The motion of bodies that are resisted in proportion to their velocity
Proposition 1, Theorem 1
If a body is resisted in proportion to its velocity, the motion lost as a result of the
resistance is as the space described in moving.
For since the motion lost in each of the equal particles of time is as the velocity, that is, as a particle of the path described, then, by composition [or componendo], the motion lost in the whole time will be as the whole path. Q.E.D.
COROLLARY. Therefore, if a body, devoid of all gravity, moves in free spaces by its inherent force alone and if there are given both the whole motion at the beginning and also the remaining motion after some space has been described, the whole space that the body can describe in an infinite time will be given. For that space will be to the space already described as the whole motion at the beginning is to the lost part of that motion.
Lemma 1
Quantities proportional to their differences are continually proportional.
Let A be to A − B as B to B − C and C to C − D, . . . ; then, by conversion [or convertendo], A will be to B as B to C and C to D, . . . . Q.E.D.
Proposition 2, Theorem 2
If a body is resisted in proportion to its velocity and moves through a homogeneous
medium by its inherent force alone and if the times are taken as equal, the velocities
at the beginnings of the individual times are in a geometric progression, and the
spaces described in the individual times are as the velocities.
CASE 1. Divide the time into equal particles; and if, at the very beginnings of the particles, a force of resistance which is as the velocity acts with a single impulse, the decrease of the velocity in the individual particles of time will be as that velocity. The velocities are therefore proportional to their differences and thus (by book 2, lem. 1) are continually proportional. Accordingly, if any equal times are compounded of an equal number of particles, the velocities at the very beginnings of the times will be as the terms in a continual progression in which some have been skipped, omitting an equal number of intermediate terms in each interval. The ratios of these terms are indeed compounded of equally repeated equal ratios of the intermediate terms, and therefore these compound ratios are also equal to one another. Therefore, since the velocities are proportional to these terms, they are in a geometric progression. Now let those equal particles of times be diminished, and their number increased indefinitely, so that the impulse of the resistance becomes continual; then the velocities at the beginnings of equal times, which are always continually proportional, will also be continually proportional in this case. Q.E.D.
CASE 2. And by separation [or dividendo] the differences of the velocities (that is, the parts of them which are lost in the individual times) are as the wholes, while the spaces described in the individual times are as the lost parts of the velocities (by book 2, prop. 1) and are therefore also as the wholes. Q.E.D.
COROLLARY. Hence, if a hyperbola BG is described with respect to the
rectangular asymptotes AC and CH and if AB and DG are perpendicular
to asymptote AC and if both the velocity of the body and the resistance of
the medium are represented, at the very beginning
of the motion, by any given line AC, but after some
time has elapsed, by the indefinite line DC, then the
time can be represented by area ABGD, and the space
described in that time can be represented by line AD.
For if the area is increased uniformly by the motion of
point D, in the same manner as the time, the straight line DC will decrease
in a geometric ratio in the same way as the velocity, and the parts of the
straight line AC described in equal times will decrease in the same ratio.
Proposition 3, Problem 1
To determine the motion of a body which, while moving straight up or down in a
homogeneous medium, is resisted in proportion to the velocity, and which is acted
on by uniform gravity.
When the body is moving up,
represent the gravity by any given
rectangle BACH, and the resistance
of the medium at the beginning of
the ascent by the rectangle BADE
taken on the other side of the
straight line AB. With respect to
the rectangular asymptotes AC and
CH, describe a hyperbola through
point B, cutting perpendiculars DE and de in G and g; then the body, by
ascending in the time DGgd, will describe the space EGge; and in the time
DGBA will describe the space of the total ascent EGB; and in the time
ABKI will describe the space of descent BFK; and in the time IKki will
describe the space of descent KFfk; and the body’s velocities (proportional
to the resistance of the medium) in these periods of time will be ABED,
ABed, null, ABFI, and ABfi respectively; and the greatest velocity that the
body can attain in descending will be BACH.
For resolve the rectangle BACH into innumerable rectangles Ak, Kl,
Lm, Mn, . . . , which are as the increases of the velocities, occurring in
the same number of equal times; then nil, Ak, Al, Am, An, . . . will be
as the total velocities, and thus (by hypothesis) as the resistances of the
medium at the beginning of each of the equal times. Make AC to AK,
or ABHC to ABkK, as the force of gravity to the resistance at the
beginning of the second time, and subtract the resistances from the force
of gravity; then the remainders ABHC,
KkHC, LlHC, MmHC, . . . will be as
the absolute forces by which the body is
urged at the beginning of each of the
times, and thus (by the second law of
motion) as the increments of the
velocities, that is, as the rectangles Ak, Kl,
Lm, Mn, . . . , and therefore (by book
2, lem. 1) in a geometric progression. Therefore, if the straight lines Kk, Ll,
Mm, Nn, . . . , produced, meet the hyperbola in q, r, s, t, . . . , areas ABqK,
KqrL, LrsM, MstN, . . . will be equal, and thus proportional both to the
times and to the forces of gravity, which are always equal. But area ABqK
(by book 1, lem. 7, corol. 3, and lem. 8) is to area Bkq as Kq to ½kq or
AC to ½AK, that is, as the force of gravity to the resistance in the middle
of the first time. And by a similar argument, areas qKLr, rLMs, sMNt, . . .
are to areas qklr, rlms, smnt, . . . as the force of gravity to the resistance in
the middle of the second time, of the third, of the fourth, . . . . Accordingly,
since the equal areas BAKq, qKLr, rLMs, sMNt, . . . are proportional to
the forces of gravity, areas Bkq, qklr, rlms, smnt, . . . will be proportional
to the resistance in the middle of each of the times, that is (by hypothesis),
to the velocities, and thus to the spaces described. Take the sums of the
proportional quantities; then areas Bkq, Blr, Bms, Bnt, . . . will be proportional
to the total spaces described, and areas ABqK, ABrL, ABsM, ABtN, . . .
will be proportional to the times. Therefore the body, while descending in
any time ABrL, describes the space Blr, and in the time LrtN describes
the space rlnt. Q.E.D. And the proof is similar for an ascending
motion. Q.E.D.
COROLLARY 1. Therefore the greatest velocity that a body can acquire in falling is to the velocity acquired in any given time as the given force of gravity by which the body is continually urged to athe force of the resistance by which it is impeded at the end of that time.a
COROLLARY 2. If the time is increased in an arithmetic progression, the sum of that greatest velocity and of the velocity in the ascent, and also their difference in the descent, decreases in a geometric progression.
COROLLARY 3. The differences of the spaces which are described in equal differences of the times decrease in the same geometric progression.
COROLLARY 4. The space described by a body is the difference of two spaces, of which one is as the time reckoned from the beginning of the descent, and the other is as the velocity; and these spaces are equal to each other at the very beginning of the descent.
Proposition 4, Problem 2
Supposing that the force of gravity in some homogeneous medium is uniform and
tends perpendicularly toward the plane of the horizon, it is required to determine
the motion of a projectile in that medium, while it is resisted in proportion to the
velocity.
From any place D let a projectile go forth along any straight line DP, and
represent its velocity at the beginning of the motion by the length DP. Drop
the perpendicular PC from point P to the horizontal line DC, and cut DC
in A so that DA is to AC as the resistance of the medium arising from the
upward motion at the beginning is to the force of gravity; or (which comes
to the same thing) so that the rectangle of DA and DP is to the rectangle
of AC and CP as the whole resistance at the beginning of the motion is to
the force of gravity. Describe any hyperbola GTBS with asymptotes DC and
CP which cuts the perpendiculars DG and AB in G and B; and complete
the parallelogram DGKC, whose side GK cuts AB in Q. Take the line N
in the same ratio to QB as DC
to CP, and at any point R of the
straight line DC erect the
perpendicular RT which meets the
hyperbola in T and the straight lines
EH, GK, and DP in I, t, and V,
and then on RT take Vr equal to
, or (which comes to the same
thing) take Rr equal to 
.
Then in the time DRTG the
projectile will arrive at point r,
describing the curved line DraF which
point r traces out, reaching its
greatest height a in the perpendicular
AB, and afterward always
approaching the asymptote PC. And its
velocity at any point r is as the tangent
rL of the curve. Q.E.I.
For N is to QB as DC to CP or DR to RV, and thus RV is equal
to 
, and Rr 
that is, RV − Vr, or 
is equal
to 
. Now represent the time by area RDGT, and (by
corol. 2 of the laws) divide the motion of the body into two parts, one upward
and the other lateral. Since the resistance is as the motion, it also will be
divided into two parts proportional to and opposite to the parts of the motion;
and thus the distance described by the lateral motion will be (by book 2,
prop. 2) as line DR, and the distance described by the upward motion will
be (by book 2, prop. 3) as the area DR × AB − RDGT, that is, as line Rr. But
at the very beginning of the motion the area RDGT is equal to the rectangle
DR × AQ, and thus that line Rr or 
is then to
DR as AB − AQ or QB to N, that is, as CP to DC, and hence as the upward
motion to the lateral motion at the beginning. Since, therefore, Rr is always
as the distance upward, and DR is always as the distance sideward, and Rr
is to DR at the beginning as the distance upward to the distance sideward,
Rr must always be to DR as the distance upward to the distance sideward,
and therefore the body must move in the line DraF, which the point r traces
out. Q.E.D.
COROLLARY 1. Rr is therefore equal to 
; and thus,
if RT is produced to X so that RX is equal to 
, that is, if the
parallelogram ACPY is completed, and DY is joined cutting CP in Z, and
RT is produced until it meets DY in X, then Xr will be equal to 
,
and therefore will be proportional to the time.
COROLLARY 2. Hence, if innumerable lines CR are taken (or, which comes to the same thing, innumerable lines ZX) in a geometric progression, then as many lines Xr will be in an arithmetic progression. And hence it is easy to draw curve DraF with the help of a table of logarithms.
COROLLARY 3. If a
parabola is constructed with vertex D and diameter
DG (produced downward) and a latus rectum that is to 2DP as the whole
resistance at the very beginning of
the motion is to the force of gravity,
then the velocity with which a body
must go forth from place D along
the straight line DP in order to
describe curve DraF in a uniform
resisting medium will be the very one
with which it must go forth from
the same place D along the same
straight line DP in order to describe
the parabola in a nonresisting space.
For the latus rectum of this parabola, at the very beginning of the
motion, is 
; and
Vr is 
or
. But the straight line that, if
it were drawn, would touch the hyperbola GTS in G is parallel to DK,
and thus Tt is 
, and
N has been taken as 
. Therefore
Vr is 
, that is (because DR is to DC as DV is to DP),
; and the latus rectum
comes out 
, that
is (because QB is to CK as DA is to AC), 
, and thus is to 2DP
as DP × DA to CP × AC—that is, as the resistance to the gravity. Q.E.D.
COROLLARY 4. Hence, if a body
is projected from any place D with
a given velocity along any straight
line DP given in position, and the
resistance of the medium at the very
beginning of the motion is given,
the curve DraF which the body will
describe can be found. For from the
given velocity the latus rectum of
the parabola is given, as is well
known. And if 2DP is taken to that
latus rectum as the force of gravity
to the force of resistance, DP is
given. Then, if DC is cut in A so
that CP × AC is to DP × DA in that
same ratio of gravity to resistance,
point A will be given. And hence
curve DraF is given.
COROLLARY 5. And conversely, if curve DraF is given, both the velocity of the body and the resistance of the medium in each of the places r will be given. For since the ratio of CP × AC to DP × DA is given, both the resistance of the medium at the beginning of the motion and the latus rectum of the parabola are also given; and hence the velocity at the beginning of the motion is also given. Then from the length of the tangent rL, both the velocity (which is proportional to it) and the resistance (which is proportional to the velocity) are given in any place r.
COROLLARY 6. The length 2DP is to the latus rectum of the parabola as the gravity to the resistance at D; and when the velocity is increased the resistance is increased in the same ratio, but the latus rectum of the parabola is increased in the square of that ratio; hence it is evident that the length 2DP is increased in the simple ratio and thus is always proportional to the velocity and is not increased or decreased when the angle CDP is changed unless the velocity is also changed.
COROLLARY 7. Hence the method is apparent for determining the
curve DraF from phenomena approximately and for obtaining thereby the
resistance and the velocity with which the body is projected. Project two
similar and equal bodies with
the same velocity from place
D along the different angles
CDP and CDp, and let the
places F and f where they fall
upon the horizontal plane DC
be known. Then, taking any
length for DP or Dp, suppose
that the resistance at D is to
the gravity in any ratio, and
represent that ratio by any
length SM. Then, by
computation, find the lengths DF and
Df from that assumed length
DP, and from the ratio 
(found by computation) take
away the same ratio (found by experiment), and represent the difference by
the perpendicular MN. Do the same thing a second and a third time, always
taking a new ratio SM of resistance to gravity, and obtain a new difference
MN. But draw the positive differences on one side of the straight line SM
and the negative differences on the other, and through points N, N, N draw
the regular curve NNN cutting the straight line SMMM in X, and then SX
will be the true ratio of the resistance to the gravity, which it was required
to find. From this ratio the length DF is to be obtained by calculation; then
the length that is to the assumed length DP as the length DF (found out by
experiment) to the length DF (just found by computation) will be the true
length DP. When this is found, there will be known both the curved line
DraF that the body describes and the body’s velocity and resistance in every
place.
Scholium
However, the hypothesis that the resistance encountered by bodies is in the
ratio of the velocity belongs more to mathematics than to nature.a In
mediums wholly lacking in rigidity, the resistances encountered by bodies are as
the squares of the velocities. For by the action of a swifter body, a motion
that is greater in proportion to that greater velocity is communicated to a
given quantity of the medium in a smaller time; and thus in an equal time,
because a greater quantity of the medium is disturbed, a greater motion is
communicated in proportion to the square of the velocity, and (by the second
and third laws of motion) the resistance is as the motion communicated. Let
us see, therefore, what kinds of motions arise from this law of resistance.
The motion of bodies that are resisted as the squares of the velocities
Proposition 5, Theorem 3
If the resistance of a body is proportional to the square of the velocity and if the
body moves through a homogeneous medium by its inherent force alone and if the
times are taken in a geometric progression going from the smaller to the greater
terms, I say that the velocities at the beginning of each of the times are inversely
in that same geometric progression and that the spaces described in each of the
times are equal.
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 is divided into
innumerable equal particles, the squares of the
velocities at each of the beginnings of the
times will be proportional to the differences
of those same velocities. Let the particles of
time be AK, KL, LM, . . . , taken in the
straight line CD, and erect perpendiculars
AB, Kk, Ll, Mm, . . . , meeting the
hyperbola BklmG (described with center C and rectangular asymptotes CD and
CH) in B, k, l, m, . . . ; then AB will be to Kk as CK to CA, and by
separation [or dividendo] AB − Kk to Kk as AK to CA, and by alternation
[or alternando] AB − Kk to AK as Kk to CA, and thus as AB × Kk to
AB × CA. Hence, since AK and AB × CA are given, AB − Kk will be as
AB × Kk; and ultimately, when AB and Kk come together, as AB2. And by
a similar argument Kk − Ll, Ll − Mm, . . . will
be as Kk2, Ll2, . . . . The
squares of lines AB, Kk, Ll, Mm, therefore, are as their differences; and on
that account, since the squares of the velocities were also as their differences,
the progression of both will be similar. It follows from what has been proved
that the areas described by these lines are also in a progression entirely
similar to that of the spaces described by the velocities. Therefore, if the velocity
at the beginning of the first time AK is represented by line AB, and the
velocity at the beginning of the second time KL by line Kk, and the length
described in the first time is represented by area AKkB, then all the
subsequent velocities will be represented by the subsequent lines Ll, Mm, . . . ,
and the lengths described will be represented by areas Kl, Lm, . . . . And by
composition [or componendo], if the whole time is represented by the sum
of its parts AM, the whole length described will be represented by the sum
of its parts AMmB. Now imagine time AM to be divided into parts AK,
KL, LM, . . . in such a way that CA, CK, CL, CM, . . . are in a geometric
progression; then those parts will be in the same progression, and the
velocities AB, Kk, Ll, Mm, . . . will be in the same progression inverted, and the
spaces described Ak, Kl, Lm, . . . will be equal. Q.E.D.
COROLLARY 1. Therefore it is evident that if the time is represented by any part AD of the asymptote, and the velocity at the beginning of the time by ordinate AB, then the velocity at the end of the time will be represented by ordinate DG, and the whole space described will be represented by the adjacent hyperbolic area ABGD; and furthermore, the space that a body in a nonresisting medium could describe in the same time AD, with the first velocity AB, will be represented by the rectangle AB × AD.
COROLLARY 2. Hence the space described in a resisting medium is given by taking that space to be in the same proportion to the space which could be described simultaneously with a uniform velocity AB in a nonresisting medium as the hyperbolic area ABGD is to the rectangle AB × AD.
COROLLARY 3. The resistance of the medium is also given by setting it to be, at the very beginning of the motion, equal to the uniform centripetal force that in a nonresisting medium could generate the velocity AB in a falling body in the time AC. For if BT is drawn, touching the hyperbola in B and meeting the asymptote in T, the straight line AT will be equal to AC and will represent the time in which the first resistance uniformly continued could annul the whole velocity AB.
COROLLARY 4. And hence the proportion of this resistance to the force of gravity or to any other given centripetal force is also given.
COROLLARY 5. And conversely, if the proportion of the resistance to any given centripetal force is given, the time AC is given in which a centripetal force equal to the resistance could generate any velocity AB; and hence point B is given, through which the hyperbola with asymptotes CH and CD must be described, as is also the space ABGD which the body, beginning its motion with that velocity AB, can describe in any time AD in a homogeneous resisting medium.
Proposition 6, Theorem 4
Equal homogeneous spherical bodies that are resisted in proportion to the square
of the velocity, and are carried forward by their inherent forces alone, will, in
times that are inversely as the initial velocities, always describe equal spaces, and
lose parts of their velocities proportional to the wholes.
Describe any hyperbola BbEe, with rectangular asymptotes CD and CH,
which cuts perpendiculars AB, ab, DE, and de in B, b, E, and e; and
represent the initial velocities by perpendiculars
AB and DE and the times by lines Aa and
Dd. Therefore Aa is to Dd as (by
hypothesis) DE is to AB, and as (from the nature of
the hyperbola) CA is to CD, and by
composition [or componendo] as Ca is to Cd.
Hence areas ABba and DEed, that is, the
spaces described, are equal to each other,
and the first velocities AB and DE are
proportional to the ultimate velocities ab and de, and therefore, by separation
[or dividendo], also to the lost parts of those velocities AB − ab and DE −
de. Q.E.D.
Proposition 7, Theorem 5
Spherical bodies that are resisted in proportion to the squares of the velocities will,
in times that are as the first motions directly and the first resistances inversely, lose
parts of the motions proportional to the wholes and will describe spaces
proportional to those times and the first velocities jointly.
For the lost parts of the motions are as the resistances and the times jointly. Therefore, for those parts to be proportional to the wholes, the resistance and time jointly must be as the motion. Accordingly, the time will be as the motion directly and the resistance inversely. Therefore, if the particles of times are taken in this ratio, the bodies will always lose particles of their motions proportional to the wholes and thus will retain velocities always proportional to their first velocities. And because the ratio of the velocities is given, they will always describe spaces that are as the first velocities and the times jointly. Q.E.D.
COROLLARY 1. Therefore, if equally swift bodies are resisted in proportion to the squares of their diameters, then homogeneous globes moving with any velocities will, in describing spaces proportional to their diameters, lose parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mass jointly, that is, as its velocity and the cube of its diameter; the resistance (by hypothesis) will be as the square of the diameter and the square of the velocity jointly; and the time (by this proposition) is in the former ratio directly and the latter ratio inversely, that is, as the diameter directly and the velocity inversely; and thus the space, being proportional to the time and the velocity, is as the diameter.
COROLLARY 2. If equally swift bodies are resisted in proportion to the 3/2 powers of the diameters, then homogeneous globes moving with any velocities will, in describing spaces that are as the 3/2 powers of the diameters, lose parts of motions proportional to the wholes.
COROLLARY 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 velocities will lose parts of their motions proportional to the wholes will be as the cubes of the diameters divided by that power. Let the diameters be D and E; and if the resistances, when the velocities are supposed equal, are as Dn and En, then the spaces in which the globes, moving with any velocities, will lose parts of their motions proportional to the wholes will be as D3−n and E3−n. And therefore homogeneous globes, in describing spaces proportional to D3−n and E3−n, will retain velocities in the same ratio to each other that they had at the beginning.
COROLLARY 4. But if the globes are not homogeneous, the space described by the denser globe must be augmented in proportion to the density. For the motion, with an equal velocity, is greater in proportion to the density, and the time (by this proposition) is increased in proportion to the motion directly, and the space described is increased in proportion to the time.
COROLLARY 5. And if the globes move in different mediums, the space in the medium that, other things being equal, resists more will have to be decreased in proportion to the greater resistance. For the time (by this proposition) will be decreased in proportion to the increase of the resistance, and the space will be decreased in proportion to the time.
Lemma 2a
The moment of a generated quantity is equal to the moments of each of the
generating roots multiplied continually by the exponents of the powers of those
roots and by their coefficients.
I call “generated” every quantity that is, without addition or subtraction, generated from any roots or terms: in arithmetic by multiplication, division, or extraction of roots; in geometry by the finding either of products and roots or of extreme and mean proportionals. Quantities of this sort are products, quotients, roots, rectangles, squares, cubes, square roots, cube roots, and the like.b I here consider these quantities as indeterminate and variable, and increasing or decreasing as if by a continual motion or flux; and it is their instantaneous increments or decrements that I mean by the word “moments,” in such a way that increments are considered as added or positive moments, and decrements as subtracted or negative moments. But take care: do not understand them to be finite particles! cFinite particles are not moments, but the very quantities generated from the moments.c They must be understood to be the just-now nascent beginnings of finite magnitudes. For in this lemma the magnitude of moments is not regarded, but only their first proportion when nascent. It comes to the same thing if in place of moments there are used either the velocities of increments and decrements (which it is also possible to call motions, mutations, and fluxions of quantities) or any finite quantities proportional to these velocities. And the coefficient of each generating root is the quantity that results from dividing the generated quantity by this root.
Therefore, the meaning of this lemma is that if the moments of any
quantities A, B, C, . . . increasing or decreasing by a continual motion, or
the velocities of mutation which are proportional to these moments are called
a, b, c, . . . , then the moment or mutation of the generated rectangle AB
would be aB + bA, and the moment of the generated solid ABC would
be aBC + bAC + cAB, and the moments of the generated powers A2, A3,
A4, A½, A3/2, A⅓, A⅔, A−1, A−2, A−½, would be 2aA, 3aA2, 4aA3,
½aA−½,
3/2aA½, ⅓aA−⅔, ⅔aA−⅓, −aA−2, −2aA−3, and −½aA−3/2
respectively. And generally, the moment of any power 
would be 
.
Likewise, the moment of the generated quantity A2B would be 2aAB + bA2,
and the moment of the generated quantity A3B4C2 would be 3aA2B4C2 +
4bA3B3C2 + 2cA3B4C, and the moment of the generated quantity 
or
A3B−2 would be 3aA2B−2 − 2bA3B−3, and so on. The lemma is proved as
follows.
CASE 1. Any rectangle AB increased by continual motion, when the halves of the moments, ½a and ½b, were lacking from the sides A and B, was A − ½a multiplied by B − ½b, or AB − ½aB − ½bA + ¼ab and as soon as the sides A and B have been increased by the other halves of the moments, it comes out A + ½a multiplied by B + ½b, or AB + ½aB + ½bA + ¼ab. Subtract the former rectangle from this rectangle, and there will remain the excess aB + bA. Therefore by the total increments a and b of the sides there is generated the increment aB + bA of the rectangle. Q.E.D.
CASE 2. Suppose that AB is always equal to G; then the moment of the solid ABC or GC (by case 1) will be gC + cG, that is (if AB and aB + bA are written for G and g), aBC + bAC + cAB. And the same is true of the solid contained under any number of sides [or the product of any number of terms]. Q.E.D.
CASE 3. Suppose that the sides A, B, and C are always equal to one another; then the moment aB + bA of A2, that is, of the rectangle AB, will be 2aA, while the moment aBC + bAC + cAB of A3, that is, of the solid ABC, will be 3aA2. And by the same argument, the moment of any power An is naAn−1. Q.E.D.
CASE 4. Hence, since 
multiplied by A is 1, the moment of
multiplied by A together with multiplied by a will be the moment of 1, that is,
nil. Accordingly, the moment of or of A−1 is 
. And in general, since
multiplied by An is 1, the moment of multiplied by An together
with multiplied by naAn−1 will be nil. And therefore the moment of
or A−n will be 
. Q.E.D.
CASE 5. And since A½ multiplied by A½ is A, the moment of A½
multiplied by 2A½ will be a, by case 3; and thus the moment of A½ will be
or ½aA−½. And in general, if is supposed equal to B, Am will be
equal to Bn, and hence maAm−1 will be equal to nbBn−1 and maA−1 will
be equal to nbB−1 or 
, and thus 
equal to b, that is, equal to
the moment of . Q.E.D.
CASE 6. Therefore the moment of any generated quantity AmBn is the moment of Am multiplied by Bn, together with the moment of Bn multiplied by Am, that is, maAm−1 Bn + nbBn−1 Am; and this is so whether the exponents m and n of the powers are whole numbers or fractions, whether positive or negative. And it is the same for a solid contained by more than two terms raised to powers. Q.E.D.
COROLLARY 1. Hence in continually proportional quantities, if one term is given, the moments of the remaining terms will be as those terms multiplied by the number of intervals between them and the given term. Let A, B, C, D, E, and F be continually proportional; then, if the term C is given, the moments of the remaining terms will be to one another as −2A, −B, D, 2E, and 3F.
COROLLARY 2. And if in four proportionals the two means are given, the moments of the extremes will be as those same extremes. The same is to be understood of the sides of any given rectangle.
COROLLARY 3. And if the sum or difference of two squares is given, the moments of the sides will be inversely as the sides.
Proposition 8, Theorem 6
If a body, acted on by gravity uniformly, goes straight up or down in a uniform
medium, and the total space described is divided into equal parts, and the absolute
forces at the beginnings of each of the parts are found (adding the resistance of
the medium to the force of gravity when the body is ascending, or subtracting it
when the body is descending), I say that those absolute forces are in a geometric
progression.
Represent the force of gravity by the given line AC; the 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 the line AP, which is
the mean proportional between AK
and AC, and thus is as the square
root of the resistance; the increment
of the resistance occurring in a given
particle of time, by the line-element
KL; and the simultaneous increment
of the velocity, by the line-element PQ; then with center C and rectangular
asymptotes CA and CH, describe any hyperbola BNS, meeting the erected
perpendiculars AB, KN, and LO in B, N, and O. Since AK is as AP2, the
moment KL of AK will be as the moment 2AP × PQ of AP2, that is, as
AP multiplied by KC, since the increment PQ of the velocity (by the second
law of motion) is proportional to the generating force KC. Compound the
ratio of KL with the ratio of KN, and the rectangle KL × KN will become
as AP × KC × KN—that is, because the rectangle KC × KN is given, as
AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle
KL × KN, when points K and L come together, is the ratio of equality.
Therefore that evanescent hyperbolic area is as AP. Hence the total
hyperbolic area ABOL is composed of the particles KNOL, which are always
proportional to the velocity AP, and therefore this area is proportional to
the space described with this velocity. Now divide that area into equal parts
ABMI, IMNK, KNOL, . . . , and the absolute forces AC, IC, KC, LC, . . . will be
in a geometric progression. Q.E.D.
And by a similar argument, if—in the ascent of the body—equal areas ABmi, imnk, knol, . . . are taken on the opposite side of point A, it will be manifest that the absolute forces AC, iC, kC, lC, . . . are continually proportional. And thus, if all the spaces in the ascent and descent are taken equal, all the absolute forces lC, kC, iC, AC, IC, KC, LC, . . . will be continually proportional. Q.E.D.
COROLLARY 1. Hence, if the space described is represented by the hyperbolic area ABNK, the force of gravity, the velocity of the body, and the resistance of the medium can be represented by lines AC, AP, and AK respectively, and vice versa.
COROLLARY 2. And line AC represents the greatest velocity that the body can ever acquire by descending infinitely.
COROLLARY 3. Therefore, if for a given velocity the resistance of the medium is known, the greatest velocity will be found by taking its ratio to the given velocity as the square root of the ratio of the force of gravity to that known resistance of the medium.a
Proposition 9, Theorem 7
Given what has already been proved, I say that if the tangents of the angles of a
sector of a circle and of a hyperbola are taken proportional to the velocities, the
radius being of the proper magnitude, the whole time aof ascending to the highest
placea will be as the sector of the circle, and the whole time bof descending from
the highest placeb will be as the sector of the hyperbola.
Draw AD perpendicular and equal to the straight line AC, which represents the force of gravity. With center D and semidiameter AD describe the quadrant AtE of a circle and the rectangular hyperbola AVZ having axis AX, principal vertex A, and asymptote DC. Draw Dp and DP, and the sector AtD of the circle will be as cthe whole time of ascending to the highest place,c and the sector ATD of the hyperbola will be as dthe whole time of descending from the highest place,d provided that the tangents Ap and AP of the sectors are as the velocities.
CASE 1. Draw Dvq cutting off the moments or the minimally small
particles tDv and qDp, described simultaneously, of the sector ADt and of
the triangle ADp. Since those particles, because of the common angle D, are
as the squares of the sides, particle tDv will be as 
, that is, because
tD is given, as 
.
But pD2 is AD2 + Ap2, that is, AD2 + AD × Ak, or
AD × Ck; and qDp is ½AD × pq. Therefore particle tDv of the sector is as
, that is, directly as the minimally small decrement pq of the velocity and
inversely as the force Ck that decreases the velocity, and thus as the particle
of time corresponding to the decrement of the velocity. And by composition
[or componendo] the sum of all the particles tDv in the sector ADt will be
as the sum of the particles of time corresponding to each of the lost particles
pq of the decreasing velocity Ap, until that velocity, decreased to nil, has
vanished; that is, the whole sector ADt is as ethe whole time of ascending to
the highest place.e Q.E.D.
CASE 2. Draw DQV cutting off the minimally small 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 separation [or dividendo] as DX2 − TX2
to DA2 − AP2. But from the nature of the hyperbola, DX2 − TX2 is AD2,
and by hypothesis AP2 is AD × AK. Therefore the particles are to each
other as AD2 to AD2 − AD × AK, that is, as AD to AD − AK or AC
to CK; and thus the particle TDV of the sector is 
, and hence,
because AC and AD are given, as 
, that is, directly as the increment of
the velocity and inversely as the force generating the increment, and thus as
the particle of time corresponding to the increment. And by composition [or
componendo] 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.
COROLLARY 1. Hence, if AB is equal to a fourth of AC, the space that a body describes by falling in any time will be in the same ratio to the space that the body can describe by progressing uniformly in that same time with its greatest velocity AC as the ratio of area ABNK (which represents the space described in falling) to area ATD (which represents the time). For, since AC is to AP as AP to AK, it follows (by book 2, lem. 2, corol. 1) that LK will be to PQ as 2AK to AP, that is, as 2AP to AC, and hence LK will be to ½PQ as AP to ¼AC or AB; KN is also to AC or AD as AB to CK; and thus, from the equality of the ratios [or ex aequo], LKNO will be to DPQ as AP to CK. But DPQ was to DTV as CK to AC. Therefore, once again by the equality of the ratios [or ex aequo], LKNO is to DTV as AP to AC, that is, as the velocity of the falling body to the greatest velocity that the body can acquire in falling. Since, therefore, the moments LKNO and DTV of areas ABNK and ATD are as the velocities, all the parts of those areas generated simultaneously will be as the spaces described simultaneously, and thus the whole areas ABNK and ATD generated from the beginning will be as the whole spaces described from the beginning of the descent. Q.E.D.
