The Mathematical Principles of Natural Philosophy

by Isaac Newton

*Of the motion of bodies that are resisted in the duplicate ratio of their velocities.*

*
If a body is resisted in the duplicate ratio of its velocity,
and moves by its innate force only through a similar medium; and
the times be taken in a geometrical progression, proceeding from
less to greater terms: I say, that the velocities at the beginning
of each of the times are in the same geometrical progression
inversely; and that the spaces are equal, which are described in
each of the times.
*

For since the resistance of the medium is proportional to the square
of the velocity, and the decrement of the velocity is proportional to
the resistance: if the time be divided into innumerable equal
particles, the squares of the velocities at the beginning of each of
the times will be proportional to the differences of the same
velocities. Let those particles of time be AK, KL, LM, &c., taken
in the right line CD; and erect the perpendiculars AB, K*k*, L*l*,
M*m*, &c., meeting the hyperbola B*klm*G, described
with the centre C, and the rectangular asymptotes CD, CH, in B, *k,
l, m,* &c.; then AB will be to K*k* as CK to CA, and,
by division, AB − K*k* to K*k* as AK to CA, and
alternately, AB − K*k* to AK as K*k* to CA; and
therefore as AB x K*k* to AB x CA. Therefore since AK and AB x
CA are given, AB − K*k* will be as AB x KA; and, lastly, when
AB and K*k* coincide, as AB². And, by the like reasoning, K*k*
− L*l*, L*l* − M*m*, &c., will be as K*k*²,
L*l*², &c. Therefore the squares of the lines AB, K*k*,
L*l*, M*m*, &c., are as their differences; and,
therefore, since the squares of the velocities were shewn above to be
as their differences, the progression of both will be alike. This
being demonstrated it follows also that the areas described by these
lines are in a like progression with the spaces described by these
velocities. Therefore if the velocity at the beginning of the first
time AK be expounded by the line AB, and the
velocity at the beginning of the second time KL by the line K*k*
and the length described in the first time by the area AK*k*B,
all the following velocities will be expounded by the following lines
L*l*, M*m*, &c. and the lengths described, by the
areas K*l*, L*m*. &c. And, by composition, if the
whole time be expounded by AM, the sum of its parts, the whole length
described will be expounded by AM*m*B the sum of its parts. Now
conceive the time AM to be divided into the parts AK, KL, LM, &c.
so that CA, CK, CL, CM, &c. may be in a geometrical progression;
and those parts will be in the same progression, and the velocities
AB, K*k*, L*l*, M*m*, &c., will be in the
same progression inversely, and the spaces described A*k*, K*l*,
L*m*, &c., will be equal. Q.E.D.

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 rectangle 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 velocity 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 continued, 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 any other given centripetal force.

Cor. 5. And, *vice versa*, if there
is given the proportion of the resistance to any given centripetal
force, the time AC is also given, in which a 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.

*
Homogeneous and equal spherical bodies, opposed by 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 the beginning, describe equal spaces, and
lose parts of their velocities proportional to the wholes.
*

To the rectangular asymptotes CD, CH describe any hyperbola B*b*E*e*,
cutting the perpendiculars AB, *ab*, DE, *de* in B, *b*,
E, *e*; let the initial velocities be expounded by the
perpendiculars AB, DE, and the times by the lines A*a*, D*d*.
Therefore as A*a* is to D*d*, so (by the hypothesis) is
DE to AB, and so (from the nature of the hyperbola) is CA to CD; and,
by composition, so is C*a* to C*d*. Therefore the areas
AB*ba*, DE*ed*, that is, the spaces described, are equal
among themselves, and the first velocities AB, DE are proportional to
the last *ab, de*; and therefore, by division, proportional to
the parts of the velocities lost, AB − *ab*, DE − *de*.
Q.E.D.

