COR. 1. Hence it appears, that if the time be expounded by any part
AD of the asymptote, and the velocity in the beginning of the time by the
ordinate AB, the velocity at the end of the time will be expounded by the
ordinate DG ; and the whole space described by the adjacent hyperbolic
area ABGD ; and the space which any body can describe in the same time
AD, with the first velocity AB, in a non-resisting medium, by the rectan
gle AB X AD.
COR 2. Hence the space described in a resisting medium is given, by
taking it to the space described with the uniform velocity AB in a nonresisting
medium, as the hyperbolic area ABGD to the rectangle AB X AD.
COR. 3. The resistance of the medium is also given, by making it equal,
in the very beginning of the motion, to an uniform centripetal force, which
could generate, in a body falling through a non-resisting medium, the ve
locity AB in the time AC. For if BT be drawn touching the hyperbola
in B. and meeting the asymptote in T, the right line AT will be equal to
AC, and will express the time in which the first resistance, uniformly con
tinned, may take away the whole velocity AB
COR. 4. And thence is also given the proportion of this resistance to the
force of gravity, or ay other given centripetal force.
COR. 5. And, vice versa, if there is given the proportion of the resist-
; nee to any given centripetal force, the time AC is also given, in which c
centripetal force equal to the resistance may generate any velocity as AB ;
and thence is given the point B. through which the hyperbola, having CH
CD for its asymptotes, is to be described : as also the space ABGD, which a
body, by beginning its motion with that velocity AB, can describe in any
time AD. in a similar resisting medium.
PROPOSITION VI. THEOREM lVr
c
Homogeneous and equal spherical bodies, opposed hy resistances that are
in the duplicate ratio of the velocities, and moving on by their innate
force only, will, in times which are reciprocally as the velocities at thr.
260 THE MATHEMATICAL PRINCIPLES |BOOK II,
v
beg-in fiing, describe equal spaces, and lose parts of their velocities pro
portional to the wholes.
To the rectangular asymptotes CD, CH de
scribe any hyperbola B6Ee, cutting the perpen
diculars AB, rib, DE, de in B, b, E, e; let the
initial velocities be expounded by the perpendicu
lars AB, DE, and the times by the lines Aa, Drf.
Therefore as Aa is to l)d, so (by the hypothesis)
. is DE to AB, and so (from the nature of the hy- C "^
perbola) is CA to CD ; and, by composition, so is
Crt to Cd. Therefore the areas ABba, DEerf, that is, the spaces described,
are equal among themselves, and the first velocities AB, DE are propor
tional to the last ab, de ; and therefore, by division, proportional to the
parts of the velocities lost, AB ab, DE de. Q.E.D.
PROPOSITION VII. THEOREM V.
If spherical bodies are resisted in the duplicate ratio of their velocities,
in times which are as the first motions directly, and the first resist
ances inversely, they will lose parts of their motions proportional to the
wholes, and will describe spaces proportional to those times and the
first
velocities conjunctIt/.
For the parts of the motions lost are as the resistances and times con
junctly. Therefore, that those parts may be proportional to the wholes,
the resistance and time conjunctly ought to be as the motion. Therefore the
time will be as the motion directly and the resistance inversely. Where
fore the particles of the times being taken in that ratio, the bodies will
always loso parts of their motions proportional to the wholes, and there
fore will retain velocities always proportional to their first velocities.
And because of the given ratio of the velocities, they will always describe
spaces which are as the first velocities and the times conjunctly. Q.E.D.
COR. 1. Therefore if bodies equally swift are resisted in a duplicate ra
tio of their diameters, homogeneous globes moving with any velocities
whatsoever, by describing spaces proportional to their diameters, will lose
parts of their motions proportional to the wholes. For the motion of each
o-lobe will be as its velocity and mass conjunctly, that is, as the velocity
and the cube of its diameter ; the resistance (by supposition) will be as the
square of the diameter and the square of the velocity conjunctly ; and the
time (by this proposition) is in the former ratio directly, and in the latter
inversely, that is, as the diameter directly and the velocity inversely ; and
therefore the space, which is proportional to the time and velocity is as
the diameter.
