284 THE MATHEMATICAL PRINCIPLES [BOOK II
2APQ, + 2BA X PU
II of this Book) the moment KL of AK will be equal to
2BPQ
or Z -, and the moment KLON of the area ANK will be equal to
2BPQ X LO BPQ, X BD 3
~~Z~~ >r 2Z X OK x~AB"
CASE 1. Now if the body ascends, and the gravity be as AB2 + BD 9
BET being a circle, the line AC, which is proportional to the gravity
AW2 i RF)2
will be - ~--, and DP 2 or AP 2 + 2BAP + AB 2 + BD2 will be
AK XZ + AC X Z or CK X Z : and therefore the area DTV will be to
the area DPQ, as DT2 or I)B 2 to CK X Z.
CASE 2. If the body ascends, and the gravity be as AB 2 BD 2
, the
A r>2 _ HI) 2
line AC will be
"--^---
, and DT2 will be to DP 2 as DF 2 or DB 2
Z
to BP2 BD 2 or AP 2 + 2BAP + AB 2 BD 2
, that is, to AK X Z +
H
AC X Z or CK X Z. And therefore the area DTV will be to the area
DPQ as DB2 to CK X Z.
CASE 3. And by the same reasoning, if the body descends, and therefore
the gravity is as BD 2 AB 2
, and the line AC becomes equal to
or) 2 AB 2
5T r ; the area DTV will be to the area DPQ as DB2 to CK Z X
Z : as above.
Since, therefore, these areas are always in this ratio, if for the area
SEC. 111^ OF NATURAL PHILOSOPHY. 285
DTY, by which the moment of the time, always equal to itself, is express
ed, there be put any determinate rectangle, as BD X m, the area DPQ,,
that is, |BD X PQ, will be to BD X mas CK X Z to BD 2
. And thence
PQ X BD 3 becomes equal to 2BD XmX CK X Z, and the moment KLON
BP X BD X tn
of the area A6NK, found before, becomes-.-^
--. From the area
DET subduct its moment DTV or BD X m, and there will remain
---
-Pp . Therefore the difference of the moments, that is, the
AP X BD X m
nt of the difference of tne areas, is equal to --7-5---
; and
therefore (because of the given quantity
---T-~ ) as the velocity AP ;
that is, as the moment of the space which the body describes in its ascent
or descent. And therefore the difference of the areas, and that space, in
creasing or decreasing by proportional moments, and beginning together or
vanishing together, are proportional. Q.E.D.
COR. If the length, which arises by applying the area DET to the line
BD, be called M ; and another length V be taken in that ratio to the length
M, which the line DA has to the line DE; the space which a body, in a
resisting medium, describes in its whole ascent or descent, will be to the
space which a body, in a non-resisting medium, falling from rest, can de
scribe in the same time, as the difference of the aforesaid areas to
BD X V2
-
-TO"""" j
an(^ therefore is given from the time given. For the space in a
A.LJ
non-resisting medium is in a duplicate ratio of the time, or as V2
; and.
BD X V 2
because BD and AB are given, as ----TTT- . This area is equal to the
DA 2 X BD x M2
area --
fvGr*~~~T~R "~ anc* ^ne moment Of M is m ; and therefore the
DA2 X BD X 2M X m
moment of this area is ---=----
^5 -. But this moment is to
"" X .A D
the moment of the difference of the aforesaid areas DET and A6NK, viz., to
AP X BD X m DA2 X BD X M x DA2
- -- . ^^ , as - -r- - to |BD X AP, or as into DET
to DAP ; and, therefore, when the areas DKT and DAP are least, in the
BD X V 2
ratio of equality. Therefore the area r^
-- and the difference of the
areas DET and A&NK, when all these areas are least, have equal moments ;
and { re therefore equal. Therefore since the velocities, and therefore also
the s] aces in both mediums described together, in the beginning of the de
scent or the end of the ascent, approach to equality, and therefore are then
286 THE MATHEMATICAL PRINCIPLES [BOOK II
BD X V2
one to another as the area Ar-D^ , and the difference of the areas DET
and A6NK ; and moreover since the space, in a non-resisting medium, is
BD X V 2
perpetually as Tu~~> an(^ tne sP ace
>
in a resisting medium, is perpetu
ally as the difference of the areas DET and A&NK ;
it necessarily follows,
that the spaces, in both mediums, described in any equal times, are one to
BD X V 2
another as that area 7-5 an(^ ^he difference of the areas DET and
A6NK. Q.E.D.
