&longs;o be per-
of 8. of
6. of
6.
6.
1.
1.
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.]
other downwards, along the Perpendicular Line, hath been &longs;hewn above in the 8 of the fir&longs;t
Book of this; &longs;o that we have judged it needle&longs;&longs;e to repeat it either in this, or in the re&longs;t
that follow.
THE TRANSLATOR.
Cone) the Line
cuntur
Archimedes
for want of a better word, name theNow that throughout this
Book
fe&longs;t both by the fir&longs;t words of each Propo&longs;ition, & by this that no Parabola
hath its Parameter double to the Line
that which is taken in a Rightangled Cone. And in any other Parabola, for
the Line
miparameter
chimed.
& Sphæroid.3. Lem.
1.
PROP. III. THEOR. III.
&longs;hall have its Axis le&longs;&longs;e than &longs;e&longs;quialter of the Se
mi-parameter, the Axis having any what ever pro
portion to the Liquid in Gravity, being demitted into
the Liquid &longs;o as that its Ba&longs;e be wholly within the
&longs;aid Liquid, and being &longs;et inclining, it &longs;hall not re
main inclined, but &longs;hall be &longs;o re&longs;tored, as that its Ax
is do &longs;tand upright, or according to the Perpendicular.
Let any Portion be demitted into the Liquid, as was &longs;aid; and
let its Ba&longs;e be in the
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
Right angled Cone: and let the Axis
of the Portion and Diameter of the
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 &longs;hall not be at equall Angles with
I S. Draw R
in P, and parallel to I S: and from the Point P and parall to O N
draw
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 becau&longs;e N O is &longs;e&longs;quialter of R
O, and is greater than &longs;e&longs;quialter of the Semi-Parameter; it is ma
nife&longs;t that
Semi-parameter. ^{*}Let therefore R
H be equall to the Semi-Parameter,
^{*} and O Fora&longs;
much therefore as N O is &longs;e&longs;quialter
of R O, and M O of O H,
Remainder N M &longs;hall be &longs;e&longs;quialter
of the Remainder R H: Therefore
the Axis is greater than &longs;e&longs;quialter
of the Semi parameter by the quan
tity of the Line M O. And let it be
&longs;uppo&longs;ed that the Portion hath not le&longs;&longs;e proportion in Gravity unto
the Liquid of equall Ma&longs;&longs;e, than the Square that is made of the
Exce&longs;&longs;e by which the Axis is greater than &longs;e&longs;quialter of the Semi
parameter hath to the Square made of the Axis: It is therefore ma
nife&longs;t that the Portion hath not le&longs;&longs;e 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
Gravity, the &longs;ame hath the
for this hath been demon&longs;trated
portion the &longs;ubmerged Portion hath to the whole
&longs;ame hath the Square of
been demon&longs;trated in
angled Conoid two
duced, the&longs;e
other as the Squares of their Axes: The Square of P F, therefore,
hath not le&longs;&longs;e 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 le&longs;&longs;e 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 &longs;hall meet with B
fall betwixt B and P; let it fall in T:
parallel to the Diameter, and H T is perpendicular unto the &longs;ame
Diameter, and R H equall to the Semi-parameter; a Line drawn
from R to T and prolonged, maketh Right Angles with the Line
maketh Right Angles with the Surface of the Liquid: and that
part of the Conoidall Solid which is within the Liquid &longs;hall move
upwards according to the Perpendicular drawn thorow B parallel
to R T; and that part which is above the Liquid &longs;hall move down
wards according to that drawn thorow G, parallel to the &longs;aid R T:
And thus it &longs;hall continue to do &longs;o long untill that the Conoid be
re&longs;tored to uprightne&longs;&longs;e, or to &longs;tand according to the Perpendicular.
fifth.
fifth.
&longs;econd Book.
noilibus &
roidibus
medes.
&longs;econd Book.
COMMANDINE.
Let therefore R H be equall to the Semi-parameter.]
read, and not R M, as
that which followeth.
And O H double to H M.]
ed,
And look what proportion the Submerged Portion hath to the whole
Portion, the &longs;ame hath the Square of P F unto the Square of N O.]
&longs;trated by26.
Wherefore P F is not le&longs;&longs;e than M O.]
that the Square of P F is not le&longs;&longs;e than the Square of M O: and therefore neither &longs;hall the
Line P F be leße than the Line M O, by 22 of the
&longs;ixth.
Nor B P than H O,]
P B, &longs;o is M O to H O: and, by Permutation, as
le&longs;&longs;e than M O as hath bin proved; (a) Therefore
neither &longs;hall B P be le&longs;&longs;e than H O.
If therefore a Right Line be drawn
from H at Right Angles unto N O, it
&longs;hall meet with B P, and &longs;hall fall be
twixt B and P.]
Tran&longs;lation of
ted. In regard that P F is not le&longs;&longs;e than O M, nor P B than O H, if we &longs;uppo&longs;e P F equall to
O M, P B &longs;hall be likewi&longs;e equall to O H: Wherefore the Line drawn thorow O, parallel to A L
&longs;hall fall without the Section, by 17 of the fir&longs;t of our Treati&longs;e of Conicks; And in regard that
B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth
al&longs;o meet with the &longs;ame beneath B, and it doth of nece&longs;&longs;ity fall betwixt B and P: But the
&longs;ame is much more to follow, if we &longs;uppo&longs;e P F to be greater than O M.
PROP. V. THEOR. V.
than the Liquid, when it &longs;hall have its Axis great
er than
not greater proportion in Gravity to the Liquid [of
equal Ma&longs;s] than the Exce&longs;&longs;e by which the Square
made of the Axis is greater than the Square made
of the Exce&longs;&longs;e by which the Axis is greater than
&longs;e&longs;quialter
Square made of the Axis being demitted into the Li
quid, &longs;o as that its Ba&longs;e be wholly within the Liquid,
and being &longs;et inclining, it &longs;hall not remain &longs;o inclined,
but &longs;hall turn about till that its Axis &longs;hall be accor
ding to the Perpendicular.
For let any Portion be demitted into the Liquid, as hath been
&longs;aid; and let its Ba&longs;e be wholly within the Liquid, And being
cut thorow its Axis by a Plain erect upon the Surface of the
Liquid; its Section &longs;hall be the Section
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 becau&longs;e the Axis
is not according to the Perpendicu
lar, N O will not be at equall angles
with I S. Draw K
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 re&longs;t as hath been &longs;aid
Now fora&longs;much as it was &longs;uppo&longs;ed that the Portion hath not greater
proportion in Gravity to the Liquid, than the Exce&longs;&longs;e by which
the Square N O is greater than the Square M O, hath to the &longs;aid
Square N O: And in regard that whatever proportion in Gravity
Magnitude of the Portion &longs;ubmerged unto the whole Portion; as
hath been demon&longs;trated in the fir&longs;t Propo&longs;ition; The Magnitude
&longs;ubmerged, therefore, &longs;hall not have greater proportion to the
whole
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
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 le&longs;&longs;e than O M, nor P B than O
H: ^{*} A Line, therefore, drawn from H at Right Angles unto N O
&longs;hall meet with B P betwixt P and B: Let it be in T: And be
cau&longs;e that in the Section of the Rectangled Cone P F is parallel unto
the Diameter N O; and H T perpendicular unto the &longs;aid Diame
ter; and R H equall to the Semi-parameter: It is manife&longs;t that
R T prolonged doth make Right Angles with K P
fore doth al&longs;o 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 &longs;hall be
perpendicular unto the Liquids Surface. The Portion, therefore,
which is above the Liquid &longs;hall move downwards in the Liquid ac
cording to the Perpendicular drawn thorow B; and that part
which is within the Liquid &longs;hall move upwards according to the
Perpendicular drawn thorow G; and the Solid Portion A P O L
&longs;hall not continue &longs;o inclined, [
move within the Liquid untill &longs;uch time that N O do &longs;tand accor
ding to the Perpendicular.
this.
fifth.
Book& Sphæroid.
COMMANDINE.
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.]
within the Liquid hath not greater proportion unto the whole Portion than the Exce&longs;&longs;e by which
the Square N O is greater than the Square M O hath to the &longs;aid Square N O; Converting of
the Proportion, by the 26. of the fifth of
Portion &longs;hall not have le&longs;&longs;er proportion unto the Magnitude &longs;ubmerged, than the Square N O
hath unto the Exce&longs;&longs;e 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 fir&longs;t Magnitude; the part of it which
is &longs;ubmerged the &longs;econd: and let the third Magnitude be the Square M O; and let the Exce&longs;&longs;e
by which the Square N O is greater than the Square M O be the fourth. Now of the&longs;e Mag
nitudes, the proportion of the fir&longs;t and &longs;econd, unto the &longs;econd, is not le&longs;&longs;e than that of the third &
fourth unto the fourth: For the Square M O together with the Exce&longs;&longs;e by which the Square
N O exceedeth the Square M O is equall unto the &longs;aid Square N O: Wherefore, by Conver&longs;i
on of Proportion, by 30 of the &longs;aid fifth Book, the proportion of the fir&longs;t and &longs;econd unto the
fir&longs;t, &longs;hall not be greater than that of the third and fourth unto the third: And, for the &longs;ame
greater than that of the Square N O unto the Square M O: Which was to be demon&longs;trated.
