as the triangle APD to the hyperbolic sector ATD. For the velocity in
a non-resisting medium Avould be as the time ATD, and in a resisting me
dium is as AP, that is, as the triangle APD. And those velocities, at the
beginning of the descent, are equal among themselves, as well as those
areas ATD, APD.
COR. 4. By the same argument, the velocity in the ascent is to the ve
locity with which the body in the same time, in a non-resisting space, would
lose all its motion of ascent, as the triangle ApD to the circular sector
AtD ;
or as the right line Ap to the arc At.
COR. 5. Therefore the time in which a body, by falling in a resisting
medium, would acquire the velocity AP, is to the time in which it would
acquire its greatest velocity AC, by falling in a non-resisting space, as the
sector ADT to the triangle ADC : and the time in which it would lose its
velocity Ap, by ascending in a resisting medium, is to the time in which
it would lose the same velocity by ascending in a non-resisting space, as
the arc At to its tangent Ap.
COR. 6. Hence from the given time there is given the space described in
the ascent or descent. For the greatest velocity of a body descending in
wfinitum is given (by Corol. 2 and 3, Theor. VI, of this Book) ; and thence
the time is given in which a body would acquire that velocity by falling
in a non-resisting space. And taking the sector ADT or ADt to the tri
angle ADC in the ratio of the given time to the time just now found,
there will be given both the velocity AP or Ap, and the area ABNK or
AB//&, which is to the sector ADT, or AD/, as the space sought to the
space which would, in the given time, be uniformly described with that
greatest velocity found just before.
COR. 7. And by going backward, from the given space of ascent or de
scent AB?A: or ABNK, there will be given the time AD* or ADT.
268 THE MATHEMATICAL PRINCIPLES [BOOK ii
PROPOSITION X. PROBLEM III.
Suppose the uniform force of gravity to tend directly to the plane of the
horizon, and the resistance to be as the density of the medium and the
square of the velocity coiijunctly : it is proposed to find the density of
the medium in each place, which shall make the body move in any
given carve line ; the velocity of the body and the resistance of the
medium in each place.
Let PQ be a plane perpendicular to
the plane of the scheme itself; PFHQ
a curve line meeting that plane in the
points P and Q ; G, H, I, K four
places of the body going on in this
\ curve from F to Q ; and GB, HC, ID,
KE four parallel ordinates let fall
p A. 33 c^D E Q from these points to the horizon, and
standing on the horizontal line PQ at the points B, C, D, E ; and let the
distances BC, CD, DE, of the ordinates be equal among themselves. From
the points G and H let the right lines GL, HN, be drawn touching the
curve in G and H, and meeting the ordinates CH, DI, produced upwards,
in L and N : and complete the parallelogram HCDM. And the times in
which the body describes the- arcs GH, HI, will be in a subduplicate ratio
of the altitudes LH, NI; which the bodies would describe in those times,
by falling from the tangents; and the velocities will be as the lengths de
scribed GH, HI directly, and the times inversely. Let the times be ex-
C*TT TTT
pounded by T and t, and the velocities by =- and ---; and the decrement
J_ L
/-^TT TTT
of the velocity produced in the time t will be expounded by -7^
.
This decrement arises from the resistance which retards the body, and from
the gravity which accelerates it. Gravity, in a falling body, which in its
fall describes the space NI, produces a velocity with which it would be able
to describe twice that space in the same time, as Galileo has demonstrated ;
2NI
that is, the velocity : but if the body describes the arc HI, it augments
MIxNl
that arc only by the length HI ; and therefore generates HI HN or
only the velocity iff- Let this velocity be added to the beforet
X H.JL
mentioned decrement, and we shall have the decrement of the velocity
GH HI SMI X Nl
arising from the resistance alone, that is, -^ : h T
SEC. II.J OF NATURAL PHILOSOPHY. 269
2NI.