SCHOLIUM.
The resistance of spherical bodies in fluids arises partly from the tena
city, partly from the attrition, and partly from the density of the medium.
And that part of the resistance which arises from the density of the fluid
is, as I said, in a duplicate ratio of the velocity ; the other part, which
arises from the tenacity of the fluid, is uniform, or as the moment of the
time ; and, therefore, we might now proceed to the motion of bodies, whicli
are resisted partly by an uniform force, or in the ratio of the moments of
the time, and partly in the duplicate ratio of the velocity. But it is suf
ficient to have cleared the way to this speculation in Prop. VIII and IX
foregoing, and their Corollaries. For in those Propositions, instead of the
uniform resistance made to an ascending body arising from its gravity,
one may substitute the uniform resistance which arises from the tenacity
of the medium, when the body moves by its vis insita alone ; and when the
body ascends in a right line, add this uniform resistance to the force of
gravity, and subduct it when the body descends in a right line. One
might also go on to the motion of bodies which are resisted in part uni
formly, in part in the ratio of the velocity, and in part in the duplicate
ratio of the same velocity. And I have opened a way to this in Prop.
XIII and XIV foregoing, in which the uniform resistance arising from the
tenacity of the medium may be substituted for the force of gravity, or be
compounded with it as before. But I hasten to other things.
SKC. -IV
.] OF NATUEAL PHILOSOPHY. 2S?
SECTION IV.
Of the circular motion of bodies in resisting mediums.
LEMMA III.
Let PQR be a spiral rutting all the radii SP, SO, SR, <J*c.,
in equal
angles. Draw tfie right line PT touching the spiral in any point P,
and cutting the radius SQ in T ;
cfo er?0 PO, QO perpendicular to
the spiral, and meeting- in, O, andjoin SO. .1 say, that if Hie points
P a/*(/ Q approach and coincide, the angle PSO vri/Z become a right
angle, and the ultimate ratio of the rectangle TQ, X 2PS to P^3
//>i//
/>e /ie ya/io o/" equality.
For from the right angles OPQ, OQR, sub
duct the equal angles SPQ, SQR, and there
will remain the equal angles OPS, OQS.
Therefore a circle which passes through the
points OSP will pass also through the point
Q. Let the points P and Q, coincide, and
this circle will touch the spiral in the place
of coincidence PQ, and will therefore cut the
right line OP perpendicularly. Therefore OP will become a diameter of
this circle, and the angle OSP, being in a semi-circle, becomes a right
one. Q.E.1).
Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lines
will be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS and
PD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as PO
to 2PS. Whence PO2 becomes equal to TO X 2PS. O.E.D.
PROPOSITION XV. THEOREM XII.
Tf the density of a medium in each place thereof be reciproniJly as the
distance of the places from an immovable centre, aud the centripetal
force be in the duplicate ratio of the density ; I say, that a body mny
revolve in a spiral which cuts all the radii drawn from that centre
in a given angle.
Suppose every thing to be as in the forego
ing Lemma, and produce SO to V so that SV
may be equal to SP. In any time let a body,
in a resisting medium, describe the least arc
PO, and in double the time the least arc PR :
and the decrements of those arcs arising from
the resistance, or their differences from the
arcs which would be described in a non-resist
ing medium in the same times, will be to each
other as the squares of the times in which they
are generated ; therefore the decrement of the
288 THE MATHEMATICAL PRINCIPLES [_BoOK 11
arc PQ is the fourth part of the decrement of the arc PR. Whence also
if the area QSr be taken equal to the area PSQ, the decrement of the arc
PQ will be equal to half the lineola Rr ; and therefore the force of resist
ance and the centripetal force are to each other as the lineola jRrandTQ
which they generate in the same time. Because the centripetal force with
which the body is urged in P is reciprocally as SP 2
, and (by Lem. X,
Book I) the lineola TQ, which is generated by that force, is in a ratio
compounded of the ratio of this force and the duplicate ratio of the time
in which the arc PQ, is described (for in this case I neglect the resistance,
as being infinitely less than the centripetal force), it follows that TQ X
SP 2
, that is (by the last Lemma), fPQ2 X SP, will be in a duplicate ra
tio of the time, and therefore the time is as PQ, X v/SP ; and the velo
city of the body, with which the arc PQ is described in that time, as
PQ 1
-p or , that is, in the subduplicate ratio of SP reciprocally.