And hence it followeth that P F is not le&longs;&longs;e than O M, nor P B
than O H.]
Euclid,
A
meet with P B betwixt P and B.]
next.
PROP. VI. THEOR. VI.
than the Liquid, when it &longs;hall have its Axis greater
than &longs;e&longs;quialter of the Semi-parameter, but le&longs;&longs;e than
to be unto the Semi-parameter in proportion as fifteen
to fower, being demitted into the Liquid &longs;o as that
its Ba&longs;e do touch the Liquid, it &longs;hall never stand &longs;o
enclined as that its Ba&longs;e toucheth the Liquid in one
Point only.
Let there be a Portion, as was &longs;aid; and demit it into the Li
quid in &longs;uch fa&longs;hion as that its Ba&longs;e do touch the Liquid in
one only Point: It is to be demon&longs;trated that the &longs;aid Portion
&longs;hall not continue &longs;o, but &longs;hall turn about in &longs;uch manner as that
its Ba&longs;e do in no wi&longs;e touch the Surface of the Liquid. For let it be
cut thorow its Axis by a Plane erect
upon the
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
A S; and the Axis of the Portion
and Diameter of the Section N O:
and let it be cut in F, &longs;o as that O
F be double to F N; and in
&longs;ame proportion as fifteen to four; and at Right Angles to N O
draw
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
a&longs;much, therefore, as in the Portion A P O L, contained betwixt
the Right
parallel to A L, and P I parallel unto the Diameter, and cut by the
P; It is nece&longs;&longs;ary that P I hath unto P H either the &longs;ame proportion
that
mon&longs;trated: But
either Se&longs;quialter of H P, or more than &longs;e&longs;quialter: Wherefore
P H is to H I either double, or le&longs;&longs;e than double.
double to T I: the Centre of Gravity of the part which is within
the Therefore draw a
to F prolonging it; and let the Centre of
Gravity of the part which is above the
be G: and from the Point B at Right Angles
unto And &longs;eeing that P I is
parallel unto the Diameter
perpendicular unto the &longs;aid Diameter, and F
B equall to the Semi-parameter; It is mani
fe&longs;t that the
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 al&longs;o make Right An
gles with A S, and with the Surface of the
drawn thorow T and G parallel unto F R &longs;hall be al&longs;o perpendicu
lar to the Surface of the
O L, the part which is within the
to the Perpendicular drawn thorow T; and the part which is above
the
The Solid A
not in the lea&longs;t touch the Surface of the
cut the
falleth betwixt
they are demon&longs;trated like as before.
COMMANDINE.
It is to be demon&longs;trated that the &longs;aid
&longs;o, but &longs;hall turn about in &longs;uch manner as that its Ba&longs;e do in no wi&longs;e
touch the Surface of the Liquid.]
omitted by
the Semi parameter.]
&longs;ame proportion as fifteen to fower: But it was &longs;uppo&longs;ed to have le&longs;&longs;e proportion unto the
Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F
fifth.
Fora&longs;much, therefore, as in the
twixt the Right
in P; It is nece&longs;&longs;ary that P I hath unto P H either the &longs;ame propor
tion that N
demon&longs;trated.]
any other, doth not appear; touching which we will here in&longs;ert a Demon&longs;tration, after that
we have explained &longs;ome things that pertaine thereto.
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
plea&longs;ure unto A B: and in the &longs;ame 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 &longs;ay that the Lines that lye betwixt the Pa
rallels unto F D have unto tho&longs;e 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 &longs;ame mutuall proportion: that is to &longs;ay,
the &longs;ame that A F hath to A E.
F E, &longs;o &longs;hall K G be to K H: Wherefore,
E, &longs;o &longs;hall A K be to K H: And, by Conver&longs;ion of proportion, as
A F is to A E, &longs;o &longs;hall A K be to K H. It is in the &longs;ame manner
proved that, as A F is to A E, &longs;o &longs;hall A N be to A M. Now A
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 &longs;hall be to
the Remainder E H, namely to D O, as A F is to A E. We might in
like manner demonstrate that &longs;o is F N to D P: Which is that that
was required to be demonstrated.
&longs;ixth.
fifth.
LEMMA II.
In the &longs;ame Line A B let there be two Points R and S, &longs;o di&longs;po
&longs;ed, that A S may have the &longs;ame 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, &longs;o, as that it may
meet with R T in the Point T. I &longs;ay that the Point T fall
eth in the Line A C.
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 demon&longs;trated, A X
&longs;hall have the &longs;ame proportion unto A R, as A F hath to A E. But A S hath al&longs;o the &longs;ame proportion to A R: Wherefore
A S
ble. The &longs;ame ab&longs;urdity will follow if we &longs;uppo&longs;e the Toint
T to fall beyond the Line A C: It is therefore nece&longs;&longs;ary that
it do fall in the &longs;aid A C. Which we propounded to be demonstrated.
fifth.
LEMMA III.
Let there be a Parabola, who&longs;e 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, &longs;o 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
being drawn thorow
to B D. I &longs;ay that H O is double to G B.
cut it in G, one and the &longs;ame Point &longs;hall 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 &longs;hall make the
H M, as A F to FC: and
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 fir&longs;t Book
of
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
H M, G L and F C and there Sides, &longs;hall al&longs;o be
Proportionals: And, therefore, as the (c)
Square H M is to the Square G L, &longs;o is the Line
is O H to A F; and as the Square H M is to
the Square G L, &longs;o 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
tando,
are equall, by 35. of our fir&longs;t Book of
Which was to be demon&longs;trated.
&longs;ixth.
&longs;ixth.
of the &longs;ixth.
LEMMA IV.
The &longs;ame things a&longs;&longs;umed again, and M Q being drawn from the
Point M unto the Diameter, let it touch the Section in the
Point M. I &longs;ay that H Q hath to Q O, the &longs;ame proportion
that G H hath to C N.
P N equall to F C, we might in like manner de
mon&longs;trate P O and F A to be equall to each other:
Wherefore P O &longs;hall be double to F B: But H O
is double to G B: Therefore the Remainder P H
is al&longs;o 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 al&longs;o 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 Conver&longs;ion of Pro
portion, B H &longs;hall 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 fir&longs;t Book of
M toucheth the Section in the Point M, H B and
B Q &longs;hall be equall, and Q H double to H B:
Therefore,
H O to G F; that is, to H R: and,
tando,
O, as H G to G R; that is, to P F; and, by the &longs;ame rea&longs;on, to C N: Whichwas to be de
mon&longs;trated.
The&longs;e things therefore being explained, we come now to that
which was propounded. I &longs;ay, therefore, fir&longs;t that
to C K the &longs;ame proportion that H G hath to G B.
mainder G Q &longs;hall be to the Remainder Q A, as
H Q to Q O: and, for the &longs;ame cau&longs;e, the Lines
A C and G L prolonged, by the things that wee
have above demonstrated, &longs;hall inter&longs;ect 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)
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 &longs;hall be equall; Therefore taking H
G from the two Lines H B and B G, there &longs;hall 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
to G B, and H P al&longs;o double to G F; that is, to C K; Therefore H G hath the &longs;ame propor
tion to N C, that G B hath to C K: And
that H G hath to G B.
fifth.
fifth.
Then take &longs;ome other Point at plea&longs;ure 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 &longs;ay
that V C hath greater proportion to C K, than T G hath
to G B.
Diameter parallel to B D: G T &longs;hall be le&longs;&longs;e than G Y, in regard that V S is leße than V X:
And, by the fir&longs;t Lemma, Y G &longs;hall be to V C, as H G to N C; that is, as G B to C K, which
was demon&longs;trated but now: And,
T G, for that it is le&longs;&longs;e than Y G, hath le&longs;&longs;e proportion to G B, than Y G hath to the &longs;ame;
Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de
mon&longs;trated. Therefore a Po&longs;ition given G K, there &longs;hall 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 &longs;hall have the &longs;ame 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 &longs;hall 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 manife&longs;t which was affirmed by
either the &longs;ame proportion that N
Wherefore P H is to H I either double, or le&longs;&longs;e than double.]
Portion that is within the Liquid &longs;hall be the
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 re&longs;t is demon&longs;trated as before. And the
&longs;ame is to be under&longs;tood in the Propo&longs;ition that
followeth.
The Solid A P O L, therefore,
&longs;hall turn about, and its Ba&longs;e &longs;hall
not in the lea&longs;t touch the Surface
of the Liquid.]
&longs;uperficiem humidi &longs;ecundum unum &longs;ignum;
nullo modo humidi &longs;uperficiem contingent,
becau&longs;e the Greekes frequently u&longs;e
e&longs;t;
PROP. VII. THE OR. VII.
than the Liquid, when it &longs;hall have its Axis greater
than Se&longs;quialter of the Semi-parameter, but le&longs;&longs;e
than to be unto the &longs;aid Semi-parameter in proportion
as fi&longs;teen to fower, being demitted into the Liquid &longs;o
as that its Ba&longs;e be wholly within the Liquid, it &longs;hall
never &longs;tand &longs;o as that its Ba&longs;e do touch the Surface
of the Liquid, but &longs;o, that it be wholly within the
Liquid, and &longs;hall not in the lea&longs;t touch its Surface.