Therefore since, in the same time, the action of gravity generates, in a fall
ing body, the velocity , the resistance will be to the gravity as 7^
HI 2MI X NI 2NI t X GH 2MI X NI
+ TTT- to or as ^ HI -f
Now for the abscissas CB, CD,
CE, put o, o, 2o. For the ordinate
CH put P ; and for MI put any series
Qo + Ro 2 + So 3 +, &c. And all
the terms of the series after the first,
that is, Ro 2 + So 3 +, (fee., will be
NI ; and the ordinates DI, EK, and
BGwill be P QoRo2 So 3 p A B c T> E
(fee., P 2Qo 4Ro 2 SSo 3
, (fee., and P -\- Qo Ro 2 + So 3
,
(fee., respectively. And by squaring the differences of the ordinates BG
CH and CH DI, and to the squares thence produced adding the squares
of BC and CD themselves, you will have oo -f- QQoo 2QRo 3 +, (fee.,
and oo -f QQoo -f 2QRo 3 +, (fee., the squares of the arcs GH, HI ; whose
QRoo QRoo
roots o y -
, and o
</! 4- QQ 4- are the
1 + QQ v/1 + QQ s/1 -f QQ
arcs GH and HI. Moreover, if from the ordinate CH there be subducted
half the sum of the ordinates BG and DI, and from the ordinate DI there
be subducted half the sum of the ordinates CH and EK, there will remain
Roo and Roo + 3So 3
, the versed sines of the arcs GI and HK. And these
are proportional to the lineolae LH and NI, and therefore in the duplicate
ratio of the infinitely small times T and t : and thence the ratio ~, is ^
R + SSo R -f
^ or
So , t X GH TTT 2MI X NI ,
R : and T^ HI H HTTIT , by substituting
the values of , GH, HI, MI and NI just found, becomes -^- J- /w-Lt/
I + QQ. Arid since 2NI is 2Roo, the resistance will be now to the
OO
gravity as -- TT Q to 2Roo
>
that is
>
as 3S r to 4RR.
And the velocity will be such, that a body going off therewith from any
place H, in the direction of the tangent HN, would describe, in vacuo, a
parabola, whose diameter is HC, and its latus rectum NT or --
^----
.
And the resistance is as the density of the medium and the square of
the velocity conjunctly ; and therefore the density of the medium is as the
resistance directly, and the square of the velocity inversely ; that is, as
270 THE MATHEMATICAL PRINCIPLES [BOOK II.
QQ __
4RR
Q.E.I.
COR. 1. If the tangent HN be produced both ways, so as to meet any HT
ordinatc AF in T - will be equal to
V/T+ QQ; and therefore in what
has gone before may be put for ^ \ -\- QQ. By this means the resistance
will be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a*
r-pj --^, and the density of the medium will be as -
AO TT-n. -v/ i Jti X H 1
COR. 2. And hence, if the curve line PFHQ be denned by the relation
between the base or abscissa AC and the ordinate CH, as is usual, and the
value of the ordinate be resolved into a converging series, the Problem
will be expeditiously solved by the first terms of the series ; as in the fol
lowing examples.
EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon the
diameter PQ, to find the density of the medium that shall make a projec
tile move in that line.
Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; and
CD, o ; then DI2 or AQ, 2 AD 2 = nn aa 2ao oo, or ec. 2ao
oo ; and the root being extracted by our method, will give DI= e
ao oo aaoo ao 3 a 3 o 3
~e~~~2e 2e?
~~~W~2?
&C* Here put nn f r ee + aa
> and
ao nnoo anno 3
DI will become = e , &c.
e 2e 3 2e 5
Such series I distinguish into successive terms after this manner : I call
that the first term in which the infinitely small quantity o is not found ;
the second, in which that quantity is of one dimension only ; the third, in
which it arises to two dimensions ; the fourth, in which it is of three ; and
so ad infinitum. And the first term, which here is e, will always denote
the length of the ordinate CH, standing at the beginning of the indefinite
quantity o. The second term, which here is , will denote the difference
between CH and DN ; that is, the lineola MN which is cut off by com
pleting the parallelogram HCDM; and therefore always determines the
CM?
position of the tangent HN ; as, in this case, by taking MN to HM as
G
to o, or a to e. The third term, which here is -, will represent the li
neola IN, which lies between the tangent and the curve ; and therefore
determines the angle of contact IHN, or the curvature which the curve line
SEC. II.] OF NATURAL PHILOSOPHY. 271
has in H. If that lineola IN is of a finite magnitude, it will be expressed
by the third term, together with those that follow in wfinitu:.:i. But if
that lineola be diminished in. infinitnm,
the terms following become in
finitely less than the third term, and
therefore may be neglected. The
fourth term determines the variation
of the curvature ; the fifth, the varia
tion of the variation ; and so on.