COROLLARY 2. The same result follows for the space described in ascent: namely, the whole space is to the space described in the same time with a uniform velocity AC as area ABnk is to sector ADt.
COROLLARY 3. The velocity of a body falling in time ATD is to the velocity that it would acquire in the same time in a nonresisting space as the triangle APD to the hyperbolic sector ATD. For the velocity in a nonresisting medium would be as time ATD, and in a resisting medium is as AP, that is, as triangle APD. And the velocities at the beginning of the descent are equal to each other, as are those areas ATD and APD.
COROLLARY 4. By the same argument, the velocity in the ascent is to the velocity with which the body in the same time in a nonresisting space could lose its whole ascending motion as the triangle ApD is to the sector AtD of the circle, or as the straight line Ap is to the arc At.
COROLLARY 5. Therefore the time in which a body, by falling in a resisting medium, acquires the velocity AP is to the time in which it could acquire its greatest velocity AC, by falling in a nonresisting space, as sector ADT to triangle ADC; and the time in which it could lose the velocity Ap by ascending in a resisting medium is to the time in which it could lose the same velocity by ascending in a nonresisting space as arc At is to its tangent Ap.
COROLLARY 6. Hence, from the given time, the space described by ascent or descent is given. For the greatest velocity of a body descending infinitely is given (by book 2, prop. 8, corols. 2 and 3), and hence the time is given in which a body could acquire that velocity by falling in a nonresisting space. And if sector ADT or ADt is taken to be to triangle ADC in the ratio of the given time to the time just found, there will be given both the velocity AP or Ap and the area ABNK or ABnk, which is to the sector ADT or ADt as the required space is to the space that can be described uniformly in the given time with that greatest velocity which has already been found.
COROLLARY 7. And working backward, the time ADt or ADT will be given from the given space ABnk or ABNK of ascent or descent.
Proposition 10, Problem 3
Let a uniform force of gravity tend straight toward the plane of the horizon, and
let the resistance be as the density of the medium and the square of the velocity
jointly; it is required to find, in each individual place, the density of the medium
that makes the body move in any given curved line and also the velocity of the
body and resistance of the medium.
EXAMPLE 1. Let line PFHQ be a semicircle described on the diameter PQ, and let it be required to find the density of the medium that would make a projectile move in this semicircle.
Bisect diameter PQ in A; call AQ, n; AC, a; CH, e; and CD, o. Then
DI2 or AQ2 − AD2 will be = n2
− a2 − 2ao − o2, or
e2 − 2ao − o2, and when
the root has been extracted by our method, DI will become 
· · · . Here write n2 for e2
+ a2, and DI will come out
· · · .
I divide series of this sort into successive terms in the following manner.
What I call the first term is the term in which the infinitely small quantity o
does not exist; the second, the term in which that quantity is of one
dimension; the third, the term in which it is of two dimensions; the fourth, the term
in which it is of three dimensions; and so on indefinitely. 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
line-element MN, which is cut off by completing the parallelogram HCDM and
thus always determines the position of the tangent HN; as, for example, in
this case by taking MN to HM as is to o, or a to e. The third term,
which here is 
, will designate the
line-element IN, which lies between
the tangent and the curve and thus
determines the angle of contact IHN
or the curvature that the curved line
has in H. If that line-element IN is
of a finite magnitude, it will be
designated by the third term along with the terms following without limit. But if
that line-element is diminished infinitely, the subsequent terms will come out
infinitely smaller than the third and thus can be ignored. The fourth term
determines the variation of the curvature, the fifth the variation of the
variation, and so on. Hence, by the way, one can see clearly the not inconsiderable
usefulness of these series in the solution of problems that depend on tangents
and the curvature of curves.
cNow compare the series 
with the series
P − Qo − Ro2 − So3 − · · · , and in the same manner for P, Q, R, and S
write e, 
, and 
,
and for √(1 +Q2) write 
or 
; then the
density of the medium will come outc
as 
, that is (because n is given), as 
,
or 
, that is, as the tangent’s length HT terminated at the semidiameter
AF, which stands perpendicularly upon PQ; and the resistance will be to the
gravity as 3a to 2n, that is, as 3AC to the diameter PQ of the circle, while
the velocity will be as √CH. Therefore, if the body goes forth from place
F with the proper velocity along a line parallel to PQ, and the density of
the medium in each place H is as the length of the tangent HT, and the
resistance, also in some place H, is to the force of gravity as 3AC to PQ,
then that body will describe the quadrant FHQ of a circle. Q.E.I.
But if the same body were to go forth from place P along a line
perpendicular to PQ and were to begin to move in an arc of the semicircle
PFQ, AC or a would have to be taken on the opposite side of center A,
and therefore its sign would have to be changed, and −a would have to be
written for +a. Thus the density of the medium would come out as 
. But
nature does not admit of a negative density, that is, a density that accelerates
the motions of bodies; and therefore it cannot naturally happen that a body
by ascending from P should describe the quadrant PF of a circle. For this
effect the body would have to be accelerated by an impelling medium, not
impeded by a resisting medium.
EXAMPLE 2. Let the line PFQ be a parabola having its axis AF perpendicular to the horizon PQ, and let it be required to find the density of the medium that would make a projectile move in that parabola.
From the nature of the parabola, the rectangle PD × DQ is equal to the
rectangle of the ordinate DI and some given straight line. Let that straight
line be called b; PC, a; PQ, c; CH, e; and CD, o.
Then the rectangle (a+o)×(c−a−o), or ac − a2 −
2ao+co ∔ o2, is equal to the rectangle b × DI, and
thus DI is equal to 
. Now
the second term 
of this series should be
written for Qo, the third term 
likewise for Ro2. But since there are
not more terms, the coefficient S of the fourth will have to vanish, and
therefore the quantity 
, to which the density of the medium is
proportional, will be nil. Therefore, if the density of the medium is null, a
projectile will move in a parabola, as Galileo once proved. Q.E.I.
EXAMPLE 3. Let line AGK be a hyperbola having an asymptote NX perpendicular to the horizontal plane AK; and let it be required to find the density of the medium that would make a projectile move in this hyperbola.
Let MX be the other asymptote, meeting in V the ordinate DG
produced; and from the nature of the hyperbola, the rectangle XV × VG will
be given. Moreover, the ratio of DN to VX is given, and therefore the
rectangle DN × VG is given also. Let this rectangle be b2. And after
completing the parallelogram DNXZ, call BN a; BD, o; NX, c; and suppose the
given ratio of VZ to ZX or DN to be 
. Then DN will be equal to a − o,
VG will be equal to 
, VZ will
be equal to (a − o), and GD or
NX − VZ − VG will be equal to
c − a + o − . Resolve the
term into the converging series
. . . ,
and GD will become equal to c −
a − 
+ o − 
o − 
o2 −
o3 . . . . The second term o − oof this series is to be used for Qo,
the third (with the sign changed) o2 for Ro2, and the fourth (with the
sign also changed) o3 for So3, and their coefficients 
, and
are to be written in the above rule for Q, R, and S. When this is done,
the density of the medium comes out as 
or
, that is (if in VZ, VY is taken equal to VG),
as 
. For a2 and 
are the squares of XZ and ZY. And
the resistance is found to have the same ratio to gravity that 3XY has to 2YG;
and the velocity is that with which the body would go in a parabola having
vertex G, diameter DG, and latus rectum 
. Therefore suppose that the
densities of the medium in each of the individual places G are inversely as
the distances XY and that the resistance in some place G is to gravity as
3XY to 2YG; then a body sent forth from place A with the proper velocity
will describe that hyperbola AGK. Q.E.I.
EXAMPLE 4. Suppose generally that line AGK is a hyperbola described with center X and asymptotes MX and NX with the condition that when the rectangle XZDN is described, whose side ZD cuts the hyperbola in G and its asymptote in V, VG would be inversely as some power DNn (whose index is the number n) of ZX or DN; and let it be required to find the density of the medium in which a projectile would progress in this curve.
For BN, BD, and NX write A, O, and C respectively, and let VZ be
to XZ or DN as d to e, and let VG be equal to 
; then DN will
be equal to A − O, VG = 
,VZ = 
(A − O), and GD or
NX − VZ − VG will be equal to 
. Resolve
the term into the infinite series
. . . , and GD will become equal to
. . . . The second term
of this series 
is to be used for Qo, the third term 
for Ro2, the fourth term 
for So3. And hence the density
of the medium, 
, in any place G, becomes
,
and thus if in VZ, VY is taken equal to n × VG, the density is inversely
as XY. For A2 and 
are the squares of XZ and
ZY. Moreover, the resistance in the same place G becomes to the gravity
as 3S × 
is to 4R2, that is, as XY to 
VG. And the velocity in
the same place is the very velocity with which a projected body would go
in a parabola having vertex G, diameter GD, and latus rectum 
or
. Q.E.I.
Since motion does not take place in a parabola except in a
nonresisting medium, but does take place in the hyperbola here described if
there is a continual resistance, it is obvious that the line which a
projectile describes in a uniformly resisting medium approaches closer to these
hyperbolas than to a parabola. At any rate, that line is of a hyperbolic
kind, but about its vertex it is more distant from the asymptotes, and in
those parts that are further from the
vertex it approaches the asymptotes
more closely, than the hyperbolas
which I have described here. But
the difference between them is not
so great that one cannot be
conveniently used in place of the other in
practice. And the hyperbolas which
I have been describing will perhaps
prove to be more useful than a
hyperbola that is more exact and at
the same time more compounded.
And they will be brought into use
as follows.
Complete the parallelogram XYGT, and the straight line GT will touch
the hyperbola in G, and thus the density of the medium in G is inversely as
the tangent GT, and the velocity in the same place is as 
, while the
resistance is to the force of gravity as GT to 
× GV.
Accordingly, if a body projected from place A along the straight line
AH describes the hyperbola AGK and if AH produced meets the asymptote
NX in H and if AI drawn parallel to NX meets the other asymptote MX
in I, then the density of the medium in A will be inversely as AH, and the
velocity of the body will be as 
, and the resistance in the same place
will be to the gravity as AH to × AI. Hence the following rules.
RULE 1. If both the density of the medium at A and the velocity with which the body is projected remain the same, and angle NAH is changed, lengths AH, AI, and HX will remain the same. And thus, if those lengths are found in some one case, the hyperbola can then be determined readily from any given angle NAH.
RULE 2. If both angle NAH and the density of the medium at A remain the same, and the velocity with which the body is projected is changed, the length AH will remain the same, and AI will be changed in the ratio of the inverse square of the velocity.
RULE 3. If 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 is increased in any ratio, the proportion of AH to AI will
be increased in the same ratio, and the latus rectum of the above parabola as
well as the length 
(proportional to it) will remain the same; and
therefore AH will be decreased in the same ratio, and AI will be decreased as
the square of that ratio. But the proportion of the resistance to the weight is
increased when the specific gravity (the volume remaining constant) becomes
smaller, or the density of the medium becomes greater, or the resistance (as
a result of the decreased volume) is decreased in a smaller ratio than the
weight.
RULE 4. The density of the medium near the vertex of the hyperbola is greater than at place A; hence, in order to have the mean density, the ratio of the least of the tangents GT to tangent AH must be found, and the density at A must be increased in a slightly greater ratio than that of half the sum of these tangents to the least of the tangents GT.
RULE 5. If lengths AH and AI are given, and it is required to describe the figure AGK, produce HN to X so that HX is to AI as n +1 to 1, and with center X and asymptotes MX and NX, describe a hyperbola through point A in such a way that AI is to any VG as XVn to XIn.
RULE 6. The greater the number n, the more exact are these “hyperbolas” in the ascent of the body from A, and the less exact in its descent to K, and conversely. A conic hyperbola holds a mean ratio between them and is simpler than the others. Therefore, if the hyperbola is of this kind, and if it is required to find point K, where the projected body will fall upon any straight line AN passing through point A, let AN produced meet asymptotes MX and NX in M and N, and take NK equal to AM.
RULE 7. And hence a ready method of determining this kind of
hyperbola from the phenomena is clear. Project two similar and equal bodies
with the same velocity in different
angles HAK and hAk, and let them fall
upon the plane of the horizon in K
and k, and note the proportion of AK
to Ak (let this be d to e). Then,
having erected a perpendicular AI of any
length, assume length AH or Ah in any way and from this determine
graphically lengths AK and Ak by rule 6. If the ratio of AK to Ak is the same
as the ratio of d to e, length AH was correctly assumed. But if not, then on
the indefinite straight line SM take a length SM equal to the assumed AH,
and erect perpendicular MN equal to the difference of the ratios, 
,
multiplied by any given straight line. From several assumed lengths AH find
several points N by a similar method
eand through them all draw a regular
curved line NNXN cutting the straight line SMMM in X. Finally, assume
AH equal to abscissa SX, and from this find length AK again; then the
lengths that are to the assumed length AI and this last length AH as that
length AK (found by experiment) is to the length AK (last found) will be
those true lengths AI and AH which it was required to find. And these being
given, the resistance of the medium in place A will also be given, inasmuch
as it is to the force of gravity as AH to 2AI. The density of the medium,
moreover, must be increased (by rule 4), and the resistance just found, if it is
increased in the same ratio, will become more exact.e
RULE 8. If the lengths AH and HX have been found, and the position of the straight line AH is now desired along which a projectile sent forth with that given velocity falls upon any point K, erect at points A and K the straight lines AC and KF perpendicular to the horizon, of which AC tends downward and is equal to AI or ½HX. With asymptotes AK and KF describe a hyperbola whose conjugate passes through point C, and with center A and radius AH describe a circle cutting that hyperbola in point H; then a projectile sent forth along the straight line AH will fall upon point K. Q.E.I.
For point H, because length AH is given, is located somewhere in the circle described. Draw CH meeting AK and KF, the former in E, the latter in F; then, because CH and MX are parallel and AC and AI are 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. Point H therefore falls upon the hyperbola described with asymptotes AK and KF whose conjugate passes through point C, and thus H is found in the common intersection of this hyperbola and the circle described. Q.E.D.
It is to be noted, moreover, that this operation is the same whether the straight line AKN is parallel to the horizon or is inclined to the horizon at any angle, and that from the two intersections H and H two angles NAH and NAH arise, and that in a mechanical operation it is sufficient to describe a circle once, then to apply the indeterminate rule CH to point C in such a way that its part FH, placed between the circle and the straight line FK, is equal to its part CE situated between point C and the straight line AK.
What has been said about hyperbolas is easily applied to parabolas. For
if XAGK designates a parabola that the straight line XV touches in
vertex X and if ordinates IA and VG are as any
powers XIn and XVn of abscissas XI and XV,
draw XT, GT, and AH, of which XT is parallel
to VG, and GT and AH touch the parabola in
G and A; then a body projected with the proper
velocity from any place A along the straight line
AH (produced) will describe this parabola,
provided that the density of the medium in each
individual place G is inversely as tangent GT. The
velocity in G, however, will be that with which a projectile would go, in a
nonresisting space, in a conic parabola having vertex G, diameter VG
produced downward, and latus rectum 
. And
the resistance in
G will be to the force of gravity as GT to 
VG. Hence, if NAK
designates a horizontal line and if, while both the density of the medium
in A and the velocity with which the body is projected remain the same,
the angle NAH is changed in any way, then lengths AH, AI, and HX will
remain the same; and hence vertex X of the parabola and the position of the
straight line XI are given, and, by taking VG to IA as XVn to XIn, all the
points G of the parabola, through which the projectile will pass, are given.
The motion of bodies that are resisted partly in the ratio of the velocity and partly in the squared ratio of the velocity
Proposition 11, Theorem 8
If a body is resisted partly in the ratio of the velocity and partly in the squared
ratio of the velocity and moves in a homogeneous medium by its inherent force
alone, and if the times are taken in an arithmetic progression, then quantities
inversely proportional to the velocities and increased by a certain given quantity
will be in a geometric progression.
With center C and rectangular asymptotes CADd and CH, describe
a hyperbola BEe, and let AB, DE, and de be parallel to asymptote CH.
Let points A and G be given in asymptote
CD. Then if the time is represented by the
hyperbolic area ABED increasing uniformly, I
say that the velocity can be represented by the
length DF, whose reciprocal GD together with
the given quantity CG composes the length
CD increasing in a geometric progression.
For let the area-element DEed be a minimally small given increment of
time; then Dd will be inversely as DE and thus directly as CD. And the
decrement of 
, which (by book 2, lem. 2) is 
, will be as 
or
, that is, as + . Therefore, when the time ABED
increases uniformly by the addition of the given particles EDde, decreases
in the same ratio as the velocity. For the decrement of the velocity is as the
resistance, that is (by hypothesis), as the sum of two quantities, of which
one is as the velocity and the other is as the square of the velocity; and the
decrement of is as the sum of the quantities and , of which the
former is itself and the latter is as 
. Accordingly, because the
decrements are analogous, is as the velocity. And if the quantity GD,
which is inversely proportional to , is increased by the given quantity
CG, then as the time ABED increases uniformly, the sum CD will increase
in a geometric progression. Q.E.D.
COROLLARY 1. Therefore if, given the points A and G, the time is
represented by the hyperbolic area ABED, the velocity can be represented by
, the reciprocal of GD.
COROLLARY 2. And by taking GA to GD as the reciprocal of the velocity at the beginning to the reciprocal of the velocity at the end of any time ABED, point G will be found. And when G has been found, then if any other time is given, the velocity can be found.
Proposition 12, Theorem 9
With the same suppositions, I say that if the spaces described are taken in an
arithmetic progression, the velocities increased by a certain given quantity will be
in a geometric progression.
Let point R be given in the
asymptote CD, and after erecting
perpendicular RS meeting the hyperbola in S,
represent the described space by the
hyperbolic area RSED; then the
velocity will be as the length GD, which
with the given quantity CG composes
the length CD decreasing in a geometric progression while space RSED is
increased in an arithmetic progression.
For, because the increment EDde of the space is given, the line-element Dd, which is the decrement of GD, will be inversely as ED and thus directly as CD, that is, as the sum of GD and the given length CG. But the decrement of the velocity, in the time inversely proportional to it in which the given particle DdeE of space is described, is as the resistance and the time jointly, that is, directly as the sum of two quantities (of which one is as the velocity and the other is as the square of the velocity) and inversely as the velocity; and thus is directly as the sum of two quantities, of which one is given and the other is as the velocity. Therefore the decrement of the velocity as well as of line GD is as a given quantity and a decreasing quantity jointly; and because the decrements are analogous, the decreasing quantities will always be analogous, namely, the velocity and the line GD. Q.E.D.
COROLLARY 1. If the velocity is represented by the length GD, the space described will be as the hyperbolic area DESR.
COROLLARY 2. And if point R is taken at will, point G will be found by taking GR to GD as the velocity at the beginning is to the velocity after any space RSED has been described. And when point G has been found, the space is given from the given velocity, and conversely.
COROLLARY 3. Hence, since (by prop. 11) the velocity is given from the given time, and by this prop. 12 the space is given from the given velocity, the space will be given from the given time, and conversely.
Proposition 13, Theorem 10
Supposing that a body attracted downward by uniform gravity ascends straight up
or descends straight down and is resisted partly in the ratio of the velocity and
partly in the squared ratio of the velocity, I say that if straight lines parallel to the
diameters of a circle and a hyperbola are drawn through the ends of the conjugate
diameters and if the velocities are as certain segments of the parallels, drawn from
a given point, then the times will be as the sectors of areas cut off by straight lines
drawn from the center to the ends of the segments, and conversely.
CASE 1. Let us suppose first that the body is ascending. With center
D and any semidiameter DB describe the quadrant BETF of a circle, and
through the end B of semidiameter DB draw the
indefinite line BAP parallel to semidiameter DF. Let
point A be given in that line, and take segment AP
proportional to the velocity. Since one part of the
resistance is as the velocity and the other part is as
the square of the velocity, let the whole resistance be
as AP2 + 2BA × AP. Draw DA and DP cutting the
circle in E and T, and represent the gravity by DA2 in such a way that the
gravity is to the resistance as DA2 to AP2 + 2BA × AP; and the time of
the whole ascent will be as sector EDT of the circle.
For draw DVQ cutting off both the moment PQ of velocity AP and the moment DTV (corresponding to a given moment of time) of sector DET; then that decrement PQ of the velocity will be as the sum of the forces of the gravity DA2 and the resistance AP2 +2BA × AP, that is (by book 2, prop. 12 of the Elements), as DP2. Accordingly, the area DPQ, which is proportional to PQ, is as DP2, and the area DTV, which is to the area DPQ as DT2 to DP2, is as the given quantity DT2. The area EDT therefore decreases uniformly as the remaining time, by the subtraction of the given particles DTV, and therefore is proportional to the time of the whole ascent. Q.E.D.
CASE 2. If the velocity in the ascent of the body is represented by the
length AP as in case 1, and the resistance is supposed to be as AP2 +
2BA × AP, and if the force of gravity
is less than what could be represented
by DA2, take BD of such a length that
AB2 — BD2 is proportional to the
gravity, and let DF be perpendicular and
equal to DB, and through the vertex F
describe the hyperbola FTVE, whose
conjugate semidiameters are DB and DF and which cuts DA in E and cuts
DP and DQ in T and V; then the time of the whole ascent will be as the
sector TDE of the hyperbola.
For the decrement PQ of the velocity occurring in a given particle of time is as the sum of the resistance AP2 + 2BA × AP and the gravity AB2 − BD2, that is, as BP2 − BD2. But area DTV is to area DPQ as DT2 to DP2 and thus, if a perpendicular GT is dropped to DF, is as GT2 or GD2 − DF2 to BD2, and as GD2 to BP2, and by separation [or dividendo] as DF2 to BP2 − BD2. Therefore, since area DPQ is as PQ, that is, as BP2 − BD2, area DTV will be as DF2, which is given. Area EDT therefore decreases uniformly in each equal particle of time, by the subtraction of the same number of given particles DTV, and therefore is proportional to the time. Q.E.D.
CASE 3. Let AP be the velocity in the descent
of the body, and AP2 + 2BA × AP the resistance,
and BD2 − AB2 the force of gravity, angle DBA
being a right angle. And if with center D and principal vertex B the rectangular hyperbola BETV is
described, cutting the produced lines DA, DP, and
DQ in E, T, and V, then sector DET of this
hyperbola will be as the whole time of descent.
For the increment PQ of the velocity, and the area DPQ proportional to it, is as the excess of the gravity over the resistance, that is, as BD2 − AB2 − 2BA × AP − AP2 or BD2 − BP2. And area DTV is to area DPQ as DT2 to DP2 and thus as GT2 or GD2 − BD2 to BP2, and as GD2 to BD2, and by separation [or dividendo] as BD2 to BD2 − BP2. Therefore, since area DPQ is as BD2 − BP2, area DTV will be as BD2, which is given. Therefore area EDT increases uniformly in each equal particle of time, by the addition of the same number of given particles DTV, and therefore is proportional to the time of descent. Q.E.D.
COROLLARY. If with center D and semidiameter DA, the arc At similar to arc ET and similarly subtending angle ADT is drawn through vertex A, then the velocity AP will be to the velocity that the body in time EDT in a nonresisting space could lose by ascending, or acquire by descending, as the area of triangle DAP to the area of sector DAt and thus is given from the given time. For in a nonresisting medium the velocity is proportional to the time and thus proportional to this sector; in a resisting medium the velocity is as the triangle; and in either medium, when the velocity is minimally small, it approaches the ratio of equality just as the sector and the triangle do.
Scholiuma
The case could also be proved in the ascent of the body, where the force
of gravity is less than what can be represented by DA2 or AB2 + BD2 and
greater than what can be represented by AB2 − BD2, and must be represented
by AB2. But I hasten to other topics.
Proposition 14, Theorem 11
With the same suppositions, I say that the space described in the ascent or descent
is as the difference between the area which represents the time and a certain
other area that increases or decreases in an arithmetic progression, if the forces
compounded of the resistance and the gravity are taken in a geometric progression.
Take AC (in the three figures) proportional to the gravity, and AK proportional to the resistance. And take them on the same side of point A if the body is descending, otherwise on opposite sides. Erect Ab, which is to DB as DB2 to 4BA × AC; and when the hyperbola bN has been described with respect to the rectangular asymptotes CK and CH, and KN has been erected perpendicular to CK, area AbNK will be increased or decreased in an arithmetic progression while the forces CK are taken in a geometric progression. I say therefore that the distance of the body from its greatest height is as the excess of area AbNK over area DET.
For since AK is as the resistance, that is, as AP2 + 2BA × AP, assume
any given quantity Z, and suppose AK equal to 
,
and (by book 2, lem. 2) the moment KL of AK will be equal to
or 
, and the moment KLON of
area AbNK will be equal to 
or 
.
CASE 1. Now, if the body is ascending and the gravity is as AB2 + BD2,
BET being a circle (in the first figure), then line AC, which is proportional to
the gravity, will be 
, and DP2 or AP2 + 2BA × AP + AB2 + BD2
will be AK × Z + AC × Z or CK × Z; and thus area DTV will be to area
DPQ as DT2 or DB2 or CK × Z.
CASE 2. But if the body is ascending and the gravity is as AB2 − BD2,
then line AC (in the second figure) will be 
, and DT2 will be to
DP2 as DF2 or DB2 to BP2 − BD2 or AP2 + 2BA × AP + AB2 − BD2, that
is, to AK × Z + AC × Z or CK × Z. And thus area DTV will be to area
DPQ as DB2 to CK × Z.
CASE 3. And by the same argument, if the body is descending and
therefore the gravity is as BD2 − AB2, and line AC (in the third figure)
is equal to 
, then area DTV will be to area DPQ as DB2 to
CK × Z, as above.
Since, therefore, those areas are always in this ratio, if for area DTV,
which represents the moment of time always equal to it, any determinate
rectangle is written, say BD × m, then area DPQ, that is, ½BD × PQ, will
be to BD × m as CK × Z to BD2. And hence PQ × BD3 becomes equal to
2BD × m × CK × Z, and the moment KLON (found above) of area AbNK
becomes 
. Take away the moment DTV or BD × m of area
DET, and there will remain 
. Therefore the difference of
the moments, that is, the moment of the difference of the areas, is equal to
, and therefore (because 
is given) is as the velocity
AP, that is, as the moment of the space that the body describes in ascending
or descending. And thus that space and the difference of the areas,
increasing or decreasing by proportional moments and beginning simultaneously or
vanishing simultaneously, are proportional. Q.E.D.
COROLLARY. If the length that results from dividing area DET by the
line BD is called M, and another length V is taken in the ratio to length M
that line DA has to line DE, then the space that a body describes in its whole
ascent or descent in a resisting medium will be to the space that the body
can describe in the same time in a nonresisting medium, by falling from a
state of rest, as the difference of the above areas to 
, and thus is
given from the given time. For the space in a nonresisting medium is in the
squared ratio of the time, or as V2, and, because BD and AB are given, as
. aThis area is equal to
area 
, and the moment of
M is m; and therefore the moment of this area is 
. But
this moment is to the moment of the difference of the above areas DET and
AbNK that is, to 
as 
is to ½BD × AP,
or as 
× DET is to DAP; and thus, when areas DET and DAP are
minimally small, in the ratio of equality. Therefore area and the
difference of areas DET and AbNK, when all these areas are minimally
small, have equal moments and thus are equal. Hence, since the velocities,
and therefore also the spaces described simultaneously in both mediums at
the beginning of the descent or the end of the ascent, approach equality and
thus are then to one another as area and the difference of areas
DET and AbNK; and furthermore since the space in a nonresisting medium
is always as , and the space in a resisting medium is always as the
difference of areas DET and AbNK; it follows that the spaces described
in both mediums in any equal times must be to one another as the area
and the difference of areas DET and AbNK. Q.E.D.a
Scholiumb
The resistance encountered by
spherical bodies in fluids arises partly from the
tenacity, partly from the friction, and partly
from the density of the medium.
And we have said that the part of the resistance that arises from the density
of the fluid is in the squared 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 thus it would now be possible to proceed to the motion of bodies
that are resisted partly by a uniform force or in the ratio of the moments of
the time and partly in the squared ratio of the velocity. But it is sufficient
to have opened the way to the examination of this subject in the preceding
props. 8 and 9 and their corollaries. In these propositions and corollaries,
in place of the uniform resistance of the ascending body, which arises from
its gravity, there can be substituted the uniform resistance that arises from
the tenacity of the medium, when the body is moved by its inherent force
alone; and when the body is ascending straight up, it is possible to add this
uniform resistance to the force of gravity, and to subtract it when the body
is descending straight down. It would also be possible to proceed to the
motion of bodies that are resisted partly uniformly, partly in the ratio of the
velocity, and partly in the squared ratio of the velocity. And I have opened
the way in the preceding props. 13 and 14, in which the uniform resistance
that arises from the tenacity of the medium can also be substituted for the
force of gravity, or can be compounded with it as before. But I hasten to
other topics.
The revolving motion of bodies in resisting mediums
Lemma 3
Let PQR be a spiral that cuts all the radii SP,
SQ, SR, . . . in equal angles.
Draw the straight line PT touching the spiral in any point P and cutting the
radius SQ in T; erect PO and QO perpendicular to the spiral and meeting in O,
and join SO. I say that if points P and Q approach each other and coincide, angle
PSO will come out a right angle, and the ultimate ratio of rectangle TQ × 2PS
to PQ2 will be the ratio of equality.
For, from the right angles OPQ
and OQR subtract the equal angles
SPQ and SQR, and the equal angles
OPS and OQS will remain.
Therefore a circle that passes through points
O, S, and P will also pass through
point Q. Let points P and Q come
together, and this circle will touch the
spiral in the place PQ where they coincide, and thus will cut the straight line
OP perpendicularly. OP will therefore become a diameter of this circle, and
OSP, an angle in a semicircle, will become a right angle. Q.E.D.
Drop perpendiculars QD and SE to OP, and the ultimate ratios of the lines will be as follows: TQ will be to PD as TS (or PS) to PE, or 2PO to 2PS; likewise, PD will be to PQ as PQ to 2PO; and from the equality of the ratios in inordinate proportion [or ex aequo perturbate] TQ will be to PQ as PQ to 2PS. Hence PQ2 becomes equal to TQ × 2PS. Q.E.D.
Proposition 15, Theorem 12
If the density of a medium in every place is inversely as the distance of places
from a motionless center and if the centripetal force is in the squared ratio of the
density, I say that a body can revolve in a spiral that intersects in a given angle
all the radii drawn from that center.
Let the same things be supposed as in lemma 3, and produce SQ to V, so
that SV is equal to SP. In any time, in a resisting medium, let a body describe
the minimally small arc PQ, and in twice the time, the minimally small
arc PR; then the decrements of these arcs arising from the resistance, that is,
the differences between these arcs and
the arcs that would be described in the
same times in a nonresisting medium,
will be to each other as the squares of
the times in which they are generated.
The decrement of arc PQ is therefore
a fourth of the decrement of arc PR.