*
If spherical bodies are resisted in the duplicate ratio of
their velocities, in times which are as the first motions
directly, and the first resistances inversely, they will lose
parts of their motions proportional to the wholes, and will
describe spaces proportional to those times and the first
velocities conjunctly.
*

For the parts of the motions lost are as the resistances and times conjunctly. 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. Wherefore the particles of the times being taken in that ratio, the bodies will always lose parts of their motions proportional to the wholes, and therefore 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 ratio 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 globe 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 whatsoever, 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 proportional 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 D^{n} and E^{n};
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, *caeteris paribus*,
resists the most, must be diminished in the ratio of the greater
resistance. For the time (by this Prop.) will be diminished in the
ratio of the augmented resistance, and the space in the ratio of the
time.

*
The moment of any genitum is equal to the moments of each of
the generating 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 increasing or decreasing, as it
were, by a perpetual motion or flux; and I understand their
momentaneous increments or decrements by the name of moments; 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 first
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 that side.

Wherefore the sense of the Lemma is, that if the moments of any
quantities 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, b, c,* &c., the moment or mutation of
the generated rectangle AB will be *a*B + *b*A; the
moment of the generated content ABC will be *a*BC + *b*AC
+ *c*AB; and the moments of the generated powers A², A³, A^{4},
A^{½}, A^{3/2}, A^{⅓}, A^{⅔},
A^{−1}, A^{−2}, A^{−½} will be 2*a*A, 3*a*A²,
4*a*A³, ½*a*A^{−½}, ^{3}/_{2}*a*A^{½},
⅓*a*A^{−⅔}, ⅔*a*A^{−⅓}, −*a*A^{−2},
−*2a*A^{−3}, −½*a*A^{−3/2}
respectively; and in general, that the moment of any power A
n

m, will be n

m *a*A
n−m

m. Also, that the moment of the
generated quantity A²B bill be 2*a*AB + bA²; the moment of the
generated quantity A³ B^{4} C² will be 3*a*A² B^{4}
C² + 4*b*A³B³C² + 2*c*A³B^{4}C; and the moment
of the generated quantity A^{3}

B^{2} or A³B^{−2} will
be 3*a*A²B^{−2}−2*b*A³B^{−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 ½*a* and ½*b*, was A−½*a* into B−½*b*,
or AB − ½*a* B − ½*b* A + ¼*ab*; but as soon as
the sides A and B are augmented by the other half moments, the
rectangle becomes A + ½*a* into B + ½*b*, or AB + ½*a*
B + ½*b* A + ¼*ab*. From this rectangle subduct the
former rectangle, and there will remain the excess *a*B + *b*A.
Therefore with the whole increments *a* and *b* of the
sides, the increment *a*B + *b*A of the rectangle is
generated. Q.E.D.

Case 2. Suppose AB always equal to G, and
then the moment of the content ABC or GC (by Case 1) will be *g*C
+ *c*G, that is (putting AB and *a*B + *b*A
for G and *g*), *a*BC + *b*AC + *c*AB.
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 themselves; and the moment *a*B + *b*A,
of A², that is, of the rectangle AB, will be 2*a*A; and the
moment *a*BC + *b*AC + *c*AB of A³, that is,
of the content ABC, will be 3*a*A². And by the same reasoning
the moment of any power A^{n} is *na*A^{n−1}.
Q.E.D

Case 4. Therefore since
1

A into A is 1, the moment of
1

A drawn into A,
together with 1

A drawn into *a*, will be the
moment of 1, that is, nothing. Therefore the moment of
1

A, or of A^{−1}, is
−a

A^{2}. And generally since
1

A^{n} into A^{n} is
1, the moment of 1

A^{n} drawn into A^{n}
together with 1

A^{n} into *na*A^{n−1}
will be nothing. And, therefore, the moment of
1

A^{n} or A^{−n} will
be −na

A^{n+1}. Q.E.D.

