these ratios, the efficacy of a particle, falling upon the globe obliquely in
the direction of the right line FBy to move the globe in the direction of its
incidence, is to the efficacy of the same particle falling in the same line
perpendicularly on the cylinder, to move it in the same direction, as BE 2
to BC 3
. Therefore if in 6E, which is perpendicular to the circular base of
the cylinder NAO, and equal to the radius AC, we take H equal to
BEa
-
; then 6H will be to 6E as the effect of the particle upon the globe t<?
\~i\j
the effect of the particle upon the cylinder. Arid therefore the solid which
is formed by all the right lines 6H will be to the solid formed by all the
right lines />E as the effect of all the particles upon the globe to the effect
of all the particles upon the cylinder. But the former of these solids is a
328 THE MATHEiAlATICAL PRINCIPLES [BooK li.
paraboloid whose vertex is C, its axis CA, and latus rectum CA, and the
latter solid is a cylinder circumscribing the paraboloid ; and it is knowr
that a paraboloid is half its circumscribed cylinder. Therefore the whole
force of the medium upon the globe is half of the entire force of the same
upon the cylinder. And therefore if the particles of the medium are at
rest, and the cylinder and globe move with equal velocities, the resistance
of the globe will be half the resistance of the cylinder. Q.E.D.
SCHOLIUM.
By the same method other figures may be compared together as to their
resistance; and those may be found which are most apt to continue their
motions in resisting mediums. As if upon the circular base CEBH from
the centre O, with thy radius OC, and the altitude OD, one would construct
a frustum CBGF of a cone, which should meet with less resistance than
any other frustum constructed with the same base and altitude, and going
forwards towards D in the direction of its axis : bisect the altitude OD in
U,, and produce OQ, to S, so that QS may be equal to Q,C, and S will be
the vertex of the cone whose frustum is sought.
r
J
Whence, by the bye, since the angle CSB is always acute, it follows, that,
if the solid ADBE be generated by the convolution of an elliptical or oval
figure ADBE about its axis AB, and the generating figure be touched bythree
right lines FG, GH, HI, in the points F, B, and I, so that GH shall
be perpendicular to the axis in the point of contact B, arid FG, HI may be
inclined to GH in the angles FGB, BHI of 135 degrees: the solid arising
from the convolution of the figure ADFGH1E about the same axis AB
will be less resisted than the former solid; if so be that both move forward
in the direction of their axis AB, and that the extremity B of each go
foremost. Which Proposition I conceive may be of use in the building of
ships.
If the figure DNFG be such a curve, that if, from any point thereof, as
N, the perpendicular NM be let fall on the axis AB, and from the given
point G there be drawn the right line GR parallel to a right line touching
the figure in N, and cutting the axis produced in R, MN becomes to GR
as GR, 3 to 4BR X GB 2
, the solid described, by the revolution of this figure
SEC. Vll.J OF NATURAL PHILOSOPHY. 32S
about its axis AB, moving in the before-mentioned rare medium from A
towards B, will be less resisted than any other circular solid whatsoever,
described of the same length and breadth.
PROPOSITION XXXV. PROBLEM VII.
If a rare medium consist of very small quiescent particles of equal mag
nitudes, and freely disposed at equal distances from one another : to
jind the resistance of a globe moving uniformly forward in this
medium.
CASE 1. Let a cylinder described with the same diameter and altitude be
conceived to go forward with the same velocity in the direction of its axis
through the same medium ; and let us suppose that the particles of the
medium, on which the globe or cylinder falls, fly back with as great a force
of reflexion as possible. Then since the resistance of the globe (by the last
Proposition) is but half the resistance of the cylinder, and since the globe
is to the cylinder as 2 to 3, and since the cylinder by falling perpendicu
larly on the particles, and reflecting them with the utmost force, commu
nicates to them a velocity double to its own; it follows that the cylinder.
in moving forward uniformly half the length of its axis, will communicate
a motion to the particles which is to the whole motion of the cylinder as
the density of the medium to the density of the cylinder ; and that the
globe, in the time it describes one length of its diameter in moving uni
formly forward, will communicate the same motion to the particles ; and
in the time that it describes twro thirds of its diameter, will communicate
a motion to the particles which is to the whole motion of the globe as the
density of the medium to the density of the globe. Arid therefore the
globe meets with a resistance, which is to the force by which its whole mo
tion may be either taken away or generated in the time in which it de
scribes two thirds of its diameter moving uniformly forward, as the den
sity of the medium to the density of the globe.
