DF, and the lines ER, ES to FB, FD respectively, and therefore QS to
be always equal to CK; and (by Cor. 2, Prop. XCVII) PD will be to QD
as M to N, and therefore as DL to DK, or FB to FK ; and by division as
DL FB or PH PD FB to FD or FQ QD ; arid by composition
as PH FB to FQ, that is (because PH and CG, QS and CE, are equal),
as CE + BG FR to CE FS. But (because BG is to CE as M
N to N) it. comes to pass also that CE + BG is to CE as M to N; and
therefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2,
Prop XCVI1) the superficies EF compels a body, falling upon it in the
direction DF, to go on in the line FR to the place B. Q.E.D.
SCHOLIUM.
,In the same manner one may go on to three or more superficies. But
of all figures the sphserical is the most proper for optical uses. If the ob
ject glasses of telescopes were made of two glasses of a sphaerical figure,
containing water between them, it is not unlikely that the errors of the
refractions made in the extreme parts of the superficies of the glasses may
be accurately enough corrected by the refractions of the water. Such ob
ject glasses are to be preferred before elliptic and hyperbolic glasses, not only
because they may be formed with more ease and accuracy, but because the
pencils of rays situate without the axis of the glass would be more accu
rately refracted by them. But the different refrangibility of different raya
is the real obstacle that hinders optics from being made perfect by sphaeri
cal or any other figures. Unless the errors thence arising can be corrected,
all the labour spent in correcting the others is quite thrown away.
BOOK II
BOOK II.
OF THE MOTION OF BODIES.
SECTION I.
Of the motion of bodies that are resisted in the ratio of the velocity.
PROPOSITION I. THEOREM I.
Tf a body is resisted in the ratio of its velocity, the motion lost by re
sistance is as the space gone over in its motion.
For since the motion lost in each equal particle of time is as the velocity,
that is, as the particle of space gone over, then, by composition, the motion
lost in the whole time will he as the whole space gone over. Q.E.D.
COR. Therefore if the body, destitute of all gravity, move by its innate
force only in free spaces, and there be given both its whole motion at the
beginning, and also the motion remaining after some part of the way is
gone over, there will be given also the whole space which the body can de
scribe in an infinite time. For that space will be to the space now de
scribed as the whole motion at the beginning is to the part lost of that
motion.
LEMMA I.
Quantities proportional to their differences are continually proportional.
Let A be to A B as B to B C and C to C D, (fee., and, by con
version, A will be to B as B to C and C to D, &c. Q.E.D.
PROPOSITION II. THEOREM II.
If a body is resisted in the ratio of its velocity, and moves, by its vis insita
only, through a similar medium, and the times be taken equal,
the velocities in the beginning of each of the times are in a geometri
cal progression, and the spaces described in each of the times are as
the velocities.
CASE 1. Let the time be divided into equal particles ; and if at the very
beginning of each particle we suppose the resistance to act with one single
impulse which is as the velocity, the decrement of the velocity in each of
THE MATHEMATICAL PRINCIPLES [BOOK II.
the particles of time will be as the same velocity. Therefore the veloci
ties are proportional to their differences, and therefore (by Lem. 1, Book
II) continually proportional. Therefore if out of an equal number of par
ticles there be compounded any equal portions of time, the velocities at the
beginning of those times will be as terms in a continued progression, which
are taken by intervals, omitting every where an equal number of interme
diate terms. But the ratios of these terms are compounded of the equaj
ratios of the intermediate terms equally repeated, and therefore are equal
Therefore the velocities, being proportional to those terms, are in geomet
rical progression. Let those equal particles of time be diminished, and
their number increased in infinitum, so that the impulse of resistance may
become continual
; and the velocities at the beginnings of equal times, al
ways continually proportional, will be also in this case continually pro
portional. Q.E.D.
CASE 2. And, by division, the differences of the velocities, that is, the
parts of the velocities lost in each of the times, are as the wholes ;
but the
spaces described in each of the times are as the lost parts of the velocities
(by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D.