Case 5. And since A^{½} into A^{½}
is A, the moment of A^{½} drawn into 2A^{½} will be *a*
(by Case 3); and, therefore, the moment of A^{½} will be
a

2A^{1}/_{2} or ½*a*A−½.
And, generally, putting A^{
m
n} equal to B, then A^{m}
will be equal to B^{n}, and therefore *ma*A^{m−1}
equal to *nb*B^{n−1}, and *ma*A^{−1}
equal to *nb*B^{−1}, or nbA^{−
m
n}; and therefore
m

naA^{m−n
n} is equal to *b*,
that is, equal to the moment of A^{
m
n}. Q.E.D.

Case 6. Therefore the moment of any generated
quantity A^{m}B^{n} is the moment of A^{m}
drawn into B^{n}, together with the moment of B^{n}
drawn into A^{m}, that is, *ma*A^{m−1} B^{n}
+ *nb*B^{n−1} A^{m}; and that whether the
indices *m* and *n* of the powers be whole numbers or
fractions, affirmative or negative. And the reasoning is the same for
contents under more powers. Q.E.D.

Cor. 1. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the same terms multiplied by the number of intervals between them nd 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 moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.

In a letter of mine to Mr. *J. 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 was
not made public, I subjoined these words: *This is one particular,
or rather a Corollary, of a general method,* *which
extends itself, without any troublesome calculation, not only to the
drawing of tangents to any curve lines, whether geometrical or
mechanical, or any how respecting right lines or other curves, but
also to the resolving other abstruser kinds of problems about the
crookedness, areas, lengths, centres of gravity of curves, &c.;
nor is it (as* Hudden's *method* de Maximis &
Minimis*) limited to equations which are free from surd quantities.
This method I have interwoven with that other of working in
equations, by reducing them to infinite series.
* So far that
letter. And these last words relate to a treatise I composed on that
subject in the year 1671. The foundation of that general method is
contained in the preceding Lemma.

*
If a body in an uniform medium, being uniformly acted upon by
the force 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 descends] you collect the absolute
forces; I say, that those absolute forces are in a geometrical progression.
*

For let the force of gravity be expounded by the given line AC; the
force of resistance by the indefinite line AK; the absolute force in
the descent of the body by the difference KC: the velocity of the body
by a line AP, which shall be a mean proportional between 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 increment 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 as
ABMI, IMNK, KNOL, &c., and the absolute forces AC, IC, KC, LC,
&c., will be in a geometrical progression. Q.E.D.
And by a like reasoning, in the ascent of the body,
taking, on the contrary side of the point A, the equal areas AB*mi,
imnk, knol,* &c., it will appear that the absolute forces
AC, *i*C, *k*C, *l*C, &c., are continually
proportional. Therefore if all the spaces in the ascent and descent
are taken equal, all the absolute forces *l*C, *k*C, *i*C,
AC, IC, KC, LC, &c., 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 resistance of the medium, may be
expounded by the lines AC, AP, and AK respectively; 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.

*
Supposing what is above demonstrated, I say, that if the
tangents of the angles of the sector of a circle, and of an
hyperbola, be taken proportional to the velocities, the radius
being of a fit magnitude, all the time of the ascent to the
highest place will be as the sector of the circle, and all the
time of descending from the highest place as the sector of the hyperbola.
*

To the right line AC, which expresses the force of gravity, let AD be
drawn perpendicular and equal. From the centre D with the
semi-diameter AD describe as well the quadrant A*t*E of a
circle, as the rectangular hyperbola AVZ, whose axis is AK, principal
vertex A, and asymptote DC. Let D*p*, DP be drawn; and the
circular sector A*t*D will be as all the time of the ascent to
the highest place; and the hyperbolic sector ATD as all the time of
descent from the highest place; if so be that the tangents A*p*,
AP of those sectors be as the velocities.