Whence, by the way, appears no con-p~" ~K B~C~D~E~
temptible use of these series in the solution of problems that depend upon
tangents, and the curvature of curves.
ao 77/700 anno 3
Now compare the series e ^ ^~ &c., with the
e Ze 3 Ze*
series P Qo - Roo So 3
&c., and for P, Q, II and S? put e, -, ^-^
and ~ , and for ^ 1 + QQ put 1 H or -
; and the density oi
2e 5 ee e
the medium will come out as ; that is (because n is given), as - or
lie e
~Yj, that is, as that length of the tangent HT, which is terminated at the
OH.
semi-diameter AF standing perpendicularly on PQ : and the resistance
will be to the gravity as 3a to
2>/, that is, as SAC to the diameter PQ of
the circle; and the velocity will be as i/ CH. Therefore if the body goes
from the place F, with a due velocity, in the direction of a line parallel to
PQ, and the density of the medium in each of the places H is as the length
of the tangent HT, and the resistance also in any place H is to the force
of gravity as SAC to PQ, that body will describe the quadrant FHQ of a
circle. Q.E.I.
But if the same body should go-frorn the place P, in the direction of a
line perpendicular to PQ, and should begin to move in an arc of the semi
circle PFQ, we must take AC or a on the contrary side of the centre A ;
and therefore its sign must be changed, and we must put a for + a.
Then the density of the medium would come out as . But nature
does not admit of a negative density, that is, a density which accelerates
the motion of bodies; and therefore it cannot naturally come to pass that
a body by ascending from P should describe the quadrant PF of a circle.
To produce such an effect, a body ought to be accelerated by an impelling
medium, and not impeded by a resisting one.
EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF per
272 THE MATHEMATICAL PRINCIPLES [BOOK IL
pendicular to the horizon PQ, to find the density of the medium, which
will make a projectile move in that line.
From the nature of the parabola, the rectangle PDQ,
is equal to the rectangle under the ordinate DI and some
given right line
;
that is, if that right line be called b ;
PC, a; PQ, c; CH, e; and CD, o; the rectangle a
A. CD ~Q + o into c a o or ac aa 2ao -{-co oo, ia
ac aa
equal to the rectangle b into DI, and therefore DI is equal to --7--h
c 2a oo c 2a
-. o r. Now the second term -, o of this series is to be put
b b b
oo
for Q,o, and the third term -r for Roo. But since there are no more
terms, the co-efficient S of the fourth term will vanish ; and therefore the
S
ouantitv - , to which the density of the medium is proper- R v i
tional, will be nothing. Therefore, where the medium is of no density,
the projectile will move in a parabola ; as Galileo hath heretofore demon
strated. Q.E.I.
EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptote
NX perpendicular to the horizontal plane AK, to find the density of the
medium that will make a projectile move in that line.