COR. 2. If bodies equally swift are resisted in a sesquiplicate ratio of
their diameters, homogeneous globes, moving with any velocities whatso
SEC. 1L] OF NATURAL PHILOSOPHY. 261
ever, by describing spaces that are in a sesquiplicate ratio of the diameters,
will lose parts of their motions proportional to the wholes.
COR. 3. And universally, if equally swift bodies are resisted in the ratio
of any power of the diameters, the spaces, in which homogeneous globes,
moving with any velocity whatsoever, will lose parts of their motions pro
portional to the wholes, will be as the cubes of the diameters applied to
that power. Let those diameters be D and E : and if the resistances, where
the velocities are supposed equal, are as T) n and E"
; the spaces in which
the globes, moving with any velocities whatsoever, will lose parts of their
motions proportional to the wholes, will be as D 3 n and E 3 n
. And
therefore homogeneous globes, in describing spaces proportional to D 3 n
and E 3 n
, will retain their velocities in the same ratio to one another as
at the beginning.
COR. 4. Now if the globes are not homogeneous, the space described by
the denser globe must be augmented in the ratio of the density. For the
motion, with an equal velocity, is greater in the ratio of the density, and
the time (by this Prop.) is augmented in the ratio of motion directly, and
the space described in the ratio of the time.
COR. 5. And if the globes move in different mediums, the space, in a
medium which, cccteris paribus, resists the most, must be diminished in the
ratio of the greater resistance. For the time (by this Prop.) will be di
minished in the ratio of the augmented resistance, and the space in the ra
tio of the time.
LEMMA II.
The moment of any genitum is equal to the moments of each of the generatinrr
sides drawn into the indices of the powers of those sides, and
into their co-efficients continually.
I call any quantity a genitum which is not made by addition or subduction
of divers parts, but is generated or produced in arithmetic by the
multiplication, division, or extraction of the root of any terms whatsoever :
in geometry by the invention of contents and sides, or of the extremes and
means of proportionals. Quantities of this kind are products, quotients,
roots, rectangles, squares, cubes, square and cubic sides, and the like.
These quantities I here consider as variable and indetermined, and increas
ing or decreasing, as it were, by a perpetual motion or flux ; and I under
stand their momentaneous increments or decrements by the name of mo
ments ; so that the increments may be esteemed as added or affirmative
moments ; and the decrements as subducted or negative ones. But take
care not to look upon finite particles as such. Finite particles are not
moments, but the very quantities generated by the moments. We are to
conceive them as the just nascent principles of finite magnitudes. Nor do
we in this Lemma regard the magnitude of the moments, but their firsf
262 THE MATHEMATICAL PRINCIPLES [BoOK 11
proportion, as nascent. It will be the same thing, if, instead of moments,
we use either the velocities of the increments and decrements (which may
also be called the motions, mutations, and fluxions of quantities), or any
finite quantities proportional to those velocities. The co-efficient of any
generating side is the quantity which arises by applying the genitum to
ihat side.
Wherefore the sense of the Lemma is, that if the moments of any quan
tities A, B, C, &c., increasing or decreasing by a perpetual flux, or the
velocities of the mutations which are proportional to them, be called a, 6,
r, (fee., the moment or mutation of the generated rectangle AB will be B
-h bA ; the moment of the generated content ABC will be aBC -f bAC 4
-1 -2. .1
cAB; and the moments of the generated powers A2
. A 3
, A4
, A 2
, A 2
. A 3
,
A*, A , A 2
, A * will be 2aA, 3aA2
, 4aA 3
11 , |A *, fA* 3
i A s
, |/iA
3
, aA 2
, 2aA 3
, aA 2
respectively; and
in general, that the moment of any power A^, will be ^ aAn
-^. Also,
that the moment of the generated quantity A 2 B will be 2aAB 4- bA~ ; the
moment of the generated quantity A 3 B 4 C2 will be 3A2 B 4 C 2 + 4/>A
3
A 3
B 3 C 2 4-2cA 3 B C; and the moment of the generated quantity or
A B 2 will be 3aA 2 B 2 2bA 3B 3
; and so on. The Lemma is
thus demonstrated.