And, by a like reasoning, the velocity with which the arc QR is described,
is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ and
QR are as the describing velocities to each other
; that is, in the subdu
plicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of the
equal angles SPQ, SQ? , and the equal areas PSQ, QSr, the arc PQ is to
the arc Qr as SQ to SP. Take the differences of the proportional conse
quents, and the arc PQ will be to the arc Rr as SQ to SP VSP X ~SQ~,
or ^VQ. For the points P and Q coinciding, the ultimate ratio of SP
X SQ to |VQ is the ratio of equality. Because the decrement of
the arc PQ arising from the resistance, or its double Rr, is as the resistance
and the square of the time conjunctly, the resistance will be &Sp-^r*1 op. X
But PQ was to Rr as SQ to |VQ, and thence SSaTXToD becomes as
Jr vst X oJr
-VQ -OS
pWxsvxSQ, or ns
ETp^TsPFor
the poillts p and a coincidin&
SP and SQ coincide also, and the angle PVQ becomes a right one; and,
because of the similar triangles PVQ, PSO, PQ. becomes to -VQ as OP
OS
to | OS. Therefore : y
-- is as the resistance, that is, in the ratio of
\J i X ol
the density of the medium in P and the duplicate ratio of the velocity
conjunc-tly. Subduct the duplicate ratio of the velocity, namely, the ratio
1 OS
^5, and there will remain the density of the medium in P. as 7^5-= OA Ur X fei
Let the spiral be given, and; because of the given ratio of OS to OP, the
density of the medium in P will be as
~-p.
Therefore in a medium whose
SEC. IV,] OF NATURAL PHILOSOPHY. 2S9
density is reciprocally as SP the distance from the centre, a body will re
volve in this spiral. Q.E.D.
COR. 1. The velocity in any place P, is always the same wherewith a
body in a non-resisting medium with the same centripetal force would re
volve in a circle, at the same distance SP from the centre.
COR. 2. The density of the medium, if the distance SP be given, is as
OS OS
TTp,
but if that distance is not given, as ^ ^5. And thence a spiral
may be fitted to any density of the medium.
COR. 3. The force of the resistance in any place P is to the centripetal
force in the same place as |OS to OP. For those forces are to each other
^VQ x PQ iPQ2
as iRr and TQ, or as 1 ^-^~- and ^-, that is, as iVQ and PQ,
ol%, ol
or |OS and OP. The spiral therefore being given, there is given the pro
portion of the resistance to the centripetal force ; and, vice versa, from that
proportion given the spiral is given.
COR. 4. Therefore the body cannot revolve in this spiral, except where
the force of resistance is less than half the centripetal force. Let the re
sistance be made equal to half the centripetal force, and the spiral will co
incide with the right line PS, and in that right line the body will descend
to the centre with a velocity that is to the velocity, with which it was
proved before, in the case of the parabola (Theor. X, Book I), the descent
would be made in a non-resisting medium, in the subduplicate ratio of
unity to the number two. And the times of the descent will be here recip
rocally as the velocities, and therefore given.
COR. 5. And because at equal distances
from the centre the velocity is the same in the
spiral PQ,R as it is in the right line SP, and
the length of the spiral is to the length of the
right line PS in a given ratio, namely, in the
ratio of OP to OS ; the time of the descent in
the spiral will be to the time of the descent in
the right line SP in the same given ratio, and
therefore given.
COR. 6. If from the centre S, with any two
given intervals, two circles are described ; and
these circles remaining, the angle which the spiral makes with the radius"
PS be any how changed ; the number of revolutions which the body can
complete in the space between the circumferences of those circles, going
PS
round in the spiral from one circumference to another, will be as 7^, or as
Ok5
ths tangent of the angle which the spiral makes with the radius PS ; and
19
290 THE MATHEMATICAL PRINCIPLES [BOOK II
OP
the time of the same revolutions will be as -^, that is, as the secant of the Uo
same angle, or reciprocally as the density of the medium.
COR. 7. If a body, in a medium whose density is reciprocally as the dis
tances of places from the centre, revolves in any curve AEB about that
centre, and cuts the first radius AS in the same
angle in B as it did before in A, and that with a
velocity that shall be to its first velocity in A re
ciprocally in a subduplicate ratio of the distances
from the centre (that is, as AS to a mean propor
tional between AS and BS) that body will con
tinue to describe innumerable similar revolution?