Case 1. Draw D*vq* cutting off the
moments or least particles *t*D*v* and *q*D*p*,
described in the same time, of the sector AD*t* and of the
triangle AD*p*. Since those particles (because of the common
angle D) are in a duplicate ratio of the sides, the particle *t*D*v*
will be as qDp x tD^{2}

pD^{2}, that is (because
*t*D is given), as qDp

pD^{2}. But *p*D² is
AD² + A*p*², that is, AD² + AD x A*k*, or AD x C*k*;
and *q*D*p* is ½AD x *pq*. Therefore *t*D*v*,
the particle of the sector, is as pq

Ck; that is, as the least decrement *pq*
of the velocity directly, and the force C*k* which diminishes
the velocity, inversely; and therefore as the particle of time
answering to the decrement of the velocity. And, by composition, the
sum of all the particles *t*D*v* in the sector AD*t*
will be as the sum of the particles of time answering to each of the
lost particles *pq* of the decreasing velocity A*p*,
till that velocity, being diminished into nothing, vanishes; that is,
the whole sector AD*t* 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 DX² to DA² or TX² to AP²; and, by division,
as DX² − TX² to DA² − AP² . But, from the nature of the hyperbola, DX²
− TX² is AD²; and, by the supposition, AP² is AD x AK. Therefore the
particles are to each other as AD² to AD² − AD x AK; that is, as AD to
AD − AK or AC to CK: and therefore the particle TDV of the sector is
PDQ x AC

CK; and therefore (because AC and AD
are given) as PQ

CK; that is, as the increment of the
velocity directly, and as the force generating the increment
inversely; 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 particles PQ of the velocity AP are generated, will be
as the sum of the particles of the sector ATD; 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 describe, by moving
uniformly on in the same time with its greatest velocity AC, as the
area ABNK, which expresses 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 ½PQ as AP to ¼AG or AB; and KN is to
AC or AD as AB to CK; and therefore, *ex
aequo*, LKNO to DPQ as AP to CK. But DPQ was to DTV as CK to AC.
Therefore, *ex aequo*, LKNO is to DTV as 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 velocities, 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 described
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 AB*uk* is to the sector AD*t*.

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, as the triangle APD to the hyperbolic sector ATD. For the velocity in a non-resisting medium would be as the time ATD, and in a resisting medium 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 velocity with which the body in the same time, in
a non-resisting space, would lose all its motion of ascent, as the
triangle A*p*D to the circular sector A*t*D; or as the
right line A*p* to the arc A*t*.

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 A*p*, 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 A*t*
if to its tangent A*p*.

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 infinitum* 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 AD*t* to the
triangle ADC in the ratio of the given time to the time just now
found, there will be given both the velocity AP or A*p*, and
the area ABNK or AB*nk*, which is to the sector ADT, or AD*t*,
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 descent AB*nk* or ABNK, there will be given
the time AD*t* or ADT.

*
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 conjunctly: it is
proposed to find the density of the medium in each place, which
shall make the body move in any given curve 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 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 described GH, HI directly, and the
times inversely. Let the times be expounded by T and *t*, and
the velocities by GH

T and HI

t; and the decrement of the velocity
produced in the time *t* will be expounded by
GH

T − HI

t . 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; that is, the velocity 2NI

t : but if the body describes the arc
HI, it augments that arc only by the length HI − HN or
MI x NI

HI; and therefore generates only the
velocity 2MI x NI

t x HI. Let this velocity be added to
the beforementioned decrement, and we shall have the decrement of the
velocity arising from the resistance alone, that is,
GH

T − HI

t + 2MI
x NI

t x HI . Therefore
since, in the same time, the action of gravity generates, in a falling
body, the velocity 2NI

t, the resistance will be to the
gravity as GH

T − HI

t + 2MI
x NI

t x HI or as
t x GH

T − HI + 2MI
x NI

HI to 2NI.