CASE 1. Any rectangle, as AB, augmented by a perpetual flux, when, as
yet, there wanted of the sides A and B half their moments \a and \b, was
A \a into B \b, or AB a B \b A + \ab ; but as soon as the
sides A and B are augmented by the other half moments, the rectangle be
comes A 4- 4-a into B 4- \b, or AB -f ^a B 4- \b A -f \ab. From this
rectangle subduct the former rectangle, and there will remain the exces.?
aE -f bA. Therefore with the whole increments a and b of the sides, tin
increment aB + f>A of the rectangle is generated. Q.K.D.
CASE 2. Suppose AB always equal to G, and then the moment of the
content ABC or GC (by Case 1) will be^C + cG, that is (putting AB and
aB + bA for G and *), aBC -h bAC 4- cAB. And the reasoning is the
same for contents under ever so many sides. Q.E.D.
CASE 3. Suppose the sides A, B, and C, to be always equal among them
selves; and the moment B + />A, of A2
, that is, of the rectangle AB,
will be 2aA ; and the moment aBC + bAC + cAB of A 3
, that is, of the
content ABC, will be 3aA 2
. And by the same reasoning the moment of
any power An
is naAn
. Q.E.D
CASE 4. Therefore since -7 into A is 1, the moment of -r- drawn into A A
SEC. 11.] OF NATURAL PHILOSOPHY. 263
A, together with A drawn into a. will be the moment of 1, that is, nothing.
Therefore the moment of -r, or of A ,
is -r . And generally since A .A
T- into An
is I, the moment of drawn into An
together with into
A n A. An
naA"
! will be nothing. And, therefore, the moment of -r- or A n
A
will be T^~7- Q-E.D.
V .
t
. i
CASE 5. And since A 2 into A2 is A, the moment of A1 drawn into 2A 2
will be a (by Case 3) ; and, therefore, the moment of A7 will be n~r~r or
^A-j
#A . And, generally, putting A~^ equal to B, then Am will be equal
to Bn
, and therefore maAm !
equal to nbBn
, and maA equal to
?tbB , or tibA ^ 5
an<i therefore ri aA ^~ is equal to &, that is, equal
to the moment of A^. Q.E.D.
CASE 6. Therefore the moment of any generated quantity AmBn
is the
moment of Am drawn into Bn
, together with the moment of Bn drawn into
A", that is, maAm
B" -f- nbBn ! Am
; and that whether the indices
in arid n of the powers be whole numbers or fractions, affirmative or neg
ative. And the reasoning is the same for contents under more powers.
Q.E.D.
COR. 1. Henoe in quantities continually proportional, if one term is
given, the moments of the rest of the terms will be as the same terms mul
tiplied by the number of intervals between them and the given term. Let
A, B, C, D; E, F, be continually proportional ; then if the term C is given,
the moments of the rest of the terms will be among themselves as 2A,
B? D, 2E, 3F.
COR. 2. And if in four proportionals the two means are given, the mo
ments of the extremes will be as those extremes. The same is to be un
derstood of the sides of any given rectangle.
COR. 3. And if the sum or difference of two squares is given, the mo
ments of the sides will be reciprocally as the sides.
SCHOLIUM.
In a letter of mine to Mr. /. Collins, dated December 10, 1672, having
described a method of tangents, which I suspected to be the same with
Slusius*s method, which at that time wag not made public, I subjoined these
words This is one particular, or rather a Corollary, of a general nte
264 THE MATHEMATICAL PRINCIPLES [BjOK II.
thod, which extends itself, without any troublesome calculation, not ojdy
to the drawing of tangents to any curve lines, whether geometrical or
mechanical, or any how respecting right lines or other cnrves, but also
to the resolving other abstrnser kinds of problems about the crookedness,
areas, lengths, centres of gravity of curves, &c. ; nor is it (as Hudd^ri s
method de Maximis & Minimia) limited to equations which are freefrom
surd quantities. This method I have interwoven with that other oj
working in equations, by reducing them to infinite serie?. So far that
letter. And these last words relate to a treatise I composed on that sub
ject in the year 1671. The foundation of that general method is contain
ed in the preceding Lemma.
