as the area of the bottom to twice the hole.
COR. 6. That part of the weight which presses upon the bottom is to
the whole weight of the water perpendicularly incumbent thereon as the
circle AB to the sum of the circles AB and EF, or as the circle AB to thf
excess of twice the circle AB above the area of the bottom. For that part
of the weight which presses upon the bottom is to the weight of the whole
water in the vessel as the difference of the circles AB and EF to the sum
of the same circles (by Cor. 4) ; and the weight of the whole water in the
vessel is to the weight of the whole water perpendicularly incumbent on
the bottom as the circle AB to the difference of the circles AB and EF.
Therefore, ex ce,quo perturbate, that part of the weight which presses upon
the bottom is to the weight of the whole water perpendicularly incumbent
OF NATURAL PHILOSOPHY. 337
thereon as the circle AB to the sum of the circles AB and EF. or the ex
cess of twice the circle AB above the bottom.
COR. 7. If in the middle of the hole EF there be placed the little circle
PQ described about the centre G, and parallel to the horizon, the weight
of water which that little circle sustains is greater than the weight of a
third part of a cylinder of water whose base is that little circle and its
height GH. For let ABNFEM be the cataract or column of falling water
whose axis is GH, as above, and let all the wa- K ^
ter, whose fluidity is not requisite for the ready
and quick descent of the water, be supposed to
be congealed, as well round about the cataract,
as above the little circle. And let PHQ be the
column of water congealed above the little cir
cle, whose vertex is H, and its altitude GH.
And suppose this cataract to fall with its whole
weight downwards, and not in the least to lie
against or to press PHQ, but to glide freely by
it without any friction, unless, perhaps, just at
the very vertex of the ice, where the cataract at the beginning of its fall
may tend to a concave figure. And as the congealed water AMEC, BNFD,
lying round the cataract, is convex in its internal superficies AME, BNF,
towards the falling cataract, so this column PHQ will be convex towards
the cataract also, and will therefore be greater than a cone whose base is
that little circle PQ and its altitude GH; that is, greater than a third
part of a cylinder described with the same base and altitude. Now that
little circle sustains the weight of this column, that is, a weight greater
than the weight of the cone, or a third part of the cylinder.
COR. 8. The weight of water which the circle PQ; when very small, sus
tains, seems to be less than the weight of two thirds of a cylinder of water
whose base is that little circle, and its altitude HG. For, things standing
as above supposed, imagine the half of a spheroid described whose base id
that little circle, and its semi-axis or altitude HG. This figure will be
equal to two thirds of that cylinder, and will comprehend within it the
column of congealed water PHQ, the weight of which is sustained by that
little circle. For though the motion of the water tends directly down
wards, the external superficies of that column must yet meet the base PQ
in an angle somewhat acute, because the water in its fall is perpetually ac
celerated, and by reason of that acceleration become narrower. Therefore,
oince that angle is less than a right one, this column in the lower parts
thereof will lie within the hemi-spheroid. In the upper parts also it will be
acute or pointed; because to make it otherwise, the horizontal motion of
the water must be at the vertex infinitely more swift than its motion to
wards the horizon. And the less this circle PQ is, the more acute will
22
338 THE MATHEMATICAL PRINCIPLES [BOOK II
the vertex of this column be
; and the circle being diminished in infinitn/n
the angle PHQ will be diminished in infinitum, and therefore the co
lumn will lie within the hemi-spheroid. Therefore that column is less than
that hemi-spheroid, or than two-third parts of the cylinder whose base is
that little circle, and its altitude GH. Now the little circle sustains a
force of water equal to the weight of this column, the weight of the ambient
water being employed in causing its efflux out at the hole.
COR. 9. The weight of water which the little circle PQ sustains, when
it is very small, is very nearly equal to the weight of a cylinder of water
whose base is that little circle, and its altitude |GH for this weight is an
arithmetical mean between the weights of the cone and the hemi-spheroid
above mentioned. But if that little circle be not very small, but on the
contrary increased till it be equal to the hole EF, it will sustain the weight
of all the water lying perpendicularly above it, that is, the weight of a
cylinder of water whose base is that little circle, and its altitude GH.