TT COROL. Hence if to the rectangular asymptotes AC, CH,
the hyperbola BG is described, and AB, DG be drawn per
pendicular to the asymptote AC, and both the velocity of
. the body, and the resistance of the medium, at the very be
ginning of the motion, be expressed by any given line AC,
and, after some time is elapsed, by the indefinite line DC ; the time may
be expressed by the area ABGD, and the space described in that time by
the line AD. For if that area, by the motion of the point D, be uniform
ly increased in the same manner as the time, the right line DC will de
crease in a geometrical ratio in the same manner as the velocity ; and the
parts of the right line AC, described in equal times, will decrease in the
same ratio.
PROPOSITION III. PROBLEM I.
To define the motion of a body which, in a similar medium, ascends or
descends in a right line, and is resisted in the ratio of its velocity, and
acted upon by an uniform force of gravity.
The body ascending, let the gravity be expound
ed by any given rectangle BACH ; and the resist
ance of the medium, at the beginning of the ascent,
by the rectangle BADE, taken on the contrary side
Jfl e B^l | L- of the right line AB. Through the point B, with
the rectangular asymptotes AC, CH, describe an
hyperbola, cutting the perpendiculars DE, de, ID
SEC. I.j OF NATURAL PHILOSOPHY. 253
G, g ; and the body ascending will in the time DGgd describe the space
EG-e; in the time DGBA, the space of the whole ascent EGB ;
in the
time ABK1, the space of descent BFK ; and in the time IKki the space of
descent KFfk; and the velocities of the bodies (proportional to the re
sistance of the medium) in these periods of time will be ABED, ABed, O,
ABFI, AB/z respectively ; and the greatest velocity which the body can
acquire by descending will be BACH.
For let the rectangle BACH be resolved into in
numerable rectangles AA , K/, Lm, M//, *fea, which
shall be as the increments of the velocities produced
in so many equal times; then will 0, AAr, AL Am, An,
ifec., be as the whole velocities, and therefore (by suppo
sition) as the resistances of the medium in the beginning
of each of the equal times. Make AC to
AJLLB
AK, or ABHC to AB/vK, as the force of gravity to the resistance in the
beginning of the second time ; then from the force of gravity subduct the
resistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the abso
lute forces with which the body is acted upon in the beginning of each of
the times, and therefore (by Law I) as the increments of the velocities, that
is, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, Book
II) in a geometrical progression. Therefore, if the right lines K, LI
M/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee..
the areas AB^K, K</rL, LrsM, MsJN, (fee., will be equal, and there
fore analogous to the equal times and equal gravitating forces. But the
area AB^K (by Corol. 3, Lem. VII and VIII, Book I) is to the area Bkq
as K^ to \kq, or AC to |AK, that is, as the force of gravity to the resist
ance in the middle of the first time. And by the like reasoning, the areas
<?KLr, rLMs, sMN/, (fee., are to the areas qklr, rims, smnt, (fee., as the
gravitating forces to the resistances in the middle of the second, third, fourth
time, and so on. Therefore since the equal areas BAKq, </KLr, rLMs,
sMN/, (fee., are analogous to the gravitating forces, the areas Bkq, qklr,
rims, smut, (fee., will be analogous to the resistances in the middle of
each of the times, that is (by supposition), to the velocities, and so to the
spaces described. Take the sums of the analogous quantities, and the areas
Bkq, Elr, Ems, But, (fee., will be analogous to the whole spaces described ;
and also the areas AB<?K, ABrL, ABsM, AB^N, (fee., to the times. There
fore the body, in descending, will in any time ABrL describe the space Blr,
and in the time Lr^N the space rlnt. Q,.E.D. And the like demonstra
tion holds in ascending motion.
COROL. 1. Therefore the greatest velocity that the body can acquire by
falling is to the velocity acquired in any given time as the iven force ol
gravity which perpetually acts upon it to the resisting force which opposes
it at the end of that time.