BFC, CGD, &c., and by its intersections will
distinguish the radius AS into parts AS, BS, CS, DS, &c., that are con
tinually proportional. But the times of the revolutions will be as the
perimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities
3 3
at the beginnings A, B, C of those orbits inversely ;
that is as AS % BS %
CS"
2
". And the whole time in which the body will arrive at the centre,
will be to the time of the first revolution as the sum of all the continued 142 proportionals AS 2
, BS 2
, CS 2
, going on ad itifinitum, to the first term
* i 3
AS 2
;
that is, as the first term AS 2 to the difference of the two first AS 2
BS% or as f AS to AB very nearly. Whence the whole time may be
easily found.
COR. 8. From hence also may be deduced, near enough, the motions of
bodies in mediums whose density is either uniform, or observes any other
assigned law. From the centre S, with intervals SA, SB, SC, &c., con
tinually proportional, describe as many circles
; and suppose the time of
the revolutions between the perimeters of any two of those circles, in the
medium whereof we treated, to be to the time of the revolutions between
the same in the medium proposed as the mean density of the proposed me
dium between those circles to the mean density of the medium whereof we
treated, between the same circles, nearly : and that the secant of the angle
in which the spiral above determined, in the medium whereof we treated,
cuts the radius AS, is in the same ratio to the secant of the angle in which
the new spiral, in the proposed medium, cuts the same radius : and also
that the number of all the revolutions between the same two circles is nearly
as the tangents of those angles. If this be done every where between every
two circles, the motion will be continued through all the circles. And by
this means one may without difficulty conceive at what rate and in what
time bodies ought to revolve in any regular medium.
SEC. IY.1 OF NATURAL PHILOSOPHY. 291
COR. 9. And although these motions becoming eccentrical should be
performed in spirals approaching to an oval figure, yet, conceiving the
several revolutions of those spirals to be at the same distances from each
other, and to approach to the centre by the same degrees as the spiral above
described, we may also understand how the motions of bodies may be per
formed in spirals of that kind.
PROPOSITION XVI. THEOREM XIII.
If the density of the medium in each of the places be reciprocally as the
distance of the>, places from the immoveable centre, and the centripetal
force be reciprocally as any power of the same distance, I say, that the
body may revolve in a spiral intersecting all the radii drawn from
that centre in a given, angle.
This is demonstrated in the same manner as
the foregoing Proposition. For if the centri
petal force in P be reciprocally as any power
SPn + 1 of the distance SP whose index is n
+ 1
;
it will be collected, as above, that the
time in which the body describes any arc PQ,
i
will be as PQ, X PS 2U
; and the resistance in
i!! x _
n; raS
"~ X SPPQ, X SP"XSQ,
, , 1 in X OS . 1
X OS .
therefore as
Qp"^~gpirqTT
tliat 1S
> (because -
~~Qp~~
1S a lven
quantity), reciprocally as SPn + !
. And therefore, since the velocity is recip
rocally as SP3
", the density in P will be reciprocally as SP.
COR. 1. The resistance is to the centripetal force as 1 ^//. X OS
to OP.
COR. 2. If the centripetal force be reciprocally as SP 3
. 1 w will be
===
; and therefore the resistance and density of the medium will be
nothing, as in Prop. IX, Book I.
COR. 3. If the centripetal force be reciprocally as any power of the ra
dius SP, whose index is greater than the number 3, the affirmative resist
ance will be changed into a negative.
SCHOLIUM.
This Proposition and the former, which relate to mediums of unequal
density, are to be understood of the motion of bodies that are so small, that
the greater density of the medium on one side of the body above that on
the other is not to be considered. I suppose also the resistance, cateris
paribus, to be proportional to its density. Whence, in mediums whose
292 THE MATHEMATICAL PRINCIPLES | BoOK II
force of resistance is not as the density, the density must be so much aug
mented or diminished, that either the excess of the resistance may be taken
away, or the defect supplied.
PROPOSITION XVII. PROBLEM IV
Tofind the centripetal force and the resisting force of the medium, by
which a body, the law of the velocity being given, shall revolve in a
given spiral.
Let that spiral be PQR. From the velocity,
with which the body goes over the very small arc
PQ,, the time will be given : and from the altitude
TQ,, which is as the centripetal force, and the
square of the time, that force will be given. Then
from the difference RSr of the areas PSQ, and
Q,SR described in equal particles of time, the re
tardation of the body will be given ; and from
the retardation will be found the resisting force
and density of the medium.
PROPOSITION XVIII. PROBLEM V.
The law of centripptal force being given, to find the density of the me