CASE 2. Let us suppose that the particles of the medium incident on
the globe or cylinder are not reflected ; and then the cylinder falling per
pendicularly on the particles will communicate its own simple velocity to
them, and therefore meets a resistance but half so great as in the former
case, and the globe also meets with a resistance but half so great.
CASE 3. Let us suppose the particles of the medium to fly back from
the globe with a force which is neither the greatest, nor yet none at all, but
with a certain mean force ; then the resistance of the globe will be in the
same mean ratio between the resistance in the first case and the resistance
in the second. Q.E.I.
COR. 1. Hence if the globe and the particles are infinitely hard, and
destitute of all elastic force, and therefore of all force of reflexion ; thf
resistance of the globe will be to the force by which its whole motion may
330 THE MATHEMATICAL PRINCIPLES [BOOK I)
be destroyed or generated, in the time that the globe describes four third
parts of its diameter, as the density of the medium to the density of the
^lobe.
Con. 2. The resistance of the globe, cceteris paribus, is in the duplicate
ratio of the velocity.
CUR. 3. The resistance of the globe, cocteris paribus, is in the duplicate
ratio of the diameter.
COR. 4. The resistance of the globe is, cceteris paribus, as the density of
the medium.
COR, 5. The resistance of the globe is in a ratio compounded of the du
plicate ratio of the velocity, arid the duplicate ratio of the diameter, and
the ratio of the density of the medium.
COR. 6. The motion of the globe and its re
sistance may be thus expounded Let AB be the
time in which the globe may, by its resistance
uniformly continued, lose its whole motion.
Erect AD, BC perpendicular to AB. J ,et BC be
that whole motion, and through the point C, the
asymptotes being AD, AB, describe the hyperbola
CF. Produce AB to any point E. Erect the perpendicular EF meeting
the hyperbola in F. Complete the parallelogram CBEG, and draw AF
meeting BC in H. Then if the globe in any time BE, with its first mo
tion BC uniformly continued, describes in a non-resisting medium the space
CBEG expounded by the area of the parallelogram, the same in a resisting
medium will describe the space CBEF expounded by the area of the hvperbola;
and its motion at the end of that time will be expounded by EF,
the ordinate of the hyperbola, there being lost of its motion the part FG.
And its resistance at the end of the same time will be expounded by the
length BH, there being lost of its resistance the part CH. All these things
appear by Cor. 1 and 3, Prop. V., Book II.
COR. 7. Hence if the globe in the time T by the resistance R uniformly
continued lose its whole motion M, the same globe in the time t in a
resisting medium, wherein the resistance R decreases in a duplicate
/M
ratio of the velocity, will lose out of its motion M the part ,.i the
TM
part rn . ; remaining ; and will describe a space which is to the space de
scribed in the same time t, with the uniform motion M, as the logarithm of
T + t
the number ^. multiplied by the number 2,302585092994 is to the
number ^ because the hyperbolic area BCFE is to the rectangle BCGE
in that proportion.
SEC. VII.] OF NATURAL PHILOSOPHY. 331
SCHOLIUM.
I have exhibited in this Proposition the resistance and retardation of
spherical projectiles in mediums that are not continued, and shewn that
this resistance is to the force by which the whole motion of the globe may be
destroyed or produced in the time in which the globe can describe two thirds
of its diameter, with a velocity uniformly continued, as the density of the
medium to the density of the globe, if so be the globe and the particles of
the medium be perfectly elastic, and are endued with the utmost force of
reflexion ; and that this force, where the globe and particles of the medium
are infinitely hard and void of any reflecting force, is diminished one half.
But in continued mediums, as water, hot oil, and quicksilver, the globe as
it passes through them does not immediately strike against all the parti
cles of the fluid that generate the resistance made to it, but presses only
the particles that lie next to it, which press the particles beyond, which
press other particles, and so on ; and in these mediums the resistance is di
minished one other half. A globe in these extremely fluid mediums meets
with a resistance that is to the force by which its whole motion may be
destroyed or generated in the time wherein it can describe, with that mo
tion uniformly continued, eight third parts of its diameter, as the density
of the medium to the density of the globe. This I shall endeavour to shew
in what follows.