Let there be a Portion as hath been &longs;aid; and let it be de
mitted into the Liquid, as we have &longs;uppo&longs;ed, &longs;o as that its
Ba&longs;e do touch the Surface in one Point only: It is to be de
mon&longs;trated that the &longs;ame &longs;hall not &longs;o
continue, but &longs;hall turn about in
&longs;uch manner as that its Ba&longs;e do in no
wi&longs;e 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, &longs;o, as that R P may be double to R F, and in
may be to R
to P F: There
fore let R H be &longs;uppo&longs;ed 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 fir&longs;t let N O cut K
I, as in the former Schemes: It &longs;hall be demon&longs;trated that N O is
to O I either &longs;e&longs;quialter, or greater than &longs;e&longs;quialter. Let O I be
le&longs;&longs;e than double to I N; and let O B be double to B N: and let
them be di&longs;po&longs;ed like as before. We might likewi&longs;e demon&longs;trate
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 al&longs;o &longs;hall be Perpendiculars to the Surface of the Liquid:
The Portion therefore which is above the Liquid &longs;hall move down
wards according to that &longs;ame Perpendicular
which pa&longs;&longs;eth thorow B; and the Portion
which is within the Liquid &longs;hall move up
wards acording to that pa&longs;&longs;ing thorow G:
From whence it is manife&longs;t that the Solid
&longs;hall turn about in &longs;uch manner, as that
its Ba&longs;e &longs;hall in no wi&longs;e 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 &longs;hould not cut
fifth.
PROP. VIII. THE OR. VIII.
&longs;hall have its Axis greater than &longs;e&longs;quialter of the Se
mi-parameter, but le&longs;&longs;e than to be unto the &longs;aid Semi
parameter, in proportion as fifteen to fower, if it
have a le&longs;&longs;er proportion in Gravity to the Liquid, than
the Square made of the Exce&longs;&longs;e by which the Axis is
greater than Se&longs;quialter of the Semi-parameter hath
to the Square made of the Axis, being demitted into
the Liquid, &longs;o as that its Ba&longs;e touch not the Liquid,
it &longs;hall neither return to Perpendicularity, nor conti
nue inclined, &longs;ave only when the Axis makes an
Angle with the Surface of the Liquid, equall to that
which we &longs;hall pre&longs;ently &longs;peak of.
Let there be a Portion as hath been &longs;aid; 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 Se&longs;quialter of B R:
C D &longs;hall be al&longs;o Sefquialter of K R. And as the Portion is to the
Liquid in Gravity, &longs;o let the Square F Q be to the Square D B;
and let F be double to Q: It is manife&longs;t, therefore, that F Q hath
to D B, le&longs;s proportion than C B hath to B D; For C B is the
Exce&longs;s by which the Axis is greater than Se&longs;quialter of the Semi
parameter: And, therefore, F Q is le&longs;s than B C; and, for the
&longs;ame rea&longs;on, F is le&longs;s than B R. Let R
half of that which the Lines K R and
draw a Line from B to E: It is to be demon&longs;trated, that the
enclined &longs;o as that its Axis do make an Angle with the Surface of
the Liquid equall unto the Angle E B
into the Liquid &longs;o as that its Ba&longs;e
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
greater: and the Portion being
cut thorow the Axis by a Plane e
rect unto [
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
be perpendicular to the Axis. Now becau&longs;e it hath been &longs;uppo&longs;ed
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 &longs;hall be
greater than the Angle B: Therefore the Square P I hath greater
proportion to the Square Y I, than the Square E
Square
Line K R unto the Line I Y; and as the Square E
to
and is the double of O I: Therefore O I is le&longs;&longs;e than
greater than
than F. And becau&longs;e that the Portion is &longs;uppo&longs;ed to be in Gra
vity unto the Liquid, as the Square F Q is to the Square B D; and
&longs;ince that as the Portion is to the Liquid in Gravity, &longs;o is the part
thereof &longs;ubmerged unto the whole Portion; and in regard that as
the part thereof &longs;ubmerged is to the whole, &longs;o 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 demon&longs;trated that P H is
greater than F: It is manife&longs;t, therefore, that P M is le&longs;&longs;e than
&longs;e&longs;quialter of P H: And con&longs;equently that P H is greater than
the double of H M. Let P Z be double to Z M: T &longs;hall 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 &longs;hall be in the Line Z T prolonged unto G. In
&longs;trate the
cular unto the Surface of the Liquid:
and that the Portion demerged with
in the
eth out of the
the Perpendicular that &longs;hall be
drawn thorow Z unto the Surface
of the Liquid; and that the part
that is above the Liquid de&longs;cendeth
into the Liquid according to that
drawn thorow G: therefore the Portion will not continue &longs;o inclined
as was &longs;uppo&longs;ed: But neither &longs;hall it return to Rectitude or Per
pendicularity; For that of the Perpendiculars drawn thorow Z and
G, that pa&longs;&longs;ing thorow Z doth fall on tho&longs;e parts which are to
wards L; and that that pa&longs;&longs;eth thorow G on tho&longs;e 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 &longs;hall move downwards, and tho&longs;e towards L up
wards. Again let the Propo&longs;ition run in other termes; and let
the Axis of the Portion make an Angle with the Surface of the
Liquid le&longs;&longs;e than that which is at B. Therefore the Square P I
hath le&longs;&longs;er Proportion unto the Square
I Y, than the Square E
Square
le&longs;&longs;er proportion to I Y, than the half
of K R hath to
&longs;ame rea&longs;on, I Y is greater than dou
ble of
Therefore O I &longs;hall be greater than
to R B, and the Remainder
than
be le&longs;&longs;e than F. And, in regard that
M P is equall to F Q, it is manife&longs;t that P M is greater than &longs;e&longs;qui
alter of P H; and that P H is le&longs;&longs;e than double of
be double to Z M. The Centre of Gravity of the whole Solid &longs;hall
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 &longs;hall 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 &longs;aid Portion &longs;hall not &longs;tay, but &longs;hall turn about till
that its Axis do make an Angle with the Waters Surface greater than
that which it now maketh. And becau&longs;e that when before we
Poriton did not re&longs;t then neither; It is manife&longs;t that it &longs;hall &longs;tay
or re&longs;t when it &longs;hall make an Angle eqnall to B. For &longs;o &longs;hall I O
be equall to
fore
and
fore &longs;ince H is the Centre of Gravity
of that part of it which is within the
Liquid, it &longs;hall move upwards along
the &longs;ame
which the whole
and along the &longs;ame al&longs;o &longs;hall the part
which is above move downwards:
The
a&longs;much as the parts are not repul&longs;ed by each other.
fifth.
COMMANDINE.
And let
of K R.]
ip&longs;ius B R: C D autem ip&longs;ius K R.
fit thus to correctit; for it is not &longs;uppo&longs;ed &longs;o to be, but from the things &longs;uppo&longs;ed is proved to
be &longs;o. For if B
&longs;e&longs;quialter of B R, it followeth that the
&longs;e&longs;quialter of the Semi-parameter.
fifth.
And therefore F Q is le&longs;&longs;e than
and hath le&longs;&longs;er proportion than the Square made of the Exce&longs;&longs;e by which the Axis
is greater than Se&longs;quialter 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 &longs;uppo&longs;ed to be
equall unto the Axis: Therefore the Square F Q &longs;hall have to the Square D B le&longs;&longs;er proporti
on than the Sqnare C B to the &longs;ame Square B D: And therefore the Square
fifth.
And, for the &longs;ame rea&longs;on, F is le&longs;&longs;e than
fifth.
Now becau&longs;e it hath been &longs;uppo&longs;ed that the Axis of the
doth make an Angle with the Surface of the Liquid greater than
the Angle
For the Line P Y being parallel to the Surface of the Liquid, that is, to XS
Line X S: And therefore &longs;hall be greater than the Angle B.
fir&longs;t.
Therefore the Square
Y I, than the Square E
and E
than the Angle E B
the Right Angle at I, is equall unto the Right Angle at
(e)
the
And, by the &longs;ame rea&longs;on, the Square T I &longs;hall have greater proportion to the Square I Y, than
&longs;ixth.
fifth.
fifth.
But as the Square P I is to the Square Y I, &longs;o is the Line K R unto
the Line I Y]
to the Rectangle contained under the Line I O, and under the Parameter; which
we &longs;uppo&longs;ed to be eqnall to the Semi-parameter; that is, the double of K R
and I Y, is equall to the Rectangle contained under the Line I O, and under the Parameter;
Square I Y, &longs;o is the Line K R unto the Line I Y: Therefore the Line K R &longs;hall have unto I
Y, the &longs;ame proportion that the Rectangle compounded of K R and I Y; that is, the Square P I
hath to the Square I Y.
&longs;ixth.
22
the tenth.
And as the Square E
Line K R unto the Line
to half the Rectangle contained under the Line K R and
the half of K R and the Line
the &longs;ame proportion to
22
the tenth.
And, con&longs;equently, I Y is le&longs;&longs;e than the double of
I Y is le&longs;&longs;e than the double of
fifth.