854 THE MATHEMATICAL PRINCIPLES [BOOK IL
COROL. 2. But the time being augmented in an arithmetical progression,
the sum of that greatest velocity and the velocity in the ascent, and also
their difference in the descent, decreases in a geometrical progression.
COROL. 3. Also the differences of the spaces, which are described in equal
differences of the times, decrease in the same geometrical progression.
COROL. 4. The space described by the body is the difference of two
spaces, whereof one is as the time taken from the beginning of the descent,
and the other as the velocity; which [spaces] also at the beginning of the
descent are equal among themselves.
PROPOSITION IV. PROBLEM II.
Supposing the force of gravity in any similar medium to be uniform,
and to tend perpendicularly to the plane of the horizon ; to define the
motion of a projectile therein, which suffers resistance proportional to
its velocity.
Let the projectile go from any place D in
the direction of any right line DP, and let
its velocity at the beginning of the motion
be expounded by the length DP. From the
point P let fall the perpendicular PC on the
horizontal line DC, and cut DC in A, so
that DA may be to AC as the resistance
of the medium arising from the motion up
wards at the beginning to the force of grav
ity; or (which comes to the same) so that
t ie rectangle under DA and DP may be to
that under AC and CP as the whole resist
ance at the beginning of the motion to the
force of gravity. With the asymptotes
DC, CP describe any hyperbola GTBS cut
ting the perpendiculars DG, AB in G and
B ; complete the parallelogram DGKC, and
let its side GK cut AB in Q,. Take
N in the same ratio to QB as DC is in to CP ; and from any point R of the
right line DC erect RT perpendicular to it, meeting the hy] erbola in T,
and the right lines EH, GK, DP in I, t, and V ; in that perpendicular
take Vr equal to ~- , or which is the same thing, take Rr equal to
(""""PIT?
^ T ; and the projectile in the time DRTG will arrive at the point r
describing the curve line DraF, the locus of the point r ; thence it will
come to its greatest height a in the perpendicular AB j and afterwards
SEC. 1.J OF NATURAL PHILOSOPHY. 255
ever approach to the asymptote PC. And its velocity in any pjint r will
be as the tangent rL to the curve. Q.E.I.
For N is to Q,B as DC to CP or DR to RV, and therefore RV is equal to
PR X QB , -..".
"v v DRXQB-/GT
^r-, and R/ (that is, RV Vr, or - --^---) is equal to
D-R X-Ap RDGT ~---. JNow let the time be expounded by the area
RDGT and (by Laws, Cor. 2), distinguish the motion of the body into
two others, one of ascent, the other lateral. And since the resistance is as
the motion, let that also be distinguished into two parts proportional and
contrary to the parts of the motion : and therefore the length described by
the lateral motion will be (by Prop. II, Book II) as the line DR, and the
height (by Prop. Ill, Book II) as the area DR X AB RDGT, that is,
as the line Rr. But in the very beginning of the motion the area RDGT
is equal to the rectangle DR X AQ, and therefore that line Rr (or
DRx AB
that is, as CP to DC ; and therefore as the motion upwards to the motion
lengthwise at the beginning. Since, therefore, Rr is always as the height,
and DR always as the length, and Rr is to DR at the beginning as the
height to the length, it follows, that Rr is always to DR as the height to
the length ; and therefore that the body will move in the line DraF. which
is the locus of the point r. QJE.D.
DR X AB RDGT
COR. 1. Therefore Rr is equal to --
^
------
^-. and therefore
if RT be produced to X so that RX may be equal to --
^
--
; that is,
if the parallelogram ACPY be completed, and DY cutting CP in Z be
drawn, and RT be produced till it meets DY in X ; Xr will be equal to
RDGT
^ , and therefore proportional to the time.