Hence also, if area QSr is taken equal
to area PSQ, the decrement of arc PQ will be equal to half of the line-element
Rr; and thus the force of resistance and the centripetal force are to each other
as the line-elements ½Rr and TQ that they simultaneously generate. Since
the centripetal force by which the body is urged in P is inversely as SP2; and
since (by book 1, lem. 10) the line-element TQ, which is generated by that
force, is in a ratio compounded of the ratio of this force and the squared
ratio of the time in which arc PQ is described (for I ignore the resistance in
this case, as being infinitely smaller than the centripetal force); then it follows
that TQ × SP2, that is (by lem. 3), ½PQ2 × SP, will be in the squared ratio
of the time, and thus the time is as PQ × √SP; and the body’s velocity with
which arc PQ is described in that time will be as 
or 
, that
is, as the square root of SP inversely. And by a similar argument, the velocity
with which arc QR is described is as the square root of SQ inversely. But
these arcs PQ and QR are as the velocities of description to each other, that
is, as √SQ to √SP, or as SQ to √(SP × SQ); and because angles SPQ and
SQr are equal and areas PSQ and QSr are equal, arc PQ is to arc Qr as SQ
to SP. Take the differences of the proportional consequents, and arc PQ will
become to arc Rr as SQ to SP − √(SP × SQ), or ½VQ. For, points P and
Q coming together, the ultimate ratio of SP − √(SP × SQ) to ½VQ is the
ratio of equality. aSince the decrement of arc PQ arising from the resistance,
or its double Rr, is as the resistance and the square of the time jointly, the
resistance will be as 
.a But PQ was to Rr as SQ to ½VQ, and
hence becomes as 
, or as 
. For, points
P and Q coming together, SP and SQ coincide, and angle PVQ becomes a
right angle; and because triangles PVQ and PSO are similar, PQ becomes to
½VQ as OP to ½OS. Therefore 
is as the resistance, that is, in the
ratio of the density of the medium at P and the squared ratio of the velocity
jointly. Take away the squared ratio of the velocity, namely the ratio 
, and
the result will be that the density of the medium at P is as 
. Let
the spiral be given, and because the ratio of OS to OP is given, the density
of the medium at P will be as . Therefore in a medium whose density
is inversely as the distance SP from the center, a body can revolve in this
spiral. Q.E.D.
COROLLARY 1. The velocity in any place P is always the velocity with which a body in a nonresisting medium, under the action of the same centripetal force, can revolve in a circle at the same distance SP from the center.
COROLLARY 2. The density of the medium, if the distance SP is given,
is as 
; but if that distance is not given, it is as . And hence a
spiral can be made to conform to any density of the medium.
COROLLARY 3. The force of resistance in any place P is to the centripetal
force in the same place as ½OS to OP. For those forces are to each other
as ½Rr and TQ or as 
and 
, that is, as ½VQ and PQ, or
½OS and OP. Given the spiral, therefore, the proportion of the resistance to
the centripetal force is given; and conversely, from that given proportion the
spiral is given.
COROLLARY 4. The body, therefore, cannot revolve in this spiral except when the force of resistance is less than half of the centripetal force. Let the resistance become equal to half of the centripetal force; then the spiral will coincide with the straight line PS, and the body will descend to the center in this straight line with a velocity that is (as we proved in book 1, prop. 34) to the velocity with which the body descends in a nonresisting medium in the case of a parabola in the ratio of 1 to √2. bAnd the times of descent will here be inversely as the velocities, and thus are given.b
COROLLARY 5. And since at equal
distances from the center the velocity
is the same in the spiral PQR as in the
straight line SP, and since the length
of the spiral is in a given ratio to the
length of the straight line PS, namely
the ratio of OP to OS, the time of
descent in the spiral will be to the time
of descent in the straight line SP in that same given ratio, and accordingly is
given.
COROLLARY 6. If, with center S and any two given radii, two circles
are described, and if—these circles remaining the same—the angle that the
spiral contains with radius PS is changed in any way, then the number of
revolutions that the body can complete between the circumferences of the
circles, by revolving in the spiral from one circumference to the other, is as
, or as the tangent of the angle that the spiral contains with radius PS.
And the time of those revolutions is as 
, that is, as the secant of that
angle, or inversely as the density of the medium.
COROLLARY 7. If a body, in a medium whose density is inversely as the distance of places from the center, has made a revolution about that center in any curve AEB and has cut the first radius AS in the same angle in B as it did previously in A, with a velocity that was to its prior velocity in A inversely as the square roots of distances from the center—that is, as AS to a mean proportional between AS and BS—then that body will make innumerable entirely similar revolutions BFC, CGD, . . . , and by the intersections will divide the radius AS into the continually proportional parts AS, BS, CS, DS, . . . . And the times of revolution will be as the perimeters of the orbits AEB, BFC, CGD, . . . , directly, and the velocities in the beginnings A, B, C, inversely—that is, as AS3/2, BS3/2, CS3/2. And the whole time in which the body will reach the center will be to the time of the first revolution as the sum of all the continually proportional quantities AS3/2, BS3/2, CS3/2, going on indefinitely, is to the first term AS3/2—that is, as that first term AS3/2 is to the difference of the first two terms AS3/2 − BS3/2, or very nearly as ⅔AS to AB. In this way the whole time is readily found.
COROLLARY 8. From what has been presented, it is also possible to determine approximately the motions of bodies in mediums whose density either is uniform or accords with any other assigned law. With center S and radii SA, SB, SC, . . . which are continually proportional, describe any number of circles. And suppose that the time of the revolutions between the perimeters of any two of these circles in the medium treated in corol. 7 is to the time of revolutions between those perimeters in the proposed medium very nearly as the mean density of the proposed medium between those circles is to the mean density of the medium in corol. 7 between those same circles; and suppose additionally that the secant of the angle by which the spiral in corol. 7, in the medium treated in that corollary, cuts the radius AS is in the same ratio to the secant of the angle by which the new spiral cuts that same radius in the proposed medium; and also that the numbers of all the revolutions between those same two circles are very nearly as the tangents of those same angles. If this is done throughout between every pair of circles, the motion will be continued through all the circles. And thus we can imagine without difficulty in what ways and in what times bodies would have to revolve in any regular medium.
COROLLARY 9. And even if the motions are eccentric, being performed in spirals approaching an oval shape, nevertheless by conceiving that the single revolutions of those spirals are the same distance apart from one another and approach the center by the same degrees as the spiral described above, we shall also understand how the motions of bodies are performed in spirals of this sort.
Proposition 16, Theorem 13
If the density of the medium in every place is inversely as the distance of places
from a motionless center and if the centripetal force is inversely as any power of
that distance, I say that a body can revolve in a spiral that intersects in a given
angle all the radii drawn from that center.
This is proved by the same method as prop. 15. For if the centripetal
force in P is inversely as any power SPn+1 (whose index is n + 1) of the
distance SP, then it will be gathered,
as above, that the time in which
the body describes any arc PQ will
be as PQ × PS½n, and the
resistance in P will be as 
,
or as 
, and thus
as 
, that is, because
is given, inversely as SPn+1. And therefore, since the velocity
is inversely as SP½n, the density in P will be inversely as SP.
COROLLARY 1. The resistance is to the centripetal force as (1 − ½n) × OS to OP.
COROLLARY 2. If the centripetal force is inversely as SP3, 1 − ½n will be = 0, and thus the resistance and density of the medium will be null, as in book 1, prop. 9.
COROLLARY 3. If the centripetal force is inversely as some power of the radius SP whose index is greater than the number 3, positive resistance will be changed to negative.
Scholium
But this proposition and the previous ones, which relate to unequally dense
mediums, are to be understood of the motion of bodies so small that no
consideration need be taken of a greater density of the medium on one side
of the body than on the other. I also suppose the resistance, other things being
equal, to be proportional to the density. Hence, in mediums whose force of
resisting is not as the density, the density ought to be increased or decreased
to such an extent that either the excess of the resistance may be taken away
or its defect supplied.
Proposition 17, Problem 4
To find both the centripetal force and the resistance of the medium by means of
which a body can revolve in a given spiral, if the law of the velocity is given.
Let the spiral be PQR. The time
will be given from the velocity with
which the body traverses the
minimally small arc PQ, and the force will
be given from the height TQ, which
is as the centripetal force and the
square of the time. Then the
retardation of the body will be given from
the difference RSr of the areas PSQ
and QSR traversed in equal particles
of time, and the resistance and density
of the medium will be found from the
retardation.
Proposition 18, Problem 5
Given the law of the centripetal force, it is required to find in every place the
density of the medium with which a body will describe a given spiral.
The velocity in every place is to be found from the centripetal force; then the density of the medium is to be sought from the retardation of the velocity, as in prop. 17.
I have presented the method of dealing with these problems in book 2, prop. 10 and lem. 2, and I do not wish to detain the reader any longer in complex inquiries of this sort. Some things must now be added on the forces of bodies in their forward motion, and on the density and resistance of the mediums in which the motions hitherto explained and motions related to these are performed.
The density and compression of fluids, and hydrostatics
Definition of
a Fluid
A fluid is any body whose parts yield to any force applied to it and yielding are
moved easily with respect to one another.
Proposition 19, Theorem 14
All the parts of a homogeneous and motionless fluid that is enclosed in any
motionless vessel and is compressed on all sides {apart from considerations of condensation,
gravity, and all centripetal forces) are equally pressed on all sides and remain in
their places without any motion arising from that pressure.
CASE 1. Let a fluid be enclosed in the spherical vessel ABC and be
uniformly compressed on all sides; I say that no part of this fluid will move
as a result of that pressure. For if some one part
D moves, all the parts of this sort, standing on all
sides at the same distance from the center, must
move simultaneously with a similar motion; and
this is so because the pressure on them all is
similar and equal, and every motion is supposed
excluded except that which arises from the pressure.
But they cannot all approach closer to the center
unless the fluid is condensed at the center, contrary to the hypothesis. They
cannot recede farther from it unless the fluid is condensed at the
circumference, also contrary to the hypothesis. They cannot move in any direction
and keep their distance from the center, since by a like reasoning they will
move in the opposite direction, and the same part cannot move in opposite
directions at the same time. Therefore no part of the fluid will move from
its place. Q.E.D.
CASE 2. I say additionally that all the spherical parts of this fluid are equally pressed on all sides. For let EF be a spherical part of the fluid; if this part is not pressed equally on all sides, let the lesser pressure be increased until this part is pressed equally on all sides; then its parts, by case 1, will remain in their places. But before the increase of the pressure they will remain in their places, also by case 1, and by the addition of new pressure they will be moved out of their places, by the definition of a fluid. These two results are contradictory. Therefore it was false to say that the sphere EF was not pressed equally on all sides. Q.E.D.
CASE 3. I say furthermore that there is equal pressure on different spherical parts. For contiguous spherical parts press one another equally in the point of contact, by the third law of motion. But by case 2, they are also pressed on all sides by the same force. Therefore any two noncontiguous spherical parts will be pressed by the same force, since an intermediate spherical part can touch both. Q.E.D.
CASE 4. I say also that all the parts of the fluid are equally pressed on every side. For any two parts can be touched by spherical parts in any points, and there they press those spherical parts equally, by case 3, and in turn are equally pressed by them, by the third law of motion. Q.E.D.
CASE 5. Since, therefore, any part GHI of the fluid is enclosed in the remaining fluid as if in a vessel and is pressed equally on all sides, while its parts press one another equally and are at rest with respect to one another, it is manifest that all the parts of any fluid GHI which is pressed equally on all sides press one another equally and are at rest with respect to one another. Q.E.D.
CASE 6. Therefore, if that fluid is enclosed in a vessel that is not rigid and is not pressed equally on all sides, it will yield to a greater pressure, by the definition of a fluid.
CASE 7. And thus in a rigid vessel a fluid will not sustain a pressure that is greater on one side than on another, but will yield to it, and will do so in an instant of time, since the rigid side of the vessel does not follow the yielding liquid. And by yielding, it will press the opposite side, and thus the pressure will tend on all sides to equality. And since, as soon as the fluid endeavors to recede from the part that is pressed more, it is hindered by the resistance of the vessel on the opposite side, the pressure will be reduced on all sides to equality in an instant of time without local motion; and thereupon the parts of the fluid, by case 5, will press one another equally and will be at rest with respect to one another. Q.E.D.
COROLLARY. Hence the motions of the parts of the fluid with respect to one another cannot be changed by pressure applied to the fluid anywhere on the external surface, except insofar as either the shape of the surface is changed somewhere or all the parts of the fluid, by pressing one another more intensely or more remissly [i.e., by pressing one another more strongly or less strongly], flow among themselves with more or less difficulty.
Proposition 20, Theorem 15
If every part of a fluid that is spherical and homogeneous at equal distances from
the center and rests upon a concentric spherical bottom gravitates toward the center
of the whole, then the bottom will sustain the weight of a cylinder whose base
is equal to the surface of the bottom and whose height is the same as that of the
fluid resting upon it.
Let DHM be the surface of the bottom, and AEI the upper surface of the
fluid. Divide the fluid into equally thick concentric spherical shellsa by
innumerable spherical surfaces BFK, CGL; and
suppose the force of gravity to act only upon
the upper surface of each spherical shell, and
the actions upon equal parts of all the
surfaces to be equal. The highest surface AE is
pressed, therefore, by the simple force of its
own gravity, by which also all the parts of the
highest spherical shell, and the second surface
BFK (by prop. 19), are equally pressed in
accordance with their measure. The second
surface BFK is pressed additionally by the force of its own gravity, which, added
to the previous force, makes the pressure double. The third surface CGL is
acted on by this pressure, in accordance with its measure, and additionally by
the force of its gravity, that is, by a triple pressure. And similarly the fourth
surface is urged by a quadruple pressure, the fifth by a quintuple, and so on.
The pressure by which any one surface is urged is therefore not as the solid
quantity of the fluid lying upon it, but as the number of spherical shells up
to the top of the fluid, and is equal to the gravity of the lowest spherical
shell multiplied by the number of shells; that is, it is equal to the gravity
of a solid whose ultimate ratio to the cylinder specified above will become
that of equality—provided that the number of shells is increased and their
thickness decreased indefinitely, in such a way that the action of gravity is
made continuous from the lowest surface to the highest. The lowest surface
therefore sustains the weight of the cylinder specified above. Q.E.D. And
by a similar argument this proposition is evident when the gravity decreases
in any assigned ratio of the distance from the center, and also when the fluid
is rarer upward and denser downward. Q.E.D.
COROLLARY 1. Therefore the bottom is not pressed by the whole weight of the incumbent fluid, but sustains only that part of the weight which is described in this proposition, the rest of the weight being sustained by the vaulted shape of the fluid.
COROLLARY 2. At equal distances from the center, moreover, the quantity of pressure is always the same, whether the pressed surface is parallel to the horizon or perpendicular or oblique, or whether the fluid—continued upward from the pressed surface—rises perpendicularly along a straight line or snakes obliquely through twisted cavities and channels, regular or extremely irregular, wide or very narrow. That the pressure is not at all changed by these circumstances is gathered by applying the proof of this theorem to the various cases of fluids.
COROLLARY 3. By the same proof it is also gathered (by prop. 19) that the parts of a heavy fluid acquire no motion with respect to one another as a result of the pressure of the incumbent weight, provided that the motion arising from condensation is excluded.
COROLLARY 4. And therefore, if another body, in which there is no condensation, of the same specific gravity is submerged in this fluid, it will acquire no motion as a result of the pressure of the incumbent weight; it will not descend, it will not ascend, and it will not be compelled to change its shape. If it is spherical, it will remain spherical despite the pressure; if it is square, it will remain square; and it will do so whether it is soft or very fluid, whether it floats freely in the fluid or lies on the bottom. For any internal part of a fluid is in the same situation as a submerged body, and the case is the same for all submerged bodies of the same size, shape, and specific gravity. If a submerged body, while keeping its weight, were to liquefy and assume the form of a fluid, then, if it were formerly ascending or descending or assuming a new shape as a result of pressure, it would also now ascend or descend or be compelled to assume a new shape, and would do so because its gravity and the other causes of motions remain fixed. But (by prop. 19, case 5) this body would now be at rest and would maintain its shape. Hence, this would also be the case under the earlier conditions.
COROLLARY 5. Accordingly, a body that is of a greater specific gravity than a fluid contiguous to it will sink, and a body that is of a lesser specific gravity will ascend, and will acquire as much motion and change of shape as that excess or deficiency of gravity can bring about. For that excess or deficiency acts like an impulse by which the body, otherwise in equilibrium with the parts of the fluid, is urged; and it can be compared with the excess or deficiency of weight in either of the scales of a balance.
COROLLARY 6. The gravity of bodies in fluids is therefore twofold: the one, true and absolute; the other, apparent, common, and relative. Absolute gravity is the whole force with which a body tends downward; relative or common gravity is the excess of gravity with which the body tends downward more than the surrounding fluid. By absolute gravity the parts of all fluids and bodies gravitate in their places, and thus the sum of the individual weights is the weight of the whole. For every whole is heavy, as can be tested in vessels full of liquids, and the weight of the whole is equal to the sum of the weights of all the parts, and thus is composed of them. By relative gravity bodies do not gravitate in their places; that is, compared with one another, one is not heavier than another, but each one opposes the endeavors of the others to descend, and they remain in their places just as if they had no gravity. Whatever is in the air and does not gravitate more than the air is not commonly considered to be heavy. Things that do gravitate more are commonly considered to be heavy, inasmuch as they are not sustained by the weight of the air. Weight as commonly conceived is nothing other than the excess of the true weight over the weight of the air. Bodies are commonly called light which are less heavy than the surrounding air and, by yielding to that air, which gravitates more, move upward. They are, however, only comparatively light and not truly so, since they descend in a vacuum. Similarly, bodies in water that descend or ascend because of their greater or smaller gravity are comparatively and apparently heavy or light, and their comparative and apparent heaviness or lightness is the excess or deficiency by which their true gravity either exceeds the gravity of the water or is exceeded by it. And bodies that neither descend by gravitating more nor ascend by yielding to water which gravitates more—even though they increase the weight of the whole by their own true weights—nevertheless, comparatively and as commonly understood, do not gravitate in water. For the demonstration of all these cases is similar.
COROLLARY 7. What has been demonstrated concerning gravity is valid for any other centripetal forces.
COROLLARY 8. Accordingly, if the medium in which some body moves is urged either by its own gravity or by any other centripetal force, and the body is urged more strongly by the same force, then the difference between the forces is that motive force which we have considered to be the centripetal force in the preceding propositions. But if the body is urged more lightly by that force, the difference between the forces should be considered a centrifugal force.
COROLLARY 9. Since fluids, moreover, do not change the external shapes of enclosed bodies that they press upon, it is evident in addition (by prop. 19, corol.) that fluids will not change the situation of the internal parts with respect to one another; and accordingly, if animals are immersed, and if all sensation arises from the motion of the parts, fluids will neither harm these immersed bodies nor excite any sensation, except insofar as these bodies can be condensed by compression. And the case is the same for any system of bodies that is surrounded by a compressing fluid. All the parts of the system will be moved with the same motions as if they were in a vacuum and retained only their relative gravity, except insofar as the fluid either resists their motions somewhat or is needed to make them cohere by compression.
Proposition 21, Theorem 16
Let the density of a certain fluid be proportional to the compression, and let its parts
be drawn downward by a centripetal force inversely proportional to their distances
from the center; I say that if the distances are taken continually proportional, the
densities of the fluid at these distances will also be continually proportional.
Let ATV designate the spherical bottom on which
the fluid lies, S the center, and SA, SB, SC, SD, SE,
SF, . . . the continually proportional distances. Erect
perpendiculars AH, BI, CK, DL, EM, FN, . . . , which
are as the densities of the medium in places A, B, C,
D, E, F; then the specific gravities in those places will
be as 
, . . . , or—which is the same—as
, . . . . Suppose first that these gravities
continue uniformly, the first from A to B, the second
from B to C, the third from C to D, . . . , the decrements thus occurring
by degrees at points B, C, D, . . . . Then these specific gravities multiplied
by the heights AB, BC, CD, . . . will give the pressures AH, BI, CK, . . . , by
which the bottom ATV (according to prop. 20) is pressed. The particle A
therefore sustains all the pressures AH, BI, CK, DL, going on indefinitely;
and the particle B, all the pressures except the first, AH; and the particle
C, all except the first two, AH and BI; and so on. And thus the density
AH of the first particle A is to the density BI of the second particle B as
the sum of all the AH + BI + CK + DL indefinitely, to the sum of all the
BI + CK + DL . . . . And the density BI of the second particle B is to the
density CK of the third particle C as the sum of all the BI + CK + DL . . . to
the sum of all the CK + DL . . . . Those sums are therefore proportional
to their differences AH, BI, CK, . . . , and thus are continually proportional
(by book 2, lem. 1); and accordingly the differences AH, BI, CK, . . . ,
proportional to those sums, are also continually proportional. Therefore, since
the densities in places A, B, C, . . . are as AH, BI, CK, . . . , these also will
be continually proportional. Proceed now by jumps, and from the equality
of the ratios [or ex aequo], at the continually proportional distances SA, SC,
SE, the densities AH, CK, EM will be continually proportional. And by the
same argument, at any continually proportional distances SA, SD, SG, the
densities AH, DL, GO will be continually proportional. Now let points A,
B, C, D, E, . . . come together so that the progression of the specific gravities
is made continual from the bottom A to the top of the fluid; and at any
continually proportional distances SA, SD, SG, the densities AH, DL, GO,
being always continually proportional, will still remain continually
proportional now. Q.E.D.
COROLLARY. Hence, if the density of a
fluid is given in two places, say A and E,
its density in any other place Q can be
determined. With center S and rectangular
asymptotes SQ and SX describe a hyperbola cutting
perpendiculars AH, EM, and QT in a, e,
and q, and also perpendiculars HX, MY, and
TZ, dropped to asymptote SX, in h, m, and
t. Make the area YmtZ be to the given area
YmhX as the given area EeqQ is to the given
area EeaA; and the line Zt produced will cut off the line QT proportional to
the density. For if lines SA, SE, and SQ are continually proportional, areas
EeqQ and EeaA will be equal, and hence the areas proportional to these,
YmtZ an XhmY, will also be equal, and lines SX, SY, and SZ—that is, AH,
EM, and QT—will be continually proportional, as they ought to be. And
if lines SA, SE, and SQ obtain any other order in the series of continually
proportional quantities, lines AH, EM, and QT, because the hyperbolic areas
are proportional, will obtain the same order in another series of continually
proportional quantities.
Proposition 22, Theorem 17
Let the density of a certain fluid be proportional to the compression, and let its
parts be drawn downward by a gravity inversely proportional to the squares of
their distances from the center; I say that if the distances are taken in a harmonic
progression, the densities of the fluid at these distances will be in a geometric
progression.
Let S designate the center, and SA, SB, SC, SD, and SE the distances in a
geometric progression. Erect perpendiculars AH, BI, CK, . . . , which are as
the densities of the fluid in places A, B, C, D, E, . . . ; then the specific
gravities in those places will be 
, . . . . Imagine these specific
gravities to be uniformly continued, the first from A to B, the second from B to C,
the third from C to D, . . . . Then these, multiplied by the heights AB, BC,
CD, DE, . . .—or, which is the same, by the distances SA, SB, SC, . . . ,
proportional to those heights—will yield 
, . . . , which represent
the pressures. Therefore, since the densities are as the sums of these pressures,
the differences (AH − BI, BI − CK, . . .) of the densities will be as the
differences 
of the sums. With center S and asymptotes SA
and Sx describe any hyperbola that cuts the perpendiculars AH, BI, CK, . . . in
a, b, c, . . . and also cuts in h, i, and k the perpendiculars Ht, Iu, and
Kw, dropped to asymptote Sx; then the differences tu, uw, . . . between the
densities will be as 
, . . . . And the rectangles
tu × th, uw × ui, . . . ,
or tp, uq, . . . , will be as
, . . . , that is, as
Aa, Bb, . . . . For, from
the nature of the hyperbola, SA is to AH or St as th to Aa,
and thus 
is equal to Aa. And by a similar argument,
is
equal to Bb, . . . . Moreover,
Aa, Bb, Cc, . . . are continually proportional,
and therefore proportional to their differences Aa − Bb, Bb − Cc, . . . ; and
thus the rectangles tp, uq, . . . are proportional to these differences, and also
the sums of the rectangles tp + uq or tp + uq + wr are proportional to the
sums of the differences Aa − Cc or Aa − Dd. Let there be as many terms of
this sort as you wish; then the sum of all the differences, say Aa − Ff, will be
proportional to the sum of all the rectangles, say zthn. Increase the number
of terms and decrease the distances of points A, B, C, . . . , indefinitely; then
these rectangles will come out equal to the hyperbolic area zthn, and thus
the difference Aa − Ff is proportional to this area. Now take any distances,
say SA, SD, SF, in a harmonic progression, and the differences Aa − Dd
and Dd − Ff will be equal; and therefore the areas thlx and xlnz which are
proportional to these differences will be equal to each other, and the
densities St, Sx, and Sz (that is, AH, DL, and FN) will be continually
proportional. Q.E.D.
COROLLARY. Hence, if any two densities of a fluid are given, say AH and BI, the area thiu corresponding to their difference tu will be given; and accordingly the density FN at any height SF will be found by taking the area thnz to be to that given area thiu as the difference Aa − Ff is to the difference Aa − Bb.
Scholium
Similarly, it can be proved that if the gravity of the particles of a fluid is
decreased as the cubes of the distances from the center, and if the reciprocals
of the squares of the distances SA, SB, SC, . . . namely 
are taken in an arithmetic progression, then the densities AH, BI,
CK, . . . will be in a geometric progression. And if the gravity is decreased as the
fourth power of the distances, and if the reciprocals of the cubes of the
distances say, 
are taken in an arithmetic progression,
the densities AH, BI, CK, . . . will be in a geometric progression. And so on
indefinitely. Again, if the gravity of the particles of a fluid is the same at all
distances, and if the distances are in an arithmetic progression, the densities
will be in a geometric progression, as the distinguished gentleman Edmond
Halley has found. If the gravity is as the distance, and if the squares of the
distances are in an arithmetic progression, the densities will be in a geometric
progression. And so on indefinitely.
These things are so when the density of a fluid condensed by compression is as the force of the compression or, which is the same, when the space occupied by the fluid is inversely as this force. Other laws of condensation can be imagined, as, for example, that the cube of the compressing force is as the fourth power of the density, or that the force ratio cubed is the same as the density ratio to the fourth power. In this case, if the gravity is inversely as the square of the distance from the center, the density will be inversely as the cube of the distance. Imagine that the cube of the compressing force is as the fifth power of the density; then, if the gravity is inversely as the square of the distance, the density will be inversely as the 3/2 power of the distance. Imagine that the compressing force is as the square of the density, and that the gravity is inversely as the square of the distance; then the density will be inversely as the distance. It would be tedious to cover all cases. But it is established by experiments that the density of air is either exactly or at least very nearly as the compressing force; and therefore the density of the air in the earth’s atmosphere is as the weight of the whole incumbent air, that is, as the height of the mercury in a barometer.
Suppose a fluid to be enclosed in the cubic space ACE, and then by
compression to be reduced into the smaller cubic space ace; then the distances
between particles maintaining similar
positions with respect to one another in
the two spaces will be as the edges AB
and ab of the cubes; and the densities
of the mediums will be inversely as the
containing spaces AB3 and ab3. On the
plane side ABCD of the larger cube take the square DP equal to the plane
side of the smaller cube db; then (by hypothesis) the pressure by which the
square DP urges the enclosed fluid will be to the pressure by which the
square db urges the enclosed fluid as the densities of the medium to each
other, that is, as ab3 to AB3. But the pressure by which the square DB urges
the enclosed fluid is to the pressure by which the square DP urges that same
fluid as the square DB to the square DP, that is, as AB2 to ab2. Therefore,
from the equality of the ratios [or ex aequo] the pressure by which the square
DB urges the fluid is to the pressure by which the square db urges the fluid
as ab to AB. Divide the fluid into two parts by planes FGH and fgh drawn
through the middles of the cubes; then these parts will press each other with
the same forces with which they are pressed by planes AC and ac, that is,
in the proportion of ab to AB; and thus the centrifugal forces [or forces
of repulsion] by which these pressures are sustained are in the same ratio.
Because in both cubes the number of particles is the same and their situation
similar, the forces that all the particles along planes FGH and fgh exert
upon all the others are as the forces that each individual particle exerts upon
every other particle. Therefore the forces that each particle exerts upon every
other particle along the plane FGH in the larger cube are to the forces that
individual particles exert on the particle next to them along the plane fgh in
the smaller cube as ab to AB, that is, inversely as the distances between the
particles are to one another. Q.E.D.
And conversely, if the forces of the individual particles are inversely as the distances, that is, inversely as the edges AB and ab of the cubes, the sums of the forces will be in the same ratio, and the pressures of the sides DB and db will be as the sums of the forces; and the pressure of the square DP will be to the pressure of the side DB as ab2 to AB2. And from the equality of the ratios [or ex aequo] the pressure of the square DP will be to the pressure of the side db as ab3 to AB3; that is, the one force of compression will be to the other force of compression as the one density to the other density. Q.E.D.
Scholium
By a similar argument, if the centrifugal
forces [or forces of repulsion] of the
particles are inversely as the squares of the distances between the centers, the
cubes of the compressing forces will be as the fourth powers of the densities.
If the centrifugal forces are inversely as the third or fourth powers of the
distances, the cubes of the compressing forces will be as the fifth or sixth
powers of the densities. And universally, if D is the distance, and E the
density of the compressed fluid, and if the centrifugal forces are inversely
as any power of the distance Dn, whose index is the number n, then the
compressing forces will be as the cube roots of the powers En+2, whose index
is the number n + 2; and conversely. In all of this, it is supposed that the
centrifugal forces of particles are terminated in the particles which are next
to them or do not extend far beyond them. We have an example of this in
magnetic bodies. Their attractive virtue [or power] is almost terminated in
bodies of their own kind which are next to them. The virtue of a magnet
is lessened by an interposed plate of iron and is almost terminated in the
plate. For bodies farther away are drawn not so much by the magnet as
by the plate. In the same way, if particles repel other particles of their own
kind that are next to them but do not exert any virtue upon more remote
particles,b particles
of this sort are the ones of which the fluids treated in this
proposition will be composed. But if the virtue of each particle is propagated
indefinitely, a greater force will be necessary for the equal condensation of a
greater quantity of the fluid.c Whether
elastic fluids consist of particles that
repel one another is, however, a question for physics. We have mathematically
demonstrated a property of fluids consisting of particles of this sort so as
to provide natural philosophers with the means with which to treat that
question.
Concerning the motion of asimple pendulumsa and the resistance to them
Proposition 24, Theorem 19
In simple pendulums whose centers of oscillation are equally distant from the
center of suspension, the quantities of matter are in a ratio compounded of the
ratio of the weights and the squared ratio of the times of oscillation in a vacuum.
For the velocity that a given force can generate in a given time in a given quantity of matter is as the force and the time directly and the matter inversely. The greater the force, or the greater the time, or the less the matter, the greater the velocity that will be generated. This is manifest from the second law of motion. Now if the pendulums are of the same length, the motive forces in places equally distant from the perpendicular are as the weights; and thus if two oscillating bodies describe equal arcs and if the arcs are divided into equal parts, then, since the times in which the bodies describe single corresponding parts of the arcs are as the times of the whole oscillations, the velocities in corresponding parts of the oscillations will be to one another as the motive forces and the whole times of the oscillations directly and the quantities of matter inversely; and thus the quantities of matter will be as the forces and the times of the oscillations directly and the velocities inversely. But the velocities are inversely as the times, and thus the times are directly, and the velocities are inversely, as the squares of the times, and therefore the quantities of matter are as the motive forces and the squares of the times, that is, as the weights and the squares of the times. Q.E.D.
COROLLARY 1. And thus if the times are equal, the quantities of matter in the bodies will be as their weights.
COROLLARY 2. If the weights are equal, the quantities of matter will be as the squares of the times.
COROLLARY 3. If the quantities of matter are equal, the weights will be inversely as the squares of the times.