And I
if from B R,
Remainder I
And, therefore, F Q is equall to P M.]
Euclids
But it hath been demon&longs;trated that P H is greater than F.]
In the &longs;ame manner we might demon&longs;trate the Line T H
to be Perpendicular unto the Surface of the Liquid.]
to K R; that is, to the Semi-parameter: And, therefore, by the things above demonstrated,
the Line T H &longs;hall be drawn Perpendicular unto the Liquids Surface.
Therefore, the Square P I hath le&longs;&longs;er proportion unto the
Square I Y, than the Square E
Propo&longs;itions, &longs;hall be demon&longs;trated by us no otherwi&longs;e than we have done above.
Therefore Perpendiculars being drawn thorow Z and G, unto
the Surface of the Liquid, that are parallel to T H, it followeth
that the &longs;aid Portion &longs;hall not &longs;tay, but &longs;hall turn about till that its
Axis do make an Angle with the Waters Surface greater than that
which it now maketh.]
larly towards tho&longs;e parts which are next to L; but that thorow Z, towards tho&longs;e next to A;
It is nece&longs;&longs;ary that the Centre G do move downwards, and Z upwards: and, therefore, the
parts of the Solid next to L &longs;hall move downwards, and tho&longs;e towards A upwards, that the
Axis may makea greater Angle with the Surface of the Liquid.
For &longs;o &longs;hall I O be equall to
P H equall to F.]
PROP. IX. THE OR. IX.
&longs;hall have its Axis greater than Se&longs;quialter of the
Semi-parameter, but le&longs;&longs;er than to be unto the &longs;aid
Semi-parameter in proportion as fifteen to four, and
hath greater proportion in Gravity to the Liquid, than
the exce&longs;s by which the Square made of the Axis is
greater than the Square made of the Exce&longs;s, by which
the Axis is greater than Se&longs;quialter of the Semi
parameter, hath to the Square made of the Axis,
being demitted into the Liquid, &longs;o as that its Ba&longs;e
be wholly within the Liquid, and being &longs;et inclining
it &longs;hall neither turn about, &longs;o as that its Axis &longs;tand
according to the Perpendicular, nor remain inclined,
&longs;ave 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 &longs;aid; and &longs;uppo&longs;e D B equall to
the Axis of the
K R equall to the Semi-parameter: and C B Se&longs;quialter of
B R. And as the Portion is to the Liquid in Gravity, &longs;o let the Ex
ce&longs;&longs;e 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 manife&longs;t, therefore,
that the Exce&longs;&longs;e by which the
Square B D is greater than the
Square B C hath le&longs;ser proportion
to the Square B D, than the Exce&longs;s
by which the Square B D is greater
than the Square F Q hath to the
Square B D; for B C is the Exce&longs;s
by which the Axis of the Portion is
greater than Se&longs;quialter of the
Semi-parameter: And, therefore,
the Square B D doth more exceed
the Square F Q, than doth the
Square B C: And, con&longs;equently, the Line F Q is le&longs;s than B C;
Let R
perpendicular to B D; which let be in power the half of that
which the Lines K R and
B to E: I &longs;ay that the Portion demitted into the Liquid, &longs;o as that
its Ba&longs;e be wholly within the Liquid, &longs;hall &longs;o &longs;tand, as that its Axis
do make an Angle with the Liquids Surface, equall to the Angle B. For let the
and let the Axis not make an Angle with the Liquids Surface, equall
to B, but fir&longs;t a greater: and the &longs;ame 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
Portion and Diameter of the Section N O; which let be cut in
the Points
touching the Section in P, and MP parallel to N O, and P S perpen
dicular to the Axis. And becau&longs;e now that the Axis of the Portion
maketh an
B, the Angle S Y P &longs;hall al&longs;o be greater than the Angle B: And,
therefore, the Square P S hath greater proportion to the Square
S Y, than the Square
cau&longs;e, K R hath greater proportion to S Y, than the half of K R
hath to
S O le&longs;s than
P H greater than F.
&longs;ame proportion in Gravity unto the Liquid, that the Exce&longs;s 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, &longs;o is the part thereof &longs;ubmerged unto the whole
Portion; It followeth that the part &longs;ubmerged, hath the &longs;ame
proportion to the whole
Square B D is greater than the Square F Q hath unto the Square
B D:
tion to that part which is above the
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, &longs;o is the Square N O unto
the Square P M: Therefore, P M
&longs;hall be equall to F Q: But it
hath been demon&longs;trated, that P H is
greater than F. And, therefore,
MH &longs;hall be le&longs;s than
greater than double of H M. Let
therefore, P Z be double to Z M:
long it unto G. The Centre of
Gravity of the whole Portion &longs;hall
be T; of that part which is above
the Liquid Z; and of the Remain
der which is within the Liquid, the
Centre &longs;hall be in the Line Z T pro
longed; let it be in G: It &longs;hall be
demon&longs;trated, 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 &longs;aid T H, are al&longs;o perpen
diculars unto the &longs;ame: Therefore, the Part which is above the
Liquid &longs;hall move downwards, along that which pa&longs;seth thorow Z;
and that which is within it, &longs;hall move upwards, along that which
pa&longs;seth thorow G: And, therefore, the Portion &longs;hall not remain
&longs;o inclined, nor &longs;hall &longs;o turn about, as that its Axis be perpendicular
unto the Surface of the Liquid; for the parts towards L &longs;hall move
downwards, and tho&longs;e towards
the things already demon&longs;trated. And, if the Axis &longs;hould make
an Angle with the Surface of the Liquid, le&longs;s than the Angle B;
it &longs;hall in like manner be demon&longs;trated, that the Portion will not
re&longs;t, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.
COMMANDINE.
And, therefore, the Square B D doth more exceed the Square
F Q, than doth the Square B C: And, con&longs;equently, the Line
F Q, is le&longs;s than B C; and F le&longs;s than B R.]
which the Square B D exceedeth the Square B C; having le&longs;s proportion unto the Square B D,
than the Exce&longs;s by which the Square B D exceedeth the Square F Q, hath to the &longs;aid Square
(a)
and, con&longs;quently, the Line F Q le&longs;s than the Line BC: But F Q hath the &longs;ameproportion
to F, that B C hath to B R; for the Antecedents are each Se&longs;quialter of their con&longs;equents:
And
fifth.
fifth.
And, for that cau&longs;e, K R hath greater proportion to S Y, than
the half of K R hath to
And S O le&longs;s than
And P H greater than F.]
And, therefore, the whole Portion &longs;hall have the &longs;ame propor
tion to that part which is above the Liquid, that the Square B D
hath to the Square F Q]
as the Exce&longs;s by which the Square B D is greater than the Square F Q, is to the Square B D;
the whole Portion, Converting, &longs;hall be to the part thereof &longs;ubmerged, as the Square B D is to
the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square,
F
F Q as is the &longs;aid Square F
For the parts towards L &longs;hall move downwards, and tho&longs;e to
wards A upwards.]
is fal&longs;ly, as I conceive, read
the Line thàt pa&longs;&longs;eth 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 tho&longs;e
parts which are towards L &longs;hall move downwards; and the Centre G, together with the parts
which are towards A upwards.
It &longs;hall in like manner be demon&longs;trated that the Portion &longs;hall not
re&longs;t, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.]
&longs;tratred, as nell from what hath been &longs;aid in the precedent Propo&longs;ition, as al&longs;o from the two
latter Figures, by us in&longs;erted
PROP. X. THEOR. X.
than the Liquid, when it &longs;hall have its Axis greater
than to be unto the Semiparameter, in proportion as
fifteen to four, being demitted into the Liquid, &longs;o as
times &longs;o inclined, as that its Ba&longs;e touch the Surface
of the Liquid in one Point only, and that in two Po-
not in the lea&longs;t touch the Surface of the Liquid;
in Gravity. Every one of which Ca&longs;es &longs;hall be anon
demon&longs;trated.
Let there be a Portion, as hath been &longs;aid; 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, &longs;o as that B K
be double of K D; and in C, &longs;o as that B D may have the &longs;ame
proportion to K C, as fifteen to four: It is manife&longs;t, therefore,
that K C is greater than the Semi-parameter: Let the Semi
let D S be Se&longs;quialter of K R: but
S B is al&longs;o Se&longs;quialter 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
&longs;ame B D: and let Sections of
Rightangled Cones be de&longs;cribed, 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, &longs;hall pa&longs;s
thorow K; and the Line drawn from R perpendicular unto B D,
&longs;hall cut the &longs;aid 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: la&longs;tly, draw P
touching the Section A P O L in the In regard,
therefore, that the three
contained betwixt Right Lines, and the Sections of Rightangled
Cones, and are right alike and unequall, touching one another, upon
one and the &longs;ame Ba&longs;e; and N X G O being drawn from the
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 &longs;ix to fifteen; that is, as two
to five: And as C B is to B D, &longs;o is
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:
proportion of two to five, and of the proportion of five to one, is
the &longs;ame with that of two to one: and two is to one, in double
proportion: Therefore, O G is double of GX: and, in the &longs;ame
manner is P Y proved to be double of Y F: Therefore, &longs;ince that
D S is Se&longs;quialter of K R;
Axis is greater than Se&longs;quialter of the Semi-parameter. If there
fore, the
Liquid, as the Square made of the Line
made of
its
For it hath been demon&longs;trated above, that the
Axis is greater than Se&longs;quialter of the Semi-parameter, if it have
not le&longs;ser proportion in Gravity unto the Liquid, than the Square
of the Semi-parameter, hath to the Square made of the Axis, being
demitted into the Liquid, &longs;o as hath been &longs;aid, it &longs;hall &longs;tand erect,
or
COMMANDINE.