Now for the abscissas CB, CD, CE, put −*o, o, 2o*. For the
ordinate CH put P; and for MI put any series Q*o* + R*o*²
+ S*o*³ +, &c. And all the terms of the series after the
first, that is, R*o*² + S*o*³ +, &c., will be NI;
and the ordinates DI, EK, and BG will be P − Q*o* − R*o*²
− S*o*³ −, &c., P − 2Q*o* − 4Ro² − 8S*o*³ −,
&c., and P + Q*o* − R*o*² + S*o*³ −, &c.,
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* + QQ*oo* − 2QR*o*³
+, &c., and *oo* + QQ*oo* + 2QR*o*³ +,
&c., the squares of the arcs GH, HI; whose roots o√(1+QQ)
− QRoo

√(1+QQ) , and o√(1+QQ)
+ QRoo

√(1+QQ) are the 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 R*oo*
and R*oo* + 3S*o*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 t

T is √(
R + 3So

R) or
R + ^{3}/_{2}So

R ; and t
x GH

T − HI + 2MI
x NI

HI , by substituting the
values of t

T, GH, HI, MI and NI just found,
becomes 3Soo

2R √(1+QQ). And since 2NI is
2R*oo*, the resistance will be now to the gravity as
3Soo

2R √(1+QQ), that is, as
3S√(1+qq) 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
HN^{2}

NI or 1+QQ

R.

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 3S√(1+QQ)

4RR directly and
1+QQ

R inversely; that is, as
S

R√(1+QQ). Q.E.I.

Cor. 1. If the tangent HN be produced both
ways, so as to meet any ordinate AF in T HT

AC will be equal to √(1+QQ);
and therefore in what has gone before may be put for √(1+QQ).
By this means the resistance will be to the gravity as 3S x HT to 4RR
x AC; the velocity will be as HT

AC√R, and the density of the medium
will be as S x AC

R x HT.

Cor. 2. And hence, if the curve line PFHQ be defined 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 following 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 projectile move in that line.

Bisect the diameter PQ in A; and call AQ, *n*; AC, *a*;
CH, *e*; and CD, *o*; then DI² or AQ² − AD² = *nn
− aa − 2ao − oo*, or *ee − 2ao − oo*; and the root being
extracted by our method, will give DI = e −
ao

e − oo

2e − aaoo

2e^{3} −
ao^{3}

2e^{3} −
a^{3}o^{3}

2e^{5} − , &c.
Here put *nn* for *ee + aa*, and DI will become
= e − ao

e − nnoo

2e^{3} −
anno^{3}

2e^{5} −, &c

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 ao

e, will denote the difference between
CH and DN; that is, the lineola MN which is cut off by completing the
parallelogram HCDM; and therefore always determines the position of
the tangent HN; as, in this case, by taking MN to HM as
ao

e to *o*, or *a* to *e*.
The third term, which here is nnoo

2e^{3}, will represent the
lineola IN, which lies between the tangent and the curve; and
therefore determines the angle of contact IHN, or the curvature which
the curve line
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 infinitum*.
But if that lineola be diminished *in infinitum*, 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 variation of the variation; and so on.
Whence, by the way, appears no contemptible use of these series in the
solution of problems that depend upon tangents, and the curvature of
curves.

Now compare the series e −
ao

e − nnoo

2e^{3} −
anno^{3}

2e^{5} − &c., with
the series P − Q*o* − R*oo* − S*o*³
− &c., and for P, Q, R and S, put *e*,
a

e, nn

2e^{3} and
ann

2e^{5}, and for √(1
+ QQ) put √(1 +
aa

ee ) or
n

e : and the density of the medium will
come out as a

ne; that is (because *n* is
given), as a

e or AC

CH, that is, as that length of the
tangent HT, which is terminated at the semi-diameter AF standing
perpendicularly on PQ: and the resistance will be to the gravity as 3*a*
to 2*n*, that is, as 3AC to the diameter PQ of the circle; and
the velocity will be as √(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 3AC to PQ, that body will
describe the quadrant FHQ of a circle. Q.E.I.