COROLLARY 4. Hence, since the squares of the times, other things being equal, are as the lengths of the pendulums, the weights will be as the lengths of the pendulums if both the times and the quantities of matter are equal.
COROLLARY 5. And universally, the quantity of matter in a bob of a simple pendulum is as the weight and the square of the time directly and the length of the pendulum inversely.
COROLLARY 6. But in a nonresisting medium also, the quantity of matter in the bob of a simple pendulum is as the relative weight and the square of the time directly and the length of the pendulum inversely. For the relative weight is the motive force of a body in any heavy medium, as I have explained above, and thus fulfills the same function in such a nonresisting medium as absolute weight does in a vacuum.
COROLLARY 7. And hence a method is apparent both for comparing bodies with one another with respect to the quantity of matter in each, and for comparing the weights of one and the same body in different places in order to find out the variation in its gravity. And by making experiments of the greatest possible accuracy, I have always found that the quantity of matter in individual bodies is proportional to the weight.
Proposition 25, Theorem 20
The bobs of simple pendulums that are resisted in any medium in the ratio of
the moments of time, and those that move in a nonresisting medium of the same
specific gravity, perform oscillations in a cycloid in the same time and describe
proportional parts of arcs in the same time.
Let AB be the arc of a cycloid, which body D describes by oscillating in a nonresisting medium in any time. Bisect the arc AB in C so that C is its lowest point; then the accelerative force by which the body is urged in any place D or d or E will be as the length of arc CD or Cd or CE. Represent that force by the appropriate arc [CD or Cd or CE], and since the resistance is as the moment of time, and thus is given, represent it by a given part CO of the arc of the cycloid, taking arc Od in the ratio to arc CD that arc OB has to arc CB; then the force by which the body at d is urged in the resisting medium (since it is the excess of the force Cd over the resistance CO) will be represented by arc Od, and thus will be to the force by which body D is urged in a nonresisting medium in place D as arc Od to arc CD, and therefore also in place B as arc OB to arc CB. Accordingly, if two bodies D and d leave place B and are urged by these forces, then, since the forces at the beginning are as arcs CB and OB, the first velocities and the arcs first described will be in the same ratio. Let those arcs be DO and Bd; then the remaining arcs CD and Od will be in the same ratio. Accordingly the forces, being proportional to CD and Od, will remain in the same ratio as at the beginning, and therefore the bodies will proceed simultaneously to describe arcs in the same ratio. Therefore the forces and the velocities and the remaining arcs CD and Od will always be as the whole arcs CB and OB, and therefore those remaining arcs will be described simultaneously. Therefore the two bodies D and d will arrive simultaneously at places C and O, the one in the nonresisting medium at place C, and the one in the resisting medium at place O. And since the velocities in C and O are as arcs CB and OB, the arcs that the bodies describe in the same time by going on further will be in the same ratio. Let those arcs be CE and Oe. The force by which body D in the nonresisting medium is retarded in E is as CE, and the force by which body d in the resisting medium is retarded in e is as the sum of the force Ce and the resistance CO, that is, as Oe; and thus the forces by which the bodies are retarded are as arcs CB and OB, which are proportional to arcs CE and Oe; and accordingly the velocities, which are retarded in that given ratio, remain in that same given ratio. The velocities, therefore, and the arcs described with those velocities are always to one another in the given ratio of arcs CB and OB; and therefore, if the whole arcs AB and aB are taken in the same ratio, bodies D and d will describe these arcs together and will simultaneously lose all motion in places A and a. The whole oscillations are therefore isochronal, and any parts of the arcs, BD and Bd or BE and Be, that are described in the same time are proportional to the whole arcs BA and Ba. Q.E.D.
COROLLARY. Therefore the swiftest motion in the resisting medium does not occur at the lowest point C, but is found in that point O by which aB, the whole arc described, is bisected. And the body, proceeding from that point to a, is retarded at the same rate by which it was previously accelerated in its descent from B to O.
Proposition 26, Theorem 21
If simple pendulums are resisted in the ratio of the velocities, their oscillations in
a cycloid are isochronal.
For if two oscillating bodies equally distant from the centers of suspension describe unequal arcs and if the velocities in corresponding parts of the arcs are to one another as the whole arcs, then the resistances, being proportional to the velocities, will also be to one another as the same arcs. Accordingly, if these resistances are taken away from (or added to) the motive forces arising from gravity, which are as the same arcs, the differences (or sums) will be to one another in the same ratio of the arcs; and since the increments or decrements of the velocities are as these differences or sums, the velocities will always be as the whole arcs. Therefore, if in some one case the velocities are as the whole arcs, they will always remain in that ratio. But in the beginning of the motion, when the bodies begin to descend and to describe those arcs, the forces—since they are proportional to the arcs—will generate velocities proportional to the arcs. Therefore the velocities will always be as the whole arcs to be described, and therefore those arcs will be described in the same time. Q.E.D.
Proposition 27, Theorem 22
If simple pendulums are resisted as the squares of the velocities, the differences
between the times of the oscillations in a resisting medium and the times of the
oscillations in a nonresisting medium of the same specific gravity will be very
nearly proportional to the arcs described during the oscillations.
For let the unequal arcs A and B be described by equal pendulums in a resisting medium; then the resistance to the body in arc A will be to the resistance to the body in the corresponding part of arc B very nearly in the squared ratio of the velocities, that is, as A2 to B2. If the resistance in arc B were to the resistance in arc A as AB to A2, the times in arcs A and B would be equal, by the previous proposition. And thus the resistance A2 in arc A, or AB in arc B, produces an excess of time in arc A over the time in a nonresisting medium; and the resistance B2 produces an excess of time in arc B over the time in a nonresisting medium. And those excesses are very nearly as the forces AB and B2 that produce them, that is, as arcs A and B. Q.E.D.
COROLLARY 1. Hence from the times of the oscillations made in a resisting medium in unequal arcs, the times of the oscillations in a nonresisting medium of the same specific gravity can be found. For the difference between these times will be to the excess of the time in the smaller arc over the time in the nonresisting medium as the difference between the arcs is to the smaller arc.
COROLLARY 2. Shorter oscillations are more isochronal, and the shortest are performed in very nearly the same times as in a nonresisting medium. In fact, the times of those that are made in greater arcs are a little greater, because the resistance in the descent of the body (by which the time is prolonged) is greater in proportion to the length described in the descent than the resistance in the subsequent ascent (by which the time is shortened). But also the time of short as well as long oscillations seems to be somewhat prolonged by the motion of the medium. For retarded bodies are resisted a little less in proportion to the velocity, and accelerated bodies a little more, than those that progress uniformly; and this is so because the medium, going in the same direction as the bodies with the motion that it has received from them, is in the first case more agitated, in the second less, and accordingly concurs to a greater or to a less degree with the moving bodies. The medium therefore resists the pendulums more in the descent, and less in the ascent, than in proportion to the velocity, and the time is prolonged as a result of both causes.
Proposition 28, Theorem 23
If a simple pendulum oscillating in a cycloid is resisted in the ratio of the moments
of time, its resistance will be to the force of gravity as the excess of the arc described
in the whole descent over the arc described in the subsequent ascent is to twice
the length of the pendulum.
Let BC designate the arc described in the descent, Ca the arc described in the ascent, and Aa the difference between the arcs; then, with the same constructions and proofs as in prop. 25, the force by which the oscillating body is urged in any place D will be to the force of resistance as arc CD to arc CO, which is half of that difference Aa. And thus the force by which the oscillating body is urged in the beginning (or highest point) of the cycloid—that is, the force of gravity—will be to the resistance as the arc of the cycloid between that highest point and the lowest point C is to arc CO, that is (if the arcs are doubled), as the arc of the whole cycloid, or twice the length of the pendulum, is to arc Aa. Q.E.D.
Proposition 29, Problem 6
Supposing that a body oscillating in a cycloid is resisted as the square of the
velocity, it is required to find the resistance in each of the individual places.
Let Ba be the arc described in an entire oscillation, and let C be the
lowest point of the cycloid, and let CZ be half of the arc of the whole cycloid
and be equal to the length of the pendulum; and let it be required to find the
resistance to the body in any place D. Cut the indefinite straight line OQ in
points O, S, P, and Q, with the conditions that—if perpendiculars OK, ST,
PI, and QE are erected; and if, with center O and asymptotes OK and OQ,
hyperbola TIGE is described so as to cut perpendiculars ST, PI, and QE
in T, I, and E; and if, through point I, KF is drawn parallel to asymptote
OQ and meeting asymptote OK in K and perpendiculars ST and QE in L
and F—the hyperbolic area PIEQ is to the hyperbolic area PITS as the arc
BC described during the body’s descent is to the arc Ca described during the
ascent, and area IEF is to area ILT as OQ to OS. Then let perpendicular
MN cut off the hyperbolic area PINM, which is to the hyperbolic area PIEQ
as arc CZ is to the arc BC described in the descent. And if perpendicular
RG cuts off the hyperbolic area PIGR, which is to area PIEQ as any arc
CD is to the arc BC described in the whole descent, then the resistance in
place D will be to the force of gravity as the area 
IEF − IGH to the
area PINM.
For, since the forces which arise from gravity and by which the body is
urged in places Z, B, D, and a are as arcs CZ, CB, CD, and Ca, and those arcs
are as areas PINM, PIEQ, PIGR, and PITS, let the arcs and the forces be
represented by these areas respectively. In addition, let Dd be the minimally
small space described by the body while descending, and represent it by the
minimally small area RGgr comprehended between the parallels RG and rg;
and produce rg to h, so that GMhg and RGgr are decrements of areas IGH
and PIGR made in the same time. And the increment GHhg − 
IEF,
or Rr × HG − IEF, of area IEF − IGH will be to the decrement
RGgr, or Rr × RG, of area PIGR as HG − 
, is to RG, and thus as
OR × HG − IEF is to OR × GR or OP × PI, that is (because OR × HG,
or OR × HR − OR × GR, ORHK − OPIK, PIHR, and PIGR + IGH
are equal), as PIGR + IGH − IEF is to OPIK. Therefore, if area
IEF − IGH is called Y, and if the decrement RGgr of area PIGR is
given, then the increment of area Y will be as PIGR − Y.
But if V designates the force arising from gravity, by which the body is urged in D and which is proportional to the arc CD to be described, and if R represents the resistance, then V − R will be the whole force by which the body is urged in D. The increment of the velocity is therefore jointly as V − R and that particle of time in which the increment is made. But furthermore the velocity itself is directly as the increment of the space described in the same time and inversely as that same particle of time. Hence, since the resistance, by hypothesis, is as the square of the velocity, the increment of the resistance (by lem. 2) will be as the velocity and the increment of the velocity jointly, that is, as the moment of the space and V − R jointly; and thus, if the moment of the space is given, as V − R; that is, as PIGR − Z, if for the force V there is written PIGR (which represents it), and if the resistance R is represented by some other area Z.
Therefore, as area PIGR decreases uniformly by the subtraction of the given moments, area Y increases in the ratio of PIGR − Y, and area Z increases in the ratio of PIGR − Z. And therefore, if areas Y and Z begin simultaneously and are equal at the beginning, they will continue to be equal by the addition of equal moments and, thereafter decreasing by moments that are likewise equal, will vanish simultaneously. And conversely, if they begin simultaneously and vanish simultaneously, they will have equal moments and will always be equal; and this is so because, if the resistance Z is increased, the velocity will be decreased along with that arc Ca which is described in the body’s ascent, and as the point in which there is a cessation of all motion and resistance approaches closer to point C, the resistance will vanish more quickly than area Y. And the contrary will happen when the resistance is decreased.
Now area Z begins and ends where the resistance is nil, that is, in the
beginning of the motion where arc CD is equal to arc CB and the straight
line RG falls upon the straight line QE, and in the end of the motion where
arc CD is equal to arc Ca and RG falls upon the straight line ST. And area
Y or IEF − IGH begins and ends where the resistance is nil, and thus
where IEF and IGH are equal; that is (by construction), where the
straight line RG falls successively upon the straight lines QE and ST. And
accordingly those areas begin simultaneously and vanish simultaneously and
therefore are always equal. Therefore area IEF − IGH is equal to area
Z (which represents the resistance) and therefore is to area PINM (which
represents the gravity) as the resistance is to the gravity. Q.E.D.
COROLLARY 1. The resistance in the lowest place C is, therefore, to the
force of gravity as area 
IEF is to area PINM.
COROLLARY 2. And this resistance becomes greatest when area PIHR is to area IEF as OR is to OQ. For in that case its moment (namely, PIGR − Y) comes out nil.
COROLLARY 3. Hence also the velocity in each of the individual places can be known, inasmuch as it is as the square root of the resistance, and at the very beginning of the motion is equal to the velocity of the body oscillating without any resistance in the same cycloid.
But because the computation by which the resistance and velocity are to be found by this proposition is difficult, it seemed appropriate to add the following proposition.a
Proposition 30, Theorem 24
If the straight line aB is equal to a cycloidal arc that is described by an oscillating
body, and if at each of its individual points D perpendiculars DK are erected
that are to the length of the pendulum as the resistance encountered by the body
in corresponding points of the arc is to the force of gravity, then I say that the
difference between the arc described in the whole descent and the arc described in
the whole subsequent ascent multiplied by half the sum of those same arcs will be
equal to the area BKa occupied by all the perpendiculars DK.
Represent the cycloidal arc described in an entire oscillation by the straight line aB equal to it, and represent the arc that would be described in a vacuum by the length AB. Bisect AB in C, and point C will represent the lowest point of the cycloid, and CD will be as the force arising from gravity (by which the body at D is urged along the tangent of the cycloid) and will have the ratio to the length of the pendulum that the force at D has to the force of gravity. Therefore represent that force by the length CD, and the force of gravity by the length of the pendulum; then, if DK is taken in DE in the ratio to the length of the pendulum that the resistance has to the gravity, DK will represent the resistance. With center C and radius CA or CB construct semicircle BEeA. And let the body describe space Dd in a minimally small time; then, when perpendiculars DE and de have been erected, meeting the circumference in E and e, they will be as the velocities that the body in a vacuum would acquire in places D and d by descending from point B. This is evident by book 1, prop. 52. Therefore represent these velocities by perpendiculars DE and de, and let DF be the velocity that the body acquires in D by falling from B in the resisting medium. And if with center C and radius CF circle Ff M is described, meeting the straight lines de and AB in f and M, then M will be the place to which the body would then ascend if there were no further resistance, and df will be the velocity that it would acquire in d. Hence also, if Fg designates the moment of velocity that body D, in describing the minimally small space Dd, loses as a result of the resistance of the medium, and if CN is taken equal to Cg, then N will be the place to which the body would then ascend if there were no further resistance, and MN will be the decrement of the ascent arising from the loss of that velocity. Drop perpendicular Fm to df, and the decrement Fg (generated by the resistance DK) of the velocity DF will be to the increment fm (generated by the force CD) of that same velocity as the generating force DK is to the generating force CD. Furthermore, because triangles Fmf, Fhg, and FDC are similar, fm is to Fm or Dd as CD is to DF and from the equality of the ratios [or ex aequo] Fg is to Dd as DK is to DF. aLikewise Fh is to Fg as DF to CF, and from the equality of the ratios in inordinate proportion [or ex aequo perturbate] Fh or MN is to Dd as DK to CF or CM; and thus the sum of all the MN × CM will be equal to the sum of all the Dd × DK. Suppose that a rectangular ordinate is always erected at the moving point M, equal to the indeterminate CM, which in its continual motion is multiplied by the whole length Aa; then the quadrilateral described as a result of that motion—or the rectangle equal to it, Aa × ½aB—will become equal to the sum of all the MN × CM, and thus equal to the sum of all the Dd × DK, that is, equal to area BKVTa. Q.E.D.a
COROLLARY. Hence from the law of the resistance and the difference Aa of arcs Ca and CB, the proportion of the resistance to the gravity can be determined very nearly.
For if the resistance DK is uniform, the figure BKTa will be equal to the rectangle of Ba and DK; and hence the rectangle of ½Ba and Aa will be equal to the rectangle of Ba and DK, and DK will be equal to ½Aa. Therefore, since DK represents the resistance, and the length of the pendulum represents the gravity, the resistance will be to the gravity as ½Aa is to the length of the pendulum, exactly as was proved in prop. 28.
If the resistance is as the velocity, the figure BKTa will be very nearly an ellipse. For if a body in a nonresisting medium were to describe the length BA in a whole oscillation, the velocity in any place D would be as the ordinate DE of a circle described with diameter AB. Accordingly, since Ba in the resisting medium, and BA in a nonresisting medium, are described in roughly equal times, and the velocities in the individual points of Ba are thus very nearly to the velocities in the corresponding points of the length BA as Ba is to BA, the velocity in point D in the resisting medium will be as the ordinate of a circle or ellipse described upon diameter Ba; and thus the figure BKVTa will be very nearly an ellipse. Since the resistance is supposed proportional to the velocity, let OV represent the resistance in the midpoint O; then ellipse BRVSa, described with center O and semiaxes OB and OV, will be very nearly equal to the figure BKVTa and the rectangle equal to it, Aa × BO. Aa × BO is therefore to OV ×BO as the area of this ellipse is to OV × BO; that is, Aa is to OV as the area of the semicircle is to the square of the radius, or as 11 to 7, roughly; and therefore 7/11Aa is to the length of the pendulum as the resistance of the oscillating body in O is to its gravity.
But if the resistance DK is as the square of the velocity, the figure BKVTa will be almost a parabola having vertex V and axis OV, and thus will be very nearly equal to the rectangle contained by ⅔Ba and OV. The rectangle contained by ½Ba and Aa is therefore equal to the rectangle contained by ⅔Ba and OV, and thus OV is equal to ¾Aa; and therefore the resistance on the oscillating body in O is to its gravity as ¾Aa is to the length of the pendulum.
And I judge that these conclusions are more than accurate enough for practical purposes. For, since the ellipse or parabola BRVSa and the figure BKVTa have the same midpoint V, if it is greater than that figure on either side BRV or VSa, it will be smaller than it on the other side, and thus will be very nearly equal to it.
Proposition 31, Theorem 25
If the resistance encountered by an oscillating body in each of the proportional
parts of the arcs described is increased or decreased in a given ratio, the difference
between the arc described in the descent and the arc described in the subsequent
ascent will be increased or decreased in the same ratio.
For that difference arises from the retardation of the pendulum by the resistance of the medium, and thus is as the whole retardation and the retarding resistance, which is proportional to it. In the previous proposition the rectangle contained under the straight line ½aB and the difference Aa of arcs CB and Ca was equal to area BKTa. And that area, if the length aB remains the same, is increased or decreased in the ratio of the ordinates DK, that is, in the ratio of the resistance, and thus is as the length aB and the resistance jointly. And accordingly the rectangle contained by Aa and ½aB is as aB and the resistance jointly, and therefore Aa is as the resistance. Q.E.D.
COROLLARY 1. Hence, if the resistance is as the velocity, the difference of the arcs in the same medium will be as the whole arc described; and conversely.
COROLLARY 2. If the resistance is in the squared ratio of the velocity, that difference will be in the squared ratio of the whole arc; and conversely.
COROLLARY 3. And universally, if the resistance is in the cubed or any other ratio of the velocity, the difference will be in the same ratio of the whole arc; and conversely.
COROLLARY 4. And if the resistance is partly in the simple ratio of the velocity and partly in the squared ratio of the velocity, the difference will be partly in the simple ratio of the whole arc and partly in the squared ratio of it; and conversely. The law and ratio of the resistance in relation to the velocity will be the same as the law and ratio of that difference of the arcs in relation to the length of the arc itself.
COROLLARY 5. And thus if a pendulum successively describes unequal arcs and there can be found the ratio of the increment and decrement of this difference [i.e., the difference of the arcs] in relation to the length of the arc described, then there will also be had the ratio of the increment and decrement of the resistance in relation to a greater or smaller velocity.
General Scholiuma
From these propositions it is possible to find the resistance of any mediums by
means of pendulums oscillating in those mediums. In fact, I have investigated
the resistance of air by the following experiments. I suspended a wooden ball
by a fine thread from a sufficiently firm hook in such a way that the distance
between the hook and the center of oscillation of the ball was 10½ feet; the
ball weighed 577/22 ounces avoirdupois and had a diameter of 67/8 London
inches. I marked a point on the thread 10 feet and 1 inch distant from
the center of suspension; and at a right angle at that point I placed a ruler
divided into inches, by means of which I might note the lengths of the arcs
described by the pendulum. Then I counted the oscillations during which the
ball would lose an eighth of its motion. When the pendulum was drawn back
from the perpendicular to a distance of 2 inches and was let go from there,
so as to describe an arc of 2 inches in its whole descent and to describe an
arc of about 4 inches in the first whole oscillation (composed of the descent
and subsequent ascent), it then lost an eighth of its motion in 164 oscillations,
so as to describe an arc of 1¾ inches in its final ascent. When it described
an arc of 4 inches in its first descent, it lost an eighth of its motion in 121
oscillations, so as to describe an arc of 3½ inches in its final ascent. When it
described an arc of 8, 16, 32, or 64 inches in its first descent, it lost an eighth
of its motion in 69, 35½, 18½, and 9⅔ oscillations respectively. Therefore the
difference between the arcs described in the first descent and the final ascent,
in the first, second, third, fourth, fifth, and sixth cases, was ¼, ½, 1, 2, 4, and
8 inches respectively. Divide these differences by the number of oscillations in
each case, and in one mean oscillation—in which an arc of 3¾, 7½, 15, 30,
60, and 120 inches was described—the difference between the arcs described
in the descent and subsequent ascent will be 1/656, 1/242, 1/69, 4/71, 8/37, and 24/29
parts of an inch respectively. In the greater oscillations, moreover, these are
very nearly in the squared ratio of the arcs described, while in the smaller
oscillations they are a little greater than in that ratio; and therefore (by book
2, prop. 31, corol. 2) the resistance of the ball when it moves more swiftly
is very nearly in the squared ratio of the velocity; when more slowly, a little
greater than in that ratio.
Now let V designate the greatest velocity in any oscillation, and let A,
B, and C be given quantities, and let us imagine the difference between the
arcs to be AV + BV3/2 + CV2. In a cycloid the greatest velocities are as halves
of the arcs described in oscillating, but in a circle they are as the chords of
halves of these arcs, and thus with equal arcs are greater in a cycloid than
in a circle in the ratio of halves of the arcs to their chords, while the times
in a circle are greater than in a cycloid in the inverse ratio of the velocity.
Hence it is evident that the differences between the arcs (differences which
are as the resistance and the square of the time jointly) would be very nearly
the same in both curves. For those differences in the cycloid would have to
be increased along with the resistance in roughly the squared ratio of the
arc to the chord (because the velocity is increased in the simple ratio of the
arc to the chord) and would have to be decreased along with the square
of the time in that same squared ratio. Therefore, in order to reduce all
of this to the cycloid, take the same differences between the arcs that were
observed in the circle, while supposing the greatest velocities to correspond
to the arcs, whether halved or entire, that is, to the numbers ½, 1, 2, 4,
8, and 16. In the second, fourth, and sixth cases, therefore, let us write the
numbers 1, 4, and 16 for V; and the difference between the arcs will come
out 
= A + B + C in the second case;
= 4A + 8B + 16C in the fourth
case; and 
= 16A + 64B + 256C in the sixth case. And by the proper
analytic reduction of these equations taken together, A becomes = 0.0000916,
B = 0.0010847, and C = 0.0029558. The difference between the arcs is
therefore as 0.0000916V + 0.0010847V3/2 + 0.0029558V2; and therefore—since
(by prop. 30, corol., applied to this case) the resistance of the ball in the middle
of the arc described by oscillating, where the velocity is V, is to its weight as
7/11AV + 7/10BV3/2 + ¾CV2 is to the length of the pendulum—if the numbers
found are written for A, B, and C, the resistance of the ball will become to
its weight as 0.0000583V + 0.0007593V3/2 + 0.0022169V2 is to the length of
the pendulum between the center of suspension and the ruler, that is, to 121
inches. Hence, since V in the second case has the value 1, in the fourth 4,
and in the sixth 16, the resistance will be to the weight of the ball in the
second case as 0.0030345 to 121, in the fourth as 0.041748 to 121, and in the
sixth as 0.61705 to 121.
The arc which in the sixth case was described by the point marked on the
thread was 120 − or 1195/29 inches. And therefore, since the radius was
121 inches, and the length of the pendulum between the point of suspension
and the center of the ball was 126 inches, the arc that the center of the ball
described was 1243/31 inches. Since, because of the resistance of the air, the
greatest velocity of an oscillating body does not occur at the lowest point
of the arc described but is located near the midpoint of the whole arc, that
velocity will be roughly the same as if the ball in its whole descent in a
nonresisting medium described half that arc (623/62 inches) and did so in
a cycloid, to which we have above reduced the motion of the pendulum;
and therefore that velocity will be equal to the velocity which the ball could
acquire by falling perpendicularly and describing in its fall a space equal to
the versed sine of that arc. But that versed sine in the cycloid is to that arc
(623/62) as that same arc is to twice the length of the pendulum (252) and
thus is equal to 15.278 inches. Therefore the velocity is the very velocity that
the body could acquire by falling and describing in its fall a space of 15.278
inches. With such a velocity, then, the ball encounters a resistance that is to its
weight as 0.61705 to 121, or (if only that part of the resistance is considered
which is in the squared ratio of the velocity) as 0.56752 to 121.
By a hydrostatic experiment, I found that the weight of this wooden ball
was to the weight of a globe of water of the same size as 55 to 97; and
therefore, since 121 is to 213.4 in the same ratio as 55 to 97, the resistance
of a globe of water moving forward with the above velocity will be to its
weight as 0.56752 to 213.4, that is, as 1 to 3761/50. The weight of the globe of
water, in the time during which the globe describes a length of 30.556 inches
with a uniformly continued velocity, could generate all that velocity in the
globe if it were falling; hence it is manifest that in the same time the force
of resistance uniformly continued could take away a velocity smaller in the
ratio of 1 to 3761/50, that is, 
of the whole velocity. And therefore in
the same time in which the globe, with that velocity uniformly continued,
could describe the length of its own semidiameter, or 37/16 inches, it would
lose 1/3,342 of its motion.
I also counted the oscillations in which the pendulum lost a fourth of its motion. In the following table the top numbers denote the length of the arc described in the first descent, expressed in inches and parts of an inch; the middle numbers signify the length of the arc described in the final ascent; and at the bottom stand the numbers of oscillations. I have described this experiment because it is more accurate than when only an eighth of the motion was lost. Let anyone who wishes test the computation.
Later, using the same thread, I suspended a lead ball with a diameter of 2 inches and a weight of 26¼ ounces avoirdupois, in such a way that the distance between the center of the ball and the point of suspension was 10½ feet, and I counted the oscillations in which a given part of the motion was lost. The first of the following tables shows the number of oscillations in which an eighth of the whole motion was lost; the second shows the number of oscillations in which a fourth of it was lost.
Select the third, fifth, and seventh observations from the first table and
represent the greatest velocities in these particular observations by the
numbers 1, 4, and 16 respectively, and generally by the quantity V as above; then
it will be the case that in the third observation 
= A + B + C, in the fifth
= 4A + 8B + 16C, in the seventh 
= 16A + 64B + 256C. The reduction
of these equations gives A = 0.001414, B = 0.000297, C = 0.000879. Hence
the resistance of the ball moving with velocity V comes out to have the ratio
to its own weight (26¼ ounces) that 0.0009V + 0.000208V3/2 + 0.000659V2
has to the pendulum’s length (121 inches). And if we consider only that part
of the resistance which is in the squared ratio of the velocity, it will be to the
weight of the ball as 0.000659V2 is to 121 inches. But in the first experiment
this part of the resistance was to the weight of the wooden ball (577/22 ounces)
as 0.002217V2 to 121; and hence the resistance of the wooden ball becomes to
the resistance of the lead ball (their velocities being equal) as 577/22 × 0.002217
to 26¼ × 0.000659, that is, as 7⅓ to 1. The diameters of the two balls were
67/8 and 2 inches, and the squares of these are to each other as 47¼, and 4, or
1113/16 and 1, very nearly. Therefore the resistances of equally swift balls were
in a smaller ratio than the squared ratio of the diameters. But we have not
yet considered the resistance of the thread, which certainly was very great
and ought to be subtracted from the resistance of the pendulum that has
been found. I could not determine this resistance of the thread accurately,
but nevertheless I found it to be greater than a third of the whole resistance
of the smaller pendulum; and I learned from this that the resistances of the
balls, taking away the resistance of the thread, are very nearly in the squared
ratio of the diameters. For the ratio of 7⅓ − ⅓ to 1 − ⅓ or 10½ to 1, is
very close to the squared ratio of the diameters 1113/16 to 1.
Since the resistance of the thread is of less significance in larger balls,
I also tried the experiment in a ball whose diameter was 18¾ inches. The
length of the pendulum between the point of suspension and the center of
oscillation was 122½ inches; between the point of suspension and a knot in
the thread, 109½ inches. The arc described by the knot in the first descent of
the pendulum was 32 inches. The arc described by that same knot in the final
ascent after five oscillations was 28 inches. The sum of the arcs, or the whole
arc described in a mean oscillation, was 60 inches. The difference between
the arcs was 4 inches. A tenth of it, or the difference between the descent
and the ascent in a mean oscillation, was ⅖ inch. The ratio of the radius
109½ to the radius 122½ is the same as the ratio of the whole arc of 60
inches described by the knot in a mean oscillation to the whole arc of 67⅛
inches described by the center of the ball in a mean oscillation, and is equal
to the ratio of the difference ⅖ to the new difference 0.4475. If the length
of the pendulum were to be increased in the ratio of 126 to 122½ while
the length of the arc described remained the same, the time of oscillation
would be increased and the velocity of the pendulum would be decreased as
the square root of that ratio, while the difference 0.4475 between the arcs
described in a descent and subsequent ascent would remain the same. Then,
if the arc described were to be increased in the ratio of 1243/31 to 67⅛, that
difference 0.4475 would be increased as the square of that ratio, and thus
would come out 1.5295. These things would be so on the hypothesis that the
resistance of the pendulum was in the squared ratio of the velocity. Therefore,
if the pendulum were to describe a whole arc of 1243/31 inches, and its length
between the point of suspension and the center of oscillation were 126 inches,
the difference between the arcs described in a descent and subsequent ascent
would be 1.5295 inches. And this difference multiplied by the weight of the
ball of the pendulum, which was 208 ounces, yields the product 318.136.
Again, when the above-mentioned pendulum (made with a wooden ball)
described a whole arc of 1243/31 inches by its center of oscillation (which was
126 inches distant from the point of suspension), the difference between the
arcs described in the descent and ascent was 
, which multiplied
by the weight of the ball (which was 577/22 ounces) yields the product 49.396.
And I multiplied these differences by the weights of the balls in order to find
their resistances. For the differences arise from the resistances and are as the
resistances directly and the weights inversely. The resistances are therefore as
the numbers 318.136 and 49.396. But the part of the resistance of the smaller
ball that is in the squared ratio of the velocity was to the whole resistance
as 0.56752 to 0.61675, that is, as 45.453 to 49.396; and the similar part of the
resistance of the larger ball is almost equal to its whole resistance; and thus
those parts are very nearly as 318.136 and 45.453, that is, as 7 and 1. But the
diameters of the balls are 18¾ and 6⅞, and the squares of these diameters,
3519/16 and 4717/64, are as 7.438 and 1, that is, very nearly as the resistances 7
and 1 of the balls. The difference between the ratios is no greater than what
could have arisen from the resistance of the thread. Therefore, those parts of
the resistances that are (the balls being equal) as the squares of the velocities
are also (the velocities being equal) as the squares of the diameters of the
balls.