It &longs;hall &longs;ometimes &longs;tand perpendicular.]
Demonstration of which he hath &longs;ubjoyned to the Propo&longs;ition.
And &longs;ometimes &longs;o inclined, as that its Ba&longs;e touch the Surface
of the Liquid, in one Point only.]
clu&longs;ion.
Sometimes, &longs;o that its Ba&longs;e be mo&longs;t &longs;ubmerged in the Liquid.]
face of the Liquid.]
the &longs;econd, and the other in the fifth Conclu&longs;ion.
According to the proportion, that it hath to the Liquid in Gra
vity. Every one of which Ca&longs;es &longs;hall be anon demon&longs;trated.]
portionem habeant ad humidum in Gravitate fingula horum demon&longs;trabuntur.
It is manife&longs;t, therefore, that K C is greater than the Semi
parameter]
hath unto the Semi-parameter greater proportion; (a) the Semi-parameter &longs;hall be le&longs;s
fifth.
Let the Semi-parameter be equall to KR.]
But S B is al&longs;o Se&longs;quialter of BR.]
B K, &longs;o is the part D S to the part K R. Therefore, the Remainder S B, is al&longs;o to the
fifth.
nome/nas a)po\
Ba&longs;e; the parallel and the Ba&longs;es have to the parts of the Diameters, cut off from
the Vertex, the &longs;ameproportion: as al&longs;o, the parts cut off, to the parts cut off.
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 Ba&longs;es, or making equall Angles with their
Ba&longs;es, have the &longs;ame proportion unto their Ba&longs;es.
Archim.
pond.
prop.
2. lib.
2.
Æquipond.
Now the Section of the Cone
thorow V. Inregard, therefore, that in the Section of the Right-angled Cone A E I, who&longs;e
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 &longs;hall have to B V, the &longs;ame proportion
bolæ:
And, therefore, the Lines
Therefore the Section of the Right-angled Cone A E I, &longs;hall pa&longs;s thorow the Point K; which
we would demonstrate.
fifth,
In regard, therefore, that the three
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 &longs;ame Ba&longs;e.]
upon one and the &longs;ame Ba&longs;e,
de&longs;ired: and for the Demon&longs;tration of the&longs;e particulars, it is requi&longs;ite in this place to
premi&longs;e &longs;ome things: which will al&longs;o be nece&longs;&longs;ary unto the things that follow.
LEMMA. I.
Let there be a Right
parallel to one another, A C and D E, &longs;o, that as
B D. &longs;o I &longs;ay that the Line that con
joyneth the Points C and B &longs;hall likewi&longs;e pa&longs;s by E.
above or below it. Let it first pa&longs;s below it,
as by F. The Triangles A B C and D B F &longs;hall
be alike: And, therefore, as
A C to D E: Therefore
ab&longs;urd. The &longs;ame ab&longs;urditie will follow, if the
Line C B be &longs;uppo&longs;ed to pa&longs;s above the Point E:
And, therefore, C B mu&longs;t of neces&longs;ity pa&longs;s thorow
E: Which was required to be demon&longs;trated.
&longs;ixth.
fifth.
LEMMA. II.
Let there be two like
and the Sections of Right-angled Cones; A B C the great
er, who&longs;e Diameter let be B D; and E F C the le&longs;ser, who&longs;e
Diameter let be F G: and, let them be &longs;o applyed to one
another, that the greater include the le&longs;ser; and let their
Ba&longs;es A C and E C be in the &longs;ame Right Line, that the &longs;ame
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 &longs;ay, that the Line H C, hath to that
part of it &longs;elf, that lyeth betwixt C and the Section E F C, the
&longs;ame proportion that A C hath to C E.
Diameters with the Ba&longs;es 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,
A C is to C E; that is,
H unto the Diameter B D, draw the Line H K, parallel to the Ba&longs;e A C: and, draw a Line
and, thorow L, unto the Section E F. G, on the
part E, draw the Line L M, parallel unto the
&longs;ame Ba&longs;e 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,
becau&longs;e the Triangles C B D and C F G are alike
(b)
Triangles C K B and C L F, are al&longs;o alike to
one another; therefore, as B C is to C F, that is,
as B D is to F G, &longs;o &longs;hall K C be to C L, and B K to F L: Wherefore, K C to C L, and,
B D is to F G, &longs;o is D C to C G; that is, A D to E G: And,
A D, &longs;o is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11
of our fir&longs;t ofBy the &longs;ame rea&longs;on, likewi&longs;e, the Square E G being equall to the Rectangle
G F O, the three other Lines F G, E G and F O, &longs;hall be al&longs;o Proportionals: And, as B D is
to A D, &longs;o is F G to E G: And, therefore, as A D is to B N, &longs;o is E G to F O:
B N to F O: But as D B is to G F, &longs;o is B K to F L: Therefore, B K is to F L, as
B N is to F O: And,
becau&longs;e the
and F L, L M, and F O &longs;hall al&longs;o be Proportionals: And, therefore,
Line F L is to the Line F O, &longs;o &longs;hall the Square F L be to the Square L M:
Therefore, becau&longs;e that as B K is to B N, &longs;o is F L to F O; as the Square
(g)
K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manife&longs;t that
the Line H C al&longs;o &longs;hall pa&longs;s thorow the Point M: As K C, therefore, is to C L, that is,
as A C to C E, &longs;o is H C to C M; that is, to the &longs;ame part of it &longs;elf, that lyeth betwixt C and
the Section E F C. And, in like manner might we demon&longs;trate, that the &longs;ame happeneth
in other Lines, that are produced from the Point C, and the Sections E B C. And, that
B C hath the &longs;ame 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.
fifth.
&longs;ixth.
fifth.
&longs;ixth.
fir&longs;t of
of the &longs;ixth.
&longs;ixth.
From whence it is manife&longs;t, that all Lines &longs;o drawn, &longs;hall be cut by the
&longs;aid Section in the &longs;ame proportion. For, by Divi&longs;ion and Conver&longs;ion,
C M is to M H, and C F to F B, as C E to E A.
LEMMA. III.
And, hence it may al&longs;o be proved, that the Lines which are
drawn in like Portions, &longs;o, as that with the Ba&longs;es, they con
tain equall Angles, &longs;hall al&longs;o 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 al&longs;o alike to
each other.
Points draw the Lines R P S and T Q V parallel to the Diameters. Of the Portion
H S C the Diameter &longs;hall be P S, and of the Portion M V C the Diameter &longs;hall be
another Line; which let be S X: And, as the Square C T is to the Square C Q &longs;o let F O
be to V Y. Now it is manife&longs;t, by the things which we have demon&longs;trated, in our Commentaries,
upon the fourth Propo&longs;ition of
Square C P is equall to the Rectangle P S X; and al&longs;o, 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 &longs;ince the Triangles C P R and C Q T are alike; C R &longs;hall have to C P, the
&longs;ame Proportion that C T hath to C Q: And, therefore, the
fore, al&longs;o, the Line B N &longs;hall 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, al&longs;o,
their halves C P and C Q, are al&longs;o to one
another, as A D and E G: And.
tando,
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,
to S X, as C Q is to V Y. And, &longs;ince 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 &longs;hall be proportionalls, and V Q, Q C and V Y &longs;hal be
Proportionalls al&longs;o: And therefore al&longs;o S P &longs;hall be to P C as V Q to Q C And as P C
is to C H, &longs;o &longs;hall Q C. be to C M: Therefore,
Portion H S C is to its Ba&longs;e C H, &longs;o is V Q the Diameter of the portion M V S the
Ba&longs;e C M; and the Angles which the Diameter with the Ba&longs;es do contain, are equall; and the
Lines S P and V Q are parallel: Therefore the Portions, al&longs;o, H S C and M V C &longs;hall be alike:
Which was propo&longs;ed to be demon&longs;trated
&longs;ixth.
LEMMA. IV.
Points E and F, &longs;o 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 &longs;ay that C F &longs;hall be to
F H as A E is E G.
as A B is to E B, &longs;o &longs;hall C D be to F D. Again, &longs;ince that as A G is to G B, &longs;o is C H, to
H D; it followeth that, by Compounding and Converting, as G B is to A B, &longs;o &longs;hall H D be
is to G B, &longs;o &longs;hall F D be to H D; And, by Conver
&longs;ion of Propo&longs;ition, as E B is to E G, &longs;o &longs;hall F D
be to F H: But as A E is to E B, &longs;o is C F to F D:
Ex æquali,
&longs;hall CF be to F H.
the Lines A B and C D be applyed to one another,
&longs;o as that they doe make an Angle at the parts A and C;
and let A and C be in one and the &longs;ame Point: then
draw Lines from D to B, from H to G, and from F to E. And &longs;ince that as A E is to E B,
&longs;o is C F, that is A F to F D; therefore F E &longs;hall be parallel to D B
and H G are parallel to each other: And con&longs;equently, as A E is to E G, &longs;o is A H, that is,
&longs;ixth.
fir&longs;t.