But if the same body should go from 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 −a

e. 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 perpendicular 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* + *o*
into *c − a − o* or *ac − aa − 2ao + co − oo*, is
equal to the rectangle *b* into DI, and therefore DI is equal
to ac − aa

b + c
− 2a

bo − oo

b . Now the second term
c−2a

bo of this series is to be put
for Q*o*, and the third term oo

b for R*oo*. But since there are
no more terms, the co-efficient S of the fourth term will vanish; and
therefore the quantity S

R√(1+QQ), to which the density of the
medium is proportional, will be nothing. Therefore, where the medium
is of no density, the projectile will move in a parabola; as *Galileo*
hath heretofore demonstrated. 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 parallelogram DNXZ, let BN be called *a*; BD, *o*;
NX, *c*; and let the given ratio of VZ to ZX or DN be
m

n. Then DN will be equal to *a − o*,
VG equal to bb

a − o, VZ equal to
m

n x (a − o), and GD or
NX − VZ − VG equal to c −
m

n a + m

no − bb

a−o . Let the term
bb

a−o be resolved into the converging
series bb

a + bb

aao + bb

a^{3}oo +
bb

a^{4}o^{3} ,
&c., and GD will become equal to c −
m

na − bb

a + m

no − bb

aao − bb

a^{3}o^{2} −
bb

a^{4}o^{3} ,
&c. The second term
m

no − bb

aao of this series is to be
used for Q*o*; the third
bb

a^{3}o^{2} ,
with its sign changed for R*o*²; and the fourth
bb

a^{4}o^{3} ,
with its sign changed also for S*o*³, and their coefficients
m

n − bb

aa ,
bb

a^{3} and
bb

a^{4} are to be put for Q, R,
and S in the former rule. Which being done, the density of the medium
will come out as bb

a^{4}

bb

a^{3} √(1 +
mm

nn − 2mbb

naa + b^{4}

a^{4}) or
1

√(aa + mm

nnaa − 2mbb

n + b^{4}

aa) , that is, if in
VZ you take VY equal to VG, as 1

XY. For *aa* and
m^{2}

n^{2}a^{2} −
2mbb

n + b^{4}

aa are the squares of XZ 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
describe a parabola, whose vertex is G, diameter DG, latus rectum
XY^{2}

VG. Suppose, 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, indefinitely, 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 DN^{n} 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, respectively, and let VZ be to XZ or DN
as *d* to *e*, and VG be equal to
bb

DN^{n}; then DN will be equal
to A − O, VG = bb

(A − O)^{n} ,
VZ = d

e (A − O), and GD or NX − VZ −
VG equal to

C − d

eA + d

eO − bb

(A − O)^{n} .

eA + d

eO − bb

(A − O)

Let the term bb

(A − O)^{n} be resolved into
an infinite series

bb

A^{n} +
nbb

A^{n + 1} x O +
nn + n

2A^{n + 2} x bb O^{2}
+ n^{3} + 3nn + 2n

6A^{n + 3} x bb O^{3},&c.,

A

A

2A

6A

And GD will be equal to

C − d

eA + bb

A^{n} +
d

e O − nbb

A^{n + 1} O −
+ nn + n

2A^{n + 2}bb O^{2} −
+ n^{3} + 3nn + 2n

6A^{n + 3} bbO^{3},
&c.

eA + bb

A

e O − nbb

A

2A

6A

The second term d

e O − nbb

A^{n+1} O of this
series is to be used for Q*o*, the third
nn+n

2A^{n+2}bb O^{2}
for R*oo*, the fourth
n^{3}+3nn+2n

6A^{n+3}bbO^{3}
for So³. And thence the density of the medium
S

R√(1+QQ), in any place G, will be

n+2

3√(A^{2} + dd

eeA^{2}−
2dnbb

eA^{n}A+
nnb^{4}

A^{2n} ),

3√(A

eeA

eA

A

and therefore if in VZ you take VY equal to *n* x VG, that
density is reciprocally as XY. For A² and
dd

eeA^{2} −
2dnbb

eA^{n}A +
nnb^{4}

A^{2n} are the squares
of XZ and ZY. But the resistance in the same place G is to the force
of gravity as 3S x XY

A to 4RR, that is, as XY to
2nn + 2n

n + 2 VG. 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 1
+ QQ

R or 2XY^{2}

(nn + n) x VG. Q.E.I.