COR. 10. Arid (as far as I can judge) the weight which this little circle
sustains is always to the weight of a cylinder of water whose base is that
little circle, and its altitude iGH, as EF2 to EF 2 |PQ2
, or as the cir
cle EF to the excess of this circle above half the little circle PQ,, very
nearly.
LEMMA IV.
If a cylinder move uniformly forward in. the direction of its length, the
resistance made thereto is not at all changed by augmenting or di
minishing- that length ; and is therefore the same with the resistance
of a circle, described with the same diameter, and moving forward
with the same velocity in the direction, of a right line perpendicular to
its plane.
For the sides are not at all opposed to the motion ; and a cylinder be
comes a circle when its length is diminished in infinitum.
PROPOSITION XXXVII. THEOREM XXIX.
If a cylinder move uninformly forward in a compressed, infinite, arid
non-elasticfinid, in the direction of its length, the resistance arising
from the magnitude of its transverse section is to the force by which
its whole motion may be destroyed or generated, in the time that it
moves four times its length, as the density of the medium to the den
sity of the cylinder, nearly.
For let the vessel ABDC touch the surface of stagnant water with its
bottom CD, and let the water run out of this vessel into the stagnant wa
ter through the cylindric canal EFTS perpendicular co the horizon ; and
let the little circle PQ, be placed parallel to the horizon any where in the
SEC. VII.] OF NATURAL PHILOSOPHY. 339
middle of the canal ; and produce CA to K, so K I JL
f that AK may be to CK in the duplicate of the
-^ jg
""
e
ratio, which the excess of the orifice of the canal
EF above the little circle PQ bears to the cir
cle AB. Then it is manifest (by Case 5, Case
6, and Cor. 1, Prop. XXXVI) that the velocity
of the water passing through the annular space
between the little circle and the sides of the ves
sel will be the very same which the water would
acquire by falling, and in its fall describing the
altitude KG or IG.
And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite,
so that the lineola HI may vanish, arid the altitudes IG, HG become equal ;
the force of the water that flows down and presses upon the circle will be
to the weight of a cylinder whose base is that little circle, and the altitude
iIG, as EF 2 to EF 2 |PQ 2
, very nearly. For the force of the water
flowing downward uniformly through the whole canal will be the same
upon the little circle PQ. in whatsoever part of the canal it be placed.
I ,et now the orifices of the canal EF, ST be closed, and let the littk
circle ascend in the fluid compressed on every side, and by its ascent let it
oblige the water that lies above it to descend through the annular space
between the little circle and the sides of the canal. Then will the velocity
of the ascending little circle be to the velocity of the descending water as
the difference of the circles EF and PQ, is to the circle PQ; and the ve
locity of the ascending little circle will be to the sum of the velocities, that
is, to the relative velocity of the descending water with which it passes by
the little circle in its ascent, as the difference of the circles EF and PQ to
the circle EF, or as EF* PQ2 to EF 2
. Let that relative velocity be
equal to the velocity with v/hich it was shewn above that the water would
pass through the annular space, if the circle were to remain unmoved, that
is, to the velocity which the water would acquire by falling, and in its fall
describing the altitude IG ; and the force of the water upon the ascendingcircle
will be the same as before (by Cor. 5, of the Laws of Motion) ; that
is, the resistance of the ascending little circle will be to the weight of a
cylinder of water whose base is that little circle, and its altitude iIG, as
EF2 to EF2 iPQ 2
, nearly. But the velocity of the little circle will
be to the velocity which the water acquires by falling, and in its fall de
scribing the altitude [G, as EF 2 PQ2 to EF 2
.
Let the breadth of the canal be increased in wfinitum ; and the ratios
between EF 2 PQ2 and EF 2
, and between EF 2 and EF 2 iPQ 2
.
will become at last ratios of equality. And therefore the velocity of the
little circle wr ill now be the same which the water would acquire in falling,
and in its fall describing the altitude IG: and the resistance will become
340 THE MATHEMATICAL PRINCIPJ ES [BOOK IT.
equal to the weight of a cylinder whose base is that little circle, and its
altitude half the altitude IG, from which the cylinder must fall to acquire
the velocity of the ascending circle
; and with this velocity the cylinder in
the time of its fall will describe four times its length. But the resistance
of the cylinder moving forward with this velocity in the direction of its
length is the same with the resistance of the little circle (by Lem. IV), and
is therefore nearly equal to the force by which its motion may be generated
while it describes four times its length.