COR. 2. Whence if innumerable lines CR, or, which is the same, innu
merable lines ZX, be taken in a geometrical progression, there will be as
many lines Xr in an arithmetical progression. And hence the curve DraF
is easily delineated by the table of logarithms.
COR. 3. If a parabola be constructed to the vertex D, and the diameter
DG produced downwards, and its latus rectum is to 2 DP as the whole
resistance at the beginning of the notion to the gravitating force, the ve
locity with which the body ought *o go from the place D, in the direction
of the right line DP, so as in an uniform resisting medium to describe the
curve DraF, will be the same as that with which it ought to go from the
same place D in the direction of the same right line DP, so as to describe
256 THE MATHEMATICAL PRINCIPLES ~ [BOOK II
I
a parabola in a non-resisting medium. For
the latus rectum of this parabola, at the very
DV2
beginning of the motion, is -y- , andVris
tGT DR x T*
-~JTor
^T
. But a right line, which,
if drawn, would touch the hyperbola GTS in
K G, is parallel to DK, and therefore T* is
CKX DR
c
QBx DC ^ , and N is ~pp Ahd there- DC
DR2 X CK x CP
fore Vr is equal to 2DC 2 X QlT~; *^at *S (Because D^ an<* ^)C, DV
DV2 x CK ~x OP
and DP are proportionals), to ^T5 Fcrr J an<* tne ^atus reeturn
DV2
- comes out -
2DP2 X QB
are proportional),
CK X CP
2DP 2 X DA
AC X CP
CP X AC ;
that is, as the resistance to the gravity.
(because
and therefore is to 2DP as DP X DA to
Q.E.D.
COR. 4. Hence if a body be projected from
any place D with a given velocity, in the
direction of a right line DP given by posi
tion, and the resistance of the medium, at
the beginning of the motion, be given, the
curve DraF, which that body will describe,
may be found. For the velocity being
given, the latus rectum of the parabola is
given, as is well known. And taking 2DP
to that latus rectum, as the force of gravity
to the resisting force, DP is also given.
Then cutting DC in A, so that GP X AC
may be to DP X DA in the same ratio of
the gravity to the resistance, the point A
will be given. And hence the curve DraF
is also given.
COR. 5. And, on the contrary, if the
curve DraF be given, there will be given
x>th the velocity of the body and the resistance of the medium in each of
the places r. For the ratio of CP X AC to DP X DA being given, there
is given both the resistance of the medium at the beginning of the motion
and the latus rectum of the parabola ; and thence the velocity at the be
ginning of the motion is given also. Then from the length of the tangent
SEC. I.]
OF NATURAL PHILOSOPHY. 257
L there is given both the velocity proportional to it, and the resistance
proportional to the velocity in any place r.
COR. 6. But since the length 2DP is to the latus rectum of the para
bola as the gravity to the resistance in D ; and, from the velocity aug
mented, the resistance is ti gmented in the same ratio, but the latus rectum
of the parabola is augmented in the duplicate of that ratio, it is plain thot
the length 2DP is augmented in that simple ratio only ; and is therefore
always proportional to the velocity ; nor will it be augmented or dimin
ished by the change of the angle CDP, unless the velocity be also changed.
COR. 7. Hence appears the method of deter
mining the curve DraF nearly from the phenomena,
and thence collecting the resistance and
velocity with which the body is projected. Let
two similar and equal bodies be projected with
the same velocity, from the place D, in differ
ent angles CDP, CDp ; and let the places F,
f. where they fall upon the horizontal plane
DC, be known. Then taking any length for D */ F
DP or Dp suppose the resistance in D to be to
the suavity in any ratio whatsoever, and let that
ratio be expounded by any length SM. Then, , _
by computation, from that assumed length DP, ^x
find the lengths DF, D/; and from the ratio
F/
-p^,
found by calculation, subduct the same ratio as found by experiment ;
and let the cKfference be expounded by the perpendicular MN. Repeat the
same a second and a third time, by assuming always a new ratio SM of the
resistance to the gravity, and collecting a new difference MN. Draw the
affirmative differences on one side of the right line SM, and the negative
on the other side ; and through the points N, N, N, draw a regular curve
NNN. cutting the right line SMMM in X, and SX will be the true ratio
of the resistance to the gravity, which was to be found. From this ratio
the length DF is to be collected by calculation
; and a length, which is to
the assumed length DP as the length DF known by experiment to the
length DF just now found, will be the true length DP. This being known,
you will have both the curve line DraF which the body describes, and also
the velocity and resistance of the body in each place.