In the same manner that the density of the medium comes out to be as
S x AC

R x HT, in Cor. 1, if the resistance is
put as any power V^{n} of the velocity V, the density of the
medium will come out to be as
S

R^{4−n/2}
x ( AC

HT)^{n−1}

And therefore if a curve can be found, such that the ratio of
S

R^{4−n/2}
to (
HT

AC )^{n−1}, or of
S^{2}

R^{4−n} to (1+QQ)^{n−1}
may be given; the body, in an uniform medium, whose resistance is as
the power V^{n} 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 parabola 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 uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described. The difference, however, is not so great between the one and the other but that these latter may be commodiously enough used in practice instead of the former. And perhaps these may prove more useful than an hyperbola that is more accurate, and at the same time more compounded. They may be made use of, then, in this manner.

Complete the parallelogram XYGT, and the right line GT will touch the
hyperbola in G, and therefore the density of the medium in G is
reciprocally as the tangent GT, and the velocity there as √
(GT^{2}

GV); and the resistance is to
the force of gravity as GT to
2nn + 2n

n + 2 x GV.

Therefore if a body projected from the place A, in the direction of
the right line AH, describes the hyperbola AGK and AH produced meets
the asymptote NX in H, and AI drawn parallel to it meets the other
asymptote MX in I; the density of the medium in A will be reciprocally
as AH, and the velocity of the body as √(
AH^{2}

AI), and the resistance there
to the force of gravity as AH to
2nn + 2n

n + 2 x AI. Hence the
following rules are deduced.

Rule 1. If the density of the medium at A, and the velocity with which the body is projected remain the same, and the angle NAH be changed, the lengths AH, AI, HX will remain. Therefore if those lengths, in any one case, are found, the hyperbola may afterwards be easily determined from any given angle NAH.

Rule 2. If the angle NAH, and the density of the medium at A, re main the same, and the velocity with which the body is projected be changed, the length AH will continue the same; and AI will be changed in a duplicate ratio of the velocity reciprocally.

Rule 3. If the angle NAH, the velocity of the
body at A, and the accelerative gravity remain the same, and the
proportion of the resistance at A to the motive gravity be augmented
in any ratio; the proportion of AH to AI will be augmented in the same
ratio, the latus rectum of the abovementioned parabola remaining the
same, and also the length AH^{2}

AI proportional to it; and therefore AH
will be diminished in the same ratio, and AI will be diminished in the
duplicate of that ratio. But the proportion of the resistance to the
weight is augmented, when either the specific gravity is made less,
the magnitude remaining equal, or when the density of the medium is
made greater, or when, by diminishing the magnitude, the resistance
becomes diminished in a less ratio than the weight.

Rule 4. Because the density of the medium is greater near the vertex of the hyperbola than it is in the place A, that a mean density may be preserved, the ratio of the least of the tangents GT to the tangent AH ought to be found, and the density in A augmented in a ratio a little greater than that of half the sum of those tangents to the least of the tangents GT.

Rule 5. If the lengths AH, AI are given, and
the figure AGK is to be described, produce HN to X, so that HX may be
to AI as *n* + 1 to 1; and with the centre X, and the
asymptotes MX, NX, describe an hyperbola through the point A, such
that AI may be to any of the lines VG as XV^{n} to XI^{n}.

Rule 6. By how much the greater the number *n*
is, so much the more accurate are these hyperbolas in the ascent of
the body from A, and less accurate in its descent to K; and the
contrary. The conic hyperbola keeps a mean ratio between these, and is
more simple than the rest. Therefore if the hyperbola be of this kind,
and you are to find the point K, where the projected body falls upon
any right line AN passing through the point A, let AN produced meet
the asymptotes MX, NX in M and N, and take NK equal to AM.