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, who&longs;e Diameter is B D; and E F C,
who&longs;e 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 al&longs;o toucheth the Section E F C in the &longs;ame
Point C, as &longs;hall be demon&longs;trated: I &longs;ay that the Line drawn
from C
the Line E H, &longs;hall be divided in the &longs;ame proportion by the
Section A B C, in which the
E F C; and the part of the
two Sections, &longs;hall an&longs;wer in proportion to the part of the Line
drawn, which al&longs;o falleth betwixt the &longs;ame Sections: that is,
as in the foregoing Figure, if D B be produced untill it meet
with C H in L, that it may inter&longs;ect the Section E F C in the
Point M, the
that C E hath to E A.
in O; and drawing a Line from B to C, which &longs;hall pa&longs;&longs;e by F, as hath been &longs;hewn, the
al&longs;o the Triangles C F N and C B L: Wherefore
(a)
to B L: Therefore G F &longs;hall be to D B, as F N
F N, as D B to B L: But D B is equall to
B L, by 35 of our Fir&longs;t Book of
eth the Section E F C in the &longs;ame 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.
Becau&longs;e, now, that the Line C H toucheth the
Section E F C in the Point C; L M &longs;hall have
the &longs;ame proportion to M D that C D hath to D E,
by the Fifth Propo&longs;ition of
Book
rea&longs;on of the Similitude of the Triangles C M D
and C P Q, as C M is to C D, &longs;o &longs;hall C P
be to C Q: And,
C P, &longs;o &longs;hall C D be to C Q: But as C M is to C P, &longs;o is C E to C A,; as we have but
even now demon&longs;trated: And therefore, as C E is to C A, &longs;o is C D to C
whole is to the whole, &longs;o 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,
is to D E, as C Q is to Q A: And L M is al&longs;o to M D, as C D to D E: Therefore L M is
to D A: It is manife&longs;t 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, &longs;o is C M to M P: Therefore L B is to B M, as C M to M P:
as C E to E A. And in like manner it &longs;hall be demonstrated that &longs;o is N O to O F; as al&longs;o the
Remainders. And that al&longs;o H K is to K E, as C E to E A, doth plainly appeare by the &longs;ame
5.
&longs;ixth.
fifth,
fifth.
LEMMA. VI.
And, therefore, let the things &longs;tand as above; and de&longs;cribe
yet another like Portion, contained betwixt a Right Line, and
the Section of the Rightangled Cone D R C, who&longs;e Diameter
is R S, that it may cut the Line F G in T; and prolong S R
unto the
X, and E F C in Y. I &longs;ay, 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.
Point C; and that L M is to M D, as al&longs;o N F to F T, and V Y to Y R, as C D is to E D.
And, becau&longs;e now that L B is to B M, as C E is to E A; therefore, Compounding and Conver
ting, B M &longs;hall be to L M, as E A to A C: And, as L M is to M D, &longs;o &longs;hall 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, &longs;hall al&longs;o be compounded of the proportion that E A hath to A C, and of
that which C D hath to D E. In the &longs;ame manner it &longs;hal be demon&longs;trated, that O F hath to
F T, and al&longs;o X Y to Y R, a proportion compounded of tho&longs;e &longs;ame proportions; and &longs;o in
the re&longs;t: Which was to be demonstrated.
By which it appeareth that the
the Sections A B C and D R C, &longs;hall be divided by the Section E F C
in the &longs;ame Proportion.
And C B is to B D, as &longs;ix to fifteen.]
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 Conver&longs;ion of proportion and Convert
ing, C B is to B D, as &longs;ix to &longs;ifteen.
And as C B is to B D, &longs;o is
E B to B A; and D Z to D A.]
alike; As C B is to B E, &longs;o &longs;hall D B be to B A:
And,
E B be to B A: Againe, as B C is to C E &longs;o
&longs;hall B D be to D A, And,
C B is to B D, &longs;o &longs;hall C E, that is, D Z
equall to it, be to D A.
And of D Z and D A, L I and
L A are double.]
double of D A, is manife&longs;t, for that B D is the Diameter of the Portion. And that L I is
dovble to D Z &longs;hall be thus demon&longs;trated. For as much as ZD is to D A, as two to five:
therefore, Converting and Dividing, A Z, that is, I Z, &longs;hall be to Z D, as three to two:
as two to five: Therefore,
Conver&longs;ion 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,
of D Z: Which was to be demon&longs;trated.
And, A D is to D I, as five to one.]
&longs;trated.
For it hath been demon&longs;trated, above, that the Portion who&longs;e
Axis is greater than Se&longs;quialter of the Semi-parameter, if it have
not le&longs;&longs;er proportion in Gravity to the Liquid, &c.]
ted this in the fourth Propo&longs;ition of this Book.
CONCLVSION II.
B D, but greater than the Square X O hath to the
Square B D, being demitted into the Liquid, &longs;o in
clined, as that its Ba&longs;e touch not the Liquid, it &longs;hall
continue inclined, &longs;o, as that its Ba&longs;e &longs;hall not in the
lea&longs;t touch the Surface of the Liquid, and its Axis
&longs;hall make an Angle with the Liquids Surface, greater
than the Angle X.
Therfore repeating the fir&longs;t figure, let the Portion have unto
the Liquid in Gravitie a proportion greater than the Square
X O hath to the &longs;quare B D, but le&longs;&longs;er than the Square made of
the Exce&longs;&longs;e by which the Axis is greater than Se&longs;quialter of the Semi
Parameter, that is, of S B, hath to
the Square B D: and as the Portion
is to the Liquid in Gravity, &longs;o let
the Square made of the Line
to the Square B D:
er than X O, but le&longs;&longs;er than the
Exce&longs;&longs;e by which the Axis is grea
ter than Se&longs;quialter 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
B D falling betwixt O X and B D,
it cut the remaining Conick Section A H I in the point H, and the
Right Line R G in V. It &longs;hall be demon&longs;trated that M H is double to
H N, like as it was demon&longs;trated that O G is double to G X.
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
la&longs;tly, having drawn A N & prolong
ed it to Q, the Lines A N & N Q &longs;hall
be equall to each other. For in
regard that in the Like Portions
A M Q L and A
and A N are drawn from the Ba&longs;es
unto the Portions, which Lines
contain equall Angles with the &longs;aid
Ba&longs;es, Q A &longs;hall have the &longs;ame 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 demon&longs;trated that the Portion being demitted into the
Liquid, and &longs;o inclined as that its Ba&longs;e touch not the Liquid, it
&longs;hall continue inclined &longs;o as that its Ba&longs;e &longs;hall not in the lea&longs;t touch
the Surface of the Liquid, and its Axis &longs;hall make an Angle with
the Liquids Surface greater than the Angle X. Let it be demitted
into the Liquid, and let it &longs;tand, &longs;o, as that its Ba&longs;e 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,
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
cut in the Points K and R as hath
been &longs;aid; al&longs;o 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 &longs;ame B D.
Now, fora&longs;much as the Portion is unto the Liquid in Gravity, as
the Square made of the Line
as the portion is unto the Liquid in Gravitie, &longs;o is the part thereof
&longs;ubmerged unto the whole Portion; and that as the part &longs;ubmerged
is to the whole, &longs;o is the Square T P to the Square B D; It follow
eth that the Line
M N and P T, as al&longs;o the Portions A M Q and A P O &longs;hall like
wi&longs;e be equall to each other. And &longs;eeing 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 Ba&longs;es, &longs;o, as that the Portions
cut off do make Equall Angles with their Diameters; as al&longs;o the
and B C and B S &longs;hall al&longs;o be equall: And therefore C R and S R,
and M V and P Z, and V N and Z T, &longs;hall be equall likewi&longs;e.
Since therefore M V is Le&longs;&longs;er than double of V N, it is manife&longs;t that
P Z is le&longs;&longs;er than double of Z T.
drawing a Now the Centre of
Gravity of the whole Portion &longs;hall be the point K; and the Centre
of that part which is in the Liquid &longs;hall be
above the Liquid &longs;hall be in the
And therefore al&longs;o the Lines drawn thorow the Points E and
lell unto K Z, &longs;hall be perpendicular sunto the &longs;ame: Therefore the
Portion &longs;hall not abide, but &longs;hall turn about &longs;o, as that its
do not in the lea&longs;t touch the Surface of the
now when it toucheth in but one Point only, it moveth upwards, on
the part towards A: It is therefore per&longs;picuous, that the Portion
&longs;hall con&longs;i&longs;t &longs;o, as that its Axis &longs;hall make an Angle with the
Surface greater than the Angle X.
COMMANDINE.
If the Portion have le&longs;&longs;er 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.]
propo&longs;ition; and the other pat is with their Demon&longs;trations, &longs;hall hereafter follow in the &longs;ame Order.
which the Axis is greater than Se&longs;quialter of the Semi-parameter,
that is than S B.]