If the length of the cylinder be augmented or diminished, its motion,
and the time in which it describes four times its lengOth,t will be aug&mented
or diminished in the same ratio, and therefore the force by which the mo
tion so increased or diminished, may be destroyed or generated, will con
tinue the same ; because the time is increased or diminished in the same
proportion ; and therefore that force remains still equal to the resistance
of the cylinder, because (by Lem. IV) that resistance will also remain the
same.
If the density of the cylinder be augmented or diminished, its motion,
and the force by which its motion may be generated or destroyed in the
same time, will be augmented or diminished in the same ratio. Therefore
the resistance of any cylinder whatsoever will be to the force by which its
whole motion may be generated or destroyed, in the time during which it
moves four times its length, as the density of the medium to the density of
the cylinder- nearly. Q..E.D.
A fluid must be compressed to become continued; it must be continued
and non-elastic, that all the pressure arising from its compression may be
propagated in an instant
; and so, acting equally upon all parts of the body
moved, may produce no change of the resistance. The pressure arising
from the motion of the body is spent in generating a motion in the parts
of the fluid, and this creates the resistance. But the pressure arising from
the compression of the fluid, be it ever so forcible, if it be propagated in an
instant, generates no motion in the parts of a continued fluid, produces no
change at all of motion therein ; and therefore neither augments nor les
sens the resistance. This is certain, that the action of the fluid arising
from the compression cannot be stronger on the hinder parts of the body
moved than on its fore parts, and therefore cannot lessen the resistance de
scribed in this proposition. And if its propagation be infinitely swifter
than the motion of the body pressed, it will not be stronger on the fore
parts than on the hinder parts. But that action will be infinitely
swifter, and propagated in an instant, if the fluid be continued and nonelastic.
COR. 1. The resistances, made to cylinders going uniformly forward in
the direction of their lengths through continued infinite mediums are in a
com A
Hi
E
SEC. VII.] OF NATURAL PHILOSOPHY- 341
ratio compounded of the duplicate ratio of the velocities and the duplicate
ratio of the diameters, and the ratio of the density of the mediums.
COR. 2. If the breadth of the canal be not infinitely increased but the
cylinder go forward in the direction of its length through an included
quiescent medium, its axis all the while coinciding with the axis of the
canal, its resistance will be to the force by which its whole motion, in the
time in which it describes four times its length,
K
............. I... ........L
may be generated or destroyed, in a ratio
pounded of the ratio of EF 2 to EF 2 i
once, and the ratio of EF2 to EF2 PQ,2
twice, and the ratio of the density of the medium
to the density of the cylinder.
COR. 3. The same thing supposed, and that a
length L is to the quadruple of the length of
the cylinder in a ratio compounded of the ratio
EF 2 -- iPQ2 to EF 2
once, and the ratio of
EF 2 PQ, 2 to EF 2
twice; the resistance of
the cylinder will be to the force by which its whole motion, in the time
during which it describes the length L, may be destroyed or generated, as
the density of the medium to the density of the cylinder.
SCHOLIUM.