Rule 7. And hence appears an expeditious
method of determining this hyperbola from the phenomena. Let two
similar and equal bodies be projected with the same velocity, in
different angles HAK, *h*A*k*, and let them fall upon
the plane of the horizon in K and *k*; and note the proportion
of AK to A*k*. Let it be as *d* to *e*. Then
erecting a perpendicular AI of any length, assume any how the length
AH or A*h*, and thence graphically, or
by scale and compass, collect the lengths AK, A*k* (by Rule 6).
If the ratio of AK to A*k* be the same with that of *d*
to *e*, the length of AH was
rightly assumed. If not, take on the indefinite right line SM, the
length SM equal to the assumed AH; and erect a perpendicular MN equal
to the difference AK

Ak − d

e of the ratios drawn into any
given right line. By the like method, from several assumed lengths AH,
you may find several points N; and draw through them all a regular
curve NNXN, cutting the right line SMMM in X. Lastly, assume AH equal
to the abscissa SX, and thence find again the length AK; and the
lengths, which are to the assumed length AI, and this last AH, as the
length AK known by experiment, to the length AK last found, will be
the true lengths AI and AH, which were to be found. But these being
given, there will be given also the resisting force of the medium in
the place A, it being to the force of gravity as AH to ^{4}/_{3}AI.
Let the density of the medium be increased by Rule 4, and if the
resisting force just found be increased in the same ratio, it will
become still more accurate.

Rule 8. The lengths AH, HX being found; let
there be now required the position of the line AH, according to which
a projectile thrown with that given velocity shall fall upon any point
K. At the joints A and K, erect the lines AC, KF perpendicular to the
horizon; whereof let AC be drawn downwards, and be equal to AI or ½HX.
With the asymptotes AK, KF, describe an hyperbola, whose conjugate
shall pass through the point C; and from the centre A, with the
interval AH, describe a circle cutting that hyperbola in the point H;
then the projectile thrown in the direction of the right line AH will
fall upon the point K. Q.E.I. For the point H,
because of the given length AH, must be somewhere in the circumference
of the described circle. Draw CH meeting AK and KF in E and F; and
because CH, MX are parallel, and AC, AI equal, AE will be equal to AM,
and therefore also equal to KN. But CE is to AE as FH to KN, and
therefore CE and FH are equal. Therefore the point H falls upon the
hyperbolic curve described with the asymptotes AK, KF whose conjugate
passes through the point C; and is therefore found in the
common intersection of this hyperbolic curve
and the circumference of the described circle. Q.E.D. It
is to be observed that this operation is the same, whether the right
line AKN be parallel to the horizon, or inclined thereto in any angle;
and that from two intersections H, *h*, there arise two angles
NAH, NA*h*; and that in mechanical practice it is sufficient
once to describe a circle, then to apply a ruler CH, of an
indeterminate length, so to the point C, that its part FH, intercepted
between the circle and the right line FK, may be equal to its part CE
placed between the point C and the right line AK

What has been said of hyperbolas may be easily applied to parabolas.
For if a parabola be represented by XAGK, touched by a right line XV
in the vertex X, and the ordinates IA, VG be as any powers XI^{n},
XV^{n}, of the abscissas XI, XV; draw XT, GT, AH, whereof let
XT be parallel to VG, and let GT, AH touch the parabola in G and A:
and a body projected from any place A, in the direction of the right
line AH, with a due velocity, will describe this parabola, if the
density of the medium in each of the places G be reciprocally as the
tangent GT. In that case the velocity in G will be the same as would
cause a body, moving in a nonresisting space, to describe a conic
parabola, having G for its vertex, VG produced downwards for its
diameter, and 2GT^{2}

(nn − n) x VG for its latus
rectum. And the resisting force in G will be to the force of gravity
as GT to 2nn −
2n

n − 2VG. Therefore if NAK
represent an horizontal line, and both the density of the medium at A,
and the velocity with which the body is projected, remaining the same,
the angle NAH be any how altered, the lengths AH, AI, HX will remain;
and thence will be given the vertex X of the parabola, and the
position of the right line XI; and by taking VG to IA as XV^{n}
to XI^{n}, there will be given all the points G of the
parabola, through which the projectile will pass.