SCHOLIUM.
But, yet, that the resistance of bodies is in the ratio of the velocity, is more
a mathematical hypothesis than a physical one. In mediums void of all te
nacity, the resistances made to bodies are in the duplicate ratio of the ve
locities. For by the action of a swifter body, a greater motion in propor-
17
THE MATHEMATICAL PRINCIPLES [BoOK IL
tion to a greater velocity is communicated to the same quantity of the
medium in a less time ; and in an equal time, by reason of a greater quan
tity of the disturbed medium, a motion is communicated in the duplicate
ratio greater ; and the resistance (by Law II and III) is as the motion
communicated. Let us, therefore, see what motions arise from this law of
resistance.
SECTION II.
If the motion of bodies that are resisted in tfie duplicate ratio of their
velocities.
PROPOSITION V. THEOREM III.
Ff a body is resisted in the duplicate ratio of its velocity, and moves by
its innate force only through a similar medium; and the times be
taken in a geometrical progression, proceeding from less to greater
terms : I say, that the velocities at the beginning of each of the times
are in the same geometrical progression inversely ; and that the spaces
are equal, which are described in each of the times.
For since the resistance of the medium is proportional to the square of
the velocity, and the decrement of the velocity is proportional to the resist
ance : if the time be divided into innumerable equal particles, the squares of
the velocities at the beginning of each of the times will be proportional to
the differences of the same velocities. Let those particles of time be AK,
KL, LM, &c., taken in the right line CD; and
erect the perpendiculars AB, Kk, L/, Mm, &c.,
meeting the hyperbola BklmG, described with the
centre C, and the rectangular asymptotes CD, CH.
in B, kj I, m, (fee.
; then AB will be to Kk as CK
to CA, and, by division, AB Kk to Kk as AK
C ARIMT to ^A>
an(1 alternate^ AB ^C to AK as Kk
to CA ; and therefore as AB X Kk to AB X CA.
Therefore since AK and AB X CA are given,* AB Kk will be as AB
X Kk ; and, lastly, when AB and KA* coincide, as AB2
. And, by the like
reasoning, KAr-U, J J-M/??, (fee., will be as Kk2
. LI2
, (fee. Therefore the
squares of the lines AB, KA", L/, Mm, (fee., are as their differences ; and,
therefore, since the squares of the velocities were shewn above to be as their
differences, the progression of both will be alike. This being demonstrated
it follows also that the areas described by these lines are in a like progres
sion with the spaces described by these velocities. Therefore if the velo
city at the beginning of the first time AK be expounded by the line AB,
SEC. II.] OF NATURAL PHILOSOPHY. 25CJ
and the velocity at the beginning of the second time KL by the line K&
and the length described in the hrst time by the area AKA*B, all the fol
lowing velocities will be expounded by the following lines \J. Mm, .fee.
and the lengths described, by the areas K/, I mi. &c. And, by compo
sition, if the whole time be expounded by AM, the sum of its parts, the
whole length described will be expounded by AM/ftB the sum of its parts.
Now conceive the time AM to be divided into the parts AK, KL, LM, (fee
so that CA, CK, CL, CM, (fee. may be in a geometrical progression ; and
those parts will be in the same progression, and the velocities AB, K/r,
L/, Mm, (fee., will be in the same progression inversely, and the spaces de
scribed Ak, K/, Lw, (fee., will be equal. Q,.E.D.