The largest ball that I used in these experiments, however, was not perfectly spherical, and therefore for the sake of brevity I have ignored certain minutiae in the above computation, being not at all worried about a computation being exact when the experiment itself was not sufficiently exact. Therefore, since the demonstration of a vacuum depends on such experiments, I wish that they could be tried with more, larger, and more exactly spherical balls. If the balls are taken in geometric proportion, say with diameters of 4, 8, 16, and 32 inches, it will be discovered from the progression of the experiments what ought to happen in the case of still larger balls.
To compare the resistances of different fluids with one another, I made the following experiments. I got a wooden box four feet long, one foot wide, and one foot deep. I took off its lid and filled it with fresh water, and I immersed pendulums in the water and made them oscillate. A lead ball weighing 166⅙ ounces, with a diameter of 3⅝ inches, moved as in the following table, that is, with the length of the pendulum from the point of suspension to a certain point marked on the thread being 126 inches, and to the center of oscillation being 134⅜ inches.
In the experiment recorded in the fourth column, equal motions were lost in 535 oscillations in air, and 1⅕ in water. The oscillations were indeed a little quicker in air than in water. But if the oscillations in water were accelerated in such a ratio that the motions of the pendulums in both mediums would become equally swift, the number 1⅕ oscillations in water during which the same motion would be lost as before would remain the same because the resistance is increased and the square of the time simultaneously decreased in that same ratio squared. With equal velocities of the pendulums, therefore, equal motions were lost, in air in 535 oscillations and in water in 1⅕ oscillations; and thus the resistance of the pendulum in water is to its resistance in air as 535 to 1⅕. This is the proportion of the whole resistances in the case of the fourth column.
Now let AV + CV2 designate the difference between the arcs described
(in a descent and subsequent ascent) by the ball moving in air with the
greatest velocity V; and since the greatest velocity in the case of the fourth
column is to the greatest velocity in the case of the first column as 1 to 8,
and since that difference between the arcs in the case of the fourth column
is to the difference in the case of the first column as 
to 
, or as 85½
to 4,280, let us write 1 and 8 for the velocities in these cases and 85½ and
4,280 for the differences between the arcs; then A + C will become = 85½
and 8A + 64C = 4, 280 or A + 8C = 535; and hence, by reduction of the
equations, 7C will become = 449½ and C = 643/14 and A = 212/7; and thus
the resistance, since it is as 7/11AV + ¾CV2, will be as 136/11V + 489/56V2.
Therefore, in the case of the fourth column, where the velocity was 1, the
whole resistance is to its part proportional to the square of the velocity as
136/11 + 489/56 or 6112/17 to 489/56; and on that account the resistance of the
pendulum in water is to that part of the resistance in air which is proportional
to the square of the velocity (and which alone comes into consideration in
swifter motions) as 6112/17 to 489/56 and 535 to 1⅕ jointly, that is, as 571 to 1.
If the whole thread of the pendulum oscillating in water had been immersed,
its resistance would have been still greater, to such an extent that the part
of the resistance of the pendulum oscillating in water which is proportional
to the square of the velocity (and which alone comes into consideration in
swifter bodies) is to the resistance of that same whole pendulum oscillating
in air, with the same velocity, as about 850 to 1, that is, very nearly as the
density of water to the density of air.
In this computation also, that part of the resistance of the pendulum in water which would be as the square of the velocity ought to be taken into consideration, but (which may perhaps seem strange) the resistance in water was increased in more than the squared ratio of the velocity. In searching for the reason, I hit upon this: that the box was too narrow in proportion to the size of the ball of the pendulum, and because of its narrowness overly impeded the motion of the water as it yielded to the oscillation of the ball. For if a ball of a pendulum whose diameter was one inch was immersed, the resistance was increased in very nearly the squared ratio of the velocity. I tested this by constructing a pendulum out of two balls, so that the lower and smaller of them oscillated in the water, and the higher and larger one was fastened to the thread just above the water and, by oscillating in the air, aided the pendulum’s motion and made it last longer. And the experiments made with this pendulum came out as in the following table.
In comparing the resistances of the mediums with one another I also caused iron pendulums to oscillate in quicksilver. The length of the iron wire was about three feet, and the diameter of the ball of the pendulum was about ⅓ inch. And to the wire just above the mercury there was fastened another lead ball large enough to continue the motion of the pendulum for a longer time. Then I filled a small vessel (which held about three pounds of quicksilver) with quicksilver and common water successively, so that as the pendulum oscillated first in one and then in the other of the two fluids I might find the proportion of the resistances; and the resistance of the quicksilver came out to the resistance of the water as about 13 or 14 to 1, that is, as the density of quicksilver to the density of water. When I used ba slightly larger pendulum ball, say one whose diameter would be about ⅓ or ⅔ inch,b the resistance of the quicksilver came out in the ratio to the resistance of the water that the number 12 or 10 has to 1, roughly. But the former experiment is more trustworthy because in the latter the vessel was too narrow in proportion to the size of the immersed ball. With the ball enlarged, the vessel also would have to be enlarged. Indeed, I had determined to repeat experiments of this sort in larger vessels and in molten metals and certain other liquids, hot as well as cold; but there is not time to try them all, and from what has already been described it is clear enough that the resistance of bodies moving swiftly is very nearly proportional to the density of the fluids in which they move. I do not say exactly proportional. For the more viscous fluids, of an equal density, doubtless resist more than the more liquid fluids—as, for example, cold oil more than hot, hot oil more than rainwater, water more than spirit of wine. But in the liquids that are sufficiently fluid to the senses—as in air, in water (whether fresh or salt), in spirits of wine, of turpentine, and of salts, in oil freed of its dregs by distillation and then heated, and in oil of vitriol and in mercury, and in liquefied metals, and any others there may be which are so fluid that when agitated in vessels they conserve for some time a motion impressed upon them and when poured out are quite freely broken up into falling drops—in all these I have no doubt that the above rule holds exactly enough, especially if the experiments are made with pendulums that are larger and move more swiftly.
Finally, since csome people are of the opinionc that there exists a certain aethereal medium, by far the subtlest of all, which quite freely permeates all the pores and passages of all bodies, and that a resistance ought to arise from such a medium flowing through the pores of bodies, I devised the following experiment so that I might test whether the resistance that we experience in moving bodies is wholly on their external surface or whether the internal parts also encounter a perceptible resistance on their own surfaces. I suspended a round firwood box by a cord eleven feet long from a sufficiently strong steel hook, by means of a steel ring. The upper arc of the ring rested on the very sharp concave edge of the hook so that it might move very freely. And the cord was attached to the lower arc of the ring. I drew this pendulum away from the perpendicular to a distance of about six feet, and did so along the plane perpendicular to the edge of the hook, so that the ring, as the pendulum oscillated, would not slide to and fro on the edge of the hook. For the point of suspension, in which the ring touches the hook, ought to remain motionless. I marked the exact place to which I had drawn back the pendulum and then, letting the pendulum fall, marked another three places: those to which it returned at the end of the first, second, and third oscillations. I repeated this quite often, so that I might find those places as exactly as possible. Then I filled the box with lead and some of the other heavier metals that were at hand. But first I weighed the empty box along with the part of the cord that was wound around the box and half of the remaining part that was stretched between the hook and the suspended box. For a stretched cord always urges with half of its weight a pendulum drawn aside from the perpendicular. To this weight I added the weight of the air that the box contained. And the whole weight was about 1/78 of the box full of metals. Then, since the box full of metals increased the length of the pendulum as a result of stretching the cord by its weight, I shortened the cord so that the length of the pendulum now oscillating would be the same as before. Then, drawing the pendulum back to the first marked place and letting it fall, I counted about 77 oscillations until the box returned to the second marked place, and as many thereafter until the box returned to the third marked place, and again as many until the box on its return reached the fourth place. Hence I conclude that the whole resistance of the full box did not have a greater proportion to the resistance of the empty box than 78 to 77. For if the resistances of both were equal, the full box, because its inherent force was 78 times greater than the inherent force of the empty box, ought to conserve its oscillatory motion that much longer, and thus always return to those marked places at the completion of 78 oscillations. But it returned to them at the completion of 77 oscillations.
Let A therefore designate the resistance of the box on its external surface, and B the resistance of the empty box on its internal parts; then, if the resistances of equally swift bodies on their internal parts are as the matter, or the number of particles that are resisted, 78B will be the resistance of the full box on its internal parts; and thus the whole resistance A + B of the empty box will be to the whole resistance A + 78B of the full box as 77 to 78, and by separation [or dividendo] A + B will be to 77B as 77 to 1, and hence A + B will be to B as 77 × 77 to 1, and by separation [or dividendo] A will be to B as 5,928 to 1. The resistance encountered by the empty box on its internal parts is therefore more than 5,000 times smaller than the similar resistance on the external surface. This argument depends on the hypothesis that the greater resistance encountered by the full box does not arise from some other hidden cause but only from the action of some subtle fluid upon the enclosed metal.
I have reported this experiment from memory. For the paper on which I had once described it is lost. Hence I have been forced to omit certain fractions of numbers which have escaped my memory.
There is no time to try everything again. The first time, since I had used a weak hook, the full box was retarded more quickly. In seeking the cause, I found that the hook was so weak as to give way to the weight of the box and to be bent in this direction and that as it yielded to the oscillations of the pendulum. I got a strong hook, therefore, so that the point of suspension would remain motionless, and then everything came out as we have described it above.
The motion of fluids and the resistance encountered by projectiles
Proposition 32, Theorem 26
Let two similar systems of bodies consist of an equal number of particles, and let
each of the particles in one system be similar and proportional to the corresponding
particle in the other system, and let the particles be similarly situated with respect
to one another in the two systems and have a given ratio of density to one another.
And let them begin to move similarly with respect to one another in proportional
times (the particles that are in the one system with respect to the particles in that
system, and the particles in the other with respect to those in the other). Then,
if the particles that are in the same system do not touch one another except in
instants of reflection and do not attract or repel one another except by accelerative
forces that are inversely as the diameters of corresponding particles and directly as
the squares of the velocities, I say that the particles of the systems will continue to
move similarly with respect to one another in proportional times.
I say that bodies which are similar and similarly situated move similarly with respect to one another in proportional times when their situations in relation to one another are always similar at the end of the times—for instance, if the particles of one system are compared with the corresponding particles of another. Hence the times in which similar and proportional parts of similar figures are described by corresponding particles will be proportional. Therefore, if there are two systems of this sort, the corresponding particles, because of the similarity of their motions at the beginning, will continue to move similarly until they meet one another. For if they are acted upon by no forces, they will, by the first law of motion, move forward uniformly in straight lines. If they act upon one another by some forces and if those forces are as the diameters of the corresponding particles inversely and the squares of the velocities directly, then, since the situations of the particles are similar and the forces proportional, the whole forces by which the corresponding particles are acted upon, compounded of the separate acting forces (by corol. 2 of the laws), will have similar directions, just as if they tended to centers similarly placed among the particles, and those whole forces will be to one another as the separate component forces, that is, as the diameters of the corresponding particles inversely and the squares of the velocities directly, and therefore they will cause corresponding particles to continue describing similar figures. This will be so (by book 1, prop. 4, corols. 1 and 8) provided that the centers are at rest. But if they move, since their situations with respect to the particles of the systems remain similar (because the transferences are similar), similar changes will be introduced in the figures which the particles describe. The motions of corresponding similar particles will therefore be similar until they first meet, and therefore the collisions will be similar and the reflections similar, and then (by what has already been shown) the motions of the particles with respect to one another will be similar until they encounter one another again, and so on indefinitely. Q.E.D.
COROLLARY 1. Hence, if any two bodies that are similar and similarly situated (in relation to the corresponding particles of the systems) begin to move similarly with respect to the particles in proportional times, and if their volumes and densities are to each other as the volumes and densities of the corresponding particles, the bodies will continue to move similarly in proportional times. For the case is the same for the larger parts of both systems as for the particles.
COROLLARY 2. And if all the similar and similarly situated parts of the systems are at rest with respect to one another, and if two of them, which are larger than the others and correspond to each other in the two systems, begin to move in any way with a similar motion along lines similarly situated, they will cause similar motions in the remaining parts of the systems and will continue to move similarly with respect to them in proportional times and thus will continue to describe spaces proportional to their own diameters.
Proposition 33, Theorem 27
If the same suppositions are made, I say that the larger parts of the systems are
resisted in a ratio compounded of the squared ratio of their velocities and the
squared ratio of the diameters and the simple ratio of the density of the parts of
the systems.
For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the systems act upon one another and partly from the collisions and reflections of the particles and the larger parts. Resistances of the first kind, moreover, are to one another as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts, that is (by hypothesis), as the squares of the velocities directly and the distances of the corresponding particles inversely and the quantities of matter in the corresponding parts directly. Thus, since the distances of the particles of the one system are to the corresponding distances of the particles of the other as the diameter of a particle or part in the first system to the diameter of the corresponding particle or part in the other, and since the quantities of matter are as the densities of the parts and the cubes of the diameters, the resistances are to one another as the squares of the velocities, the squares of the diameters, and the densities of the parts of the systems. Q.E.D.
Resistances of the second kind are as the numbers and forces of corresponding reflections jointly. The number of reflections in any one case, moreover, is to the number in any other as the velocities of the corresponding parts directly and the spaces between their reflections inversely. And the forces of the reflections are as the velocities and volumes and densities of the corresponding parts jointly, that is, as the velocities, the cubes of the diameters, and the densities of the parts. And if all these ratios are compounded, the resistances of the corresponding parts are to one another as the squares of the velocities, the squares of the diameters, and the densities of the parts, jointly. Q.E.D.
COROLLARY 1. Therefore, if the systems are two elastic fluids such as air and if their parts are at rest with respect to one another, and if two bodies which are similar and are proportional (with regard to volume and density) to the parts of the fluids and are similarly situated with respect to those parts are projected in any way along lines similarly situated, and if the accelerative forces with which the particles of the fluids act upon one another are as the diameters of the projected bodies inversely and the squares of the velocities directly, then the bodies will cause similar motions in the fluids in proportional times and will describe spaces that are similar and are proportional to their diameters.
COROLLARY 2. Accordingly, in the same fluid a swift projectile encounters a resistance that is very nearly in the squared ratio of the velocity. For if the forces with which distant particles act upon one another were increased in the squared ratio of the velocity, the resistance would be exactly in the squared ratio of the velocity; and thus, in a medium whose parts act upon one another with no forces because they are far apart, the resistance is exactly in the squared ratio of the velocity. Let A, B, and C, therefore, be three mediums consisting of parts that are similar and equal and regularly distributed at equal distances. Let the parts of mediums A and B recede from one another with forces that are to one another as T and V, and let the parts of medium C be entirely without forces of this sort. Then, let four equal bodies D, E, F, and G move in these mediums, the first two bodies D and E in the first two mediums A and B respectively, and the other two bodies F and G in the third medium C; and let the velocity of body D be to the velocity of body E, and let the velocity of body F be to the velocity of body G, as the square root of the ratio of the forces T to the forces V [i.e., as √T to √V]; then the resistance of body D will be to the resistance of body E, and the resistance of body F to the resistance of body G, in the squared ratio of the velocities; and therefore the resistance of body D will be to the resistance of body F as the resistance of body E to the resistance of body G. Let bodies D and F have equal velocities, and also bodies E and G; then, if the velocities of bodies D and F are increased in any ratio and the forces of the particles of medium B are decreased in the same ratio squared, medium B will approach the form and condition of medium C as closely as is desired, and on that account the resistances of the equal and equally swift bodies E and G in these mediums will continually approach equality, in such a way that their difference finally comes out less than any given difference. Accordingly, since the resistances of bodies D and F are to each other as the resistances of bodies E and G, these also will similarly approach the ratio of equality. Therefore, the resistances of bodies D and F, when they move very swiftly, are very nearly equal, and therefore, since the resistance of body F is in the squared ratio of the velocity, the resistance of body D will be very nearly in the same ratio.
COROLLARY 3. The resistance of a body moving very swiftly in any elastic fluid is about the same as if the parts of the fluid lacked their centrifugal forces and did not recede from one another, provided that the elastic force of the fluid arises from the centrifugal forces of the particles and that the velocity is so great that the forces do not have enough time to act.
COROLLARY 4. Accordingly, since the resistances of similar and equally swift bodies, in a medium whose parts (being far apart) do not recede from one another, are as the squares of the diameters, the resistances of equally swift and very quickly moving bodies in an elastic fluid are also very nearly as the squares of the diameters.
COROLLARY 5. And since similar, equal, and equally swift bodies, in mediums which have the same density and whose particles do not recede from one another, impinge upon an equal quantity of matter in equal times (whether the particles are more and smaller or fewer and larger) and impress upon it an equal quantity of motion and in turn (by the third law of motion) undergo an equal reaction from it (that is, are equally resisted), it is manifest also that in elastic fluids of the same density, when the bodies move very swiftly, the resistances they encounter are very nearly equal, whether those fluids consist of coarser particles or are made of the most subtle particles of all. The resistance to projectiles moving very quickly is not much diminished as a result of the subtlety of the medium.
COROLLARY 6. These statements all hold for fluids whose elastic force originates in the centrifugal forces [i.e., forces of repulsion] of the particles. But if that force arises from some other source, such as the expansion of the particles in the manner of wool or the branches of trees, or from any other cause which makes the particles move less freely with respect to one another, then the resistance will be greater than in the preceding corollaries because the medium is less fluid.
Proposition 34, Theorem 28
In a rare medium consisting of particles that are equal and arranged freely at
equal distances from one another, let a sphere and a cylinder—described with
equal diameters—move with equal velocity along the direction of the axis of the
cylinder; then the resistance of the sphere will be half the resistance of the cylinder.
For since the action of a medium on a body is (by corol. 5 of the
laws) the same whether the body moves in a medium at rest or the
particles of the medium impinge with the
same velocity on the body at rest, let
us consider the body to be at rest and
see with what force it will be urged
by the moving medium. Let ABKI,
therefore, designate a spherical body
described with center C and
semidiameter C A, and let the particles of the
medium strike the spherical body with a given velocity along straight lines
parallel to AC; and let FB be such a straight line. On FB take LB equal
to the semidiameter CB, and draw BD touching the sphere in B. To KC
and BD drop the perpendiculars BE and LD; then the force with which a
particle of the medium, obliquely incident along the straight line FB, strikes
the sphere at B will be to the force with which the same particle would
strike the cylinder ONGQ (described with axis ACI about the sphere)
perpendicularly at b as LD to LB or BE to BC. Again, the efficacy of this force
to move the sphere along the direction FB (or AC) of its incidence is to its
efficacy to move the sphere along the direction of its determination—that
is, along the direction of the straight line BC in which it urges the sphere
directly [a direction through the center of the sphere]—as BE to BC. And,
compounding the ratios, if a particle strikes the sphere obliquely along the
straight line FB, its efficacy to move the sphere along the direction of its
incidence is to the efficacy of the same particle to move the cylinder in the
same direction, when striking the cylinder perpendicularly along the same
straight line, as BE2 to BC2. Therefore, if in bE, which is perpendicular to
the circular base NAO of the cylinder and equal to the radius AC, bH is
taken equal to 
, then bH will be to bE as the effect of a particle upon
the sphere to the effect of the particle upon the cylinder. And therefore
the solid that is composed of all the straight lines bH will be to the solid
that is composed of all the straight lines bE as the effect of all the
particles upon the sphere to the effect of all the particles upon the cylinder. But
the first solid is a paraboloid described with vertex C, axis CA, and latus
rectum CA, and the second solid is a cylinder circumscribed around the
paraboloid; and it is known that a paraboloid is half of the circumscribed
cylinder. Therefore the whole force of the medium upon the sphere is half
of its whole force upon the cylinder. And therefore, if the particles of the
medium were at rest and the cylinder and the sphere were moving with
equal velocity, the resistance of the sphere would be half the resistance of the
cylinder. Q.E.D.
Scholium
By the same method other figures can be compared with one another with
respect to resistance, and those that are more suitable for continuing their
motions in resisting mediums can be found. For example, let it be required
to construct a frustum CBGF of a cone with the
circular base CEBH (which is described with
center O and radius OC) and with the height
OD, which is resisted less than any other
frustum constructed with the same base and height
and moving forward along the direction of the
axis toward D; bisect the height OD in Q, and
produce OQ to S so that QS is equal to QC,
and S will be the vertex of the cone whose frustum is required.
Note in passing that since the angle CSB is always acute, it follows
that if the solid ADBE is generated by a revolution of the elliptical or
oval figure ADBE about the axis
AB, and if the generating figure
is touched by the three straight
lines FG, GH, and HI in points
F, B, and I, in such a way that
GH is perpendicular to the axis
in the point of contact B, and FG
and HI meet the said line GH at
the angles FGB and BHI of 135 degrees, then the solid that is generated by
the revolution of the figure ADFGHIE about the same axis AB is less
resisted than the former solid, provided that each of the two moves forward
along the direction of its axis AB, and the end B of each one is in front.
Indeed, I think that this proposition will be of some use for the construction
of ships.
But suppose the figure DNFG to be a curve of such a sort that if the perpendicular NM is dropped from any point N of that curve to the axis AB, and if from the given point G the straight line GR is drawn, which is parallel to a straight line touching the figure in N and cuts the axis (produced) in R, then MN would be to GR as GR3 to 4BR × GB2. Then, in this case, the solid that is described by a revolution of this figure about the axis AB will, in moving in the aforesaid rare medium from A toward B, be resisted less than any other solid of revolution described with the same length and width.
Proposition 35a, Problem 7
If a rare medium consists of minimally small equal particles that are at rest and
arranged freely at equal distances from one another, it is required to find the
resistance encountered by a sphere moving forward uniformly in this medium.
CASE 1. Let a cylinder described with the same diameter and height as before move forward with the same velocity along the length of its own axis in the same medium. And let us suppose that the particles of the medium upon which the sphere or cylinder impinges rebound with the greatest possible force of reflection. Then the resistance of the sphere (by prop. 34) is half the resistance of the cylinder, and the sphere is to the cylinder as 2 to 3, and the cylinder in impinging perpendicularly upon the particles and reflecting them as greatly as possible communicates twice its own velocity to them. Therefore, the cylinder, in the time in which it describes half the length of its axis by moving uniformly forward, will communicate to the particles a motion which is to the whole motion of the cylinder as the density of the medium is to the density of the cylinder; and the sphere, in the time in which it describes the whole length of its diameter by moving uniformly forward, will communicate the same motion to the particles, and in the time in which it describes ⅔ of its diameter it will communicate to the particles a motion which is to the whole motion of the sphere as the density of the medium to the density of the sphere. And therefore the sphere encounters a resistance that is to the force by which its whole motion could be either destroyed or generated, in the time in which it describes ⅔ of its diameter by moving uniformly forward, as the density of the medium is to the density of the sphere.
CASE 2. Let us suppose that the particles of the medium impinging upon the sphere or cylinder are not reflected; then the cylinder, in impinging perpendicularly upon the particles, will communicate its whole velocity to them and thus encounters half the resistance which it met in the former case, and the resistance encountered by the sphere will also be half of what it was before.
CASE 3. Let us suppose that the particles of the medium rebound from the sphere with a force of reflection that is neither the greatest nor nil but some intermediate force; then the resistance encountered by the sphere will also be intermediate between the resistance in case 1 and the resistance in case 2. Q.E.I.
COROLLARY 1. Hence, if the sphere and the particles are infinitely hard without any elastic force and therefore also without any force of reflection, the resistance encountered by the sphere will be to the force by which its whole motion could be either destroyed or generated, in the time in which the sphere describes 4/3 of its diameter, as the density of the medium is to the density of the sphere.
COROLLARY 2. The resistance encountered by the sphere, other things being equal, is in the squared ratio of the velocity.
COROLLARY 3. The resistance encountered by the sphere, other things being equal, is in the squared ratio of the diameter.
COROLLARY 4. The resistance encountered by the sphere, other things being equal, is as the density of the medium.
COROLLARY 5. The resistance encountered by the sphere is in a ratio that is compounded of the squared ratio of the velocity and the squared ratio of the diameter, and the simple ratio of the density of the medium.
COROLLARY 6. And the motion of the sphere with the resistance it
encounters can be represented as follows. Let AB be the time in which the
sphere can lose its whole motion when the
resistance is continued uniformly. Erect AD and BC
perpendicular to AB. And let BC be the whole
motion, and through point C with asymptotes
AD and AB describe the hyperbola CF.
Produce AB to any point E. Erect the perpendicular
EF meeting the hyperbola in F. Complete the parallelogram CBEG, and
draw AF meeting BC in H. Then, if the sphere, in any time BE, when its
first motion BC is continued uniformly, in a nonresisting medium, describes
the space CBEG represented by the area of the parallelogram, it will in a
resisting medium describe the space CBEF represented by the area of the
hyperbola, and its motion at the end of that time will be represented by the
ordinate EF of the hyperbola, with loss of part FG of its motion. And the
resistance at the end of the same time will be represented by the length BH,
with loss of part CH of the resistance. All of this is evident by book 2, prop.
5, corols. 1 and 3.
COROLLARY 7. Hence, if in time T, when the resistance R is continued
uniformly, the sphere loses its whole motion M, then in time t in a resisting
medium, when the resistance R decreases in the squared ratio of the velocity,
the sphere will lose part 
of its motion M without loss of part 
;
and the sphere will describe a space that is to the space described by the
uniform motion M, in the same time t, as the logarithm of the number 
multiplied by the number 2.302585092994 is to the number 
, because the
hyperbolic area BCFE is in this proportion to the rectangle BCGE.
Scholium
In this proportion I have set forth the resistance and retardation encountered
by spherical projectiles in noncontinuous mediums, and I have shown that
this resistance is to the force by which the whole motion of a sphere could
be either destroyed or generated in the time in which the sphere describes
⅔ of its diameter, with a velocity continued uniformly, as the density of
the medium is to the density of the sphere, provided that the sphere and
the particles of the medium are highly elastic and possess the greatest force
of reflecting, and I have shown that this force is half as great when the
sphere and the particles of the medium are infinitely hard and devoid of all
force of reflecting. Moreover, in continuous mediums such as water, hot oil,
and quicksilver, in which the sphere does not impinge directly upon all the
particles of the fluid which generate resistance but presses only the nearest
particles, and these press others and these still others, the resistance is half
as great as in the second case. In extremely fluid mediums of this sort the
sphere encounters a resistance that is to the force by which its whole motion
could be either destroyed or generated, in the time in which it describes 8/3
of its diameter with the motion continued uniformly, as the density of the
medium is to the density of the sphere. We will try to show this in what
follows.
Proposition 36, Problem 8
To determine the motion of water flowing out of a cylindrical vessel through a
hole in the bottom.
Let ACDB be the cylindrical vessel, AB its upper opening, CD its bottom
parallel to the horizon, EF a circular hole in the middle of the bottom, G the
center of the hole, and GH the cylinder’s axis perpendicular to the horizon.
And imagine that a cylinder of ice APQB is
of the same width as the interior of the vessel,
has the same axis, and descends continually with
a uniform motion. Imagine also that its parts
liquefy as soon as they touch the surface AB,
that when they have turned into water they flow
down into the vessel as a result of their gravity,
and that in falling these parts form a cataract or
column of water ABNFEM and pass through
the hole EF and fill it exactly. And let the
uniform velocity of the descending ice, as well as that of the contiguous water
in the circle AB, be the velocity which the water can acquire in falling and
describing by its fall the space IH, and let IH and HG lie in a straight
line, and through point I draw the straight line KL parallel to the
horizon and meeting the sides of the ice in K and L. Then the velocity of the
water flowing out through the hole EF will be that which the water can
acquire in falling from I and describing by its fall the space IG. And thus,
by Galileo’s theorems, IG will be to IH as the square of the ratio of the
velocity of the water flowing out through the hole to the velocity of the
water in the circle AB, that is, as the square of the ratio of the circle AB
to the circle EF, for these circles are inversely as the velocities of the water
passing through them in the same time and with an equal quantity, filling
them both exactly. Here it is the velocity of the water toward the horizon that
is of concern. And the motion parallel to the horizon by which the parts of
the falling water approach one another is not considered here, since it does
not arise from gravity or change the motion perpendicular to the horizon
that does arise from gravity. Indeed, we are supposing that the parts of the
water cohere somewhat and that by their cohesion they approach one another
with motions parallel to the horizon as they fall, so that they form only one
single cataract and are not divided into several cataracts, but here we are not
considering the motion parallel to the horizon arising from that cohesion.
CASE 1. Now suppose that the interior of the vessel around the falling water ABNFEM is filled with ice, so that the water passes through the ice as if through a funnel. Then, if the water does not quite touch the ice, or (what comes to the same thing) if it touches it and, because of the great smoothness of the ice, slides through it with the greatest possible freedom and without any resistance, the water will flow down through the hole EF with the same velocity as before, and the whole weight of the column of water ABNFEM will be used in generating its downflow as before, and the bottom of the vessel will sustain the weight of the ice surrounding the column.
Now let the ice liquefy in the vessel; then the flow of the water will remain the same as before with respect to velocity. It will not be less, since the melted ice will endeavor to descend; and not greater, since the melted ice cannot descend without impeding an equal descent of the original water. The same force ought to generate the same velocity in the flowing water [i.e., since the force is the same, the velocity that it generates will also be the same].
But the hole in the bottom of the vessel, because of the oblique motions of the particles of the flowing water, ought to be a little larger than before. For now the particles of water do not all pass through the hole perpendicularly but, flowing together from all the sides of the vessel and converging into the hole, pass through with oblique motions and, turning their course downward, unite into a stream of water gushing out which is narrower a little below the hole than in the hole itself, its diameter being to the diameter of the hole as 5 to 6, or 5½ to 6½ very nearly, provided that I measured the diameters correctly. At any rate, I obtained a very thin flat plate perforated in the middle, the diameter of the circular hole being ⅝ inch. And so that the stream of water gushing out might not be accelerated in falling and made narrower by the acceleration, I fastened this plate not to the bottom but to the side of the vessel in such a way that the stream went out along a line parallel to the horizon. Then, when the vessel was full of water, I opened the hole so that the water might flow out, and the diameter of the stream, measured as accurately as possible at a distance of about ½ inch from the hole, came out 21/40 inch. The diameter of this circular hole, therefore, was to the diameter of the stream very nearly as 25 to 21. Therefore the water in passing through the hole converges from all directions, and after flowing out of the vessel the stream is made narrower by converging and is accelerated by narrowing until it has reached a distance of ½ inch from the hole and at that distance becomes narrower and swifter than it is in the hole itself in the ratio of 25 × 25 to 21 × 21 or very nearly 17 to 12, that is, roughly as the square root of the ratio of 2 to 1. And experiments prove that the quantity of water that flows out in a given time through a circular hole in the bottom of a vessel is the quantity that ought to flow out in the same time, with the velocity mentioned above, not through that hole but through a circular hole whose diameter is to the diameter of that hole as 21 to 25. And thus the flowing water has the downward velocity in the hole itself that a heavy body can acquire very nearly in falling and describing by its fall a space equal to half the height of the water standing in the vessel. But after the water has gone out of the vessel, it is accelerated by converging until it has reached a distance from the hole almost equal to the diameter of the hole and has acquired a velocity that is greater approximately as the square root of the ratio of 2 to 1, which is, as a matter of fact, very nearly the velocity that a heavy body can acquire in falling and describing by its fall a space equal to the whole height of the water standing in the vessel.