In this proposition we have investigated that resistance alone which
arises from the magnitude of the transverse section of the cylinder, neg
lecting that part of the same which may arise from the obliquity of the
motions. For as, in Case 1, of Prop. XXXVL, the obliquity of the mo
tions with which the parts of the water in the vessel converged on every
side to the hole EF hindered the efflux of the water through the hole, so,
in this Proposition, the obliquity of the motions, with which the parts of
the water, pressed by the antecedent extremity of the cylinder, yield to the
pressure, and diverge on all sides, retards their passage through the places
that lie round that antecedent extremity, toward the hinder parts of the
cylinder, and causes the fluid to be moved to a greater distance; which in
creases the resistance, and that in the same ratio almost in which it dimin
ished the efflux of the water out of the vessel, that is, in the duplicate ratio
of 25 to 21, nearly. And as, in Case 1, of that Proposition, we made the
parts of the water pass through the hole EF perpendicularly and in the
greatest plenty, by supposing all the water in the vessel lying round the
cataract to be frozen, and that part of the water whose motion was oblique,
and useless to remain without motion, so in this Proposition, that the
obliquity of the motions may be taken away, and the parts of the water
may give the freest passage to the cylinder, by yielding to it witli the most
direct and quick motion possible, so that only so much resistance may re
542 THE MATHEMATICAL PRINCIPLES [BoOK II.
main as arises from the magnitude of the transverse section, and which is
incapable of diminution, unless by diminishing the diameter of the cylinder ;
we must conceive those parts of the fluid whose motions are oblique and
useless, and produce resistance, to be at rest among themselves at both ex
tremities of the cylinder, and there to cohere, and be joined to the cylinder.
Let ABCD be a rectangle, and let
AE and BE be two parabolic arcs, i 1
described with the axis AB, and g j^
with a latus rectum that is to the .----""
space HG, which must be described
by the cylinder in falling, in order
to acquire the velocity with which it moves, as HG to ^AB. Let CF and
DF be two other parabolic arcs described with the axis CD, and a latus
rectum quadruple of the former; and by the convolution of the figure
about the axis EF let there be generated a solid, whose middle part ABDC
is the cylinder we are here speaking of, and whose extreme parts ABE and
CDF contain the parts of the fluid at rest among themselves, and concreted
into two hard bodies, adhering to the cylinder at each end like a head and
tail. Then if this solid EACFDB move in the direction of the length of
its axis FE toward the parts beyond E, the resistance will be the same
which we have here determined in this Proposition, nearly ; that is, it will
have the same ratio to the force with which the whole motion of the cyl
inder may be destroyed or generated, in the time that it is describing the
length 4AC with that motion uniformly continued, as the density of the
fluid has to the density of the cylinder, nearly. And (by Cor. 7, Prop.
XXXVI) the resistance must be to this force in the ratio of 2 to 3, at the
least.
LEMMA V.
If a cylinder, a sphere, and a spheroid, of equal breadths be placed suc
cessively in the middle of a cylindric canal, so that their axes may
coincide with the axis of the canal, these bodies will equally hinder t^e
passage of the water through the canal.
For the spaces lying between the sides of the canal, and the cylinder,
sphere, and spheroid, through which the water passes, are equal ; and the
water will pass equally through equal spaces.
This is true, upon the supposition that all the water above the cylinder,
sphere, or spheroid, whose fluidity is not necessary to make the passage of
the water the quickest possible, is congealed, as was explained above in Cer
7, Prop. XXXVI.
SEC. VII.] OF NATURAL PHILOSOPHY 343
LEMMA VI.
The same supposition remaining, the fore- mentioned bodies are equally
acted OIL by the water Jlowing through the canal.
This appears by Lein. V and the third Law. For tht water and the
bodies act upon each other mutually and equally.
LEMMA VIL
If the water be at rest in the canal, and these bodies move with equil ve
locity and the contrary way through the canal, their resistances will
be equal among themselves.
This appears from the last Lemma, for the relative motions remain the
same among themselves.
SCHOLIUM.
The case is the same of all convex and round bodies, whose axes coincide
with the axis of the canal. Some difference may arise from a greater or
less friction; but in these Lemmata we suppose the bodies to be perfectly
smooth, and the medium to be void of all tenacity and friction
; and that
those parts of the fluid which by their oblique and superfluous motions may
disturb, hinder, and retard the flux of the water through the canal, are at
rest amorg themselves ; being fixed like water by frost, and adhering to
the fore and hinder parts of the bodies in the manner explained in the
Scholium of the last Proposition : for in what follows we consider the very
least resistance that round bodies described with the greatest given trans
verse sections can possibly meet with.