PROPOSITION XXXVI. PROBLEM VIII.
To define the motion of water running out of a cylindrical vessel through
a hole made at the bottom.
Let AC DB be a cylindrical vessel, AB the mouth p = Q:
of it, CD the bottom p irallel to the horizon, EF a
circular hole in the middle of the bottom, G the
c-?ritre of the hole, and GH the axis of the cylin- Kj
cler perpendicular to the horizon. And suppose a
cylinder of ice APQ,B to be of the same breadth
with the cavity of the vessel, and to have the same
axis, and to descend perpetually with an uniform
motion, and that its parts, as soon as they touch the
superficies AB, dissolve into water, and flow
( wn by their weight into the vessel, and in their
fall compose the cataract or column of water
ABNFEM, passing through the hole EF, and filling up the same exactly.
Let the uniform velocity of the descending ice and of the contiguous water
in the circle AB be that which the water would acquire by falling through
the space IH ; and let IH and HG lie in the same right line ; and through
332 THE MATHEMATICAL PRINCIPLES [BOOK Jl
the point I let there be drawn the right line KL parallel to the horizon
and meeting the ice on both the sides thereof in K and L. Then the ve
locity of the water running out at the hole EF will be the same that it
would acquire by falling from I through the space IG. Therefore, by
Galih cJ s Theorems, IG will be to IH in the duplicate ratio of the velo
city of the water that runs out at the hole to the velocity of the wrater in
the circle AB, that is, in the duplicate ratio of the circle AB to the circle
EF ; those circles being reciprocally as the velocities of the water which
in the same time and in equal quantities passes severally through each of
them, and completely fills them both. We are now considering the velo
city with which the water tends to the plane of the horizon. But the mo
tion parallel to the same, by which the parts of the falling water approach to
each other, is not here taken notice of; since it is neither produced by
gravity, nor at all changes the motion perpendicular to the horizon which the
gravity produces. We suppose, indeed, that the parts of the water cohere
a little, that by their cohesion they may in falling approach to each othei
with motions parallel to the horizon in order to form one single cataract,
and to prevent their being divided into several : but the motion parallel to
the horizon arising from this cohesion does not come under our present
consideration.
CASE 1. Conceive now the w^hole cavity in the vessel, wrhich encompasses
the falling water ABNFEM, to be full of ice, so that the water may pass
through the ice as through a funnel. Then if the water pass very near to
the ice only, without touching it; or, which is the same tiling, if by rea
son of the perfect smoothness of the surface of the ice, the water, though
touching it. glides over it writh the utmost freedom, and without the le-ast
resistance; the water will run through the hole EF with the same velocity
as before, and the whole weight of the column of water ABNFEM will be
all taken up as before in forcing out the water, and the bottom of the vessel
will sustain the weight of the ice encompassing that column.
Let now the ice in the vessel dissolve into water ; yet will the efflux of
the water remain, as to its velocity, the same as before. It will not be
less, because the ice now dissolved will endeavour to descend ;
it will not
be greater, because the ice. now become water, cannot descend without hin
dering the descent of other water equal to its own descent. The same force
ought always to generate the same velocity in the effluent water.