In what follows, therefore, let the diameter of the stream be designated by
that smaller hole which we have called EF. And suppose that another higher
plane VW is drawn parallel to the plane of
the hole EF at a distance about equal to the
diameter of the hole and pierced by a larger
hole ST, and through this let a stream fall that
exactly fills the lower hole EF and thus has a
diameter which is to the diameter of this lower
hole as about 25 to 21. For thus the stream will
pass perpendicularly through the lower hole,
and the quantity of the water flowing out, depending on the size of this
hole, will be very nearly that which the solution of the problem demands.
Now, the space which is enclosed by the two planes and the falling stream
can be considered to be the bottom of the vessel. But so that the solution of
the problem may be simpler and more mathematical, it is preferable to use
only the lower plane for the bottom of the vessel and to imagine that the
water which flowed down through the ice as if through a funnel and came
out of the vessel through the hole EF in the lower plane keeps its motion
continually and that the ice keeps its state of rest. In what follows, therefore,
let ST be the diameter of a circular hole described with center Z, through
which a cataract flows out of the vessel when all the water in the vessel is
fluid. And let EF be the diameter of the hole which the cataract fills exactly
when falling through it, whether the water comes out of the vessel through
the upper hole ST or falls through the middle of the ice in the vessel as
if through a funnel. And let the diameter of the upper hole ST be to the
diameter of the lower hole EF as about 25 to 21, and let the perpendicular
distance between the planes of the holes be equal to the diameter of the
smaller hole EF. Then the downward velocity of the water coming out of
the vessel through the hole ST will in the hole itself be that which a body
can acquire in falling from half of the height IZ; and the velocity of both
falling cataracts will, in the hole EF, be that which a body will acquire in
falling from the whole height IG.
CASE 2. If the hole EF is not in the middle of the bottom of the vessel, but the bottom is perforated elsewhere, the water will flow out with the same velocity as before, provided that the size of the hole is the same. For a heavy body does descend to the same depth in a greater time along an oblique line than along a perpendicular line, but in descending it acquires the same velocity in either case, as Galileo proved.
CASE 3. The velocity of the water flowing out through a hole in the side of the vessel is the same. For if the hole is small, so that the distance between the surfaces AB and KL vanishes, so far as the senses can tell, and the stream of water gushing out horizontally forms a parabolic figure, it will be found from the latus rectum of this parabola that the velocity of the water flowing out is that which a body could have acquired by falling from the height HG or IG of the water standing in the vessel. Indeed, by making an experiment I found that when the height of the standing water above the hole was 20 inches and the height of the hole above a plane parallel to the horizon was also 20 inches, the stream of water gushing forth would fall upon the plane at a distance of about 37 inches, taken from a perpendicular that was dropped to the plane from the hole. For in the absence of resistance the stream would have had to fall upon the plane at a distance of 40 inches, the latus rectum of the parabolic stream being 80 inches.
CASE 4. Further, if the water flowing out has an upward motion, it comes out with the same velocity. For a small stream of water gushing out ascends with a perpendicular motion to the height GH or GI of the water standing in the vessel, except insofar as its ascent is somewhat impeded by the resistance of the air; and accordingly it flows out with the velocity that it could have acquired in falling from that height. Any one particle of the standing water (by book 2, prop. 19) is pressed equally from all sides and, yielding to the pressure, goes with equal force in every direction, whether it descends through a hole in the bottom of the vessel or flows out horizontally through a hole in its side or comes out into a channel and ascends from there through a small hole in the upper part of the channel. And that the velocity with which the water flows out is that which we have designated in this proposition is not only found by reason but is also manifest from the well-known experiments already described.
CASE 5. The velocity of the water flowing out is the same whether the hole is circular or square or triangular or of any other shape equal in area to the circular one. For the velocity of the water flowing out does not depend on the shape of the hole but on the height of the water in relation to the plane KL.
CASE 6. If the lower part of the vessel ABDC is immersed in standing
water, and the height of the standing water above the bottom of the vessel is
GR, the velocity with which the water in the
vessel will flow out through the hole EF into
the standing water will be that which the water
can acquire in falling and describing by its fall
the space IR. For the weight of all the water in
the vessel that is lower than the surface of the
standing water will be sustained in equilibrium
by the weight of the standing water and thus
will not at all accelerate the motion of the descending water in the vessel.
This case can also be shown by experiments, by measuring the times in which
the water flows out.
COROLLARY 1. Hence, if the height CA of the water is produced to K, so that AK is to CK in the squared ratio of the area of a hole made in any part of the bottom to the area of the circle AB, the velocity of the water flowing out will be equal to the velocity that the water can acquire in falling and describing by its fall the space KC.
COROLLARY 2. And the force by which the whole motion of the water gushing out can be generated is equal to the weight of a cylindrical column of water whose base is the hole EF and whose height is 2GI or 2CK. For the gushing water, in the time in which it equals this column, can acquire in falling (by its weight) from the height GI the very velocity with which it gushes out.
COROLLARY 3. The weight of all the water in the vessel ABDC is to the part of the weight that is used in making the water flow down as the sum of the circles AB and EF to twice the circle EF. For let IO be a mean proportional between IH and IG; then the water coming out through the hole EF, in the time in which a drop could describe a space equal to the height IG in falling from I, will be equal to a cylinder whose base is the circle EF and whose height is 2IG, that is, to a cylinder whose base is the circle AB and whose height is 2IO, for the circle EF is to the circle AB as the square root of the ratio of the height IH to the height IG, that is, in the simple ratio of the mean proportional IO to the height IG, and in the time in which a drop can describe a space equal to the height IH in falling from I, the water coming out will be equal to a cylinder whose base is the circle AB and whose height is 2IH, and in the time in which a drop describes a space equal to the difference HG between the heights in falling from I through H to G, the water coming out—that is, all the water in the solid ABNFEM—will be equal to the difference between the cylinders, that is, equal to a cylinder whose base is AB and whose height is 2HO. And therefore all the water in the vessel ABDC is to all the water falling in the solid ABNFEM as HG to 2HO, that is, as HO + OG to 2HO, or IH + IO to 2IH. But the weight of all the water in the solid ABNFEM is used in making the water flow down, and accordingly the weight of all the water in the vessel is to the part of the weight that is used in making the water flow down as IH + IO to 2IH and thus as the sum of the circles EF and AB to twice the circle EF.
COROLLARY 4. And hence the weight of all the water in the vessel ABDC is to the part of the weight sustained by the bottom of the vessel as the sum of the circles AB and EF is to the difference between these circles.
COROLLARY 5. And the part of the weight sustained by the bottom of the vessel is to the part of the weight used in making the water flow down as the difference between the circles AB and EF is to twice the smaller circle EF, or as the area of the bottom to twice the hole.
COROLLARY 6. And the part of the weight which alone presses upon the bottom is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the sum of the circles AB and EF, or as the circle AB is to the amount by which twice the circle AB exceeds the bottom. For the part of the weight which alone presses upon the bottom is to the weight of all the water in the vessel as the difference between the circles AB and EF is to the sum of these circles, by corol. 4; and the weight of all the water in the vessel is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the difference between the circles AB and EF. Therefore, from the equality of the ratios in inordinate proportion [or ex aequo perturbate], the part of the weight which alone presses upon the bottom is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the sum of the circles AB and EF, or to the amount by which twice the circle AB exceeds the bottom.
COROLLARY 7. If in the middle of the hole EF there is placed a little
circle PQ described with center G and parallel to the horizon, the weight
of the water which that little circle sustains
is greater than the weight of ⅓ of a
cylinder of water whose base is that little circle
and whose height is GH. For let ABNFEM
be a cataract or column of falling water, with
axis GH as above, and suppose that there has
been a freezing of all the water in the vessel
(around the cataract as well as above the little
circle) whose fluidity is not required for the
very ready and very swift descent of the water. And let PHQ be the frozen
column of water above the little circle, having vertex H and height GH.
And imagine that this cataract falls with its whole weight and does not rest
or press on PHQ but slides past freely and without friction, except perhaps
at the very vertex of the ice, where at the very beginning of falling the
cataract begins to be concave. And just as the frozen water (AMEC and
BNFD) which is around the cataract is convex on the inner surface (AME
and BNF) toward the falling cataract, so also this column PHQ will be
convex toward the cataract, and therefore will be greater than a cone whose
base is the little circle PQ and whose height is GH, that is, greater than ⅓
of a cylinder described with the same base and height. And the little circle
sustains the weight of this column, that is, a weight that is greater than the
weight of the cone or of ⅓ of the cylinder.
COROLLARY 8. The weight of the water sustained by the little circle PQ, when it is extremely small, appears to be less than the weight of ⅔ of a cylinder of water whose base is that little circle and whose height is HG. Keeping the same suppositions, imagine that half a spheroid is described, whose base is the little circle and whose semiaxis or height is HG. Then this figure will be equal to ⅔ of that cylinder and will comprehend the column of frozen water PHQ whose weight that little circle sustains. In order that the motion of the water may be straight down, the outer surface of this column must meet the base PQ in a somewhat acute angle, because the water in falling is continually accelerated and the acceleration makes the column become narrower; and since that angle is less than a right angle, the lower parts of this column will lie within the half-spheroid. But higher up, the column will be acute or pointed, for otherwise the horizontal motion of the water at the vertex of the spheroid would be infinitely swifter than its motion toward the horizon. And the smaller the little circle PQ, the more acute the vertex of the column; and if the little circle is diminished indefinitely, the angle PHQ will be diminished indefinitely, and therefore the column will lie within the half-spheroid. That column is therefore less than the half-spheroid, or less than ⅔ of a cylinder whose base is that little circle and whose height is GH. Moreover, the little circle sustains the water’s force equal to the weight of this column, since the weight of the surrounding water is used in making it flow down.
COROLLARY 9. The weight of the water sustained by the little circle PQ, when it is extremely small, is very nearly equal to the weight of a cylinder of water whose base is that little circle and whose height is ½GH. For this weight is an arithmetical mean between the weights of the cone and the said half-spheroid. If, however, the little circle is not extremely small but is increased until it equals the hole EF, it will sustain the weight of all the water resting perpendicularly on it, that is, the weight of a cylinder of water whose base is that little circle and whose height is GH.
COROLLARY 10. And (as far as I can tell) the weight that the little circle sustains always has the proportion to the weight of a cylinder of water whose base is that little circle and whose height is ½GH that EF2 has to EF2 − ½PQ2, or that the circle EF has to the excess of this circle over half of the little circle PQ, very nearly.
Lemma 4
The resistance of a cylinder moving uniformly forward in the direction of its length
is not changed by an increase or decrease in length and thus is the same as the
resistance of a circle described with the same diameter and moving forward with
the same velocity along a straight line perpendicular to its plane.
For the sides of a cylinder offer no opposition to its motion, and a cylinder is turned into a circle if its length is decreased indefinitely.
Proposition 37, Theorem 29
If a cylinder moves uniformly forward in a compressed, infinite, and nonelastic
fluid in the direction of its own length, its resistance arising from the magnitude
of its transverse section is to the force by which its whole motion can be either
destroyed or generated, while it is describing four times its length, very nearly as
the density of the medium is to the density of the cylinder.
For if the bottom CD of the vessel ABDC touches the surface of stagnant
water, and if water flows out of this vessel into the stagnant water through
the cylindrical channel EFTS perpendicular to
the horizon, and if the little circle PQ is placed
parallel to the horizon anywhere in the middle
of the channel, and if CA is produced to K so
that CK is to AK in the squared ratio of the
circle AB to the amount by which the opening
of the channel EF exceeds the little circle PQ,
then it is obvious (by prop. 36, case 5, case 6,
and corol. 1) that the velocity of the water
passing through the annular space between the little
circle and the sides of the vessel will be that which the water can acquire in
falling and describing by its fall a space equal to the height KC or IG.
And (by prop. 36, corol. 10) if the width of the vessel is infinite, so that the line-element HI vanishes and the heights IG and HG are equal, then the force of the water flowing down into the little circle will be to the weight of a cylinder whose base is that little circle, and whose height is ½IG, very nearly as EF2 to EF2 − ½PQ2. For the force of the water flowing down through the whole channel with uniform motion will be the same upon the little circle PQ in whatever part of the channel it is placed.
Now let the openings EF and ST of the channel be closed, and let the little circle ascend in the fluid compressed on all sides, and by its ascent let it make the upper water descend through the annular space between the little circle and the sides of the channel; then the velocity of the ascending little circle will be to the velocity of the descending water as the difference between the circles EF and PQ is to the circle PQ, and the velocity of the ascending little circle will be to the sum of the velocities (that is, to the relative velocity of the descending water, with which it flows past the ascending little circle) as the difference between the circles EF and PQ is to the circle EF, or as EF2 − PQ2 to EF2. Let that relative velocity be equal to the velocity with which (as shown above) the water passes through the same annular space while the little circle remains unmoved, that is, to the velocity that the water can acquire in falling and describing by its fall a space equal to the height IG; then the force of the water upon the ascending little circle will be the same as before (by corol. 5 of the laws), that is, the resistance of the ascending little circle will be to the weight of a cylinder of water whose base is that little circle, and whose height is ½IG, very nearly as EF2 to EF2 − ½PQ2. And the velocity of the little circle will be to the velocity that the water acquires in falling, and describing by its fall a space equal to the height IG, as EF2 − PQ2 to EF2.
Let the breadth of the channel be increased indefinitely; then those ratios between EF2 − PQ2 and EF2 and between EF2 and EF2 − ½PQ2 will ultimately approach ratios of equality. And therefore the velocity of the little circle will now be that which the water can acquire in falling and describing by its fall a space equal to the height IG, and its resistance will come out equal to the weight of a cylinder whose base is that little circle and whose height is half of the height IG from which the cylinder must fall in order to acquire the velocity of the ascending little circle, and with this velocity the cylinder will, in the time of falling, describe four times its own length. And the resistance of the cylinder, moving forward with this velocity in the direction of its length, is the same as the resistance of the little circle (by lem. 4) and thus is very nearly equal to the force by which its motion can be generated while it is describing four times its length.
If the length of the cylinder is increased or decreased, its motion, and also the time in which it describes four times its length, will be increased or decreased in the same ratio; and thus that force by which the increased or decreased motion, in a time equally increased or decreased, could be generated or destroyed will not be changed and accordingly is under these circumstances still equal to the resistance of the cylinder; for this also remains unchanged, by lem. 4.
If the density of the cylinder is increased or decreased, its motion, and also the force by which the motion can be generated or destroyed in the same time, will be increased or decreased in the same ratio. The resistance, therefore, of any cylinder to the force by which its whole motion could be either generated or destroyed, while it is describing four times its length, will be very nearly as the density of the medium to the density of the cylinder. Q.E.D.
A fluid must be compressed in order to be continuous, and it must be continuous and nonelastic in order that every pressure arising from its compression may be propagated instantaneously and, by acting equally upon all parts of a moving body, not change the resistance. The pressure arising from the body’s motion is of course used in generating the motion of the parts of the fluid and creates resistance. But the pressure arising from the compression of the fluid, however strong it may be, if it is propagated instantaneously, generates no motion in the parts of a continuous fluid, introduces no change of motion at all, and thus neither increases nor decreases the resistance. Certainly the action of a fluid that arises from its compression cannot be stronger upon the back of a moving body than upon the front and thus cannot decrease the resistance described in this proposition; and the action will not be stronger upon the front than upon the back provided that its propagation is infinitely swifter than the motion of the body pressed. And the action will be infinitely swifter and will be propagated instantaneously provided that the fluid is continuous and nonelastic.
COROLLARY 1. The resistances to cylinders that move uniformly forward in the direction of their lengths in infinite and continuous mediums are in a ratio compounded of the squared ratio of the velocities and the squared ratio of the diameters and the ratio of the density of the mediums.
COROLLARY 2. If the breadth of the channel is not increased
indefinitely, but the cylinder moves forward in the direction of its own length
in an enclosed medium at rest, and meanwhile
its axis coincides with the axis of the channel,
then the resistance to the cylinder will be to
the force by which its whole motion could be
either generated or destroyed, in the time in
which it describes four times its length, in a
ratio compounded of the simple ratio of EF2 to
EF2 − ½PQ2 and the squared ratio of EF2 to
EF2 − PQ2 and the ratio of the density of the
medium to the density of the cylinder.
COROLLARY 3. With the same suppositions, let the length L be to four times the length of the cylinder in a ratio compounded of the simple ratio of EF2 − ½PQ2 to EF2 and the squared ratio of EF2 − PQ2 to EF2; then the resistance of the cylinder will be to the force by which its whole motion could be either destroyed or generated, while it is describing the length L, as the density of the medium to the density of the cylinder.
Scholium
In this proposition we have investigated the resistance arising solely from the
magnitude of the transverse section of a cylinder, without considering the
part of the resistance that can arise from the obliquity of the motions. In
prop. 36, case 1, the flow of the water through the hole EF was impeded
by the obliquity of the motions with which the parts of the water in the
vessel converged from all sides into the hole. Similarly, in this proposition,
the obliquity of the motions with which the parts of the water pressed by
the front end of the cylinder yield to the pressure and diverge on all sides
has these effects: it retards the passage of those motions through the places
around that front end toward the back of the cylinder, it makes the fluid
move to a greater distance, and it increases the resistance in nearly the ratio
with which it decreases the flow of the water from the vessel, that is, in the
squared ratio of 25 to 21, roughly.
In case 1 of prop. 36 we made the parts of the water pass through the
hole EF perpendicularly and in the greatest abundance by supposing that
all the water in the vessel that had been frozen around the cataract, and
whose motion was oblique and useless, remained without motion. Similarly,
in this proposition, in order that the obliquity of the motions may be
annulled, and the parts of the water, by
yielding with the most direct and rapid
motion, may provide the easiest passage
to the cylinder, and in order that only
the resistance may remain that arises
from the magnitude of the transverse section and that cannot be decreased
except by decreasing the diameter of the cylinder, it must be understood
that the parts of the fluid whose motions are oblique and useless and
create resistance are at rest with respect to one another at both ends of the
cylinder and cohere and are joined to the cylinder. Let ABDC be a
rectangle, and let AE and BE be two parabolic arcs described with axis AB
and with a latus rectum that is to the space HG, which is to be described
by the falling cylinder while it is acquiring its velocity, as HG to ½AB.
Additionally, let CF and DF be two other parabolic arcs, described with
axis CD and a latus rectum that is four times the former latus rectum; and
by the revolution of the figure about its axis EF, let a solid be generated
whose middle ABDC is the cylinder with which we are dealing, and whose
extremities ABE and CDF contain the parts of the fluid which are at rest
with respect to one another and solidified into two rigid bodies that adhere
to the cylinder at the ends as head and tail. Then the resistance to the solid
EACFDB moving forward in the direction of its axis FE from F toward E
will be very nearly that which we have described in this proposition. That
is, the density of the fluid is to the density of the cylinder very nearly in
the ratio of this resistance to the force by which the whole motion of the
cylinder could be either destroyed or generated, while the length 4AC is
being described with that motion continued uniformly. And with this force the
resistance cannot be less than in the ratio of 2 to 3, by prop. 36, corol. 7.
Lemma 5
If a cylinder, a sphere, and a spheroid, whose widths are equal, are placed
successively in the middle of a cylindrical channel in such a way that their axes coincide
with the axis of the channel, these bodies will equally impede the flow of water
through the channel.
For the spaces through which the water passes between the channel and the cylinder, sphere, and spheroid are equal; and water passes equally through equal spaces.
This is so on the hypothesis that all the water is frozen which is above the cylinder, sphere, or spheroid, and whose fluidity is not required for the very swift passage of the water, as I have explained in prop. 36, corol. 7.
Lemma 6
With the same suppositions, these bodies are equally urged by the water flowing
through the channel.
This is evident by lem. 5 and the third law of motion. Of course, the water and the bodies act equally upon one another.
Lemma 7
If the water is at rest in the channel, and these bodies go through the channel with
equal velocity in opposite directions, the resistances will be equal to one another.
This is clear from lem. 6; for the relative motions remain the same with respect to one another.
Scholium
It is the same for all convex round bodies whose axes coincide with the axis
of the channel. Some difference can arise from a greater or lesser friction;
but in these lemmas we are supposing that the bodies are very smooth, that
the tenacity and friction of the medium are nil, and that the parts of the
fluid which by their oblique and superfluous motions can perturb, impede,
and retard the flow of the water through the channel are at rest with respect
to one another as if icebound and adhere to the front and back of the bodies,
as I have explained in the scholium to prop. 37. For what follows deals with
the least possible resistance of round bodies described with the greatest given
transverse sections.
Bodies moving straight ahead in fluids make the fluid ascend in front of them and subside in back of them, especially if they are blunt in shape; and hence they encounter a slightly greater resistance than if they had pointed heads and tails. And bodies moving in elastic fluids, if they are blunt in front and in back, condense the fluid a little more at the front and make it a little less dense at the back; and hence they encounter a slightly greater resistance than if they had pointed heads and tails. But in these lemmas and propositions we are not dealing with elastic fluids but with nonelastic fluids, not with bodies floating on the surface of the fluid but with bodies deeply immersed. And once the resistance of bodies in nonelastic fluids is known, this resistance will have to be increased somewhat for elastic fluids such as air as well as for the surfaces of stagnant fluids such as seas and swamps.
Proposition 38, Theorem 30
The resistance to a sphere moving uniformly forward in an infinite and nonelastic
compressed fluid is to the force by which its whole motion could either be destroyed
or generated, in the time in which it describes 8/3 of its diameter, very nearly as
the density of the fluid to the density of the sphere.
For a sphere is to the circumscribed cylinder as 2 to 3, and therefore the force that could take away all the motion of a cylinder, while the cylinder is describing a length of four diameters, will take away all the motion of the sphere while the sphere describes ⅔ of this length, that is, 8/3 of its own diameter. And the resistance of the cylinder is to this force very nearly as the density of the fluid to the density of the cylinder or sphere, by prop. 37, and the resistance of the sphere is equal to the resistance of the cylinder, by lems. 5, 6, and 7. Q.E.D.
COROLLARY 1. The resistances of spheres in infinite compressed mediums are in a ratio compounded of the squared ratio of the velocity and the squared ratio of the diameter and the ratio of the density of the mediums.
COROLLARY 2. The greatest velocity with which a sphere, by the force of its own relative weight, can descend in a resisting fluid is that which the same sphere with the same weight can acquire in falling without resistance and describing by its fall a space that is to 4/3 of its diameter as the density of the sphere to the density of the fluid. For the sphere in the time of its fall, with the velocity acquired in falling, will describe a space that will be to 8/3 of its diameter as the density of the sphere to the density of the fluid; and the force of its weight generating this motion will be to the force that could generate the same motion, in the time in which the sphere describes 8/3 of its diameter with the same velocity, as the density of the fluid to the density of the sphere; and thus, by this proposition, the force of its weight will be equal to the force of resistance and therefore cannot accelerate the sphere.
COROLLARY 3. Given both the density of the sphere and its velocity at the beginning of the motion, and also the density of the compressed fluid at rest in which the sphere moves, then by prop. 35, corol. 7, the velocity of the sphere, its resistance, and the space described by it are given for any time.
COROLLARY 4. A sphere moving in a compressed fluid at rest, having the same density as itself, will, by the same corol. 7, lose half of its motion before it has described the length of two of its diameters.
Proposition 39, Theorem 31
The resistance to a sphere moving uniformly forward through a fluid enclosed and
compressed in a cylindrical channel is to the force by which its whole motion could
be either generated or destroyed, while it describes 8/3 of its diameter, in a ratio
compounded of three ratios, very nearly: the ratio of the opening of the channel
to the excess of this opening over half of a great circle of the sphere, the squared
ratio of the opening of the channel to the excess of this opening over a great circle
of the sphere, and the ratio of the density of the fluid to the density of the sphere.
This is evident by prop. 37, corol. 2, and the proof proceeds as in prop. 38.
Scholium
In the last two propositions (as in lem. 5) I assume that all the water which
is in front of the sphere, and whose fluidity increases the resistance to the
sphere, is frozen. If all that water liquefies, the resistance will be somewhat
increased. But in these propositions the increase will be small and can be
ignored because the convex surface of the sphere has almost the same effect
as ice.
Proposition 40, Problem 9
To find from phenomena the resistance of a sphere moving forward in a compressed,
very fluid medium.
Let A be the weight of the sphere in a vacuum, B its weight in a resisting medium, D the diameter of the sphere, F a space that is to 4/3D as the density of the sphere to the density of the medium (that is, as A to A − B), G the time in which the sphere in falling by its weight B without resistance describes the space F, and H the velocity that the sphere acquires by this fall. Then H will be the greatest velocity with which the sphere can descend by its weight B in the resisting medium, by prop. 38, corol. 2, and the resistance that the sphere encounters while descending with this velocity will be equal to its weight B; and the resistance that it encounters with any other velocity will be to the weight B as the square of the ratio of this velocity to the greatest velocity H, by prop. 38, corol. 1.
This is the resistance that arises from the inertia of matter of the fluid. And that which arises from the elasticity, tenacity, and friction of its parts can be investigated as follows.
Drop the sphere so that it descends in the fluid by its own weight B;
and let P be the time of falling, in seconds if the time G is in seconds. Find
the absolute number N that corresponds to the logarithm 0.4342944819
,
and let L be the logarithm of the number 
, then the velocity
acquired in falling will be 
H, and the space described will be
− 1.3862943611F + 4.605170186LF.
If the fluid is sufficiently deep, the term 4.605170186LF can be ignored,
and 
− 1.3862943611F will be the space described, very nearly. These
things are evident by book 2, prop. 9 and its corollaries, on the hypothesis
that the sphere encounters no other resistance than that which arises from
the inertia of matter. But if it encounters another resistance in addition, the
descent will be slower, and the quantity of this resistance can be found from
the retardation.
So that the velocity and descent of a body falling in a fluid may be
found more easily, I have put together the accompanying table, in which the
first column denotes the times of descent, the second shows the velocities
acquired in falling (the greatest velocity being 100,000,000), the third shows
the spaces described in falling in those times (2F being the space that the body
describes in the time G with the greatest velocity), and the fourth shows the
spaces described in the same times with the greatest velocity. The numbers
in the fourth column are 
, and by subtracting the number 1.3862944 −
4.6051702L, the numbers in the third column are found, and these numbers
must be multiplied by the space F in order to get the spaces described in
falling. There has been added to these a fifth column, which contains the
spaces described in the same times by a body falling in a vacuum by the
force of its relative weight B.
Scholium
In order to investigate the resistances of fluids by experiments, I got a square
wooden vessel, with an internal length and width of 9 inches (of a London
foot), and a depth of 9½ feet, and I filled it with rainwater; and making
balls of wax with lead inside, I noted the times of descent of the balls, the
space of the descent being 112 inches. A solid cubic London foot contains
76 Roman pounds [troy] of rainwater, and a solid inch of this foot contains
19/36 ounce of this pound or 253⅓ grains; and a sphere of water described
with a diameter of 1 inch contains 132.645 grains in air, or 132.8 grains in a
vacuum; and any other ball is as the excess of its weight in a vacuum over
its weight in water.
EXPERIMENT 1. A ball which weighed 156¼ grains in air and 77 grains in water described the whole space of 112 inches [when dropped in water] in a time of 4 seconds. And when the experiment was repeated, the ball again fell in the same time of 4 seconds.
The weight of the ball in a vacuum is 15613/38 grains, and the excess of this weight over the weight of the ball in water is 7913/38 grains. And hence the diameter of the ball comes out 0.84224 inch. That excess is to the weight of the ball in a vacuum as the density of water to the density of the ball, and as 8/3 of the diameter of the ball (that is, 2.24597 inches) to the space 2F, which accordingly will be 4.4256 inches. In a time of 1 second the ball will fall in a vacuum by its whole weight of 15613/38 grains through 193⅓ inches; and by a weight of 77 grains falling in water without resistance, it will in the same time describe 95.219 inches; and in the time G, which is to 1 second as the square root of the ratio of the space F or 2.2128 inches to 95.219 inches, it will describe 2.2128 inches and will attain the greatest velocity H with which it can descend in water. Therefore the time G is 0.15244 seconds. And in this time G, with that greatest velocity H, the ball will describe a space 2F of 4.4256 inches; and thus in the time of 4 seconds it will describe a space of 116.1245 inches. Subtract the space 1.3862944F or 3.0676 inches and there will remain a space of 113.0569 inches which the ball will describe in falling in water in a very wide vessel in the time of 4 seconds. This space, because of the narrowness of the wooden vessel, must be decreased in a ratio which is compounded of the square root of the ratio of the opening of the vessel to the excess of this opening over a great semicircle of the ball, and of the simple ratio of that same opening to its excess over a great circle of the ball, that is, in the ratio of 1 to 0.9914. When this has been done, the result will be a space of 112.08 inches which the ball should, according to the theory, have very nearly described in falling in water in this wooden vessel in the time of 4 seconds. And it described 112 inches in the experiment.
EXPERIMENT 2. Three equal balls, each of which weighed 76⅓ grains in air and 51/16 grains in water, were dropped successively in water, and in a time of 15 seconds each one fell through 112 inches.
By computation the weight of a ball in a vacuum is 765/12 grains; the excess of this weight over the weight in water is 711748 grains; the diameter of the ball is 0.81296 inch; 8/3 of this diameter is 2.16789 inches; the space 2F is 2.3217 inches; the space that a ball describes in falling by a weight of 51/16 grains in the time of 1 second without resistance is 12.808 inches; and the time G is 0.301056 second. The ball, therefore, with the greatest velocity with which it can descend in water by the force of the weight of 51/16 grains, will describe in a time of 0.301056 second a space of 2.3217 inches, and in the time of 15 seconds a space of 115.678 inches. Subtract the space 1.3862944F or 1.609 inches, and there will remain a space of 114.069 inches which accordingly the ball ought to describe in falling in the same time in a very wide vessel. Because of the narrowness of our vessel a space of roughly 0.895 inch must be taken away. And thus there will remain a space of 113.174 inches which the ball, according to the theory, should have very nearly described in falling in this vessel in the time of 15 seconds. And it described 112 inches in the experiment. The difference is imperceptible.
EXPERIMENT 3. Three equal balls, each of which weighed 121 grains in air and 1 grain in water, were dropped successively in water, and in times of 46 seconds, 47 seconds, and 50 seconds, fell 112 inches.
According to the theory, these balls should have fallen in a time of roughly 40 seconds. I am uncertain whether their falling more slowly is to be attributed to the smaller proportion of the resistance that arises from the force of inertia in slow motions to the resistance that arises from other causes, or rather to some little bubbles adhering to the ball, or to the rarefaction of the wax from the heat either of the weather or of the hand dropping the ball, or even to imperceptible errors in weighing the balls in water. And thus the weight of the ball in water ought to be more than 1 grain, so that the experiment may be made certain and trustworthy.
EXPERIMENT 4. I began the experiments thus far described in order to investigate the resistances of fluids before formulating the theory set forth in the immediately preceding propositions. Afterward, in order to examine that theory, I obtained a wooden vessel with an internal width of 8⅔ inches and a depth of 15⅓ feet. Then I made four balls out of wax with lead inside, each one weighing 139¼ grains in air and 7⅛ grains in water. And I let them fall in water in order to measure the times of falling, using a pendulum oscillating in half-seconds. When the balls were being weighed, and afterward when they were falling, they were cold and had remained cold for some time, because heat rarefies the wax and by the rarefaction diminishes the weight of the ball in water, and the rarefied wax is not immediately brought back to its original density by chilling. Before they fell, they were entirely immersed in water, so that their descent might not be accelerated at the beginning by the weight of some part projecting out of the water. And when totally immersed and at rest, they were let fall as carefully as possible, so as not to receive some impulse from the hand letting them fall. And they fell successively in the times of 47½, 48½, 50, and 51 oscillations, describing a space of 15 feet 2 inches. But the weather was now a little colder than when the balls were weighed, and so I repeated the experiment on another day, and the balls fell in the times of 49, 49½, 50, and 53 oscillations, and on a third day in the times of 49½, 50, 51, and 53 oscillations. The experiment was made quite often, and the balls for the most part fell in the times of 49½ and 50 oscillations. When they fell more slowly, I suspect that they were retarded by hitting against the sides of the vessel.