PROPOSITION VIII. THEOREM VI.
If a body in an uniform medium, being uniformly acted upon by theforce
of gravity, ascends or descends in a right line ; and the whole space
described be distinguished into equal parts, and in the beginning of
each of the parts (by adding or subducting the resisting force of the
medium to or from the force of gravity, when the body ascends or de
scends] yon collect the absolute forces ; Isay, that those absolute forces
ire in a geometrical progression.
For let the force of gravity be expounded by the
given line AC ; the force of resistance by the indefi
nite line AK ; the absolute force in the descent of the
body by the difference KC : the velocity of the I tody
<^LKJL&i>F/ by a line AP, which shall be a mean proportional be
tween AK and AC, and therefore in a subduplicate ratio of the resistance;
the increment of the resistance made in a given particle of time by the lineola
KL, and the contemporaneous increment of the velocity by the lineola
PQ ; and with the centre C, and rectangular asymptotes CA, CH,
describe any hyperbola BNS meeting the erected perpendiculars AB, KN,
LO in B, N and O. Because AK is as AP2
, the moment KL of the one will
be as the moment 2APQ of the other, that is, as AP X KC ; for the in
crement PQ of the velocity is (by Law II) proportional to the generating
force KC. Let the ratio of KL be compounded with the ratio KN, and
the rectangle KL X KN will become as AP X KC X KN ; that is (because
the rectangle KC X KN is given), as AP. But the ultimate ratio of the
hyperbolic area KNOL to the rectangle KL X KN becomes, when the
points K and L coincide, the ratio of equality. Therefore that hyperbolic
evanescent area is as AP. Therefore the whole hyperbolic area ABOL
is composed of particles KNOL which are always proportional to the
velocity AP; and therefore is itself proportional to the space described
with that velocity. Let ,that area be now divided into equal parts
SEC. IJ.J OF NATURAL PHILOSOPHY. 265
as ABMI, IMNK, KNOL, (fee., and the absolute forces AC, 1C, KC, LC,
(fee., will be in a geometrical progression. Q,.E.D. And by a like rea
soning, in the ascent of the body, taking, on the contrary side of the point
A, the equal areas AB?m, i/nnk, knol, (fee., it will appear that the absolute
forces AC. iG, kC, 1C, (fee., are continually proportional. Therefore if all
the spaces in the ascent and descent are taken equal, all the absolute forces
1C, kC, iC, AC, 1C, KC, LC, (fee., will be continually proportional. Q,.E.D.
COR. 1. Hence if the space described be expounded by the hyperbolic
area ABNK, the force of gravity, the velocity of the body, and the resist
ance of the medium, may be expounded by the lines AC, AP, and AK re
spectively and vice versa.
COR. 2. And the greatest velocity which the body can ever acquire in
an infinite descent will be expounded by the line AC.
COR. 3. Therefore if the resistance of the medium answering to any
given velocity be known, the greatest velocity will be found, by taking it
to that given velocity in a ratio subduplicate of the ratio which the force
of gravity bears to that known resistance of the medium.
PROPOSITION IX. THEOREM VII.
Supposing ivhat is above demonstrated, I say, that if the tangents of t-he
angles of the sector of a circle, and of an hyperbola, be taken propor
tional to the velocities, the radius being of a fit magnitude, all the time
of the ascent to the highest place icill be as the sector of the circle, and
all the time of descending from the highest place as the sector of t/ie
hyperbola.