But the hole at the bottom of the vessel, by reason of the oblique mo
tions of the particles of the effluent water, must be a little greater than before,
For now the particles of the water do not all of them pass through the
hole perpendicularly, but, flowing down on all parts from the sides of the
vessel, and converging towards the hole, pass through it with oblique mo
tions : r,r,d in tending downwards meet in a stream whose diameter is a little
smaller below the hole than at the hole itself : its diameter being to the
SEC. V1L! OF NATURAL PHILOSOPHY. 333
diameter of the hole as 5 to 6, or as 5^ to 6|, very nearly, if I took the
measures of those diameters right. I procured a very thin flat plate, hav
ing a hole pierced in the middle, the diameter of the circular hole being
f parts of an inch. And that the stream of running waters might not be
accelerated in falling, and by that acceleration become narrower, I fixed
this plate not to the bottom, but to the side of the vessel, so us to make the
water go out in the direction of a line parallel to the horizon. Then, when
the vessel was full of water, I opened the hole to let it run out ; and the
diameter of the stream, measured with great accuracy at the distance of
about half an inch from the hole, was f J- of an inch. Therefore the di
ameter of this circular hole was to the diameter of the stream very nearly
as 25 to 21. So that the water in passing through the hole converges on
all sides, and, after it has run out of the vessel, becomes smaller by converg
ing in that manner, and by becoming smaller is accelerated till it comes to
the distance of half an inch from the hole, and at that distance flows in a
smaller stream and with greater celerity than in the hole itself, and this
in the ratio of 25 X 25 to 21 X 21, or 17 to 12, very nearly ; that is, in
about the subdaplicate ratio of 2 to 1. Now it is certain from experiments,
that the quantity of water running out in a given time through a circular
hole made in the bottom of a vessel is equal to the quantity, which, flow
ing with the aforesaid velocity, would run out in the same time through
another circular hole, whose diameter is to the diameter of the former as
21 to 25. And therefore that running water in passing through the
hole itself has a velocity downwards equal to that which a heavy body
would acquire in falling through half the height of the stagnant water in
the vessel, nearly. But, then, after it has run out, it is still accelerated by
converging, till it arrives at a distance from the hole that is nearly equal to
its diameter, and acquires a velocity greater than the other in about the
subduplicate ratio of 2 to 1
; which velocity a heavy body would nearly
acquire by falling through the whole height of the stagnant water in the
vessel.
Therefore in what follows let the diameter of
the stream be represented by that lesser hole which
we called EF. And imagine another plane VW
above the hole EF, and parallel to the plane there
of, to be placed at a distance equal to the diame
ter of the same hole, and to be pierced through
with a greater hole ST, of such a magnitude that
a stream which will exactly fill the lower hole EF
may pass through it
; the diameter of which hole
will therefore be to the diameter of the lower hole as 25 to 21, nearly. By
this means the water will run perpendicularly out at the lower hole ; and
the quantity of the water running out will be, according to the magnitude
334 THE MATHEMATICAL PRINCIPLES [BOOK 11
of this last hole, the same, very nearly, which the solution of the Problem
requires. The space included between the two planes and the falling stream
may be considered as the bottom of the vessel. But, to make the solution
more simple and mathematical, it is better to take the lower plane alone
for the bottom of the vessel, and to suppose that the water which flowed
through the ice as through a funnel, and ran out of the vessel through the
hole EF made in the lower plane, preserves its motion continually, and that
the ice continues at rest. Therefore in what follows let ST be the diame
ter of a circular hole described from the centre Z, and let the stream run
out of the vessel through that hole, when the water in the vessel is all
fluid. And let EP be the diameter of the hole, which the stream, in fall
ing through, exactly fills up, whether the water runs out of the vessel by
that upper hole ST, or flows through the middle of the ice in the vessel,
as through a funnel. And let the diameter of the upper hole ST be to the
diameter of the lower EF as about 25 to 21, and let the perpendicular di&
tance between the planes of the holes be equal to the diameter of the lesser
hole EF. Then the velocity of the water downwards, in running out of
the vessel through the hole ST, will be in that hole the same that a body
may acquire by falling from half the height IZ ; and the velocity of both
the falling streams will be in the hole EF, the same which a body would
acquire by falling from the Avhole height IG.
CASE 2. If the hole EF be not in the middle of the bottom of the ves
sel, but in some other part thereof, the water will still run out with the
same velocity as before, if the magnitude of the hole be the same. For
though an heavy body takes a longer time in descending to the same depth,
by an oblique line, than by a perpendicular line, yet in both cases it acquires
in its descent the same velocity ; as Galileo has demonstrated.
CASE 3. The velocity of the water is the same when it runs out through
a hole in the side of the vessel. For if the hole be small, so that the in
terval between the superficies AB and KL may vanish ns to sense, and the
stream of water horizontally issuing out may form a parabolic figure; from
the latus rectum of this parabola may be collected, that the velocity of the
effluent water is that which a body may acquire by falling the height IG
or HG of the stagnant water in the vessel. For, by making an experi
ment, I found that if the height of the stagnant water above the hole were
20 inches, and the height of the hole above a plane parallel to the horizon
were also 20 inches, a stream of water springing out from thence wrould
fall upon the plane, at the distance of 37 inches, very nearly, from a per
pendicular let fall upon that plane from the hole. For without resistance
the stream would have fallen upon the plane at the distance of 40 inches,
the latus rectum of the parabolic stream being 80 inches.