Now by computation according to the theory, the weight of a ball in a vacuum is 139⅖ grains; the excess of this weight over the weight of the ball in water is 13211/40 grains; the diameter of the ball is 0.99868 inch; 8/3 of the diameter is 2.66315 inches; the space 2F is 2.8066 inches; the space that a ball describes in falling with a weight of 7⅛ grains in the time of 1 second without resistance is 9.88164 inches; and the time G is 0.376843 second. The ball, therefore, with the greatest velocity with which it can descend in water by a force of weight of 7⅛ grains, describes in the time of 0.376843 second a space of 2.8066 inches; in the time of 1 second a space of 7.44766 inches; and in the time of 25 seconds, or 50 oscillations, a space of 186.1915 inches. Subtract the space 1.386294F, or 1.9454 inches, and there will remain the space of 184.2461 inches which the ball will describe in the same time in a very wide vessel. Because of the narrowness of our vessel, decrease this space in a ratio that is compounded of the square root of the ratio of the opening of the vessel to the excess of this opening over a great semicircle of the ball, and the simple ratio of that same opening to its excess over a great circle of the ball, and the result will be the space of 181.86 inches which the ball, according to the theory, should very nearly have described in this vessel in the time of 50 oscillations. And in the experiment it described a space of 182 inches in the time of 49½ or 50 oscillations.
EXPERIMENT 5. Four balls weighing 154⅜ grains in air and 21½ grains in water were dropped often and fell in the times of 28½, 29, 29½, and 30 oscillations, and sometimes 31, 32, and 33, describing a space of 15 feet 2 inches.
By the theory they ought to have fallen in the time of very nearly 29 oscillations.
EXPERIMENT 6. Five balls weighing 212⅜ grains in air and 79½ in water were dropped often and fell in the times of 15, 15½, 16, 17, and 18 oscillations, describing a space of 15 feet 2 inches.
By the theory they ought to have fallen in the time of very nearly 15 oscillations.
EXPERIMENT 7. Four balls weighing 293⅜ grains in air and 35⅞ grains in water were dropped often and fell in the times of 29½, 30, 30½, 31, 32, and 33 oscillations, describing a space of 15 feet 1½ inches.
By the theory they ought to have fallen in the time of very nearly 28 oscillations.
In investigating the reason why some of the balls which were of the same weight and size fell more quickly and others more slowly, I hit upon this: that when the balls were first dropped and were beginning to fall, the side which happened to be heavier descended first and generated an oscillatory motion, so that they oscillated around their centers. For by its oscillations a ball communicates a greater motion to the water than if it were descending without oscillations, and in the process loses part of its own motion with which it should descend; and it is retarded more or retarded less in proportion to the greatness or smallness of the oscillation. Further, the ball always recedes from that side which is descending in the oscillation and, by receding, approaches the sides of the vessel and sometimes strikes against the sides. In the case of heavier balls, this oscillation is stronger, and with larger balls, it agitates the water more. Therefore, in order to reduce the oscillation of the balls, I constructed new balls of wax and lead, fixing the lead into one side of the ball near its surface; and I dropped the ball in such a way that the heavier side, as far as possible, was lowest at the beginning of the descent. Thus the oscillations became much smaller than before, and the balls fell in less unequal times, as in the following experiments.
EXPERIMENT 8. Four balls, weighing 139 grains in air and 6½ in water, were dropped often and fell in the times of not more than 52 oscillations, and not fewer than 50, and for the most part in the time of roughly 51 oscillations, describing a space of 182 inches.
By the theory they ought to have fallen in the time of roughly 52 oscillations.
EXPERIMENT 9. Four balls, weighing 273¼ grains in air and 140¾ in water, were dropped often and fell in the times of not fewer than 12 oscillations and not more than 13, describing a space of 182 inches.
And by the theory these balls ought to have fallen in the time of very nearly 11⅓ oscillations.
EXPERIMENT 10. Four balls, weighing 384 grains in air and 119½ in water, were dropped often and fell in the times of 17¾, 18, 18½, and 19 oscillations, describing a space of 181½ inches. And when they fell in the time of 19 oscillations, I sometimes heard them strike the sides of the vessel before they reached the bottom.
And by the theory they ought to have fallen in the time of very nearly 155/9 oscillations.
EXPERIMENT 11. Three equal balls, weighing 48 grains in air and 329/32 in water, were dropped often and fell in the times of 43½, 44, 44½, 45, and 46 oscillations, and for the most part 44 and 45, describing a space of very nearly 182½ inches.
By the theory they ought to have fallen in the time of roughly 465/9 oscillations.
EXPERIMENT 12. Three equal balls, weighing 141 grains in air and 4⅜ in water, were dropped several times and fell in the times of 61, 62, 63, 64, and 65 oscillations, describing a space of 182 inches.
And by the theory they ought to have fallen in the time of very nearly 64½ oscillations.
From these experiments it is obvious that when the balls fell slowly (as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments), the times of falling are shown correctly by the theory, but that when the balls fell more quickly (as in the sixth, ninth, and tenth experiments), the resistance was a little greater than in the squared ratio of the velocity. For the balls oscillate somewhat while falling, and this oscillation—in balls that are lighter and fall more slowly—ceases swiftly because the motion is weak, while in heavier and larger balls, because the motion is strong, the oscillation lasts longer and can be checked by the surrounding water only after more oscillations. Additionally, the swifter the balls, the less they are pressed by the fluid in back of them; and if the velocity is continually increased, they will at length leave an empty space behind, unless the compression of the fluid is simultaneously increased. The compression of the fluid, moreover, ought (by props. 32 and 33) to be increased in the squared ratio of the velocity in order for the resistance also to be in a squared ratio. Since this does not happen, the swifter balls are pressed a little less from behind, and because of this diminished pressure their resistance becomes a little greater than in the squared ratio of the velocity.
The theory therefore agrees with the phenomena of bodies falling in water; it remains for us to examine the phenomena of bodies falling in air.
EXPERIMENT 13. aFrom the top of St. Paul’s Cathedral in Londona in June 1710, glass balls were dropped simultaneously in pairs, one full of quicksilver, the other full of air; and in falling they described a space of 220 London feet. A wooden platform was suspended at one end by iron pivots, and at the other was supported by a wooden peg. The two balls were placed upon this platform and were let fall simultaneously by pulling out the peg by means of an iron wire extending to the ground, so that the platform, resting on the iron pivots alone, might swing downward upon the pivots and at the same moment a seconds pendulum, pulled by that iron wire, might be released and begin to oscillate. The diameters and weights of the balls and the times of falling are shown in the following table.
However, the observed times need to be corrected. For balls filled with mercury will (by Galileo’s theory) describe 257 London feet in 4 seconds, and 220 feet in only 3 seconds 42 thirds. The wooden platform, when the peg was withdrawn, swung downward more slowly than it should have [i.e., more slowly than in free fall] and as a result impeded the descent of the balls at the start. For the balls were lying upon the platform near its center, and were in fact a little closer to the pivots than to the peg. And hence the times of falling were prolonged by roughly 18 thirds and so need to be corrected by taking away those thirds, especially in the larger balls, which because of the magnitude of their diameters remained a little longer upon the platform as it swung downward. When this has been done, the times in which the six larger balls fell will come out 8 sec. 12 thirds, 7 sec. 42 thirds, 7 sec. 42 thirds, 7 sec. 57 thirds, 8 sec. 12 thirds, and 7 sec. 42 thirds.
Therefore the fifth of those balls filled with air, with a diameter of 5
inches and a weight of 483 grains, fell in the time of 8 sec. 12 thirds,
describing the space of 220 feet. The weight of water equal to this ball is 16,600
grains; and the weight of air equal to it is 
grains, or 193/10 grains, and
thus the weight of the ball in a vacuum is 5023/10 grains, and this weight is
to the weight of air equal to the ball as 5023/10 to 193/10, as is the ratio of 2F
to 8/3 of the diameter of the ball (that is, 2F to 13⅓ inches). And hence 2F
comes out 28 feet 11 inches. The ball in falling in a vacuum, with its whole
weight of 5023/10 grains, in the time of one second describes 193⅓ inches as
above, and with a weight of 483 grains describes 185.905 inches, and with
the same weight of 483 grains also in a vacuum describes the space F, or 14
feet 5½ inches, in the time of 57 thirds 58 fourths, and attains the greatest
velocity with which it could descend in air. With this velocity the ball, in the
time of 8 sec. 12 thirds, will describe a space of 245 feet 5⅓ inches. Take
away 1.3863F, or 20 feet ½ inch, and there will remain 225 feet 5 inches. It
is this space, therefore, that the ball should, by the theory, have described in
falling in the time of 8 sec. 12 thirds. And it described a space of 220 feet in
the experiment. The difference is negligible.
Applying similar computations also to the remaining balls filled with air, I constructed the following table.
EXPERIMENT 14. In July 1719, Dr. Desaguliers made experiments of this sort again, making hogs’ bladders into a round shape by means of a concave wooden sphere, which the moist bladders, inflated with air, were forced to fill; after they were dried and taken out, they were dropped bfrom the lantern at the top of the cupola of the same cathedral, that is, from a height of 272 feet,b and at the same moment a lead ball was also dropped, whose weight was roughly two pounds troy. And meanwhile some people standing in the highest part of St. Paul’s where the balls were released noted the whole times of falling, and others standing on the ground noted the difference between the times of fall of the lead ball and of the bladder. And the times were measured by half-second pendulums. And one of those who were standing on the ground had a clock with an oscillating spring, vibrating four times per second; someone else had another machine ingeniously constructed with a pendulum also vibrating four times per second. And one of those who were standing in the gallery of the cupola had a similar device. And these instruments were so constructed that their motions might begin or be stopped at will. The lead ball fell in a time of roughly 4¼ seconds. And by adding this time to the aforesaid difference between the times, the whole time in which the bladder fell was determined. The times in which the five bladders continued to fall after the lead ball had completed its fall were 14¾ sec., 12¾ sec., 14⅝ sec., 17¾ sec., and 16⅞ sec. the first time, and 14½ sec., 14¼ sec., 14 sec., 19 sec., and 16¾ sec. the second time. Add 4¼ sec., the time in which the lead ball fell, and the whole times in which the five bladders fell were 19 sec., 17 sec., 18⅞ sec., 22 sec., and 21⅛ sec. the first time, and 18¾ sec., 18½ sec., 18¼ sec., 23¼ sec., and 21 sec. the second time. And the times noted from the cupola were 19⅜ sec., 17¼ sec., 18¾ sec., 22⅛ sec., and 21⅝ sec. the first time, and 19 sec., 18⅝ sec., 18⅜ sec., 24 sec., and 21¼ sec. the second time. But the bladders did not always fall straight down, but sometimes flew about and oscillated to and fro while falling. And the times of falling were prolonged and increased by these motions, sometimes by one-half of one second, sometimes by a whole second. The second and fourth bladders, moreover, fell straighter down the first time, as did the first and third the second time. The fifth bladder was wrinkled and was somewhat retarded by its wrinkles. I calculated the diameters of the bladders from their circumferences, measured by a very thin thread wound round them twice. And I compared the theory with the experiments in the following table, assuming the density of air to be to the density of rainwater as 1 to 860, and calculating the spaces that the balls should, by the theory, have described in falling.
Therefore almost all the resistance encountered by balls moving in air as well as in water is correctly shown by our theory, and is proportional to the density of the fluids—the velocities and sizes of the balls being equal.
In the scholium at the end of sec. 6, we showed by experiments with
pendulums that the resistances encountered by equal and equally swift balls
moving in air, water, and quicksilver are as the densities of the fluids. We
have shown the same thing here more accurately by experiments with bodies
falling in air and water. For pendulums in each oscillation arouse in the fluid
a motion always opposite to the motion of the pendulum when it returns; and
the resistance arising from this motion, and also the resistance to the cord
by which the pendulum was suspended, made the whole resistance to the
pendulum greater than the resistance found by the experiments with falling
bodies. For by the experiments with pendulums set forth in that scholium, a
ball of the same density as water ought, in describing the length of its own
semidiameter in air, to lose 
of its motion. But by the theory set forth in
this seventh section and confirmed by experiments with falling bodies, that
same ball ought, in describing that same length, to lose only 
of its
motion, supposing that the density of water is to the density of air as 860
to 1. The resistances therefore were found to be greater by the experiments
with pendulums (for the reasons already described) than by the experiments
with falling balls, and in a ratio of roughly 4 to 3. But since the resistances
to pendulums oscillating in air, water, and quicksilver are increased similarly
by similar causes, the proportion of the resistances in these mediums will be
shown correctly enough by the experiments with pendulums as well as by
the experiments with falling bodies. And hence it can be concluded that the
resistances encountered by bodies moving in any fluids that are very fluid,
other things being equal, are as the densities of the fluids.
On the basis of what has been established, it is now possible to predict
very nearly what part of the motion of any ball projected in any fluid will
be lost in a given time. Let D be the diameter of the ball, and V its velocity
at the beginning of the motion, and T the time in which the ball will—with
velocity V in a vacuum—describe a space that is to the space 8/3D as the
density of the ball to the density of the fluid; then the ball projected in that
fluid will, in any other time t, lose the part 
of its velocity the part
remaining 
and will describe a space that is to the space described in
a vacuum in the same time with the uniform velocity V as the logarithm of
the number 
multiplied by the number 2.302585093 is to the number
t/T, by prop. 35, corol. 7. In slow motions the resistance can be a little
less, because the shape of a ball is a little more suitable for motion than the
shape of a cylinder described with the same diameter. In swift motions the
resistance can be a little greater, because the elasticity and the compression
of the fluid are not increased in the squared ratio of the velocity. But here I
am not considering petty details of this sort.
And even if air, water, quicksilver, and similar fluids, by some infinite division of their parts, could be subtilized and become infinitely fluid mediums, they would not resist projected balls any the less. For the resistance which is the subject of the preceding propositions arises from the inertia of matter; and the inertia of matter is essential to bodies and is always proportional to the quantity of matter. By the division of the parts of a fluid, the resistance that arises from the tenacity and friction of the parts can indeed be diminished, but the quantity of matter is not diminished by the division of its parts; and since the quantity of matter remains the same, its force of inertia—to which the resistance discussed here is always proportional—remains the same. For this resistance to be diminished, the quantity of matter in the spaces through which bodies move must be diminished. And therefore the celestial spaces, through which the globes of the planets and comets move continually in all directions very freely and without any sensible diminution of motion, are devoid of any corporeal fluid, except perhaps the very rarest vapors and rays of light transmitted through those spaces.
Projectiles, of course, arouse motion in fluids by going through them, and this motion arises from the excess of the pressure of the fluid on the front of the projectile over the pressure on the back, and cannot be less in infinitely fluid mediums than in air, water, and quicksilver in proportion to the density of matter in each. And this excess of pressure, in proportion to its quantity, not only arouses motion in the fluid but also acts upon the projectile to retard its motion; and therefore the resistance in every fluid is as the motion excited in the fluid by the projectile, and it cannot be less in the most subtle aether, in proportion to the density of the aether, than in air, water, and quicksilver, in proportion to the densities of these fluids.
Motion propagated through fluids
Proposition 41, Theorem 32
Pressure is not propagated through a fluid along straight lines, unless the particles
of the fluid lie in a straight line.
If the particles a, b, c, d, and e lie in a straight line, a pressure can indeed
be propagated directly from a to e; but the particle e will urge the obliquely
placed particles f and g obliquely, and those particles
f and g will not sustain the pressure brought upon
them unless they are supported by the further particles
h and k; but to the extent that they are supported,
they press the supporting particles, and these will not
sustain the pressure unless they are supported by the
further particles l and m and press them, and so on indefinitely. Therefore,
as soon as a pressure is propagated to particles which do not lie in a straight
line, it will begin to spread out and will be obliquely propagated indefinitely;
and after the pressure begins to be propagated obliquely, if it should impinge
upon further particles which do not lie in a straight line, it will spread out
again, and will do so as often as it impinges upon particles not lying exactly
in a straight line. Q.E.D.
COROLLARY. If some part of a pressure propagated through a fluid from a given point is intercepted by an obstacle, the remaining part (which is not intercepted) will spread out into the spaces behind the obstacle. This can be proved as follows. From point A let a pressure be propagated in any direction and, if possible, along straight lines; and by the obstacle NBCK, perforated in BC, let all the pressure be intercepted except the cone-shaped part APQ, which passes through the circular hole BC. By transverse planes de, fg, and hi, divide the cone APQ into frusta; then, while the cone ABC, by propagating the pressure, is urging the further conic frustum degf on the surface de, and this frustum is urging the next frustum fgih on the surface fg, and that frustum is urging a third frustum, and so on indefinitely, obviously (by the third law of motion) the first frustum defg will be as much urged and pressed on the surface fg by the reaction of the second frustum fghi as it urges and presses the second frustum. Therefore the frustum degf between the cone Ade and the frustum fhig is compressed on both sides, and therefore (by book 2, prop. 19, case 6) it cannot keep its figure unless it is compressed by the same force on all sides. With the same force, therefore, with which it is pressed on the surfaces de and fg, it will endeavor to yield at the sides df and eg; and there (since it is not rigid, but altogether fluid) it will run out and expand, unless a surrounding fluid is present to restrain that endeavor. Accordingly, by the endeavor to run out, it will press the surrounding fluid at the sides df and eg, as well as the frustum fghi, with the same force; and therefore the pressure will be no less propagated from the sides df and eg into the spaces NO on one side and KL on the other, than it is propagated from the surface fg toward PQ. Q.E.D.
Proposition 42, Theorem 33
All motion propagated through a fluid diverges from a straight path into the
motionless spaces.
CASE 1. Let a motion be propagated from point A through a hole BC, and let it proceed, if possible, in the conic space BCQP along straight lines diverging from point A. And let us suppose first that this motion is that of waves on the surface of stagnant water. And let de, fg, hi, kl, . . . be the highest parts of the individual waves, separated from one another by the same number of intermediate troughs. Therefore, since the water is higher in the crests of the waves than in the motionless parts LK and NO of the fluid, it will flow down from e, g, i, l, . . . , and d, f, h, k, . . . , the ends of the crests, toward KL on one side and NO on the other; and since it is lower in the troughs of the waves than in the motionless parts KL and NO of the fluid, it will flow down from those motionless parts into the troughs of the waves. In one case the crests of the waves, and in the other their troughs, are expanded and propagated toward KL on one side and NO on the other. And since the motion of the waves from A toward PQ takes place by the continual flowing down of the crests into the nearest troughs, and thus is not quicker than in proportion to the quickness of the descent, and since the descent of the water toward KL on one side and NO on the other ought to occur with the same velocity, the expansion of the waves will be propagated toward KL on one side and NO on the other with the same velocity with which the waves themselves progress directly from A toward PQ. And accordingly the whole space toward KL on one side and NO on the other will be occupied by the expanded waves rfgr, shis, tklt, vmnv, . . . . Q.E.D. Anyone can test this in stagnant water.
CASE 2. Now let us suppose that de, fg, hi, kl, and mn designate pulses successively propagated from point A through an elastic medium. Think of the pulses as propagated by successive condensations and rarefactions of the medium, in such a way that the densest part of each pulse occupies a spherical surface described about the center A, and that the spaces which come between successive pulses are equal. And let de, fg, hi, kl, . . . designate the densest parts of the pulses, parts which are propagated through the hole BC. And since the medium is denser there than in the spaces toward KL on one side and NO on the other, it will expand toward those spaces KL and NO situated on both sides as well as toward the rarer intervals between the pulses; and thus, always becoming rarer next to the intervals and denser next to the pulses, the medium will participate in their motion. And since the progressive motion of the pulses arises from the continual slackening of the denser parts toward the rarer intervals in front of them, and since the pulses ought to slacken with nearly the same speed into the medium’s parts KL on one side and NO on the other, which are at rest, those pulses will expand on all sides into the motionless spaces KL and NO with nearly the same speed with which they are propagated straight forward from the center A, and thus will occupy the whole space KLON. Q.E.D. We find this by experience in the case of sounds, which are heard when there is a mountain in the way or which expand into all parts of a room when let in through a window and are heard in all corners, being not so much reflected from the opposite walls as propagated directly from the window, as far as the senses can tell.
CASE 3. Finally, let us suppose that a motion of any kind is propagated from A through the hole BC. That propagation does not occur except insofar as the parts of the medium that are nearer to the center A urge and move the further parts; and the parts that are urged are fluid and thus recede in every direction into regions where they are less pressed, and so will recede toward all the parts of the medium that are at rest, the parts KL and NO on the sides as well as the parts PQ in front. And therefore all the motion, as soon as it has passed through the hole BC, will begin to spread out and to be propagated directly from there into all parts as if from an origin and center. Q.E.D.
Proposition 43, Theorem 34
Every vibrating body in an elastic medium will propagate the motion of the
pulses straight ahead in every direction, but in a nonelastic medium will produce
a circular motion.
CASE 1. For the parts of a vibrating body, by going forward and returning alternately, will in their going urge and propel the parts of the medium that are nearest to them and by that urging will compress and condense them; then in their return they will allow the compressed parts to recede [i.e., to move apart from one another] and expand. Thus the parts of the medium that are nearest to the vibrating body will go and return alternately, like the parts of the vibrating body; and just as the parts of this body acted upon the parts of the medium, so the latter, acted upon by similar vibrations, will act upon the parts nearest to them, and these, similarly acted upon, will act upon further parts, and so on indefinitely. And just as the first parts of the medium condense in going and rarefy in returning, so the remaining parts will condense whenever they go and will expand [i.e., rarefy] whenever they return. And therefore they will not all go and return at the same time (for thus, by keeping determined distances from one another, they would not rarefy and condense alternately), but by approaching one another when they condense and moving apart when they rarefy, some of them will go while others return, and these conditions will alternate indefinitely. And the parts that are going and that condense in going (because of their forward motion with which they strike obstacles) are pulses; and therefore successive pulses will be propagated straight ahead from every vibrating body, and they will be so propagated at roughly equal distances from one another, because of the equal intervals of time in which the body produces each pulse by each of its vibrations. And even if the parts of the vibrating body go and return in some fixed and determined direction, nevertheless the pulses propagated from there through the medium will (by prop. 42) expand sideways and will be propagated in all directions from the vibrating body as if from a common center, in surfaces almost spherical and concentric. We have an example of this in waves, which, if they are produced by a wagging finger, not only will go to and fro according to the finger’s motion but will immediately surround the finger like concentric circles and will be propagated in all directions. For the gravity of the waves takes the place of the elastic force.
CASE 2. But if the medium is not elastic, then, since its parts, pressed by the oscillating parts of the vibrating body, cannot be condensed, the motion will be propagated instantly to the parts where the medium yields most easily, that is, to the parts that the vibrating body would otherwise leave empty behind it. The case is the same as the case of a body projected in any medium. A medium, in yielding to projectiles, does not recede indefinitely, but goes with a circular motion to the spaces that the body leaves behind it. Therefore, whenever a vibrating body goes toward any place [or in any direction], the medium, in yielding, will go with a circular motion to the spaces that the body leaves; and whenever the body returns to its former place, the medium will be forced out and will return to its former place. And even though the vibrating body is not rigid but completely pliant, if it nevertheless remains of a fixed size, then, since it cannot urge the medium by its vibrations in any one place without simultaneously yielding to it in another, that body will make the medium, by receding from the parts where it is pressed, go always with a circular motion to the parts that yield to it. Q.E.D.
COROLLARY. Therefore it is a delusion to believe that the agitation of the parts of flame conduces to the propagation of a pressure along straight lines through a surrounding medium. A pressure of this sort must be derived not only from the agitation of the parts of the flame but from the dilation of the whole.
Proposition 44, Theorem 35
If water ascends and descends alternately in the vertical arms KL and MN of
a tube, and if a pendulum is constructed whose length between the point of
suspension and the center of oscillation is equal to half of the length of the water
in the tube, then I say that the water will ascend and descend in the same times
in which the pendulum oscillates.
I measure the length of the water along the axes of the tube and the arms and make it equal to the sum of these axes, and I do not here consider the resistance of the water that arises from the friction of the tube. Let AB and CD therefore designate the mean height of the water in the two arms, and when the water in the arm KL ascends to the height EF, the water in the arm MN will have descended to the height GH. Moreover, let P be a pendulum bob, VP the cord, V the point of suspension, RPQS the cycloid described by the pendulum, P its lowest point, and PQ an arc equal to the height AE. The force by which the motion of the water is alternately accelerated and retarded is the amount by which the weight of the water in one of the two arms exceeds the weight in the other. And thus, when the water in the arm KL ascends to EF, and in the other arm descends to GH, that force is twice the weight of the water EABF and therefore is to the weight of all the water as AE or PQ to VP or PR. Furthermore, the force by which the weight P in any place Q is accelerated and retarded in the cycloid is (by book 1, prop. 51, corol.) to its whole weight as its distance PQ from the lowest place P to the length PR of the cycloid. Therefore the motive forces of the water and the pendulum, describing the equal spaces AE and PQ, are as the weights that are to be moved; and thus, if the water and the pendulum are at rest in the beginning, those forces will move them equally in equal times and will cause them to go and return synchronously with an alternating motion. Q.E.D.
COROLLARY 1. Therefore all the alternations of the ascending and descending water are isochronous, whether the motion is of greater intension or greater remission.a
COROLLARY 2. If the length of all the water in the tube is 61/9 Paris feet, the water will descend in the time of one second and will ascend in another second and will continue to alternate in this way indefinitely. For a pendulum 31/18 feet long oscillates in the time of one second.
COROLLARY 3. When the length of the water is increased or decreased, moreover, the time of alternation is increased or decreased as the square root of the length.
Proposition 45, Theorem 36
The velocity of waves is as the square roots of the lengths.
This follows from the construction of the following proposition.
Proposition 46, Problem 10
To find the velocity of waves.
Set up a pendulum whose length between the point of suspension and the center of oscillation is equal to the length of the waves; and in the same time in which the pendulum performs each of its oscillations, the waves as they move forward will traverse nearly their own lengths.
By length of a wave I mean the transverse distance either between bottoms of troughs or between tops of crests. Let ABCDEF designate the surface of stagnant water ascending and descending in successive waves; and let A, C, E, . . . be the crests of the waves, and B, D, F, . . . the troughs in between. Since the motion of the waves is caused by the successive ascent and descent of the water, in such a way that its parts, A, C, E, . . . , which now are highest, soon become lowest, and since the motive force by which the highest parts descend and the lowest ascend is the weight of the elevated water, the alternate ascent and descent will be analogous to the alternating motion of the water in the tube and will observe the same laws with respect to times; and therefore (by prop. 44), if the distances between the highest places A, C, and E of the waves and the lowest, B, D, and F, are equal to twice the length of a pendulum, the highest parts A, C, and E will in the time of one oscillation come to be the lowest, and in the time of a second oscillation will ascend once again. Therefore there will be a time of two oscillations between successive waves; that is, a wave will describe its own length in the time in which the pendulum oscillates twice; but in the same time a pendulum whose length is four times as great, and thus equals the length of the waves, will oscillate once. Q.E.I.
COROLLARY 1. Therefore waves with a length of 31/18 Paris feet will move forward through their own length in the time of one second and thus in the time of one minute will traverse 183⅓ feet, and in the space of an hour very nearly 11,000 feet.
COROLLARY 2. And the velocity of waves of greater or smaller length will be increased or decreased as the square root of the length.
What has been said is premised on the hypothesis that the parts of the water go straight up or straight down; but this ascent and descent takes place more truly in a circle, and thus I admit that in this proposition the time has been determined only approximately.
Proposition 47, Theorem 37
If pulses are propagated through a fluid, the individual particles of the fluid, going
and returning with a very short alternating motion, are always accelerated and
retarded in accordance with the law of an oscillating pendulum.
Let AB, BC, CD, . . . designate the
equal distances between successive pulses;
ABC the line of motion of the pulses,
propagated from A toward B; E, F, and
G three physical points in the medium
at rest, situated at equal intervals along
the straight line AC; Ee, Ff, and Gg
very short equal spaces through which
those points go and return in each
vibration with an alternating motion; ε, φ, γ
any intermediate positions of those same
points; and EF and FG physical
line-elements or linear parts of the medium,
put between those points and successively
transferred into the places εφ, φγ and
ef, fg. Draw the straight line PS equal
to the straight line Ee. Bisect PS in O,
and with center O and radius OP
describe the circle SIPi.
Let the whole circumference of this circle with its parts represent the whole time of one vibration with its proportional parts, in such a way that when any time PH or PHSh is completed, if the perpendicular HL or hl is dropped to PS, and if Eε is taken equal to PL or Pl, then the physical point E is found in ε. By this law any point E, in going from E through ε to e and returning from there through ε to E, will perform each vibration with the same degrees of acceleration and retardation as the oscillating pendulum. It is to be proved that each of the physical points of the medium must move in such a way. Let us imagine, therefore, that there is such a motion in the medium, arising from any cause, and see what follows.
In the circumference PHSh take the equal arcs HI and IK or hi and
ik, having the ratio to the whole circumference that the equal straight lines
EF and FG have to the whole interval BC between pulses. Drop the
perpendiculars IM and KN and also im and kn. Then the points E, F, and G
are successively agitated with similar motions and carry out their complete
vibrations (consisting of a going and returning) while a pulse is transferred
from B to C; accordingly, if PH or PHSh is the time from the beginning
of the motion of point E, PI or PHSi will be the time from the beginning
of the motion of point F, and PK or PHSk will be the time from the
beginning of the motion of point G; and therefore Eε, Fφ, and Gγ will be
equal respectively to PL, PM, and PN in the going of the points, or to Pl,
Pm, and Pn in the returning of the points. Hence εγ or EG + Gγ − Eε
will be equal to EG − LN in the going of the points, and will be equal to
EG + ln in their returning. But εγ is the width or expansion of the part of
the medium EG in the place εγ; and therefore the expansion of that part in
the going is to its mean expansion as EG − LN to EG, and in the returning
is as EG + ln or EG + LN to EG. Therefore, since LN is to KH as IM to
the radius OP, and KH is to EG as the circumference PHShP to BC, that
is (if V is put for the radius of a circle having a circumference equal to the
interval between the pulses BC), as OP to V, and since, from the equality of
the ratios [or ex aequo], LN is to EG as IM to V, the expansion of the part
EG or of the physical point F in the place εγ will be to the mean expansion
which that part has in its own first place EG as V − IM to V in the going,
and as V + im to V in the returning. Hence the elastic force of point F in
the place εγ is to its mean elastic force in the place EG as 
to 
in
the going, and as 
to in the returning. And by the same argument
the elastic forces of the physical points E and G in the going are as 
and 
to ; and the difference between the forces is to the mean
elastic force or the medium as 
to
, that is, as 
to , or as HL − KN to V, provided that
(because of the narrow limits of the vibrations) we suppose HL and KN to
be indefinitely smaller than the quantity V. Therefore, since the quantity V
is given, the difference between the forces is as HL — KN, that is, as OM
(because HL − KN is proportional to HK and OM to OI or OP; and HK
and OP are given)—that is, if Ff is bisected in Ω, as Ωφ. And by the same
argument the difference between the elastic forces of the physical points ε
and γ, in the returning of the physical line-element εγ, is as Ωφ. But that
difference (that is, the amount by which the elastic force of point ε exceeds
the elastic force of point γ) is the force by which the intervening physical
line-element εγ of the medium is accelerated in the going and retarded in
the returning; and therefore the accelerative force of the physical line-element
εγ is as its distance from the midpoint Ω of the vibration. Accordingly, the
time (by book 1, prop. 38) is correctly represented by the arc PI, and the
linear part εγ of the medium moves by the law previously mentioned, that
is, by the law of an oscillating pendulum; and the same is true of all the
linear parts of which the whole medium is composed. Q.E.D.