Bodies swimming upon fluids, when they move straight forward, cause
the fluid to ascend at their fore parts and subside at their hinder parts,
especially if they are of an obtuse figure ; and thence they meet with a
little more resistance than if they were acu*-e at the head and tail. And
bodies moving in elastic fluids, if they are obtuse behind and before, con
dense the fluid a little more at their fore parts, and relax the same at theii
hinder parts ; and therefore meet also with a little more resistance than ii
they were acute at the head and tail. But in these Lemmas and Proposi
tions we are not treating of elastic but non-elastic fluids; not of bodies
floating on the surface of the fluid, but deeply immersed therein. And
when the resistance of bodies in non-elastic fluids is once known, we may
then augment this resistance a little in elastic fluids, as our air; and in
the surfaces of stagnating fluids, as lakes and seas.
PROPOSITION XXXVIII. THEOREM XXX.
If a globe move uniformly forward in a compressed, infinite, and no?t
elastic fluid, its resistance is to the force by which its whole
544 THE MATHEMATICAL PRINCIPLES [BOOK II
may be destroyed or generated, in the time that it describes eight third
parts of its diameter, as the density of the fluid to the density of the
globe, very nearly.
For the globe is to its circumscribed cylinder as two to three ; and there
fore the force which can destroy all the motion of the cylinder, while the
same cylinder is describing the length of four of its diameters, will destroy
all the motion of the globe, while the globe is describing two thirds of this
length, that is, eight third parts of its own diameter. Now the resistance
of the cylinder is to this force very nearly as the density of the fluid to the
density of the cylinder or globe (by Prop. XXXVII), and the resistance of
the globe is equal to the resistance of the cylinder (by Lem. V, VI, and
VII). Q.E.D.
COR. I. The resistances of globes in infinite compressed mediums are in
a ratio compounded of the duplicate ratio of the velocity, and the dupli
cate ratio of the diameter, and the ratio of the density of the mediums.
COR. 2. The greatest velocity, with which a globe can descend by its
comparative weight through a resisting fluid, is the same which it may
acquire by falling with the same weight, and without any resistance, and
in its fall describing a space that is, to four third parts of its diameter as
the density of the globe to the density of the fluid. For the globe in the
time of its fall, moving with the velocity acquired in falling, will describe
a space that will be to eight third parts of its diameter as the density of
the globe to the density of the fluid ; and the force of its weight which
generates this motion will be to the force that can generate the same mo
tion, in the time that the globe describes eight third parts of its diameter,
with the same velocity as the density of the fluid to the density of the
globe; and therefore (by this Proposition) the force of weight will be equal
to the force of resistance, and therefore cannot accelerate the globe.
COR. 3. If there be given both the density of the globe and its velocity
at the beginning of the motion, and the density of the compressed quiescent
fluid in which the globe moves, there is given at any time both the velo
city of the globe and its resistance, and the space described by it (by Cor.
7, Prop. XXXV).
COR. 4. A globe moving in a compressed quiescent fluid of the same
density with itself will lose half its motion before it can describe the length
of two of its diameters (by the same Cor. 7).
PROPOSITION XXXIX. THEOREM XXXI.
If a S lobe move uniformly forward through a fluid inclosed and com
pressed in a cylindric canal, its resistance is to the force by which its
whole motion may be generated or destroyed, in the time in which it
describes eight third parts of its dia?netert in a ratio compounded of
EC. VII.] OF NATURAL PHILOSOPHY. 345
the ratio of the orifice of the canal to the excess of that orifice above
half the greatest circle of the globe ; and the duplicate ratio of the
orifice of the canal, to the excess of that orifice above the greatest circle
of the globe ; and t/ie ratio of the density of thefluid to the density of
the globe, nearly.
This appears by Cor. 2, Prop. XXXVII, and the demonstration pro
ceeds in the same manner as in the foregoing Proposition.
SCHOLIUM.
In the last two Propositions we suppose (as was done before in Lem. V)
that all the water which precedes the globe, and whose fluidity increases
the resistance of the same, is congealed. Now if that water becomes fluid,
it will somewhat increase the resistance. But in these Propositions that
increase is so small, that it may be neglected, because the convex superfi
cies of the globe produces the very same effect almost as the congelation
of the water.
PROPOSITION XL. PROBLEM IX.
Tofind by phenomena the resistance of a globe moving through a per
fectly fluid compressed medium.
Let A be the weight of the globe in vacua, B its weight in the resisting
medium, D the diameter of the globe. F a space which is to fD as the den