CASE 4. If the effluent water tend upward, it will still issue forth with
the same velocity. For the small stream of water springing upward, as
SEC. V11.J OF NATURAL PHILOSOPHY. 335
cends with a perpendicular motion to GH or GI, the height of the stagnant
water in the vessel
; excepting in so far as its ascent is hindered a little by
the resistance of the air : and therefore it springs out with the same ve
locity that it would acquire in falling from that height. Every particle of
the stagnant water is equally pressed on all sides (by Prop. XIX., Book II),
and, yielding to the pressure, tends always with an equal force, whether it
descends through the hole in the bottom of the vessel, or gushes out in an
horizontal direction through a hole in the side, or passes into a canal, and
springs up from thence through a little hole made in the upper part of the
canal. And it may not only be collected from reasoning, but is manifest
also from the well-known experiments just mentioned, that the velocity
with which the water runs out is the very same that is assigned in this
Proposition.
CASE 5. The velocity of the effluent water is the same, whether the
figure of the hole be circular, or square, or triangular, or any other figureequal
to the circular ; for the velocity of the effluent water does not depend
upon the figure of the hole, but arises from its depth below the plane
KL.
CASE 6. If the lower part of the vessel ABDC
B be immersed into stagnant water, and the height
of the stagnant water above the bottom of the ves
sel be GR, the velocity with which the water that
is in the vessel will run out at the hole EF into
the stagnant water will be the same which the
water would acquire by falling from the height
IR ; for the weight of all the water in the vessel
that is below the superficies of the stagnant water
will be sustained in equilibrio by the weight of the stagnant water, and
therefore does riot at all accelerate the motion of the descending water in
the vessel. This case will also appear by experiments, measuring the times
in which the water will run out.
COR. 1. Hence if CA the depth of the water be produced to K, so that
AK may be to CK in the duplicate ratio of the area of a hole made in any
part of the bottom to the area of the circle AB, the velocity of the effluent
water will be equal to the velocity which the water would acquire by falling
from the height KC.
COR. 2. And the force with which the whole motion of the effluent watei
may be generated is equal to the weight of a cylindric column of water r
whose base is the hole EF, and its altitude 2GI or 2CK. For the effluent
water, in the time it becomes equal to this column, may acquire, by falling
by its own weight from the height GI, a velocity equal to that with which
it runs out.
COR. 3. The weigb t of all the water in the vessel ABDC is to that part
\
336 THE MATHEMATICAL PRINCIPLES [BOOK II
of the weight which is employed in forcing out the water as the sum of
the circles AB and EF to twice the circle EF. For let IO be a mean pro
portional between IH and IG, and the water running out at the hole EF
will, in the time that a drop falling from I would describe the altitude IG,
become equal to a cylinder whose base is the circle EF and its altitude
2IG; that is, to a cylinder whose base is the circle AB, and whose altitude
is 2IO. For the circle EF is to the circle AB in the subduplicate ratio cf
the altitude IH to the altitude IG ; that is, in the simple ratio of the mean
proportional IO to the altitude IG. Moreover, in the time that a drop
falling from I can describe the altitude IH, the water that runs out will
hare become equal to a cylinder whose base is the circle AB, and its alti
tude 2IH ; and in the time that a drop falling from I through H to G de
scribes HG, the difference of the altitudes, the effluent water, that is, the
water contained within the solid ABNFEM, will be equal to the difference
of the cylinders, that is, to a cylinder whose base is AB, and its altitude
2HO. And therefore all the water contained in the vessel ABDC is to the
whole falling water contained in the said solid ABNFEM as HG to2HO,
that is, as HO + OG to 2HO, or IH + K ) to 2IH. But the weight of all
the water in the solid ABNFEM is employed in forcing out the water ;
and therefore the weight of all the water in the vessel is to that part of
the weight that is employed in forcing out the water as IH + IO to 2IH,
and therefore as the sum of the circles EF and AB to twice the circle
EF.
COR. 4. And hence the weight of all the water in the vessel ABDC is
to the other part of the weight which is sustained by the bottom of the
vessel as the sum of the circles AB and EF to the difference of the same
circles.
COR. 5. And that part of the weight which the bottom of the vessel sus
tains is to the other part of the weight employed in forcing out the water
as the difference of the circles AB and EF to twice the lesser circle EF, or