COROLLARY. Hence it is evident that the number of pulses propagated is the same as the number of vibrations of the vibrating body and does not increase as the pulses move forward. For as soon as the physical line-element εγ has returned to its first place, it will be at rest and will not move afterward unless it receives a new motion either by the impact of the vibrating body or by the impact of pulses that are propagated from the vibrating body. It will be at rest, therefore, as soon as the pulses cease to be propagated from the vibrating body.
Proposition 48, Theorem 38
The velocities of pulses propagated in an elastic fluid are as the square root of the
elastic force directly and the square root of the density inversely, provided that the
elastic force of the fluid is proportional to its condensation.
CASE 1. If the mediums are homogeneous and the distances between pulses in these mediums are equal to one another, but the motion in one medium is more intense, then the contractions and expansions of corresponding parts will be as the motions. In fact, this proportion is not exact. Even so, unless the contractions and expansions are extremely intense, the error will not be perceptible, and thus the proportion can be considered physically exact. But the motive elastic forces are as the contractions and expansions; and the velocities—generated in the same time—of equal parts are as the forces. And thus equal and corresponding parts of corresponding pulses will go and return together through spaces proportional to the contractions and expansions, with velocities that are as the spaces; and therefore the pulses, which advance through their own length in the time of one going and returning and which always succeed into the places of the immediately preceding pulses, will progress in both mediums with an equal velocity, because of the equality of the distances.
CASE 2. But if the distances between pulses, or their lengths, are greater in one medium than in the other, let us suppose that the corresponding parts by going and returning in each alternation describe spaces proportional to the lengths of the pulses; then their contractions and expansions will be equal. And thus if the mediums are homogeneous, those motive elastic forces by which they are agitated with an alternating motion will also be equal. But the matter to be moved by these forces is as the length of the pulses; and the space through which they must move by going and returning in each alternation is in the same ratio. And the time of going and returning is jointly proportional to the square root of the matter and the square root of the space and thus is as the space. But the pulses advance through their own lengths in the times of one going and returning, that is, traverse spaces proportional to the times, and therefore have equal velocities.
CASE 3. In mediums of the same density and elastic force, therefore, all pulses have equal velocities. But if either the density or the elastic force of the medium is intended [i.e., increased], then, since the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density, the time in which the same motions as before can be performed will be increased as the square root of the density and will be decreased as the square root of the elastic force. And therefore the velocity of the pulses will be jointly proportional to the square root of the density of the medium inversely and the square root of the elastic force directly. Q.E.D.
This proposition will be clearer from the construction of the following proposition.
Proposition 49, Problem 11
Given the density and elastic force of a medium, it is required to find the velocity
of the pulses.
Let us imagine the medium to be compressed, as our air is, by an
incumbent weight and let A be the height of a homogeneous medium whose
weight is equal to the incumbent weight and whose density is the same as
the density of the compressed medium in which the pulses are propagated.
And suppose that a pendulum is set up,
whose length between the point of
suspension and the center of oscillation is
A; then, in the same time in which that
pendulum performs an entire oscillation
composed of a going and a returning, a
pulse will advance through a space equal
to the circumference of a circle described
with radius A.
For with the same constructions as
in prop. 47, if any physical line EF,
describing the space PS in each single
vibration, is urged in the extremities P
and S of each going and returning by an
elastic force that is equal to its weight,
it will perform each single vibration
in the time in which it could oscillate
in a cycloid whose whole perimeter is
equal to the length PS; and this is so
because equal forces will simultaneously
impel equal corpuscles through equal
spaces. Therefore, since the times of the
oscillations are as the square root of the
length of the pendulums, and since the
length of the pendulum is equal to half
the arc of the whole cycloid, the time of
one vibration would be to the time of
oscillation of a pendulum whose length
is A as the square root of the length
½PS or PO to the length A. But the elastic force by which the physical
line-element EG is urged in its extremities P and S was (in the proof of
prop. 47) to its whole elastic force as HL − KN to V, that is (since point K
now falls upon P), as HK to V; and that whole force, that is, the incumbent
weight by which the line-element EG is compressed, is to the weight of the
line-element as the height A of the incumbent weight to the length EG of
the line-element; and thus from the equality of the ratios [or ex aequo] the
force by which the line-element EG is urged in its places P and S is to the
weight of that line-element as HK × A to V × EG, or as PO × A to V2
(for HK was to EG as PO to V). Therefore, since the times in which equal
bodies are impelled through equal spaces are inversely as the square root of
the forces, the time of one vibration under the action of that elastic force will
be to the time of the vibration, under the action of the force of weight, as the
square root of V2 to PO × A, and thus will be to the time of oscillation of a
pendulum having a length A as 
and 
jointly, that is, as V
to A. But in the time of one vibration, composed of a going and returning,
a pulse advances through its own length BC. Therefore the time in which
the pulse traverses the space BC is to the time of one oscillation (composed
of a going and returning) as V to A, that is, as BC to the circumference of
a circle whose radius is A. But the time in which the pulse will traverse the
space BC is in the same ratio to the time in which it will traverse a length
equal to this circumference; and thus in the time of such an oscillation the
pulse will traverse a length equal to this circumference. Q.E.D.
COROLLARY 1. The velocity of the pulses is that which heavy bodies acquire in falling with a uniformly accelerated motion and describing by their fall half of the height A. For in the time of this fall, with the velocity acquired in falling, the pulse will traverse a space equal to the whole height A; and thus in the time of one oscillation (composed of a going and returning) it will traverse a space equal to the circumference of a circle described with radius A; for the time of fall is to the time of oscillation as the radius of the circle to its circumference.
COROLLARY 2. Hence, since that height A is as the elastic force of the fluid directly and its density inversely, the velocity of the pulses will be as the square root of the density inversely and the square root of the elastic force directly.
Proposition 50, Problem 12
To find the distances between pulses.
In a given time, find the number of vibrations of the body by whose vibration the pulses are excited. Divide by that number the space that a pulse could traverse in the same time, and the part found will be the length of one pulse. Q.E.I.
Scholium
The preceding propositions apply to the motion of light and of sounds. For
since light is propagated along straight lines, it cannot consist in action alone
(by props. 41 and 42). And because sounds arise from vibrating bodies, they
are nothing other than propagated pulses of air (by prop. 43). This is
confirmed from the vibrations that they excite in bodies exposed to them,
provided that they are loud and deep, such as the sounds of drums. For swifter
and shorter vibrations are excited with more difficulty. But it is also well
known that any sounds impinging upon strings in unison with the sonorous
bodies excite vibrations in them. It is confirmed also from the velocity of
sounds. For since the specific weights of rainwater and quicksilver are to
each other as roughly 1 to 13⅔, and since, when the mercury in a barometer
reaches a height of 30 English inches, the specific weight of the air and that
of rainwater are to each other as roughly 1 to 870, the specific weights of
air and quicksilver will be as 1 to 11,890. Accordingly, since the height of
the quicksilver is 30 inches, the height of uniform air whose weight could
compress our air lying beneath it will be 356,700 inches, or 29,725 English
feet. And this height is the very one that we called A in the construction
of prop. 49. The circumference of a circle described with a radius of 29,725
feet is 186,768 feet. And since a pendulum 39⅕ inches long completes an
oscillation composed of a going and returning in the time of 2 seconds, as is
known, a pendulum 29,725 feet or 356,700 inches long must complete an
entirely similar oscillation in the time of 190¾ seconds. In that time, therefore,
sound will advance 186,768 feet, and thus in the time of one second, 979 feet.
Furthermore, it is evident from book 2, prop. 47, corol., why sounds immediately cease when the motion of the sonorous body ceases, and why they are not heard for a longer time when we are very far distant from the sonorous bodies than when we are very close. Why sounds are very much increased in megaphones is also manifest from the principles set forth. For every reciprocal motion is increased at each reflection by the generating cause. And the motion is lost more slowly and is reflected more strongly in tubes that impede the expansion of sounds, and therefore is more increased by the new motion impressed at each reflection. And these are the major phenomena of sounds.
Hypothesis
The resistance that arises from the friction [lit. lack of lubricity or slipperiness] of
the parts of a fluid is, other things being equal, proportional to the velocity with
which the parts of the fluid are separated from one another.
Proposition 51, Theorem 39
If an infinitely long solid cylinder revolves with a uniform motion in a uniform
and infinite fluid about an axis given in position, and if the fluid is made to
revolve by only the impulse of the cylinder, and if each part of the fluid perseveres
uniformly in its motion, then I say that the periodic times of the parts of the fluid
are as their distances from the axis of the cylinder.
Let AFL be the cylinder made to revolve uniformly about the axis S,
and divide the fluid into innumerable concentric solid cylindrical orbsa of
the same thickness by the concentric
circles BGM, CHN, DIO, EKP, . . . . Then,
since the fluid is homogeneous, the
impressions that contiguous orbs make upon one
another will (by hypothesis) be as their
relative displacements and the contiguous
surfaces on which the impressions are made.
If the impression upon some orb is greater
or less on its concave side than on its
convex side, the stronger impression will
prevail and will either accelerate or retard the motion of the orb, according as it
is directed the same way as its motion or the opposite way. Consequently, so
that each orb may persevere uniformly in its motion, the impressions on each
of the two sides should be equal and be made in opposite directions. Hence,
since the impressions are as the contiguous surfaces and their relative
velocities, the relative velocities will be inversely as the surfaces, that is, inversely as
the distances of the surfaces from the axis. And the differences between the
angular motions about the axis are as these relative velocities divided by the
distances, or as the relative velocities directly and the distances
inversely—that is, if the ratios are compounded, as the squares of the distances inversely.
Therefore, if the perpendiculars Aa, Bb, Cc, Dd, Ee, . . . , inversely
proportional to the squares of SA, SB, SC, SD, SE, . . . , are erected to each of
the parts of the infinite straight line SABCDEQ and if a hyperbolic curve
is understood to be drawn through the ends of the perpendiculars, then the
sums of the differences, that is, the whole angular motions, will be as the
corresponding sums of the lines Aa, Bb, Cc, Dd, Ee; that is, if, in order to
make the medium uniformly fluid, the number of orbs is increased and their
width decreased indefinitely, as the hyperbolic areas AaQ, BbQ, CcQ, DdQ,
EeQ, . . . , corresponding to these sums. And the times, which are inversely
proportional to the angular motions, will also be inversely proportional to
these areas. The periodic time of any particle D, therefore, is inversely as
the area DdQ, that is (by the known quadratures of curves), directly as the
distance SD. Q.E.D.
COROLLARY 1. Hence the angular motions of the particles of the fluid are inversely as the distances of the particles from the axis of the cylinder, and the absolute velocities are equal.
COROLLARY 2. If the fluid is contained in a cylindrical vessel of an infinite length and contains another inner cylinder, and if both cylinders revolve about a common axis, and the times of the revolutions are as the semidiameters of the cylinders, and each part of the fluid perseveres in its motion, then the periodic times of the individual parts will be as their distances from the axis of the cylinders.
COROLLARY 3. If any common angular motion is added to, or taken away from, the cylinder and the fluid moving in this way, then, since the mutual friction of the parts of the fluid is not changed by this new motion, the motions of the parts with respect to one another will not be changed. For the relative velocities of the parts depend upon the friction. Any part will persevere in that motion which is not more accelerated than retarded by the friction on opposite sides in opposite directions.
COROLLARY 4. Hence, if all the angular motion of the outer cylinder is taken away from the whole system of the cylinders and fluid, the result will be the motion of the fluid in the cylinder at rest.
COROLLARY 5. Therefore, if, while the fluid and outer cylinder are at rest, the inner cylinder revolves uniformly, a circular motion will be communicated to the fluid and will be propagated little by little through the whole fluid, and it will not cease to be increased until the individual parts of the fluid acquire the motion defined in corol. 4.
COROLLARY 6. And since the fluid endeavors to propagate its own motion even further, its force will make the outer cylinder also revolve, unless that cylinder is forcibly held in place, and the motion of that cylinder will be accelerated until the periodic times of both cylinders become equal. But if the outer cylinder is forcibly held in place, it will endeavor to retard the motion of the fluid, and unless the inner cylinder conserves that motion by some force impressed from outside, the outer cylinder will cause the motion to cease little by little.
All of this can be tested in deep stagnant water.
Proposition 52, Theorem 40
If a solid sphere revolves with a uniform motion in a uniform and infinite fluid
about an axis given in position, and if the fluid is made to revolve by only the
impulse of this sphere, and if each part of the fluid perseveres uniformly in its
motion, then I say that the periodic times of the parts of the fluid will be as the
squares of the distances from the center of the sphere.
CASE 1. Let AFL be a sphere made to
revolve uniformly about the axis S, and
divide the fluid into innumerable concentric
orbsa of the same thickness by means of
the concentric circles BGM, CHN, DIO,
EKP, . . . . And imagine the orbs to be
solid; then, since the fluid is homogeneous,
the impressions that the contiguous orbs
make upon one another will (by the
hypothesis) be as their relative velocities and
the contiguous surfaces on which the impressions are made. If the impression
upon some orb is greater or less on the concave side than on the convex side,
the stronger impression will prevail and will either accelerate or retard the
velocity of the orb, according as it is directed the same way as the motion of
the orb or the opposite way. Consequently, so that each orb may persevere
uniformly in its motion, the impressions on each of the two sides will have to
be equal and to be made in opposite directions. Hence, since the impressions
are as the contiguous surfaces and their relative velocities, the relative
velocities will be inversely as the surfaces, that is, inversely as the squares of the
distances of the surfaces from the center. But the differences in the angular
motions about the axis are as these relative velocities divided by the distances,
or as the relative velocities directly and the distances inversely—that is, if the
ratios are compounded, as the cubes of the distances inversely. Therefore, if
to each of the parts of the infinite straight line SABCDEQ there are erected
the perpendiculars Aa, Bb, Cc, Dd, Ee, . . . , inversely proportional to the
cubes of SA, SB, SC, SD, SE, . . . , then the sums of the differences, that is,
the whole angular motions, will be as the corresponding sums of the lines
Aa, Bb, Cc, Dd, Ee—that is (if, to make the medium uniformly fluid, the
number of orbs is increased and their width decreased indefinitely), as the
hyperbolic areas AaQ, BbQ, CcQ, DdQ,
EeQ, . . . , corresponding to these
sums. And the periodic times, inversely proportional to the angular motions,
will also be inversely proportional to these areas. Therefore the periodic time
of any orb DIO is inversely as the area DdQ, that is (by the known methods
of quadratures of curves), directly as the square of the distance SD. And this
is what I wanted to prove in the first place.
CASE 2. From the center of the sphere draw as many infinite straight lines as possible which with the axis contain given angles exceeding one another by equal differences, and imagine the orbs to be cut into innumerable rings by the revolution of these straight lines about the axis; then each ring will have four rings contiguous to it, one inside, another outside, and two at the sides. Each ring cannot be urged equally and in opposite directions by the friction of the inner ring and of the outer ring, except in a motion made according to the law of case 1. This is evident from the proof of case 1. And therefore any series of rings proceeding straight from the sphere indefinitely will be moved in accordance with the law of case 1, except insofar as it is impeded by the friction of the rings at the sides. But in motion made according to this law the friction of the rings at the sides is nil, and thus it will not impede the motion from being made according to this law. If rings equally distant from the center revolved either more quickly or more slowly near the poles than near the ecliptic, the slower rings would be accelerated and the swifter would be retarded by mutual friction, and thus the periodic times would always tend toward equality, in accordance with the law of case 1. This friction, therefore, does not prevent the motion from being made according to the law of case 1, and therefore that law will hold good; that is, the periodic time of each of the rings will be as the square of its distance from the center of the sphere. This is what I wanted to prove in the second place.
CASE 3. Now let each ring be divided by transverse sections into innumerable particles constituting an absolutely and uniformly fluid substance; then, since these sections have no relation to the law of circular motion but contribute only to the constitution of the fluid, the circular motion will continue as before. As a result of this sectioning, all the minimally small rings either will not change the unevenness and the force of their mutual friction or will change them equally. Furthermore, since the proportion of the causes remains the same, the proportion of the effects—that is, the proportion of the motions and periodic times—will also remain the same. Q.E.D.
But since the circular motion, along with the centrifugal force arising from it, is greater at the ecliptic than at the poles, there must be some cause by which each of the particles is kept in its circle; otherwise the matter at the ecliptic would always recede from the center and move on the outside of the vortex to the poles, and return from there along the axis to the ecliptic with a continual circulation.
COROLLARY 1. Hence the angular motions of the parts of the fluid about the axis of the sphere are inversely as the squares of the distances from the center of the sphere, and the absolute velocities are inversely as those same squares divided by the distances from the axis.
COROLLARY 2. If a sphere, in a homogeneous and infinite fluid at rest, revolves with a uniform motion about an axis given in position, it will communicate a motion to the fluid like that of a vortex, and this motion will be propagated little by little without limit, and this motion will not cease to be accelerated in each part of the fluid until the periodic time of each of the parts is as the squares of the distances from the center of the sphere.
COROLLARY 3. Since the inner parts of a vortex, because of their greater velocity, rub and push the outer parts and continually communicate motion to them by this action, and since those outer parts simultaneously transfer the same quantity of motion to others still further out and by this action conserve the quantity of their motion completely unaltered, it is evident that the motion is continually transferred from the center to the circumference of the vortex and is absorbed in that limitless circumference. The matter between any two spherical surfaces concentric with the vortex will never be accelerated, because all of the motion it receives from the inner matter is continually transferred to the outer matter.
COROLLARY 4. Accordingly, for a vortex to conserve the same state of motion constantly, some active principle is required from which the sphere may always receive the same quantity of motion that it impresses on the matter of the vortex. Without such a principle, it is necessary for the sphere and the inner parts of the vortex, always propagating their motion to outer parts and not receiving any new motion, to slow down little by little and cease to be carried around.
COROLLARY 5. If a second sphere were to be immersed in this vortex at a certain distance from the center, and meanwhile by some force were to revolve constantly about an axis given in inclination, then the fluid would be drawn into a vortex by the motion of this sphere; and first this new and tiny vortex would revolve along with the sphere about the center of the first vortex, and meanwhile its motion would spread more widely and little by little would be propagated without limit, in the same way as the first vortex. And for the same reason that the sphere of the new vortex was drawn into the motion of the first vortex, the sphere of the first vortex would also be drawn into the motion of this new vortex, in such a way that the two spheres would revolve about some intermediate point and because of that circular motion would recede from each other unless constrained by some force. Afterward, if the continually impressed forces by which the spheres persevere in their motions were to cease, and everything were left to the laws of mechanics, the motion of the spheres would weaken little by little (for the reason assigned in corols. 3 and 4), and the vortices would at last be completely at rest.
COROLLARY 6. If several spheres in given places revolved continually with certain velocities around axes given in position, the same number of vortices, going on without limit, would be made. For all of the spheres, for the same reason that any one of them propagates its motion without limit, will also propagate their motions without limit, in such a way that each part of the infinite fluid is agitated by that motion which results from the actions of all the spheres. Hence the vortices will not be limited by fixed bounds but will little by little run into one another, and the spheres will be continually moved from their places by the actions of the vortices upon one another, as was explained in corol. 5; nor will they keep any fixed position with respect to one another, unless constrained by some force. And when those forces, which conserve the motions by being continually impressed upon the spheres, cease, the matter—for the reason assigned in corols. 3 and 4—will little by little come to rest and will no longer be made to move in vortices.
COROLLARY 7. If a homogeneous fluid is enclosed in a spherical vessel and is made to revolve in a vortex by the uniform rotation of a sphere placed in the center, and if the sphere and the vessel revolve in the same direction about the same axis, and if their periodic times are as the squares of the semidiameters, then the parts of the fluid will not persevere in their motions without acceleration and retardation until their periodic times are as the squares of the distances from the center of the vortex. No other constitution of a vortex can be stable.
COROLLARY 8. If the vessel, the enclosed fluid, and the sphere conserve this motion and additionally revolve with a common angular motion about any given axis, then, since the friction of the parts of the fluid upon one another is not changed by this new motion, the motions of the parts with respect to one another will not be changed. For the relative velocities of the parts with respect to one another depend upon friction. Any part will persevere in that motion by which the friction on one side does not retard it more than the friction on the other accelerates it.
COROLLARY 9. Hence, if the vessel is at rest, and if the motion of the sphere is given, the motion of the fluid will be given. For imagine that a plane passes through the axis of the sphere and revolves with an opposite motion, and suppose that the sum of the time of the revolution of the plane and the revolution of the sphere is to the time of the revolution of the sphere as the square of the semidiameter of the vessel to the square of the semidiameter of the sphere; then the periodic times of the parts of the fluid with respect to the plane will be as the squares of their distances from the center of the sphere.
COROLLARY 10. Accordingly, if the vessel moves with any velocity either about the same axis as the sphere or about some different axis, the motion of the fluid will be given. For if the angular motion of the vessel is taken away from the whole system, all the motions with respect to one another will remain the same as before, by corol. 8. And these motions will be given by corol. 9.
COROLLARY 11. If the vessel and the fluid are at rest, and if the sphere revolves with a uniform motion, then the motion will be propagated little by little through the whole fluid to the vessel, and the vessel will be driven around unless forcibly constrained, and the fluid and vessel will not cease to be accelerated until their periodic times are equal to the periodic times of the sphere. But if the vessel is constrained by some force or revolves with any continual and uniform motion, the medium will little by little come to the state of the motion defined in corols. 8, 9, and 10, nor will it ever persevere in any other state. But then if, when those forces cease by which the vessel and the sphere were revolving with fixed motions, the whole system is left to the laws of mechanics, the vessel and the sphere will act upon each other by means of the intervening fluid and will not cease to propagate their motions to each other through the fluid until their periodic times are equal and the whole system revolves together like one solid body.
Scholium
In the preceding propositions, I have been supposing the fluid to consist of
matter which is uniform in density and fluidity. The fluid is such that a given
sphere, set anywhere in it, would with a given motion in a given interval of
time be able to propagate similar and equal motions, at distances always
equal from itself. Indeed, matter endeavors by its circular motion to recede
from the axis of a vortex and therefore presses all the further matter. From
this pressure the friction of the parts becomes stronger and their separation
from one another more difficult, and consequently the fluidity of the matter is
decreased. Again, if there is any place where the parts of the fluid are thicker
or larger, the fluidity will be less there, because the surfaces separating the
parts from one another are fewer. In cases of this sort, I suppose the deficiency
in fluidity to be supplied either by the slipperiness of the parts or by their
pliancy or by some other condition. If this does not happen, the matter will
cohere more and will be more sluggish where it is less fluid, and thus will
receive motion more slowly and will propagate it further than according
to the ratio assigned above. If the shape of the vessel is not spherical, the
particles will move in paths which are not circular but correspond to the
shape of the vessel, and the periodic times will be very nearly as the squares
of the mean distances from the center. In the parts between the center and
the circumference where the spaces are wider, the motions will be slower,
and where the spaces are narrower the motions will be swifter, and yet the
swifter particles will not seek the circumference. For they will describe
less-curved arcs, and the endeavor to recede from the center will not be less
decreased by the decrement of this curvature than it will be increased by the
increment of the velocity. In going from the narrower spaces into the wider,
they will recede a little further from the center, but they will be retarded
by this receding, and afterward in approaching the narrower spaces from the
wider ones they will be accelerated, and thus each of the particles will forever
alternately be retarded and accelerated. All of this will be so in a rigid vessel.
For the constitution of vortices in an infinite fluid can be found by corol. 6
of this proposition.
Moreover, in this proposition I have tried to investigate the properties of vortices in order to test whether the celestial phenomena could be explained in any way by vortices. For it is a phenomenon that the periodic times of the secondary planets that revolve about Jupiter are as the 3/2 powers of the distances from the center of Jupiter; and the same rule applies to the planets that revolve about the sun. Moreover, these rules apply to both the primary and the secondary planets very exactly, as far as astronomical observations have shown up to now. And thus if those planets are carried along by vortices revolving about Jupiter and the sun, the vortices will also have to revolve according to the same law. But the periodic times of the parts of a vortex turned out to be in the squared ratio of the distances from the center of motion, and that ratio cannot be decreased and reduced to the 3/2 power, unless either the matter of the vortex is the more fluid the further it is from the center, or the resistance arising from a deficiency in the slipperiness of the parts of the fluid (as a result of the increased velocity by which the parts of the fluid are separated from one another) is increased in a greater ratio than the ratio in which the velocity is increased. Yet neither of these seems reasonable. The thicker and less-fluid parts, if they are not heavy toward the center, will seek the circumference; and although—for the sake of the proofs—I proposed at the beginning of this section a hypothesis in which the resistance would be proportional to the velocity, it is nevertheless likely that the resistance is in a lesser ratio than that of the velocity. If this is conceded, then the periodic times of the parts of a vortex will be in a ratio greater than the squared ratio of the distances from its center. But if vortices (as is the opinion of some) move more quickly near the center, then more slowly up to a certain limit, then again more quickly near the circumference, certainly neither the 3/2 power nor any other fixed and determinate ratio can hold. It is therefore up to philosophers to see how that phenomenon of the 3/2 power can be explained by vortices.
Proposition 53, Theorem 41
Bodies that are carried along in a vortex and return in the same orbit have the
same density as the vortex and move according to the same law as the parts of the
vortex with respect to velocity and direction.
For if some tiny part of the vortex is composed of particles or physical points which preserve a given situation with respect to one another and is supposed to be frozen, then this part will move according to the same law as before, since it is not changed with respect to its density, or its inherent force or figure. And conversely, if a frozen and solid part of the vortex has the same density as the rest of the vortex and is resolved into a fluid, this part will move according to the same law as before, except insofar as its particles, which have now become fluid, move with respect to one another. Therefore, the motion of the particles with respect to one another may be ignored as having no relevance to the progressive motion of the whole, and the motion of the whole will be the same as before. But this motion will be the same as the motion of other parts of the vortex that are equally distant from the center, because the solid resolved into a fluid becomes a part of the vortex similar in every way to the other parts. Therefore, if a solid is of the same density as the matter of the vortex, it will move with the same motion as the parts of the vortex and will be relatively at rest in the immediately surrounding matter. But if the solid is denser, it will now endeavor to recede from the center of the vortex more than before; and thus, overcoming that force of the vortex by which it was formerly kept in its orbit as if set in equilibrium, it will recede from the center and in revolving will describe a spiral and will no longer return into the same orbit. And by the same argument, if the solid is rarer, it will approach the center. Therefore, the solid will not return into the same orbit unless it is of the same density as the fluid. And it has been shown that in this case the solid would revolve according to the same law as the parts of the fluid that are equally distant from the center of the vortex. Q.E.D.
COROLLARY 1. Therefore a solid that revolves in a vortex and always returns into the same orbit is relatively at rest in the fluid in which it is immersed.
COROLLARY 2. And if the vortex is of a uniform density, the same body can revolve at any distance from the center of the vortex.
Scholium
Hence it is clear that the planets are not carried along by corporeal
vortices. For the planets, which—according to the Copernican hypothesis—move
about the sun, revolve in ellipses having a focus in the sun, and by radii
drawn to the sun describe areas
proportional to the times. But the parts
of a vortex cannot revolve with such
a motion. Let AD, BE, and CF
designate three orbits described about the
sun S, of which let the outermost CF
be a circle concentric with the sun, and
let A and B be the aphelia of the two inner ones, and D and E their perihelia. Therefore, a body that revolves in
the orbit CF, describing areas proportional to the times by a radius drawn
to the sun, will move with a uniform motion. And a body that revolves in
the orbit BE will, according to the laws of astronomy, move more slowly
in the aphelion B and more swiftly in the perihelion E, although according
to the laws of mechanics the matter of the vortex ought to move more swiftly
in the narrower space between A and C than in the wider space between
D and F, that is, more swiftly in the aphelion than in the perihelion. These
two statements are contradictory. Thus in the beginning of the sign of Virgo,
where the aphelion of Mars now is, the distance between the orbits of Mars
and Venus is to the distance between these orbits in the beginning of the sign
of Pisces as roughly 3 to 2, and therefore the matter of the vortex between
these orbits in the beginning of Pisces must move more swiftly than in the
beginning of Virgo in the ratio of 3 to 2. For the narrower the space through
which a given quantity of matter passes in the given time of one revolution,
the greater the velocity with which it must pass. Therefore, if the earth,
relatively at rest in this celestial matter, were carried by it and revolved along
with it about the sun, its velocity in the beginning of Pisces would be to its
velocity in the beginning of Virgo as 3 to 2. Hence the apparent daily motion
of the sun in the beginning of Virgo would be greater than 70 minutes, and
in the beginning of Pisces less than 48 minutes, although (as experience bears
witness) the apparent motion of the sun is greater in the beginning of Pisces
than in the beginning of Virgo, and thus the earth is swifter in the beginning
of Virgo than in the beginning of Pisces. Therefore the hypothesis of vortices
can in no way be reconciled with astronomical phenomena and serves less
to clarify the celestial motions than to obscure them. But how those motions
are performed in free spaces without vortices can be understood from book
1 and will now be shown more fully in book 3 on the system of the world.
to the horizon and standing upon
the horizontal line PQ at points
B, C, D, and E; and let BC, CD,
and DE be distances between the
ordinates equal to one another. From
points G and H draw the straight
lines GL and HN touching the
curve in G and H, and meeting
in L and N the ordinates CH and DI produced upward; and complete
the parallelogram HCDM. Then the times in which the body describes
arcs GH and HI will be as the square roots of the distances LH and
NI which the body could describe in those times by falling from the
tangents; and the velocities will be directly as GH and HI (the lengths
described) and inversely as the times. Represent the times by T and t,
and the velocities by 
and 
; and the decrement of the velocity
occurring in time t will be represented by 
; but in a body
describing arc HI, gravity increases the arc by only the length HI − HN or
, and thus generates only the velocity 
. Add this velocity
to the above decrement, and the result is the decrement of the velocity
arising from the resistance alone, namely 
. And
accordingly, since gravity generates the velocity 
to 2NI.
by adding to the resulting squares the
squares of BC and CD, there will
result the squares of the arcs GH and
HI: o2 + Q2o2 −
2QRo3 + · · · and
o2 + Q2o2 +
2QRo3 + · · · . The roots
of these, o√(1 + Q2) − 
and
o√(1 + Q2) + 
is 
or 
; and if the values just found of 
, the
result will be 
√(1 + Q2). And since 2NI is 2Ro2, the resistance will now
be to the gravity as 
or 
.
directly and 
inversely, that is, as 
. Q.E.I.
will be equal to √(1 + Q2) and thus can
be written for √(1 + Q2) above. And so the resistance will be to the gravity
as 3S × HT to 4R2 × AC, the velocity will be as 
, and the density of
the medium will be as 
.a
in corol. 1, if the resistance is supposed to be as any power
Vn of the velocity V, the density of
the medium will turn out to be as
. And therefore if
a curve can be found under the
condition that there would be given the
ratio of 
to 
, or 
to (1 + Q2)n−1, body will move in this curve in a uniform medium with a
resistance that is as the power Vn of the velocity. But let us return to simpler
curves.d
feet or roughly 109
feet may be added, because of the density of the particles of air; and thus
sound will travel roughly 1,088 feet in the time of 1 second.