It &longs;hall be demon&longs;trated, that M H is double to H N, like as it
was demon&longs;trated, that O G is double to G X.]
of this Propo&longs;ition, and from what we have but even now written, thereupon appeareth:
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 Ba&longs;es unto the Portions,
which Lines contain equall Angles with the &longs;aid Ba&longs;es, Q A &longs;hall
have the &longs;ame proportion to A N, that L A hath to A D.]
Therefore A N is equall to N Q]
A D; Dividing and Converting, A N &longs;hall be to N Q as A D to D L: But A D
is equall to D L; for that D B is &longs;uppo&longs;ed to be the Diameter of the Portion: Therefore
fifth.
And A Q parallel to M Y.]
And let B D be cut in the Points K and R as hath been &longs;aid.]
K D, and in R &longs;o, as that K R may be equall to the Semi-parameter.
And, &longs;eeing that in the Equall and Like Portions A P O L and
A
of their Ba&longs;es, &longs;o, as that the Portions cut off, do make equall Angles
Therefore, the Lines Y B and G B, & B C & B S, &longs;hall al&longs;o be equall.]
from the Extremities of their Ba&longs;es, A O and
A Q are drawn, that contain equall Angles with
tho&longs;e Ba&longs;es; and &longs;ince the Angles at D, are both
Right; Therefore, the Remaining Angles A
and A
the Line P G is parallel unto the Line A O; al&longs;o
M Y is parallel to A
A D: Therefore the Triangles P G S and M Y C,
as al&longs;o the Triangles A
alike to each other
A D and A D are equall to each other: Therefore,
A
A Q are equall to each other; as al&longs;o their halves
A T and A N: Therefore the Remainders TAnd, as
Y C: and
G S to Y C: And, therefore, G S and Y C are
equall; as al&longs;o their halves B S and B C: From
whence it followeth, that the Remainders S R and C R
are al&longs;o equall: And, con&longs;equently, that P Z and
M V, and V N and Z T, are lkiewi&longs;e equall to one
another.
&longs;ixth.
fir&longs;t,
Since, therefore, that N V is le&longs;&longs;er
than double of V N.]
H N, and M V is le&longs;&longs;er than M H: Therefore, M V
is le&longs;&longs;er than double of H N, and much le&longs;&longs;er than
double of V N.
Therefore, the Portion &longs;hall not abide, but &longs;hall turn about,
&longs;o, as that its Ba&longs;e do not in the lea&longs;t 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
tion hath it thus,
unum tangat Superficiem Humidi, quon am nunc &longs;ecundum unum tacta ip&longs;a reclina
tur
Archimedes, For in the &longs;ixth Propo&longs;ition of this,
he thus writeth (as we al&longs;o have it in the Tran&longs;lation,)
turn about, and its Ba&longs;e &longs;hall not in the lea&longs;t touch the Surface of the Liquid.
in the &longs;eventh Propo&longs;ition
&longs;uch manner, a that its Ba&longs;e doth in no wi&longs;e 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.
pear: For &longs;ince that the Perpendiculars unto the Surface of the Liquid, that pa&longs;s thorow
fall on the part towards A, and tho&longs;e that pa&longs;s thorow E, on the part towards L; it is nece&longs;&longs;ary
that the Centre
It is therefore per&longs;picuous, that the Portion &longs;hall con&longs;i&longs;t, &longs;o, as that
its Axis &longs;hall make an Angle with the Liquids Surface greater than
the Angle
A
as in the first Figure: And the
is equall to the Angle at
A
without it: Therefore the Angle at Y &longs;hall be great
er than that at X. And becau&longs;e now the Portion
turneth about, &longs;o, as that the Ba&longs;e doth not touch
the Liquid, the Axis &longs;hall make an Angle with its
Surface greater than the Angle G; that is, than the
Angle Y: And, for that rea&longs;on, much greater than
the Angle X.
fir&longs;t.
fir&longs;t.
CONCLUSION III.
Liquid, that the Square X O hath to the Square
BD,
its Ba&longs;e touch not the Liquid, it &longs;hall &longs;tand and
continue inclined, &longs;o, as that its Ba&longs;e touch the Sur
face of the Liquid, in one Point only, and its Axis &longs;hall
make an Angle with the Liquids Surface equall to the
Angle X. And, if the Portion have the &longs;ame proportion
in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid,
& &longs;et &longs;o inclined, as that its Ba&longs;e touch not the Liquid,
it &longs;hall &longs;tand inclined, &longs;o, as that its Ba&longs;e touch the
Surface of the Liquid in one Point only, & its Axis &longs;hall
make an Angle with it, equall to the Angle
Let the Portion have the &longs;ame proportion in Gravity to tho
Liquid that the Square
it be demitted into the Liquid &longs;o inclined, as that its Ba&longs;e touch
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
dicular unto B D. It is to be demon&longs;trated that the Portion &longs;hall
not &longs;tand &longs;o, but &longs;hall encline until
that the Ba&longs;e touch the Surface of
the Liquid, in one Point only, for let
the &longs;uperior figure &longs;tand as it was,
and draw O C, Perpendicular to B D;
and drawing a
prolong it to Q: A X &longs;halbe equall
to
to A
is &longs;uppo&longs;ed to have the &longs;ame pro
portion in Gravity to the Liquid
that the &longs;quare X O hath to the
Square B D; the part thereof &longs;ubmerged &longs;hall al&longs;o have the &longs;ame
proportion to the whole; that is, the Square T P to the Square
B D; and &longs;o T P &longs;hall be equal to
I P M and A O Q the Diameters are equall, the portions &longs;hall al&longs;o be
equall.
and AP ML the Lines A Q and I M, which cut off equall
tions, are drawn, that, from the Extremity of the
not from the Extremity; it appeareth that that which is drawn from
the end or Extremity of the
the Diameter of the whole
being le&longs;&longs;e than the Angle at N, B C &longs;hall be greater than B S; and
C R le&longs;&longs;er than S R:
and G
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
Gravity of the whole Portion &longs;hall 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
&longs;hall be demon&longs;trated, both, that K H is perpendicular to the Surface
of the Liquid, and tho&longs;e Lines al&longs;o that are drawn thorow the Points
Hand
&longs;hall encline untill that its Ba&longs;e do touch the Surface of the Liquid
in one Point; and &longs;o it &longs;hall continue. For in the Equall Portions
A O Q L and A P M L, the
Lines A Q and A M, that cut off
equall Portions, &longs;hall be dawn
from the Ends or Terms of the Ba&longs;es;
and A O Q and A P M &longs;hall be
demon&longs;trated, 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
And, therefore, if drawing HK,
it be prolonged to
be K; of the part which is within the Liquid H; and of the part which
is above the Liquid in K
the Surface of the Liquid. Therfore
along the &longs;ame Right Lines &longs;hall the
part which is within the Liquid move
upwards, and the part above it down
wards: And therfore the Portion
&longs;hall re&longs;t with one of its Points
touching the Surface of the Liquid,
and its Axis &longs;hall make with the
&longs;ame an Angle equall to X. It is
to be demon&longs;trated in the &longs;ame
manner that the Portion that hath
the &longs;ame proportion in Gravity to the Liquid, that the Square P F hath
to the Square B D, being demitted into the Liquid, &longs;o, as that its
Ba&longs;e touch not the Liquid, it &longs;hall &longs;tand inclined, &longs;o, as that its Ba&longs;e
touch the Surface of the Liquid in one Point only; and its Axis &longs;hall
make therwith an Angle equall to the Angle
COMMANDINE.
That is the Square T P to the Square B D.]
&longs;hall be equall to the Square X O: And for that rea&longs;on, the Line T P equall to the
Line X O.
fifth.
The Portions &longs;hall al&longs;o be equall.]
Again, becau&longs;e that in the Equall and Like Portions, A O Q L
and A P M L.]
to the Portion I P M: The Point Q falleth beneath M; for otherwi&longs;e, the Whole would be
equall to the Part. Then draw I V parallel to A Q, and cutting the Diameter is
let I M cut the &longs;ame
that the Angle AFor the Angle I
Angle A
fir&longs;t.
fir&longs;t.
And the Angle at X, being le&longs;&longs;e
than the Angle at N.]
Lines, O C perpendicular to the Diameter B D, and
O X touching the Section in the Point O, and cutting
to A
that at N: And, con&longs;equently, X &longs;hall fall beneath N: Therefore, the Line X B is greater than
N B. And, &longs;ince B C is equall to X B, and B S equall to N B; B C &longs;hall be greater than B S.
cond of
fir&longs;t.
fir&longs;t of
Therefore, A Q and A M do make equall Acute Angles with
the Diameters of the Portions.]
upon the &longs;econd Conclu&longs;ion.