Let MX be the other asymptote, meeting
the ordinate DG produced in V ; and from
the nature of the hyperbola, the rectangle of
XV into VG will be given. There is also
given the ratio of DN to VX, and therefore
the rectangle of DN into VG is given. Let
that be bb : and, completing the parallelo
gram DNXZ, let BN be called a; BD, o;
NX, c; and let the given ratio of VZ to
MA. BD K N ZX or DN be -. Then DN will be equal
bb
to a o} VG equal to , VZ equal to X a o. and GD or NX
a o n
m m
-VZ VG equal to c a + o . Let the term - be
n n a o a o
bb bb bb bb ,
resolved into the converging series
~^"
+ ^ + ^l
00 + ^4 > &c and
m bb m bb bb bb
GD will become equal to c - a - + -o ~ o ^ o 2
51
SEC. II.] OF NATURAL PHILOSOPHY. 273
&c. The second term o o of this series is to be u?ed for Qo; the
n aa
third o 2
, with its sign changed for Ro2
; and the fourth o 3
, with its
m bb bb bb
sign changed also for So 3
, and their coefficients , and are to
n aa a a
be pat for Q, R, and S in the former rule. Which being done, the denbb
a*
sity of the medium will come out as , ,
bb
a
mm
nn
2mbb
naa
I
mm -
, that is, if in VZ you take VY equal to
aa aa
1 m2
VG, as YT7- For aa and ^ a 2
2mbb b
nn n aa
2mbb b 4
H are the squares of XZ
n aa
and ZY. But the ratio of the resistance to gravity is found to be that of
3XY to 2YG ; and the velocity is that with which the body would de-
XY2
scribe a parabola, whose vertex is G, diameter DG, latus rectum
^
v . Sup
pose, therefore, that the densities of the medium in each of the places G
are reciprocally as the distances XY, and that the resistance in any place
G is to the gravity as 3XY to 2YG ; and a body let go from the place A,
with a due velocity, will describe that hyperbola AGK. Q.E.I.
EXAMPLE 4. Suppose, mdeMtely, the line AGK to be an hyperbola
described with the centre X, and the asymptotes MX, NX, so that, having
constructed the rectangle XZDN, whose side ZD cuts the hyperbola in G
and its asymptote in V, VG may be reciprocally as any power DNn of the
line ZX or DN, whose index is the number n : to find the density of the
medium in which a projected body will describe this curve.
For BN, BD, NX, put A, O, C, respec- ^
tively, and let VZ be to XZ or DN as d to
bb
e, and VG be equal to
be equal to A O, VG == ^=
then DN will
VZ =
O, and GD or NX VZ VG equal
274
term
THE MATHEMATICAL PRINCIPLES [BOOK H
nbb
nn -f- n
bb
U ! J x *
=rr be resolved into an infinite series -r- +
A Of A" A.n
3 -- 3nn + 2/i
X O +
n
" ~ x bb O 3 g^TT-T ,&c,,andGD will be equal X 00 O 2 +
c bb d nbb + ?m -
toC -A--T-+-O- -r O - ~ e A" e A" + l 2An
-f
+ H
i^T t "\
bb 3
> &c- The second tcrm - O - -T 6An + e An
4- l
series is to be used for 0,0, the third
^
66O 2 for Roo, the fourth
-\~r~3 bbO 5 for So 3
. And thence the density of the medium
Oof this
-, in any place G7 will be
2dnbb nub*
and therefore if in VZ you take VY equal to n X VG, that density is ren
w IT- j ^ *
2rf//66 /mfi 4
ciprocally as XY. For A 2 and A 2 -- A + r are the
tc/ o^x ./\_
"
squares of XZ and ZY. Hut the resistance in the same place G is to the
force of gravity as 3S X - to 4RR, that is, as XY to
And the velocity there is the same wherewith the projected body would
move in a parabola, whose vertex is G, diameter GD, and latus rectum
2XY 2
or --------- --. Q.E.I. R nn VG
AC
HT
SCHOLIUM.
In the same manner that the den
sity of the medium comes out to be as
S X AC .
Tjr m ^ r- 1) if the resistance
lx X HI
is put as any power V" of the velocity
V, the density of the medium will
come out to be as
B C D E Q
. x
S
And therefore if a curve can be found, such that the ratio of to
4 o
R i
SEC. II.J OF NATURAL PHILOSOPHY, 275
n 1
, or ofgr^ to may be given ; the body, in an uniz
HT
AC
form medium, whose resistance is as the power V" of the velocity V, will
move in this curve. But let us return to more simple curves.
Because there can be no motion in a para
bola except in a non-resisting medium, but
in the hyperbolas here described it is produced
by a perpetual resistance ;
it is evident that
the line which a projectile describes in an