To the right line AC, which ex
presses the force of gravity, let AD
drawn perpendicular and equal. From
the centre D with the semi-diameter
AD describe as well the cmadrant A^E -t
of a circle, as the rectangular hyper
bola AVZ, whose axis is AK, principal
vertex A, and asymptote DC. Let Dp,
DP be drawn ; and the circular sector
AtD will be as all the time of the as
cent to the highest place ; and the hy
perbolic sector ATD as all the time of descent from the highest place; ii
BO be that the tangents Ap, AP of those sectors be as the velocities.
CASE 1. Draw Dvq cutting off the moments or least particles tDv and
^
?, described in the same time, of the sector ADt and of the triangle
AD/?. Since those particles (because of the common angle D) are in a du
plicate ratio of the sides, the particle tDv will be as -^-^-
, that is
266 THE MATHEMATICAL PRINCIPLES [BOOK li.
(because tD is given), as ^f. But joD
8 is AD 3 + Ap 2
, that is, AD 2 -h
AD X AA-, or AD X Gk ; and (/Dp is 1 AD X pq. Therefore tDv, the
BO
particle of the sector, is as ^ ,
; that is, as the least decrement pq of the
velocity directly, and the force Gk which diminishes the velocity, inversely ;
and therefore as the particle of time answering to the decrement of the ve
locity. And, by composition, the sum of all the particles tDv in the sector
AD/ will be as the sum of the particles of time answering to each of the
lost particles pq of the decreasing velocity Ap, till that velocity, being di
minished into nothing, vanishes; that is, the whole sector AD/ is as the
whole time of ascent to the highest place. Q.E.D.
CASE 2. Draw DQV cutting off the least particles TDV and PDQ of
the sector DAV, and of the triangle DAQ ; and these particles will be to
each other as DT2 to DP2
, that is (if TX and AP are parallel), as DX 2
to DA2 or TX 2 to AP 2
; and, by division, as DX2 TX2 to DA2 -
AP 2
. But. from the nature of the hyperbola, DX2 TX2
is AD 2
; and, by
the supposition, AP 2 is AD X AK. Therefore the particles are to each
other as AD 2 to AD2 AD X AK ; that is, as AD to AD AK or AC
to CK ; and : and therefore the particle TDV of the sector is -
PQ
therefore (because AC and AD are given) as
CK
that is, as the increment
of the velocity directly, and as the force generating the increment inverse
ly ; and therefore as the particle of the time answering to the increment.
And, by composition, the sum of the particles of time, in which all the par
ticles PQ of the velocity AI
3 are generated, will be as the sum of the par
ticles of the sector ATI) ; that is, the whole time will be as the whole
sector. Q.E.D.
COR. 1. Hence if AB be equal to a
fourth part of AC, the space which a body
will describe by falling in any time will
be to the space which the body could de
scribe, by moving uniform]} on in the
same time with its greatest velocity
AC, as the area ABNK, which ex
presses the space described in falling to
the area ATD, which expresses the
time. For since AC is to AP as AP
_ to AK, then (by Cor. 1, Lem. II, of this
Book) LK is to PQ as 2AK to AP, that is, as 2AP to AC, and thence
LK is to ^PQ as AP to JAC or AB ; and KN is to AC or AD as AB tc
. II.] OF NATURAL PHILOSOPHY. 267
UK ; and therefore, ex cequo, LKNO to DPQ, as AP to CK. But DPQ
was to DTV as CK to AC. Therefore, ex aquo, LKNO is to DTV r,?
AP to AC ; that is, as the velocity of the falling body to the greatest
velocity which the body by falling can acquire. Since, therefore, the
moments LKNO and DTV of the areas ABNK and ATD are as the ve
locities, all the parts of those areas generated in the same time will be as
the spaces described in the same time ; and therefore the whole areas ABNK
and ADT, generated from the beginning, will be as the whole spaces de
scribed from the beginning of the descent. Q.E.D.
COR. 2. The same is true also of the space described in the ascent.
That is to say, that all that space is to the space described in the same
time, with the uniform velocity AC, as the area ABttk is to the sector ADt.
COR. 3. The velocity of the body, falling in the time ATD, is to the
velocity which it would acquire in the same time in a non-resisting space,