It is to be demon&longs;trated in the &longs;ame manner, that the Portion
that hath the &longs;ame proportion in Gravity to the Liquid, that the
Square P F hath to the Square B D,
being demitted into the Liquid, &longs;o,
as that its Ba&longs;e touch not the Li
quid, it &longs;hall &longs;tand inclined, &longs;o, as
that its Ba&longs;e touch the Surface of the
Liquid in one point only; and its Axis
&longs;hall make therewith an angle equall
to the Angle
Liquid in Gravity, as the Square P F to the
Square B D: and being demitted into the
Liquid, &longs;o inclined, as that its Ba&longs;e 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 &longs;ame parts as
we &longs;aid 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 &longs;ame. It is to be demon
strated, that the Portion &longs;hall not re&longs;t, but &longs;hall incline, &longs;o, as that it touch the Liquids
Surface, in one Point of its Ba&longs;e only. For,
a Line from A to F, prolong it till it meet with
the Section in
rallel to A Q: Now, by the things allready de
mon&longs;trated by us, A F and F Q &longs;hall be equall
to one another. And being that the Portion hath
the &longs;ame proportion in Gravity unto the Liquid,
that the Square P F hath to the Square B D; and
&longs;eeing that the part &longs;ubmerged, hath the &longs;ame pro-
M T to the Square B D; (g) the Square M T
&longs;hall be equall to the Square P F; and, by the
&longs;ame rea&longs;on, 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
fir&longs;t from the Extreme term of the Ba&longs;e, the la&longs;t not from the Extremity; it followeth, that
A Q drawn from the Extremity, containeth a le&longs;&longs;er Acute Angle with the Diameter of the
Portion, than I O: But the Line P
fore, the Angle at
and S R, that is, M X, greater than C R, that is, than P Y: and, by the &longs;ame rea&longs;on, X T
le&longs;&longs;er than Y F. And, &longs;ince P Y is double to Y F, M X &longs;hall 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
&longs;hall 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 &longs;uppo&longs;e in
manner, as before, that both the Line K H and tho&longs;e that are drawn thorow the Points H
and
Portion therefore, &longs;hall not re&longs;t; but when it &longs;hall be enclined &longs;o far as to touch the Sur
face of the Liquid in one Point and no more, then it &longs;hall &longs;tay. For the Angle at N
equall to the Line B C; and S R to C R: Where
fore, M H &longs;hall be likewi&longs;e equall to P Y. There
fore, having drawn HK and prolonged it; the
Centre of Gravity of the whole Portion &longs;hall be
K; of that which is in the Liquid H; and of
that which is above it, the Centre &longs;hall be in
the Line prolonged: let it be in
fore, along that &longs;ame Line K H, which is per
pendicular to the Surface of the Liquid, &longs;hall
the part which is within the Liquid move up
wards, and that which is above the Liquld
downwards: And, for this cau&longs;e, the Portion,
&longs;hall be no longer moved, but &longs;hall &longs;tay, and
re&longs;t, &longs;o, as that its Ba&longs;e do touch the Liquids Surface in but one Point; and its Axis
maketh an Angle therewith equall to the Angle
demon&longs;trate.
fifth.
CONCLVSION IV.
to the Liquid, than the Square F P to the Square
B D, but le&longs;&longs;er than that of the Square X O to the
Square B D, being demitted into the Liquid,
and inclined, &longs;o, as that its Ba&longs;e touch not the
Liquid, it &longs;hall &longs;tand and re&longs;t, &longs;o, as that its Ba&longs;e
&longs;hall be more &longs;ubmerged 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 le&longs;&longs;er than that of the Square X O to
the Square B D; and as the Portion is in Gravity to the Liquid,
&longs;o let the Square made of the Line
&longs;hall be greater than F P, and le&longs;&longs;er 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
the Remaining Section in Y: V Y &longs;hall al&longs;o be proved double
to Y I, like as it hath been demon&longs;trated, that O G is double off
G X. And, draw from V, the Line V
A V Q L in V; and drawing a Line from A to I, prolong it unto
to I
that the Portion being demitted into the Liquid, and &longs;o inclined,
as that its Ba&longs;e touch not the Liquid, &longs;hall &longs;tand, &longs;o, that its Ba&longs;e
&longs;hall be more &longs;ubmerged in the Liquid, than to touch it Surface in
For let it be de
mitted into the Liquid, as hath been
&longs;aid; and let it fir&longs;t be &longs;o inclined, as
that its Ba&longs;e do not in the lea&longs;t
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
B D. Now, &longs;eeing that the Por
tion is in Gravity unto the Liquid,
as the Square made of the Line
is to the Square B D;
be equall to N T: Which is to
be demon&longs;trated as above: And,
therefore, N T is al&longs;o equall to
V I: The Portions, therefore,
A V Q and E N Z are equall to
one another. And, &longs;ince 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
Extremity of the Ba&longs;e, this not
from the Extreme, that which is
drawn from the Extremity of the
Ba&longs;e, &longs;hall make the Acute Angle
with the Diameter of the Portion
le&longs;&longs;er: and in the Triangles N L S
and V
greater than the Angle at
Therefore, B S &longs;hall be le&longs;&longs;er
than B C; and S R le&longs;&longs;er than
C R: and, con&longs;equently, N X
greater than V H; and X T le&longs;&longs;er than H I. Seeing, therefore,
that V Y is double to Y I; It is manife&longs;t, that N X is greater than
double to X T. Let N M be double to M T: It is manife&longs;t, from what
hath been &longs;aid, that the Portion &longs;hall not re&longs;t, 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
&longs;tanding as before, we will again
demon&longs;trate, 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 &longs;hall with the Diameters
of the Portions, contain equall
Angles. Therefore, in the Triangles
N L S and V
the Points
B C; S R to C R; N X to V H; and X T to H I: And, &longs;ince
V Y is double to Y I, N X &longs;hall be greater than double of X T. Let therefore, N M be double to M T.
It is hence again manife&longs;t,
that the Portion will not remain, but &longs;hall incline on the part
towards A: But it was &longs;uppo&longs;ed, that the &longs;aid Portion did
touch the Surface of the Liquid in one &longs;ole Point: Therefore,
its Ba&longs;e mu&longs;t of nece&longs;&longs;ity &longs;ubmerge farther into the Liquid.
CONCLVSION V.
the Liquid, than the Square F P to the Square
B D, being demitted into the Liquid, and in
clined, &longs;o, as that its Ba&longs;e touch not the Liquid,
it &longs;hall &longs;tand &longs;o inclined, as that its Axis &longs;hall
make an Angle with the Surface of the Liquid,
le&longs;&longs;e than the Angle
not in the lea&longs;t touch the Liquids Surface.
Finally, let the Portion have le&longs;&longs;er 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, &longs;o let the
Square made of the Line
le&longs;&longs;er 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
We
&longs;hall demon&longs;trate G H to be double
to H I, as it hathbeen demon&longs;tra
ted, that O G is double to G X. Then draw G
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
A I &longs;hall be equall to I
A Q parallel to G
demon&longs;trated, that the Portion being
demitted into the Liquid, and inclined, &longs;o, as that its Ba&longs;e touch
the Liquid, it &longs;hall &longs;tand &longs;o incli
ned, as that its Axis &longs;hall make
an Angle with the Surface of the
Liquid le&longs;&longs;e than the Angle
and its Ba&longs;e &longs;hall not in the lea&longs;t
touch the Liquids Surface. For
let it be demitted into the Liquid,
and let it &longs;tand, &longs;o, as that its Ba&longs;e
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
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 &longs;aid 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 &longs;ame. Be
cau&longs;e, now, that the Portion is in Gravity to the Liquid, as
the Square made of
Portion is to the Liquid in Gravity, &longs;o is the Square N T to the
Square B D, by the things that have been &longs;aid; it is plain, that
N T is equall to the Line
A N Z and A G Q are equall. And, &longs;eeing that in the Equall and
Like Portions A G Q L and A N Z L; there are drawn from the
Extremities of their Ba&longs;es, A Q and A Z which cut off equall Porti
ons: It is obvious, that with the Diameters of the Portions they
the Angles at F and
S R and C R are equall to one another: And, therefore, N X and
G Y are al&longs;o equall; and X T and Y I. And &longs;ince G H is double
to H I, N X &longs;hall be le&longs;&longs;er 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 &longs;hall 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 &longs;uppo&longs;e in E. Therefore, by what was even now demon&longs;trated
it is manife&longs;t that the Portion &longs;hall not &longs;tay thus, but &longs;hall incline, &longs;o
as that its Ba&longs;e do in no wi&longs;e touch the Surface of the Liquid
And that the Portion will &longs;tand, &longs;o, as to make an Angle with the
Surface of the Liquid le&longs;&longs;er than
the Angle
&longs;trated. Let it, if po&longs;&longs;ible, &longs;tand,
&longs;o, as that it do not make an Angle
le&longs;&longs;er than the Angle
all things el&longs;e in the &longs;ame manner a
before; as is done in the pre&longs;et
Figure. We are to demon&longs;trat
in the &longs;ame method, that N T is e
quall to
equall al&longs;o to G I. And &longs;ince that in
the Triangles P
Angle
&longs;hall S R be le&longs;&longs;er than C R; nor N X than P Y: But &longs;ince P F is
greater than N T, let P F be Se&longs;quialter of P Y: N T &longs;hall be le&longs;&longs;er
than Se&longs;quialter of N X: And, therefore, N X &longs;hall 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 manife&longs;t, now, by what hath
been &longs;aid, that the Portion &longs;hall not continue in this po&longs;ition, but &longs;hall
turn about, &longs;o, as that its Axis do make an Angle with the Surface
of the Liquid, le&longs;&longs;er than the Angle